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