Issue on Compression Thermodinamics n<k?

[ChemicalColEng]

We all now that compression thermodinamics is described as:

PV^n=constant

Where

n=1 Isothermal
n=k=Cp/Cv Isentropic
n>k Polytropic

Is possible to have a n<k compression process?

 

[ione]

If it was 1<n<k then you would have a gas with a negative specific heat (you supply heat and gas temperature drops)

 

[zdas04]

There is no closed-form definition of the polytropic exponent. It is a function of gas properties and equipment configuration. Basically for a given dynamic compressor, you back into the exponent from a series of experiments. I've never heard of a case where n<k, but with the flexible definition of "n" I would say it is far from impossible.

David

 

[chicopee]

You forgot your isobaric n value in your OP.

 

[gxmplx]

Is possible to have a n<k compression process?

Of course! In particular n=1<k is one of such processes.

Having 1<n<k only means that your process is neither isothermal nor isentropic, allow temperature variation and energy flow through the borders and you have it!

n<1 are all possible even with negative n.

It is you in your experiment who make your process undergo quasi equilibrium stages such that P v^n = constant.

Just buy moving P and V holding that relation true at all stages.

Rgrds

 

[sheiko]

ione,

Could you please elaborate what you meant?

Thanks

"We don't believe things because they are true, things are true because we believe them."

 

[iainuts]

For n = 1, the relationship has to hold: PV = constant for all points during the compression cycle.

For this to happen, temperature has to remain constant. For temperature to remain constant, heat must be removed during the compression cycle. A completely isothermal compressor would have PV = constant and therefore n=1.

For a perfectly isentropic process, n = k.

So for 1 < n < k, heat must be removed during the compression process. This isn't so unusual for reciprocating compressors, especially very small, slow moving ones. And there are numerous patents for other types of isothermal compressors which use a fluid such as water as the 'piston' for the machine. So yes, n can be between 1 and k for any compression process that removes heat during compression, but does not remove so much heat that the temperature of the gas ever drops below it's initial starting point.

 

[ione]

I’ll try to expand what I meant with my previous post.
What reported below stands for a perfect gas.

The specific heat Cn for a polytropic process is

Cn = [(n-1)*Cv –R]/(n-1)

With:
Cv = specific heat at constant volume
N = polytropic index
R = gas constant

Doing the math to have Cn>0 it must be n<1 or n> Cp/Cv = k

where Cp is the specific heat at constant pressure.

 

[gxmplx]

For a general equation for heat capacity, and not just for ideal gases look here:

http://www.manchesteruniversitypress.co.uk/uploads/docs/290227.pdf

n can take any real value. The integrals you usually see are not valid for n=1 only because this is a special case, not that the integral would not exist.

P V^n = constant, can be carried out for any real value of n.

You can find lots of insight from eng-tips searching for “politropic process”.

 

[ione]

I agree in a polytropic process n could range from - infinity to + infinity, but here we have to focus on a compression process.

We should see it from another point of view, that is from the efficiency point of view.

Polytropic efficiency for compression could be defined as:

Eff,comp = (k-1)*n/[k*(n-1)].

Now, in order to have a polytropic efficiency lower than 1 it must be: n<1 or n>k.

 

[gxmplx]

A compression process is a transformation that reduces the volume of the fluid you are compressing.

You can reduce the volume following the rule that P V^(-3.1416) remains constant.

You can also reduce the volume of the same fluid following the rule P V^(+3.1416) remains constant.

For that to happen you will have to transfer energy (in the form of work or heat) to your system (it may be + or -), but all these processes ARE PHYSICALLY POSSIBLE.

You can always reduce the volume following a P V^n=constant rule for an arbitrary (real) value of n.