Modelling the mean temperatures used for calculating heat and mass transfer in sprays

Abstract

This paper deals with a variation on the approach to the modelling of sprays developed by Beck, J.C., Watkins, A.P., 2003a. On the development of a spray model based on drop-size moments. Proc. R. Soc. Lond. A 459 1365. In that model the size information concerning the spray is obtained by solving transport equations for two moments of the drop number size distribution, and their respective mean velocities, and obtaining two other moments from an assumed size distribution. The sub-models required for hydrodynamic phenomena, such as drop drag, break-up and collisions, have been presented in Beck, J.C., Watkins, A.P., 2002. On the development of spray sub-models based on droplet size moments. J. Comp. Phys. 182 586–621. The heat and mass transfer sub-models are described in Beck, J.C., Watkins, A.P., 2003b. The droplet number moments approach to spray modelling: the development of heat and mass transfer sub-models. Int. J. Heat and Fluid Flow 24 242–259. A single transport equation for the energy of the liquid phase is solved from which the local value of the volume-average liquid temperature is obtained. Consequently all the heat and mass transfer phenomena are related to this temperature. In practice it is the drop surface temperature that determines the heat and mass transfer rates. This paper explores the development and application of a simple model that assumes a parabolic temperature profile within individual drops to calculate a surface-area-average temperature for the spray locally. The models are applied to the simulation of sprays having assumed drop size distributions with Sauter mean radii at inlet in the 7–70 μm range. The use of surface-area-averaged temperatures, in place of volume-averaged ones, results in substantially higher levels of mass being evaporated from the drops. This is predominantly because of the increased film mean temperature and resulting increases in the vapour pressure in the film. The effects of the fuel used are also explored. The role of the vapour pressure in the film is again found to be the important parameter determining the different mass transfer rates for the different fuels. Comparisons made with spray penetration data of diesel fuel for a wide range of injector and ambient gas conditions indicate that the model reacts correctly to changes in these parameters. Comparisons are also made with measured liquid velocities and mean drop sizes. The importance of break-up and collision models on both the hydrodynamics and the heat and mass transfer in simulations made with the larger drops at inlet is demonstrated.