Modeling the flow of non-Newtonian melts in single-screw extruders generally requires numerical methods. In this study, we analyze viscous dissipation for fully developed flow of non-Newtonian fluids within three-dimensional metering channels of single-screw extruders. Applying the theory of similarity we identified a set of four independent dimensionless influencing parameters that fully describe the underlying physics of the considered problem: the channel height-to-width ratio h/w, the screw-pitch ratio t/D-b, the power-law exponent n, and the dimensionless down-channel pressure gradient Pi(p, z). Based on a comprehensive numerical parametric design study we developed two generalized symbolic regression models predicting dimensionless viscous dissipation - one for given pressure gradient, and the other one for given throughput. As a result of the three-dimensional modeling approach our regression models take the effects of flight flanks, transverse flow, and non-Newtonian fluid behavior into account. The accuracy of these developed models was proven by means of an error analysis using an independent data set. These models allow fast, stable and accurate predictions of viscous dissipation, thus contributing to the optimal design of extruder screws without the need for numerical simulations. Furthermore they are applicable to injection molding machines and for predicting the required drive power.