High reynolds number fluid dynamics and heat and mass transfer in real concentrated particulate two-phase systems

Formation of an air‐water two‐phase flow

“…General but untractable forms for these terms have been known for a long time (Vernier and Delhaye, 1968), and some expressions are available (Zuber, 1964;Yaron and Galor, 1973). Furthermore, they must include a differential part (Bourk, 1973;RCocreux, 1974), as explained below.…”

“…and virtual mass force given by Instead of considering a spherical bubble moving in a liquid sphere (Zuber, 1964;Yaron and Galor, 1973), the bubble is assumed to move in a mixture of approximate density p~ (1 -a). Hence, the virtual mass force is modified as which is a differential interaction term.…”

Discussion: “Mixing and Reacting Flow Characterization” (Morganthaler, J. H., 1975, ASME J. Fluids Eng., 97, pp. 608–614)

“…It should first be stressed that Soo's reasoning hold only for small volume fraction of the dispersed phase, i.e., dilute systems, as he states in his explanation following his equation (1). Indeed, the pressure gradient had been neglected in our approximate solutions for very dilute systems.…”

“…A resolution of the problem of whether or not to include pressure gradients in the equation of motion should be considered from more general and fundamental points of view than those presented by 800. These should include detailed microscopic analyses of the flows around all individual particles as we had previously done in analyses of ensembles of particles [1,2]. 4 Under certain operating conditions involving high mass transfer rates one must also take into account contribution of "diffusion drag" to the total drag [3].…”

“…Such analyses are categorically different from Soo's or our latest macroscopic approaches in which one does not go into details of the flow around the particles. Thus in our equation (3) the pressure term is an average pressure gradient that is roughly proportional to the mass concentration of the particles in the mixture and it does not take into account the local flluid accelerations in the vincinity of each particle in the ensemble (see equations (29) and (3) in Reference [1]).…”