The droplet number moments approach to spray modelling: The development of heat and mass transfer sub-models

In the past most poly-disperse spray models have been based on either discretising the liquid flow field into groups of equally sized droplets, as in the discrete droplet model (DDM) in which parcels of drops are tracked in space in a Lagrangian framework, or by solving separate Eulerian conservation equations for a number of size ranges. Both of these approaches can result in very long computing times, although the DDM is generally regarded as being superior in this respect under most conditions of interest. Recently an alternative approach to the modelling of sprays has been developed by Beck [Ph.D. Thesis, UMIST, 2000] and Beck and Watkins [Proc. R. Soc. Lond. A, 2003]. In this approach the size information concerning the spray is obtained by solving conservation and transport equations for two moments of the drop number distribution, and their respective mean velocities, and obtaining two other moments from an assumed size distribution function. The sub-models required in this approach, for hydrodynamic phenomena in sprays, such as drop drag, break-up and collisions, have been presented elsewhere [Beck, Ph.D. Thesis, UMIST, 2000; Beck and Watkins, J. Comp. Phys., 2002]. The purpose of this paper is to present those sub-models relating to the mass and heat transfer processes in sprays. As part of this, an equation for the energy of the liquid phase is required. Standard gas phase equations, including a k–ε turbulence model, are also solved. All the equations are solved in a Eulerian framework using the finite-volume approach. The inter-phase heat and mass transfers are captured through the use of source terms, and all the source terms for these aspects of the spray model are derived in this paper in terms of the four moments of the droplet number distribution in order to find the net effect on the whole spray flow field. The model has been applied to a wide variety of different sprays, including high-pressure diesel sprays, wide-angle solid cone water sprays, hollow cone sprays and evaporating sprays [Beck, Ph.D. Thesis, UMIST, 2000]. The comparisons of the results with experimental data show that the model generally performs well. In this paper the evaporation effects are examined and compared with experimental data wherever possible. Again this aspect of the spray model is shown to be generally successful.