Momentum and energy balances for dispersed two-phase flow

Summary The equations of motion and the mechanical energy balances for two-phase flow systems are derived by integration over a volume containing a large number of elements of the dispersed phase.List of symbols A, A boundary of volumes V, V - dA, dA surface element of A, A - A_s boundary of particles in V - dA_s surface element of A_s - F force per unit volume of the system - g –gz=gravity vector - g acceleration by gravity - I unit tensor - p pressure - Q dissipation in the continuous phase - Q_s dissipation in the dispersed phase - R compression work in the continuous phase - R_s compression work in the dispersed phase - t time - u velocity of continuous phase - u_s velocity of dispersed phase - u magnitude of u - u_s magnitude of u_s - V volume in the two-phase system - V part of V occupied by the continuous phase - W work done by F - z vertical coordinate - local volume fraction of the dispersed phase - pI–=stress tensor of the continuous phase - _s turbulent particle stress tensor - density of the continuous phase - _s density of the dispersed phase - shearing-stress tensor of the continuous phase - _s turbulent particle shearing-stress tensor - nabla operator - u, u_s velocity gradient tensor - substantial derivative(Shell Internationale Research Maatschappij N.V.)(Bataafse Internationale Petroleum Maatschappij N.V.)