A set of empirical equations has been developed which defines the peak efficiency and shape of the efficiency curve for hydraulic turbines as a function of the commissioning date for the unit, rated head, rated flow, runner speed, and runner throat or impulse turbine jet diameter. The equations are based on an analysis of peak efficiency data from 56 Francis, 33 axial-flow, and eight impulse runners dating from 1908 to the present, with runner diameters ranging from just under 0.6 m to almost 9.5 m. The metric specific speeds (n(q)) ranged from 5.3 to 294. The root mean square error of the calculated peak efficiency for Francis and axial-flow runners was found to be 0.65%. The shape of the efficiency curves was derived from eight Francis, five Kaplan, three propeller, and four impulse turbines. Charts showing the relationship between calculated and actual efficiency curves for these 20 runners are provided. A good match between calculated and measured or guaranteed efficiency was obtained. The equations were also used to determine the relative increase in peak efficiency for new reaction runners installed in existing casings at 22 powerplants, with a root mean square accuracy of 1.0%. The equations can be used to (i) develop efficiency curves for new and old runners; (ii) compare the energy output of alternative types of turbines, where this choice is available; and (iii) calculate the approximate incremental energy benefit from installing a new runner in an existing reaction turbine casing, or onto the shaft of an impulse unit.
