A polytropic process is a thermodynamic process that obeys the relation:
where p is the pressure, v is specific volume, n, the polytropic index, is any real number, and C is a constant. This equation can be used to accurately characterize processes of certain systems, notably the compression or expansion (including with heat transfer) of a gas and in some cases liquids and solids.
The following derivation is adapted from Christians. Consider a gas in a closed system undergoing an internally reversible process with negligible changes in kinetic and potential energy. The First Law of Thermodynamics is
Define the energy transfer ratio, K, as δq/δw. For an internally reversible process the only type of work interaction is moving boundary work, given by Pdv. Also assume the gas is calorically perfect (constant specific heat) so du = cvdT. The First Law can then be written
Consider the Ideal Gas equation of state with the well-known compressibility factor, Z: Pv = ZRT. Assume the compressibility factor is constant for the process. Assume the gas constant is also fixed (i.e. no chemical reactions are occurring). The PV = ZRT equation of state can be differentiated to give
Based on the well-known specific heat relationship arising from the definition of enthalpy, the term ZR can be replaced by cp - cv. With these observations First Law becomes
where γ is the ratio of specific heats. This equation will be important for understanding the basis of the polytropic process equation. Now consider the polytropic process equation itself:
Taking the natural log of both sides (recognizing that the exponent n is constant for a polytropic process) gives
which can be differentiated and re-arranged to give
By comparing this result to the result obtained from the First Law, it is concluded that the polytropic exponent is constant (and therefore the process is polytropic) when the energy transfer ratio is constant for the process. In fact the polytropic exponent can be expressed in terms of the energy transfer ratio:
This derivation can be expanded to include polytropic processes in open systems, including instances where the kinetic energy (i.e. Mach Number) is significant. It can also be expanded to include irreversible polytropic processes (see Ref ).
The polytropic process equation is usually applicable for reversible or irreversible processes of ideal or near-ideal gases involving heat transfer and/or work interactions when the energy transfer ratio (δq/δw) is constant for the process. The equation may not be applicable for processes in an open system if the kinetic energy (i.e. Mach Number) is significant. The polytropic process equation may also be applicable in some cases to processes involving liquids, or even solids.
Polytropic Specific Heat Capacity
It is denoted by and it is equal to
Relationship to ideal processes
For certain values of the polytropic index, the process will be synonymous with other common processes. Some examples of the effects of varying index values are given in the table.
|—||Negative exponents reflect a process where the amount of heat being added is large compared to the amount of work being done (i.e. the energy transfer ratio > γ/(γ-1)). Negative exponents can also be meaningful in some special cases not dominated by thermal interactions, such as in the processes of certain plasmas in astrophysics.|
|Equivalent to an isobaric process (constant pressure)|
|Equivalent to an isothermal process (constant temperature)|
|—||A quasi-adiabatic process such as in an internal combustion engine during expansion, or in vapor compression refrigeration during compression. Also a "polytropic compression" process like gas through a centrifugal compressor where heat loss from the compressor (into environment) is greater than the heat added to the gas through compression.|
|—||is the isentropic exponent, yielding an isentropic process ( no heat and no entropy transferred). It is also widely referred as adiabatic index, yielding an adiabatic process (no heat transferred). However the term adiabatic does not adequately describe this process, since it only implies no heat transfer. A reversible adiabatic process is an isentropic process.|
|—||Normally polytropic index is greater than specific heat ratio (gamma) within a "polytropic compression" process like gas through a centrifugal compressor. The inefficiencies of centrifugal compression and heat added to the gas outweigh the loss of heat into the environment.|
|—||Equivalent to an isochoric process (constant volume)|
Note that , since .
An isothermal ideal gas is also a polytropic gas. Here, the polytropic index is equal to one, and differs from the adiabatic index .
In order to discriminate between the two gammas, the polytropic gamma is sometimes capitalized, .
To confuse matters further, some authors refer to as the polytropic index, rather than . Note that
- Adiabatic process
- Isentropic process
- Isobaric process
- Isochoric process
- Isothermal process
- vapor compression refrigeration
- gas compressor
- internal combustion engine
- Quasistatic equilibrium
- Christians, Joseph, "Approach for Teaching Polytropic Processes Based on the Energy Transfer Ratio, International Journal of Mechanical Engineering Education, Volume 40, Number 1 (January 2012), Manchester University Press
- G. P. Horedt Polytropes: Applications In Astrophysics And Related Fields, Springer, 10/08/2004, pp.24.
- GPSA book section 13