A mathematical model of a three-dimensional mixing layer laden with evaporating fuel drops composed of many chemical species has been derived. The study is motivated by the fact that typical real petroleum fuels contain hundreds of chemical species. Previously, for the sake of computational efficiency, spray studies were performed using either models based on a single representative species or models based on surrogate fuels of at most 15 species. The present multicomponent model makes it possible to perform more realistic simulations by accounting for hundreds of chemical species in a computationally efficient manner.

Figure 1. The Evolution of Residual Droplet Areas was computed for the single-component and multicomponent cases. Here t* is time in units of a characteristic time calculated from initial parameters of the mixing layer, Do is the initial drop diameter, D is the drop diameter as a function of time, and denotes an ensemble average.

The model is used to perform Direct Numerical Simulations in continuing studies directed toward understanding the behavior of liquid petroleum fuel sprays. The model includes governing equations formulated in an Eulerian and a Lagrangian reference frame for the gas and the drops, respectively. This representation is consistent with the expected volumetrically small loading of the drops in gas (of the order of 10–3), although the mass loading can be substantial because of the high ratio (of the order of 103) between the densities of liquid and gas. The drops are treated as point sources of mass, momentum, and energy; this representation is consistent with the drop size being smaller than the Kolmogorov scale. Unsteady drag, added-mass effects, Basset history forces, and collisions between the drops are neglected, and the gas is assumed calorically perfect.

The model incorporates the concept of continuous thermodynamics, according to which the chemical composition of a fuel is described probabilistically, by use of a distribution function. Distribution functions generally depend on many parameters. However, for mixtures of homologous species, the distribution can be approximated with acceptable accuracy as a sole function of the molecular weight. The mixing layer is initially laden with drops in its lower stream, and the drops are colder than the gas. Drop evaporation leads to a change in the gas-phase composition, which, like the composition of the drops, is described in a probabilistic manner.

Figure 2. Drop Number Density is shown in a representative plane at the final computation time: (a) single-component drop layer and (b) multicomponent drop layer.

The advantage of the probabilistic description is that while a wide range of individual species can be accommodated in the mixture, the number of governing equations is increased minimally over that necessary for a single species because the composition is represented only by the parameter(s) necessary to construct the distribution function. Here the distribution function is entirely defined by the mean and variance. For this choice of distribution function, the model accounts for evaporation-induced changes in the composition of fuel drops and the surrounding gas, yet involves only two more conservation equations (one for the mean and one for the variance) than does an equivalent model for a single-component fuel. The initial mathematical form of the distribution function is postulated to be retained during the drop lifetime, but with evolving mean and variance as the drops evaporate.

In a test, a mixing-layer simulation was performed for drops of single-component-fuel and another such simulation for drops of a multicomponent fuel. Analysis of the results revealed that although the global layer characteristics were similar in the single-component and multicomponent cases, the drops evaporated more slowly in the multicomponent than in the single-component case (see Figure 1). The slower evaporation of the multicomponent drops was primarily attributed to the lower volatility of higher molar-weight species and to condensation of these species on drops transported in regions of different gas composition. The more volatile species released in the gas phase earlier during the drop lifetime were found to be entrained in the mixing layer, whereas the heavier species that evaporated later during the drop lifetime tended to reside in regions of high drop-number density. This behavior was found to lead to segregation of species in the gas phase on the basis of the relative evaporation time from the drops. The slower evaporation of multicomponent fuel drops was found to lead to regions of higher drop-number density in the drop-laden layer and to permit greater interaction of the drops with the flow, resulting in a more developed small-scale structure (see Figure 2).

This work was done by Josette Bellan and Patrick Le Clercq of Caltech for NASA's Jet Propulsion Laboratory. For further information, access the Technical Support Package (TSP) free on-line at www.techbriefs.com/tsp under the Physical Sciences category.


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Model of Mixing Layer With Multicomponent Evaporating Drops

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This article first appeared in the March, 2004 issue of NASA Tech Briefs Magazine.

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