Interaction Between the Jet and Pelton Wheel

Abstract

The impingement of a round jet on a flat plate at an angle θ represents a basic model of jet mechanics (Fig. 5.1). To calculate the spreading of the water sheet on the plate surface, the mass, momentum, and energy conservation laws need to be used. Under the assumptions of frictionless deflection of the jet and frictionless spreading of the water sheet, it can be deduced from the conservation law of energy that the spreading speed of the water sheet on the plate surface is equal to the jet speed. The flow distribution along a periphery circle as well as in the radial extent can then be calculated by means of the law of momentum. The mass conservation law determines that the integrated mass flow along each periphery circle should reflect the mass flow of the jet. The first accurate calculation of such a flow distribution was conducted by Hasson and Peck (1964). The distribution of the water-sheet height along the periphery of an arbitrary circle is given by

$$ \frac{2r\cdot h}{R^2}=\frac{{ \sin}^3\theta }{{\left(1- \cos \theta \cos \varphi \right)}^2}. $$

The center of the arbitrary circle coincides with the stagnation point of the round jet on the flat plate and is eccentric to the jet axis at a distance s which is given by

$$ \frac{s}{R}= \cos \theta . $$

The interaction force between the jet at a speed C and the flat plate can be determined again by the law of momentum. Since the flow is assumed to be free of friction and hence no force component in the plane of the plate exists, the resulting interaction force is perpendicular to the flat plate. Using the momentum law in the normal direction of the flat plate, then it is calculated as

$$ {F}_{\mathrm{jet}}=\pi {R}^2\cdot \rho {C}^2 \sin \theta . $$

This force is referred to as . Near the stagnation point beneath the water sheet, some overpressure exists. Its integration over the plate must be equal to the jet impact force from Eq. (5.3). The pressure distribution in the immediate vicinity of the stagnation point has been determined in a study by G.I. Taylor (1960), among others.