Universal Form of Radial Hydraulic Machinery Four-Quadrant Equations for Calculation of Transient Processes

Abstract

Suter curves for the Wh and Wm characteristics and four-quadrant (4Q) diagrams of 11 radial pump–turbine models with different specific speeds (nq = 24.34, 24.8, 27, 28.6, 38, 41.6, 41.9, 43.83, 50, 56, and 64.04) are presented for the first time in this paper, as well as Suter curves for two pump models (nq = 25 and 41.8) previously published in the literature. All of these curves were analyzed to establish a certain universal law of behavior, depending on the specific speed. To determine such a law, a fitting procedure using regression and spline methods was carried out. This paper provides details of a research plan and structures (including data collection for four-quadrant diagrams for pump–turbine and pump models under different specific speeds nq), a procedure for re-calculating four-quadrant diagrams of the models as Suter curves for the Wh and Wm characteristics, definitions of the optimal points for pump and turbine operating modes in pump–turbine models under different specific speeds, and the development of numerical models in MATLAB to obtain a universal equation for the Wh and Wm characteristics. The scientific contribution of this paper is that it is the first to publish original mathematical curves using universal equations for the Wh and Wm characteristics of radial pumps and pump–turbines. The applicability of the equations is demonstrated by considering a pumping station in which two radial pumps were installed, for which the calculation of transient processes was performed using a numerical model developed in MATLAB by the authors. The transition process results are compared for two cases: first, when input data in the numerical model are used with the values of the Suter curves for the Wh and Wm characteristics obtained by re-calculating the four-quadrant operating characteristics (Q11, n11, M11) at a given specific speed, and second, when the values of the Suter curves for the Wh and Wm characteristics are obtained from the universal equations.