On Scaling Down Turbines to Millimeter Size

The purpose of this work is to establish the maximum theoretical efficiency that a viscous flow turbine (such as a Tesla turbine) can achieve. This is very much in the spirit of the Betz limit for wind turbines. The scaling down of viscous flow turbines is thought not to alter this result, whereas the scaling down of conventional turbines, whether axial or radial flow, results in an ever lowering of their efficiencies. A semiempirical scaling law is developed for conventional gas turbines using published machine performance data, which is fitted to a simple boundary layer model of turbine efficiency. An analytical model is developed for a viscous flow turbine. This is compared to experimental measurements of the efficiency of a Tesla turbine using compressed air. The semiempirical scaling law predicts that below a rotor diameter of between about $11mm$ and $4mm$⁠, a practical Brayton cycle is not possible. Despite that, however, and for rotor diameters less than between about $7mm$ and $2mm$⁠, a viscous flow turbine, compressor, or pump will be more efficient than a conventional design. This may have a significant impact on the design of microelectromechanical system devices.