Isothermal Flow

The earliest reference to isothermal flow was found in Shapiro's Book. The model suggests that the choking occurs at $1/\sqrt{k}$ and it appears that Shapiro was the first one to realize this difference compared to the other models. In reality, the flow is choked somewhere between $1/\sqrt{k}$ to one for cases that are between Fanno (adiabatic) and isothermal flow. This fact was evident in industrial applications where the expectation of the choking is at Mach one, but can be explained by choking at a lower Mach number. No experimental evidence, known by the undersigned, was ever produced to verify this finding.