This document summarizes the development of an integral perturbation solution of the equations governing momentum transport in microchannels between disks of multiple-disk drag turbines such as the Tesla turbine. This analysis allows a parametric study of turbine performance based on several nondimensional parameters. The results of this analysis are then compared to two sets of test data published in previous work and by other projects. The results are further compared to Computational Fluid Dynamics (CFD) simulations. Finally, expected performance and potential applications of these devices are discussed in light of the results developed.
Analysis of this type of flow problem is a key element in the optimal design of Tesla drag-type turbines for geothermal, waste heat, energy harvesting, or solar alternative energy applications. In multiple-disk turbines, high speed flow enters tangentially at the outer radius of cylindrical microchannels formed by closely spaced parallel disks, spiraling through the channel to an exhaust at a small radius or at the center of the disk. Previous investigations have generally developed models based on simplifying idealizations of the flow in these circumstances. Here, beginning with the momentum and continuity equations for incompressible and steady flow in cylindrical coordinates, an integral solution scheme is developed that leads to a dimensionless perturbation series solution that retains the full complement of momentum and viscous effects to consistent levels of approximation in the series solution. This more rigorous approach indicates all dimensionless parameters that affect flow and transport, and allows a direct assessment of the relative importance of viscous, pressure, and momentum effects in different directions in the flow. The resulting lowest-order equations are solved explicitly and higher order terms in the series solutions are determined numerically.
Enhancement of rotor drag in this type of turbine enhances energy conversion efficiency. A modified version of the integral perturbation analysis is presented that incorporates the effects of enhanced drag due to surface microstructuring. Results of the model analysis for smooth disk walls are shown to agree well with experimental performance data for two prototype Tesla turbines, and predictions of performance models developed in earlier investigations. Specifically, experimental efficiencies corelate well with those predicted by the integral perturbation solution, deviating by an average of 29% and a maximum of 52%. Model predictions indicate that enhancement of disk drag by strategic microstructuring of the disk surfaces can significantly increase turbine efficiency. Exploratory calculations with the model indicate that turbine efficiencies exceeding 75% can be achieved by designing for optimal ranges of the governing dimensionless parameters.
The same parametric trends in performance are compared to test data for a micro-scale Tesla turbine with water as a working fluid. Experimental efficiencies again correlate well with those predicted by the integral perturbation solution. Exerimental efficiencies show a mean deviation of 52% with efficiencies predicted by the model, and a max deviation of 65%. A Computational Fluid Dynamics (CFD) model is then compared to both the analytical and experimental turbine efficiencies. The CFD solutions of the flow field are then used to help reconcile areas where the analytical predictions do not match experimental data. CFD predicted efficiencies match the efficiencies predicted by the integral perturbation solution very closely, deviating by an average of only 18%.
Based on the results of the CFD simulations and experimental data, conclusions are made about the validity of the integral perturbation solution. The model accurately predicts the flow inside the rotor, but a better treatment of the flow in the inlet to the turbine is necessary. Despite this, the integral perturbation solution is shown to be capable of directing high efficiency turbine design, and design strategies and parameter ranges that result in high efficiency devices are outlined.