Simply put, the turbine equation of Euler states that energy is conserved in an adiabatic rotating machine. “Adiabatic” means no heat is added, no heat is lost. Then Euler’s equation states that gas parcel at radial position “1” has the same energy at radial position “2”. Throughout this presentation, we write Euler’s equation in terms of temperature.

The turbine equation of Euler written in vectorial notation is:

This is Euler’s turbine equation for the simple case when the flow exits at the center and the relative velocity of the flow v’ is orthogonal to the vector product (omega x r):

In engineering textbooks, Euler’s turbine equation is not written in vectorial form; it usually looks like

(as it appears in Z. S. Spakovszky, Unified: Thermodynamics and Propulsion (MIT Lecture Notes), ch.12.3 (MIT, 2007)). Set the “b” radial position at the center r=0 and c= omega * r at the “c” radial position and this equation reduces to the simple form of Euler’s turbine equation shown immediately above.

It is important to have an understanding of Euler’s turbine equation, since it is the governing physical law of the vortex tube effect.

The information contained in this site is based on the following research articles written by Jeliazko G Polihronov and collaborators:

• “Thermodynamics of Angular Propulsion”
• Physical Review Letters 109, 054504, 2012.
• “Vortex Tube Effect Without Walls”
• Canadian Journal of Physics, doi:10.1139/cjp-2014-0227 (2015).
• “Angular Propulsion - The Rotational Analog of Rocket Motion”
• Canadian Journal of Physics, doi:10.1139/cjp-2014-0297 (2014).
• “On the Thermodynamics of Angular Propulsion",
• Proceedings of the 10th International Conference on HEFAT, 14-16 July 2014, Orlando (2014).