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Registered Member #190
Joined: Fri Feb 17 2006, 12:00AM
Location:
Posts: 1567
This question is for someone knowledgeable in thermodynamics.
The polytropic law states:
(1) P1V1^n = P2V2^n
The perfect gas equation states:
PV = mRT --> P1V1/T1 = P2/V2/T2
If T1 = T2 then (2) P1V1 = P2V2
So, how can equation 1 and 2 both be true for the same gas? If the gas follows a polytropic process, where n ≠1, how can 2 be correct when there is no temperature change?
Registered Member #190
Joined: Fri Feb 17 2006, 12:00AM
Location:
Posts: 1567
Dr. Slack wrote ...
Try wikipedia -> on polytropic processes
Doesn't seem to answer the question. On page 116 of Rayner Joel's Engineering Thermodynamics, both equations are used to derive another set of equations. One equation is setting n = 1; the other is just leaving it as n. This does not make sense.
Registered Member #72
Joined: Thu Feb 09 2006, 08:29AM
Location: UK St. Albans
Posts: 1659
IamSmooth wrote ...
Dr. Slack wrote ...
Try wikipedia -> on polytropic processes
Doesn't seem to answer the question. On page 116 of Rayner Joel's Engineering Thermodynamics, both equations are used to derive another set of equations. One equation is setting n = 1; the other is just leaving it as n. This does not make sense.
This is a classic model/system confusion. The systems behave as they behave, under the conditions pertaining at the time. Their behaviour does not follow simple equations, their behaviour can be approximately described by simple equations. If you change the conditions, the describing equations will often change. So if you compress gas adiabatically, or isothermally, you need a different value of n, because you are operating the system under different conditions.
But I understand your pain, it would be nice if all systems followed simple rules, rather than did what they did and left us trailing in their wake trying to figure out how best to describe them.
Registered Member #2099
Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1716
Dr. Slack wrote ...
This is a classic model/system confusion. The systems behave as they behave, under the conditions pertaining at the time. Their behaviour does not follow simple equations, their behaviour can be approximately described by simple equations. If you change the conditions, the describing equations will often change. So if you compress gas adiabatically, or isothermally, you need a different value of n, because you are operating the system under different conditions.
But I understand your pain, it would be nice if all systems followed simple rules, rather than did what they did and left us trailing in their wake trying to figure out how best to describe them.
Well said, and not just applicable to thermodynamics. It brings to mind: "All models are wrong, but some models are useful."
Registered Member #2099
Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1716
Adding words to questions already answered. Suppose you have an ideal gas initially at P1, V1, T1 state. And you expand it to P2 < P1.
If you do that isothermally, volume goes up in same ratio as pressure goes down. Some heat must be supplied, and some work can be done.
Suppose you expand it adiabiatically and reversibly (isentropically) instead. Then the temperature drops. The final volume (and work done) are less than in the isothermal case. The pressure ratio is split between temperature ratio and volume ratio, according to a polytropic exponent called gamma. (gases following the gamma law can still be ideal).
Simply saying "adiabatic expansion" is NOT sufficient to figure the final temperature and volume. We're still talking about ideal gas and simple equations of state. But if the expansion is, for example, through a porous plug, no work is done, though the final temperature and volume are greater than with reversible expansion.
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Polytropic process vs characteristic equation for a perfect gas
