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Wednesday, December 21, 2022

Adiabatic Cooling

An adiabatic process is one in which no heat enters or leaves the system.


The thermdynamic first law takes the form

dU=PdV,


since dQ=0.

Note that Cv=(U/T)V , but the internal energy of an ideal gas depends only on the temperature and is independent of the volume (because there are no intermolecular forces). Thus for an ideal gas, we have

dU=CvdT=PdV


PV=RT=CpCv)T

γ=Cp/Cv


Then we have

TVγ1=const
dln(T)/dt=(γ1)u
where we have use 
1VdVdt=u
where u is the flow speed.

Since KBTE, here E is thermal energy. We have 
dln(E)/dt=(γ1)u

For non-relativistic particle, γ=5/3, E=p2/m.
For relativistic particle, γ=4/3, E=pc,
we have 
dpdt=p3u

The bulk of particles lose energy by adiabatic cooling due to the work that protons exert on the expanding material. Particles (pressure ?) do (positive) work on the expanding medium cause their bulk energy reduce.


Reference:

https://github.com/bolverk/theoretical_physics_digest

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