Gross thermodynamics of heat engines in deep interior of Earth

From the gross conservation laws of thermodynamics in a convecting material we derive a bound on the ratio of the rate of production of mechanical or magnetic energy to the rate of internal radioactive heating which drives the convection. Our bound for this "efficiency" depends on the temperatures in the material, and can exceed unity. Whether the bound can be attained by "efficiencies" in real fluids is not known, but a simple machine shows that "efficiencies" larger than unity are physically realizable. Our bound gives upper limits on the viscous dissipation in the earth's mantle and ohmic heating in the core, but these limits are too large to be physically interesting.