Polytropic process

[sailoday28]

Can a polytropic process define a problem in which flow occurs?

pv^n = constant

I raise this question because I have been under the impression that this process is where the mass remains constant and there is no flow. For example, the compression or expansion stroke of a reciprocating compressor.
Further, it is for use only with a perfect gas.

For example, in a fanno process, which is adiabatic, pv^n, will imply heat transfer.

Please provide polytropic flow examples for perfect and non perfect gases or comments in general.

 

[Onno]

Depending on the value of n you have different processes taking place: isothermic, isobaric, adiabatic. (

http://www.taftan.com/thermodynamics/POLYTROP.HTM)

These are "reversible" processes. I wonder if your question ia roused by one of the processes in special. Perhaps you can add a little more light to your question.

 

[sailoday28]

Onno (Mechanical) The website that you have referenced gives examples for non-flow perfect gases. My question is the validity of pv^n =constant for
processes in which
cp, cv are not constant
real gases
and those in which flow takes place.

 

[imok2]

These comments were taken from "Principles of Refrigeration"R.J. Dosset and T. Horan.5th edition
" Broadly defined, a polytropic process is any process in which the specific heat of the gas remains constant. by this defination, all five thermodynamic processes are polyropic. Gewnerally the term polytropic is only applied to those processes whose path falls between those of the isothermal and adiabaticprocesses. In other words, any expansion process in which the energy to do the work of expansion is supplied partly from the surroundings and partly from the gas within the system will follow a path that falls between those of isothermal and adiabatic. In processes in which the greater part of the energy to do the work comes from the surroundings, the process path approaches the isothermal, Similarly, when the geater part of the energy to do the work on the surroundingscomes from the gas itself, the process path approaches that of the adiabatic process. These relationships also apply to the compression process. When the quantity of heat that a gas loses to its surroundings during a compression process is inadequate, it's temperature cannot be constantly maintained. Consequentlythe compression process is polytropic so that the the greater the transfer of heat the
the closer this polytropic compression process approaches the isothermal process path. Conversely, the smaller the trasnfer of heat the closer the polytropic compression processapproaches that of an adibatic process. In Fact, the actual compression process of a as very nearly approaches adibatic compression because the time of compression is very short".

 

[sailoday28]

imok2 (Mechanical) Following your text reference, with specific heat a constant, pv^n, applies only to a perfect gas.
This is how I was introduced to the polytropic process. Yet others have stated that the process can be applied to a flow process.
I am still waiting for input with regard to application to flow and possibly non perfect gases.

For info only, both specif heats for non perfect gases cannot remain constant. Only for the perfect gas is
Cp-Cv=R, a constant.

 

[gtsim]

Hello Sailoday28,

The web reference given by “Onno” is good one, but it does not cover the polytropic processes between the isothermal and isentropic processes. There can be infinite such processes and are all considered adiabatic – the heat transfer is insignificant compared with the work transfer. This is implied by “imok2”. We can model these processes using the relative ship PV^n=Constant i.e. polytropic and applicable to both non-flow and steady flow processes and real gases.

For a non-flow process n is defined as:

For a compression process

n-1 = (?-1)/?p where ?p is the polytropic efficiency

For an expansion process

n-1=(?-1) ?p

and T2=T1(V1/V2)^(n-1)

For a steady state process n is defined as:

For a compression process

(n-1)/n=(?-1)/(??p)

For a expansion process

(n-1)/n=?p(?-1) ?

and T2=T1(P2/P1)^(n-1)/n

? should be calculated at the average temperature for respective processes (i.e. compression or expansion) using the appropriate polynomial describing the cp with temperature variation for the gas.

Regards

 

[gtsim]

Please note ? is gamma and ?p and ??p are the polytropic efficiencies.

Regards

 

[gtsim]

There is an excellent paper on this subject “Polytropic Processes in the Performance of Prediction of Centrifugal Compressors” Mallen and Saville. C183/77 IMechE 1977.

Regards

 

[Onno]

I would like to tell something about the flow of gases as calcualted from the perfect gas law.

p.V=n*R*T

p=pa
V=m^3
n=moles
R=Joule/mole*K
T=K

Suppose 1kg of this gas

p*(V/1*kg)= R*T with R=[Joule/kg*K]

p/Rho = R*T with Rho= kg/m^3 specific weight gas.
If gas flows the state for a perfect one is described by p/Rho = R*T for a real gas. this is the same as for a non flow condition.

Now real flow. In real flow there is pressure loss, temperature change and so the specific weight of the flowing gas is changed. But it still means that for real flow problems the state equation p/Rho = R*T can be used along the line of flowdirection.

p*V^n describes a whole lot of phenomena (adiabatic, isothermal etc) For real flow problems p*V^n is not constant because friction in flow is not considered. Of course you can neglect friction but that is kind of silly to do if you are wondering about the flow for real gasses.
Theoretically, wandering around in the p-V diagram is done by assuming compression and expansion due to some change but without considering mass flow.

 

[gtsim]

PV^n can be constant, but n can vary depending in the losses (e.g. friction etc).

Regards

 

[sailoday28]

Onno (Mechanical)I think you are stating that the polytropic process is reversible. I am not disagreeing. If so then it can not be applied to flow processes where friction is taking place.

gtsim (Mechanical) If pv^n =constant, what is the basis for T2=T1(V1/V2)^(n-1) being applicable to real gases.Unless pv=ZRt where z is constant.

 

[gtsim]

Real gases are accounted in the computation of n, where the polytropic efficiency represents the effects of irreversibilities and described above. In fact, the reference I quoted takes this further by applying the principle of corresponding states to allow for the non-spherical nature of molecules (acentric factor) in calculating n, thus increasing accuracy particularly at very high pressures.

Regarding the equation of state for real gases (PV=ZRT), it is used to calculate the volumetric flow and/or density. My understanding is that compressibility factor Z is not constant but depends on the pressure and temperature.

Regards,

 

[sailoday28]

gtsim (Mechanical)
If pv^n=constant (1)
and pv=zRT or p=zRT/v (2)

Then substituting (2) into (1) yields

zRT v^(n-1)=constant (3)
now your previous formulation T2=T1(V1/V2)^(n-1)
is only true if z in (3) is a constant.

 

[gtsim]

Strictly speaking yes, but at low pressures (<=60Bar) Z is often omitted in the temperature calculation. It is at high pressures that we need to allow for varying Z in order to avoid large errors in calculation the discharge temperature. This is discussed in detail in the reference above (Mallen and Saville). Another reference that deals with polytropic analysis in flow processes is JM Schultz, "Polytropic analysis of centrifugal compressors" Trans. ASME, J for Power, 1962, Pg 84-69.