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polytrope
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 |
(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.
![\begin{pspicture}(-0.5,-0.5)(7,5.7) \psaxes[Dx=1,Dy=1]{->}(0,0)(5.4,5.4) \rput(5... ...{$\displaystyle y \,=\, \frac{1}{x^n}$\ \;for\; $n = 0,\,1,\,2$} \end{pspicture}](http://images.planetmath.org/cache/objects/12110/js/img2.png)
![\begin{pspicture}(-7,-0.5)(7,5.7) \psaxes[Dx=1,Dy=1]{->}(0,0)(-5.4,0)(5.4,5.4) \... ....5,3){Cubic hyperbola\;\; $\displaystyle y \,=\, \frac{1}{x^2}$} \end{pspicture}](http://images.planetmath.org/cache/objects/12110/js/img3.png)
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 $n$ .