# 11.7.5.1: Maximum Length for the Supersonic Flow

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It has to be noted and recognized that as opposed to subsonic branch the supersonic branch has a limited length. It also must be recognized that there is a maximum length for which only supersonic flow can exist. The maximum length of the supersonic can be evaluated when $$M=\infty$$ as follows:

$\dfrac{4\, f\,L_{max} }{ D} = \dfrac{1 - M^2 }{ k\, M^2} + \dfrac{k+1 }{ 2\,k}\ln \dfrac{\dfrac{k+1 }{2}\,M^2} {2\, \left(1+ \dfrac{k-1 }{ 2}\,M^2 \right)} = \\ ld \left( M\rightarrow\infty \right) \sim \dfrac{- \infty }{ k \times \infty} + \dfrac{k + 1 }{ 2\,k} \ln \dfrac{ (k+1)\, \infty }{ (k-1)\, \infty} = \\ \dfrac{-1 }{ k} + \dfrac{k + 1 }{ 2\,k} \,\ln \dfrac{ (k+1) }{ (k-1) } = ld ( M\rightarrow\infty , k=1.4) = 0.8215$

$\dfrac{4 \,f\,L_{max} }{ D} = ld ( M\rightarrow\infty , k=1.4) = 0.8215$ The maximum length of the supersonic flow is limited by the above number. From the above analysis, it can be observed that no matter how high the entrance Mach number will be the tube length is limited and depends only on specific heat ratio, $$k$$.

## Contributors and Attributions

• Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.

This page titled 11.7.5.1: Maximum Length for the Supersonic Flow is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

This page titled 11.7.5.1: Maximum Length for the Supersonic Flow is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Genick Bar-Meir via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.