
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”. Before we write the actual equation, we need to explain what is meant by “energy”. To be more precise, Euler’s equation does not deal with energy, but with enthalpy of gas, which has units of energy. Some definitions of enthalpy are: Static enthalpy: Total enthalpy (stagnation enthalpy):
Rotational enthalpy (rothalpy): where v’ is the relative velocity of the gas with respect to the rotating frame; omega is the angular velocity (it is constant throughout this presentation) and r is the radial position. Now we are ready to write Euler’s turbine equation. It is: Rewrite this by plugging into it the velocity addition formula in other words, Plug this into the definition of rothalpy and get Because c_{p} and m are constants, they are absorbed in the constant at the Rhs of the equation; raising to the second power in the above expression yields which yields the turbine equation of Euler written in vectorial notation: Consider a rotating flow confined to a plane and 2 radial positions: (1) at some nonzero radius where we set v=c and and also (2) at the center of rotation r=0. Then Euler’s turbine equation yields: or the difference in total temperatures is This is Euler’s turbine equation for the case when the flow exits at the center and the relative velocity v’ is orthogonal to the vector product (omega x r). In 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:
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