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# Mathematical concept

Let $n$ be a nonnegative constant. A
*polytropic equation* expresses that the real variable
$y$ is inversely proportional to the power $x^{n}$ of the real
variable $x$. So, it is a question of the equation

$\displaystyle y\;=\;\frac{c}{x^{n}}$ | (1) |

where $c$ is another constant.

The graph of a polytropic equation is called a
*polytrope*. It has the coordinate axes as asymptotes
for $n\neq 0$. Special cases of polytrope are the
hyperbola^{} $y=\frac{c}{x}$ and the
*cubic hyperbola*

$y=\frac{c}{x^{2}}.$ |

Below one sees the right halves of three polytropes given by the integer $n$ values 0 (green), 1 (cyan) and 2 (blue); farther below a whole cubic hyperbola.

(-0.5,-0.5)(7,5.7) \psaxes[Dx=1,Dy=1]-¿(0,0)(5.4,5.4) \rput(5.6,-0.2)$x$ \rput(-0.2,5.6)$y$ \psplot[linecolor=green]051 \psplot[linecolor=cyan]0.251 x div \psplot[linecolor=blue]0.4551 x div x div \psdot(1,1) \rput(3.5,3)$\displaystyle y\,=\,\frac{1}{x^{n}}$ for $n=0,\,1,\,2$

(-7,-0.5)(7,5.7) \psaxes[Dx=1,Dy=1]-¿(0,0)(-5.4,0)(5.4,5.4) \rput(5.6,-0.2)$x$ \rput(-0.2,5.6)$y$ \psdot(0,0) \psplot[linecolor=blue]-5-0.451 x div x div \psplot[linecolor=blue]0.4551 x div x div \rput(3.5,3)Cubic hyperbola $\displaystyle y\,=\,\frac{1}{x^{2}}$

# An application to thermodynamics: The reversible polytropic process

When a gas undergoes a reversible process in which there is heat transfer, the process frequently takes place in such a manner that a plot of $\log{p}$ vs. $\log{v}$ is a straight line. Here $p$ denotes absolute pressure (e.g. in psia) and $v$ specific volume (e.g. in $ft^{3}/lbm$). For such a process

$pv^{n}=\mbox{constant}.$ |

This is called a *polytropic process*. Indeed this equation represents a *constitutive equation* since it can be verifiable experimentally in a lab for different processes of heat transfer, i.e. for distinct rational values of $n$. It is obvious, from the above equation, that

$\frac{d\log{p}}{d\log{v}}=-n,$ |

where $-n$ is the slope of the straight line on the mentioned logarithmic chart. Typical polytropic processes are tabulated as follow.

Some Typical Polytropic Processes | |||
---|---|---|---|

Process Name | Constant Property | $n$ | Physical Units |

Isobaric | $p$ | 0 | lbf/in${}^{2}$ abs. |

Isothermal | $T$ | 1 | ${}^{{\circ}}\mathrm{R}$ |

Isentropic | $s$ | $k$ | Btu/lbm-${}^{{\circ}}\mathrm{R}$ |

Isochoric(Isovolumetric) | $v$ | $\infty$ | ft${}^{3}$/lbm |

Legend: $p$ $=$ abs. pressure; $T$ $=$ abs. temperature; $s$ $=$ specific entropy; $v$ $=$ specific volume

In general, one may have more complex processes of heat transfer where $n$ is any rational number^{} (e.g. $n=-2,\,-1,\,-0.5$, etc.). Specific examples are the expansion of the combustion gases in the cylinders of water-cooled internal combustion engines and reciprocating machines. In such cases the pressure and volume during a polytropic process are measured, as might be done with an *engine indicator*, so that the logarithm of the pressure and volume are plotted in order to know the *polytropic constant* $n$.

## Mathematics Subject Classification

26E05*no label found*26A15

*no label found*26A09

*no label found*26A03

*no label found*

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