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Maximum Length for the Supersonic Flow

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^9.8. These results were obtained from the mathematical derivations but were verified by numerous experiments^9.9. The maximum length of the supersonic can be evaluated when M = ∞ as follows:

$\displaystyle {4 fL_{max}\over D} = {1 - M^2 \over k M^2} + {k+1 \over 2k}\ln {{k+1 \over 2}M^2 \over 2\left(1+ {k-1 \over 2}M^2 \right)}$

$\displaystyle \fld \left( M\rightarrow\infty \right) \sim {- \infty \over k \times \infty} + {k + 1 \over 2k} \ln { (k+1) \infty \over (k-1) \infty}$

$\displaystyle = {-1 \over k} + {k + 1 \over 2k} \ln { (k+1) \over (k-1) }$

$\displaystyle {4 fL_{max}\over D} = \fld ( M\rightarrow\infty , k=1.4) = 0.8215$

(9.50)

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, as shown in Figure (9.5).

**Figure 9.5:** The maximum length as a function of specific heat,

Next: Working Conditions Up: Fanno Flow Previous: Supersonic Branch Index

Created by:Genick Bar-Meir, Ph.D.
On: 2007-11-21