Hello
I am self-learning the DeLaval nozzle, or Convergent-Divergent nozzle, and have read much from many different sources.
Have one question about the C-D nozzle with air.

The standard math shows choking at Mach 1, and the math derivations describe this. In my readings, I have found a single reference that does not agree with the standard formulas, taking into consideration real gas effects. This fellow, Abramov, is saying that considering air as a real gas, the throat critical flow is not M=1, but is instead much lower at M=0.5345 for air. His conclusion is that the effect of increasing velocity at the diverging section can be achieved with all velocities less than M=1 in the pipe, thereby avoiding Mach waves and shock. This seems odd to me.

QUESTION: Is this a legitimate statement, or is this false? If false, can you point to the falsity?

This is an interesting question because I can see two possible answers, and both contain amazing implications.
1. Abramov is correct that in practice, the DeLaval effect can occur at M=0.5345 at throat, which implies all other reference materials and texts are erroneously assuming ideal gas behavior for air in a simple C-D nozzle. For this case, it leads to many energy-efficient applications of devices that could avoid shock entropy generation.
2. Abramov is not correct. He has generated a steady stream of patent applications and some granted patents over the last decade, all of which are based on this core idea of when the Delaval effect occurs. I doubt he has made a mistake, which then leads to the possibility of a complicated, long-term Scam or ruse. Equally amazing possibility here.

Abramov does supply a derivation of his math, and I can see the algebra is correct there, but is there some error buried within this derivation that you can identify? Possibly the very first step of his math contains an assumption that is wrong? If Abramov's documents are part of a Scam, it should be possible to reveal that via his math derivation steps, but my ability level is not high enough to see if there is an error buried there, especially if the error was purposely buried.

Attached is an edited pdf excerpt from his original lengthy document. Equations 6.1 to 6.9 show his derivation.

Abramov also states that "in practice" the DeLaval effect is seen at M<1. If a person has hands-on experience with C-D nozzles, then this is another path that could be used to reveal if Abramov is telling the truth or not.

Any help would be much appreciated. I hope that you see this as an interesting problem. Of course, I can send you any additional material, if needed.
Thanks again for your time,

• https://files.engineering.com/getfile.aspx?folder=01feb80d-d90b-4d6c-a6a4-3