A general theory of isotopes separation in a gas centrifuge based on a radial averaging method has been developed. The ultimate upper limit for a centrifuge separative power is expressed as a function of its external parameters. This is a more accurate definition of the centrifuge efficiency upper limit than the well-known Dirac's estimation or estimations of his followers, because it takes into account the feed flow value (throughput) and it provides the way to a one-dimensional diffusion equation without a lot of assumptions. For the first time the problem of an energy efficient centrifuge is formulated and the solution is obtained. Two cases of a centrifuge internal flow optimization are compared. The optimal internal circulations for both optimization cases are calculated. The results help us to understand how far the modern centrifuges are from their highest possible effectiveness limit and to identify ways to improve centrifuge performance.