Re: [Phys-l] Temperature profile in a gravitational field
Well, it has been a spectacle to see so much hand-wringing in attempting
to apply the usual physics models to the mind experiment of a towering
atmospheric insulated cylinder of air with no mixing, no energy flux, no
thermal flux - so no matter how inane my suggestion, I can hardly do worse:
This is a situation where pressure decreases with height and
temperature decreases with height and entropy varies with height.
There; I said it! :-)
Brian W
On 1/26/2012 11:38 AM, Folkerts, Timothy J wrote:
> OK -- a few more thought experiments.
>
> Consider an infinitely long insulated cylinder. There are two insulated
> pistons placed far apart with some gas in between (say 1 km apart, with 1 atm
> of N2 @ 300 K). This makes any processes adiabatic within the tube adiabatic.
>
> If I accelerate one piston inward (say at 9.8 m/s^2), there will be an
> adiabatic compression at that end (and that compression will be occurring
> faster and faster). The gas at that end will warm. In the quasistatic
> limit, the gas throughout would be the same temperature, but does a
> quasi-static approximation apply in this continuously changing situation? I
> conclude there will a definite (and continuously changing) temperature
> gradient -- hottest near the moving "back piston", and coolest near the
> stationary "front piston".
>
> I could also pull out on the far side with the same sort of acceleration.
> Same questions (but with cooling rather than warming, of course). I conclude
> there will a definite (and continuously changing) temperature gradient --
> hottest near the "back piston", and coolest near the moving "front piston".
>
> I could also move BOTH sides with the same acceleration (maintaining a
> constant volume) , so that there would be a continued adiabatic compression
> at one end and a continued adiabatic expansion at the other. Could this be
> considered quasi-static, so that we can assume the gas will relax to a
> uniform temperature, or does the fact that the ends are continuously changing
> (accelerating) mean we might never reach a quasi-static situation and the
> compression& expansion would maintain a temperature gradient across the tube?
>
> Of course, I could ALSO do this with a 1 km long tube mounted in a spaceship
> accelerating at 9.8 m/s^2. Or do it with a 1 km long tube standing on the
> earth. John Denker's previous analysis concludes that we would indeed
> achieve a uniform temperature in any of these cases.
>
>
>
> FOLLOW-UP # 1: How would the analysis change if either or both of the
> pistons was a thermal reservoir held at the original temperature (eg 300 K),
> rather than an insulated piston?
>
>
> FOLLOW-UP #2: For the one piston moving outward, at first the motion is slow
> and we could treat this as a typical adiabatic expansion, which cools the
> gas. But by the time the piston is moving very fast, there will be
> essentially no molecules hitting the piston, and we have approximately an
> adiabatic free expansion, which would NOT cool the gas. Is the amount of
> cooling a function of the speed that the piston is expanding? Presumably it
> must be. If could be interesting to see what that function is.
>
>
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