Analysis of energy–size reduction relationships in batch tumbling ball mills

A theoretical energy–size reduction relationship is derived for tumbling ball mills based on a solution of the integro-differential equation of comminution kinetics, in which the proportional relationship is applied between the grinding rate constant and the net mill power. The derived formula is similar to an empirical energy law, dW ∝ dxr/xri, where W is the specific energy input, xr is the particle size of product and the exponent i is shown to be a variable depending upon the ground material, the type of mill and the method to measure energy. Derived results are confirmed with reported data in reasonable agreement. Also, the Bond's energy law is examined and a method for the correction of the Bond work index is discussed.

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► A theoretical energy-size reduction relationship is derived for tumbling ball mills.
► The theory finds the formula of empirical energy laws to be valid.
► Derived results are confirmed with reported data in reasonable agreement.
► A method for the correction of the Bond work index is proposed.