Time averaging of volume-averaged conservation equations of multiphase flow

Volume-averaging of equations of conservation for a multiphase system yields equations in terms of volume-averaged products of density, velocity, energy, stresses and field forces, together with interface transfer integrals. Solutions of these equations call for expressing these volume-averaged products in terms of products of averages. The present study shows that such a determination can be made via time-averaging over a duration T such that /tau//SUB HF/ << T << /tau/SUB LF/ where /tau/SUB HF/ is the characteristic time of high frequency fluctuations as in turbulent flow and /tau/SUB LF/ is the characteristic time of low frequency change as in a blow-down process. Time-averaging reduces the volume- and areaaveraged products to products of averages plus terms representing eddy diffusivities of mass, Reynolds stresses, eddy conductivities of heat, etc., arising from high frequency fluctuations. The interface transfer integrals, after time-averaging, give transfer integrals of low and high frequency components. This procedure of time-averaging after volume-averaging, but not the reverse order, leads to a complete set of differential-integral equations of conservation