Conservation of energy equation for a rotating component

A number of rotating components are in common use in process plants, e.g. turbines, compressors, pumps, centrifuges and stirring paddles, and it is important to understand how such pieces of equipment are affected by the conservation of energy. For a rotating component, the principle of conservation of energy states that 

A formal differentiation of (3.48) with respect to time gives  

The term in dJ/dt has been retained, since it is needed routinely in, for example, the modelling of solids-liquid separation centrifuges and it may be needed to model other rotating components in fault situations where change of geometry is possible. 

Power lost to friction is likely to have a component proportional to for motion against a lubricated surface and a component proportional to for motion through fluids. 

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