Re: [Phys-l] Temperture profile in a graviational field

[John Denker <jsd@av8n.com>]

On 01/26/2012 10:38 AM, Folkerts, Timothy J wrote:

> 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.

The word "adiabatic" is ambiguous.  Sometimes the intended meaning
can be inferred from context, but in this case I'm not sure whether
it means "isentropic" or simply "thermally insulated".  For now I
will assume it means *both* isentropic and thermally insulated.  If
this is not what was intended, please clarify.

Also, given that the system (gas + pistons) is meant to be thermally 
insulated from the rest of the world, it's not clear whether the two
pistons are meant to be thermally insulated from each other.

> 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? 

Two answers:

It is possible in principle for things to be quasi-static as well as
isentropic and thermally insulated.  This implies that the pistons
accelerate slowly compared to dynamical timescales (e.g. round-trip 
propagation of sound) but not so slowly that we need to worry about
leakage through the thermal insulation.

OTOH I doubt the example of 9.8 m/s/s is within the quasi-static limit.
I suspect it is much too fast.

>  I conclude there will a definite (and
> continuously changing) temperature gradient -- hottest near the
> moving "back piston", and coolest near the stationary "front piston".

Three answers:

1) In the quasi-static limit, the column of gas will look like a rigid 
 push-rod.
  1a) if it started out isothermal it will remain isothermal.
  1b) if it started out with some wacky non-equilibrium temperature 
    profile, it will keep that profile.

2) For fast acceleration, there will be heating due to compression.
 The heating will not be confined to the region near the piston,
 because *sound waves* will be emitted.  To a good approximation, 
 an ordinary sound wave is considered both isentropic and thermally 
 insulated, i.e. we neglect dissipation due to viscosity and due
 to thermal conductivity ... so the peaks of the pressure wave have
 high temperature and the troughs have low temperature.

> 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".

Analogous question, analogous answers.

> 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?

Again, it depends on timescales.  There will be a timescale where
sound waves will be rattling around in the air column.  On a longer
timescale the sound waves will dissipate ... but at this point we
will have violated several of the previously-made assumptions.

> 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.

I was discussing the /equilibrium/ state.  Thermally insulating the
two ends of the system from each other and then disturbing things 
pretty much guarantees a non-equilibrium situation.

> 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?

That violates several of the previous assumptions.  It would make
the problem even more complicated and even more ill-posed, because
there would be additional timescales in the problem, and you get
to decide which timescales are dominant, and which timescales you
are interested in.

> 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.

OK.

Conversely at the other end (the pushing end) if the piston is going
fast enough to be supersonic, you get a /shock/ rather than ordinary
sound waves.  A shock is rather well understood, albeit complicated. 
It is guaranteed to be dissipative, violating many of the previously-
made assumptions.