In this paper we formulate a multiphase model with nonequilibrated temperatures but with equal velocities and pressures for each species. Turbulent mixing is driven by diffusion in these equations. The closure equations are defined in part by reference to a more exact chunk mix model developed by the authors and coworkers which has separate pressures, temperatures, and velocities for each species. There are two main results in this paper. The first is to identify a thermodynamic constraint, in the form of a process dependence, for pressure equilibrated models. The second is to determine one of the diffusion coefficients needed for the closure of the equilibrated pressure multiphase flow equations, in the incompressible case. The diffusion coefficients depend on entrainment times derived from the chunk mix model. These entrainment times are determined here first via general formulas and then explicitly for Rayleigh-Taylor and Richtmyer-Meshkov large time asymptotic flows. We also determine volume fractions for these flows, using the chunk mix model.