We consider the mixing between two miscible liquids of slightly different density (< 10%) when one of them (cargo) is injected into a tank partially filled with the other (inventory). The injection of the cargo is such that buoyancy and inertia act in concert on the plume produced by the cargo. The two basic processes that govern the mixing of the two liquids in the tank are the entrainment of tank liquid by the plume and the tank circulation set up by this entrainment and by the plume discharge. Unlike plumes in an environment of infinite extent, the plume in this case changes its environment continuously, which, in turn, has a continuously-varying effect on the plume. A mathematical model for the mixing of the two liquids is presented, from which one can compute the tank stratification that may result when given amounts of cargo and inventory are thus mixed. Plume entrainment theory is used for the plume dynamics and a ‘filling-box’ model is used for the tank circulation. The partial differential equations of the model are integrated by an original and unique numerical method. The problem was also treated experimentally. The tank stratification is expressed in terms of a normalized density-difference variable δ. Except for some very localized large discrepancies, due to certain local effects not included in the model, computed and experimental profiles of δ agree very well, their maximum and average deviations being within 4 and 2%, respectively. It is found that values of the empirical plume parameters α and λ that are used commonly for steady plumes in environments of infinite extent are approximately right for the time-dependent plumes under consideration too.
A time dependent, two-layer frontal model of buoyant plume dynamics