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These equations govern the power, efficiencies and other factors that contribute to the design of turbomachines. With the help of these equations the head developed by a pump and the head utilised by a turbine can be easily determined. As the name suggests these equations were formulated by Leonhard Euler in the eighteenth century. [1]
In Euler's original work, the system of equations consisted of the momentum and continuity equations, and thus was underdetermined except in the case of an incompressible flow. An additional equation, which was called the adiabatic condition , was supplied by Pierre-Simon Laplace in 1816.
Warman centrifugal pump in a coal preparation plant application A pair of centrifugal pumps for circulating hot water within a hydronic heating system. Centrifugal pumps are used to transport fluids by the conversion of rotational kinetic energy to the hydrodynamic energy of the fluid flow. The rotational energy typically comes from an engine ...
According to a form of Euler's fluid dynamics equation, known as the pump and turbine equation, the energy input to the fluid is proportional to the flow's local spinning velocity multiplied by the local impeller tangential velocity. In many cases, the flow leaving the centrifugal impeller is traveling near the speed of sound. It then flows ...
These two types of machines are governed by the same basic relationships including Newton's second Law of Motion and Euler's pump and turbine equation for compressible fluids. Centrifugal pumps are also turbomachines that transfer energy from a rotor to a fluid, usually a liquid, while turbines and compressors usually work with a gas. [1]
The Euler number (Eu) is a dimensionless number used in fluid flow calculations. It expresses the relationship between a local pressure drop caused by a restriction and the kinetic energy per volume of the flow, and is used to characterize energy losses in the flow, where a perfect frictionless flow corresponds to an Euler number of 0.
Otherwise, Euler's equation may refer to a non-differential equation, as in these three cases: Euler–Lotka equation, a characteristic equation employed in mathematical demography; Euler's pump and turbine equation; Euler transform used to accelerate the convergence of an alternating series and is also frequently applied to the hypergeometric ...
Leonhard Euler is credited with introducing both specifications in two publications written in 1755 [3] and 1759. [4] [5] Joseph-Louis Lagrange studied the equations of motion in connection to the principle of least action in 1760, later in a treaty of fluid mechanics in 1781, [6] and thirdly in his book Mécanique analytique. [5]