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In a memoir titled Théorie plus complette des machines qui sont mises en mouvement par la réaction de l'eau (“A more complete theory of machines which are put in motion by the reaction of water”) [L. Euler, Mémoires de l'académie des sciences de Berlin, 10, 227–295 (1756)], Euler (1707–1783) consolidates what is considered to be the first general theory of rotating hydraulic machines and proposes a first model of a hydraulic turbine. In a time span of about nine years, Euler had addressed the core of the theory in four publications, which cover basic topics such as to find the moment applied to the rotor of the machine—from which the shaft torque could be found—to find the pressure acting on the internal channels of the machine, and to find the power that the machine is capable to deliver. Here, Euler correctly finds that the power depends only on the discharge and on the water head. From the pressure equation, Euler also anticipated the phenomenon of cavitation—later known to be a major hindrance to the performance of hydraulic machines. At Euler's time, the conservation equations of Fluid Mechanics had not been proposed yet, and, therefore, he could not count on the form of the equation that would be readily applicable to the modeling, that is, the integral form of the moment of momentum equation in relation to an axis. As we shall see, this makes his developments quite lengthy and intricate. The present work is based on primary sources and has the goal of examining the developments of the 1756 memoir, to show Euler's intuition and skills in the development of the theory which was applied to an archaic form of a hydraulic turbine (the Segner–Euler turbine), showing the pervasiveness of the theory which still undergirds the modeling of pumps and turbines today.
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