The Antoine Equation is a vapor pressure equation and describes the relation between vapor pressure and temperature for pure water between 0°C and 373.946°C. [Wikipedia]
For T in Kelvins and P in kPa:
273.150 < T ≤ 373.150: A = 16.5699, B = 3984.92, C = −39.724
Antoine(T) = exp(A − B / (T + C))
For temperatures above the critical temperature (373.946°C / 647.096K), where water is a superfluid, the increase in vapor pressure appears essentially linear† with the same slope (i.e. dP/dT ≈ 235.88 kPa/°C). This behavior appears consistent with the Ideal Gas Law and statements that supercritical water behaves like an ideal gas. Now if we define Tc = 647.096 K, we can write
P(T ≤ Tc) = Antoine(T)
Here I've plotted the vapor pressure of a fixed volume of water (in red) and the pressure of a fixed volume of air (in green) between 0°C and 373.946°C for comparision.
It was this graph that led me to ask the question “What if I replace all the air in a fluidyne with water and operate it above the critical temperature?” I decided to call such an engine a “hydrodyne” to avoid confusion with fluidynes.
For readers who have no experience of supercritical fluids, this video shows a fluid being heated through its critical point to become a superfluid, and then cooling to change back to a fluid. The video shows Xenon, but water would look exactly the same.
The formula for the difference in fluid levels within a U-tube manometer is
h = (Pa − Po) / (g · ρ), and when I used...
Pa = 21718 kPa applied pressure,
...I obtained h ≈ 2.2 km, as the head of a hydrodyne pump operating at Tc.
† Data from S. L. Rivkin and T. C. Akhundov plotted in “Steam Tables” by Keenan, Keys, Hill, and Moore. The authors cited “Teploenerg.”, No. 1, 57-65 (1962), and No. 9, 66-68 (1963).