Polytropic change of state

In thermodynamics , a change of state of a system in which the equation applies to pressure and specific volume is called polytropic . The exponent is called the polytropic exponent . In technical processes, the polytropic exponent can be viewed as constant. In the pv diagram, a polytrope takes the form of a power function with a negative slope. ${\displaystyle p}$ ${\displaystyle v}$ ${\displaystyle pv^{n}=\mathrm {const} }$ ${\displaystyle n}$

Special cases of the polytropic change of state are:

Special cases of the polytropic change of state

• ${\displaystyle n=0}$ : isobaric
• ${\displaystyle n=1}$ : isothermal
• ${\displaystyle n=\infty }$ : isochoric
• ${\displaystyle n=\kappa ={\frac {c_{\mathrm {p} }}{c_{\mathrm {v} }}}}$ : isentropic or adiabatic - reversible

The heat supplied to a gas during this change of state is given by:

${\displaystyle Q_{12}=m\ c_{\mathrm {v} }{\frac {n-\kappa }{n-1}}\ (T_{2}-T_{1})}$

Here referred to the mass , and start and end temperature of the process. The polytropy is characterized by a fixed heat capacity , which is from , and results. ${\displaystyle m}$ ${\displaystyle T_{1}}$ ${\displaystyle T_{2}}$ ${\displaystyle c_{\mathrm {p} }}$ ${\displaystyle c_{\mathrm {v} }}$ ${\displaystyle n}$

One also speaks of a polytropic equation of state :

${\displaystyle p=K\cdot \rho ^{\gamma }}$

with the pressure p, the density , the polytropic constant K and the polytropic index m in . It is used, for example, in astrophysics ( Lane-Emden equation ). ${\displaystyle \rho }$ ${\displaystyle \gamma =1+{\frac {1}{m}}}$

Ideal gases

For ideal gases with isentropic changes of state, the following relationships also apply:

${\displaystyle {\frac {T_{2}}{T_{1}}}=\left({\frac {p_{2}}{p_{1}}}\right)^{\frac {n-1}{n}}=\left({\frac {V_{1}}{V_{2}}}\right)^{n-1}}$ or.

${\displaystyle {\frac {p_{2}}{p_{1}}}=\left({\frac {T_{2}}{T_{1}}}\right)^{\frac {n}{n-1}}=\left({\frac {V_{1}}{V_{2}}}\right)^{n}}$

With

${\displaystyle T}$: absolute temperature

${\displaystyle p}$: Pressure

${\displaystyle V}$: Volume.

For the isentropic change of state of an ideal gas, the following applies . With the isobaric heat capacity and the isochoric heat capacity . In the case of diatomic gases (for example air as a gas mixture ) and monatomic gases ( noble gases ) can be used. ${\displaystyle n=c_{\mathrm {p} }/c_{\mathrm {v} }}$ ${\displaystyle c_{\mathrm {p} }}$ ${\displaystyle c_{\mathrm {v} }}$ ${\displaystyle n=1{,}403}$ ${\displaystyle n=1{,}66}$

literature

• Dieter Winklmair - Energy and heating technology (script, PDF) (1.08 MB)

Individual evidence

1. ↑ Fran Bosniakovic, "Technical Thermodynamics", 7th edition, Steinkopf-Verlag Darmstadt; Chapter 4.5 "Polytropic state change"
2. ↑ Peter Stephan u. a .: Thermodynamics. Basics and technical applications, Vol. 1: One-component systems . 18th edition Springer, Berlin 2013, p. 115, ISBN 3-642-30097-9 .

See also

Wiktionary: polytropic  - explanations of meanings, word origins, synonyms, translations

• Thermal equation of state of ideal gases