Baumgarten

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mass transport into the gas. Due to the enhanced diffusion an increased amount of the energy (transferred from the gas to the liquid) is needed for evaporation, and a quasi-steady state is reached where the droplet temperature stays constant until it is completely vaporized.

It must be pointed out that the use of single-component model fuels is the standard approach today because of its simplicity and low consumption of computational time. Nevertheless, research work today concentrates on more sophisticated evaporation models, especially with respect to modeling more realistic multicomponent fuels and to replace the single-component fuel calculations, see next section.

4.5.2 Evaporation of Multi-Component Droplets

The basic challenge when describing the evaporation process of fuel droplets is the choice of an appropriate reference fuel that represents the relevant behavior of the fuel with sufficient accuracy. For example, in standard CFD simulations, tetradecane (n-C14H30) is normally used to represent the relevant properties of diesel. However, diesel consists of more than 300 different components, and it is obvious that a single reference fuel cannot predict all of the relevant sub-processes during evaporation, ignition, and combustion with sufficient accuracy. Hence, multicomponent model fuels are desirable. The goal is to achieve a high degree of accuracy while keeping the consumption of computational time small.

Initial studies on multi-component fuels concentrated on binary fuels consisting of two discrete hydrocarbons. Both discrete components were implemented in CFD codes using different fuel properties and transport equations for each of the two hydrocarbons. For example, Kneer [65] investigated the evaporation behavior of mixtures consisting of heptane and dodecane as well as hexane and tetradecane. Due to an increased evaporation of the more volatile component, especially during the initial stage of evaporation, these investigations have shown a significantly larger fuel vapor concentration. Further investigations have been published by Jin and Borman [60] and Klingsporn [63], for example. The investigations show that the overall mixture formation process, and thus also ignition and combustion, may be significantly influenced by the composition of the fuel vapor, and that it may be important to describe the multi-component character of fuels more accurately. However, the computational effort describing mixtures of ten and more components is enormous. For example, Rosseel and Sierens [121] developed a model including ten discrete components. According to the authors, the consumption of computational time for simulating the evaporation of a single droplet was twenty times higher than that of a single-component fuel.

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components belonging to different groups, like alkanes and aromatic compounds, with a single distribution function.

Besides the distribution variable, the correct choice of an appropriate mathematical distribution function is necessary. On the one hand, this function must be able to describe the mixture with sufficient accuracy, but on the other hand, the mathematical description should be as simple as possible. In order to describe hydrocarbon mixtures, the so-called gamma-function is used [24, 137, 100]. This distribution function is also known as Pearson type III distribution and is defined as [102]:

where and Ε and are parameters describing the shape of the curve, and ϑis responsible for the displacement of the origin of the curve, Fig. 4.40.

The quantity I is the distribution variable (molar mass of the fuel components), and ( ) is given by

Because Eq. 4.157 is the normalized gamma-function, the area below the curve is unity:

Thus, the first and the second moment of the distribution are given by

Fig. 4.40. Gamma-function: effect of parameters , Ε, ϑ

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the transport equation for the species mass fraction, and by weighting the equation by I and by I2 before integrating in order to get the transport equations for the first and the second moment [78, 137, 46]:

side of Eqs. 4.165–4.167 is due to mass diffusion, while the second and the third ones are source terms due to spray evaporation and combustion.

The source term due to spray evaporation is determined by the evaporation of the multi-component fuel droplets. If the droplet interior is assumed to be well mixed, which is usually done in most of the 3D CFD calculations, the molar flux

where Adrop = 4ΣR2, Vdrop = (4/3)ΣR3, and cl = Υl /Τl is the liquid molar density in mol/cm3. In contrast to the fuel vapor distribution function, the relevant parame-

ters of the liquid fuel distribution inside the droplet are denoted by the subscript l. It should also be mentioned that all liquid-phase composition calculations are performed on a molar basis, since the vapor-liquid equilibrium is solved in a molar frame of reference. Using the expressions for drop surface area and volume in Eq. 4.168, the time-dependent change of droplet radius becomes

The mole flux n, which is needed to determine the mean and the second moment of the droplet composition, can be calculated equating the fluxes at the liquid/gas interface. In the case of a single component k of the liquid phase, Eq. 4.168 yields

where xk is the mole fraction of component k in the liquid phase. Because a wellmixed droplet interior is assumed, it is also the mole fraction inside the whole droplet. The mole flux yf,R in the gas phase at the liquid-vapor interface (index R) is given by assuming equal molar diffusion from droplet to gas,

The resulting equations for the liquid composition are derived by integrating over the complete distribution function and by several further transformations [137]:

The index R denotes the droplet surface. Next, equilibrium is assumed at the interface, resulting in

where Ι represents yΤ and y , respectively. The subscripts “R” and “φ” indicate the gas phase properties at the droplet surface and in the undisturbed surroundings, respectively. B is the Spalding transfer number,

Using Eqs. 4.176 and 4.177 in Eqs. 4.174 and 4.175 results in expressions that are only dependent on quantities calculated by assuming equilibrium conditions at the liquid-gas interface (R) and on those of the surrounding (φ). The final expressions describing the change of the mean and the second moment of the droplet composition are

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The time-dependent development of droplet temperature can be calculated by an energy balance in analogy to Eq. 4.139 as

where h fg and c v = cv·Τ are the molar heat of evaporation and the molar specific heat of the liquid. The convective heat flux and the molar flux may be approximated by

The appropriate Nusselt and Sherwood numbers including corrections for the influence of a relative velocity between droplet and gas are

where Pr and Sc are the Prandtl and the Schmidt numbers, respectively.

As already mentioned, phase equilibrium is assumed to determine the vapor mole fraction in the gas phase at the droplet’s surface, and Raoult’s law is applied. In the case of a single component, this relation between the mole fraction of vaporized fuel yk at the interface and the mole fraction xk inside the liquid drop is

where p is the static pressure in the gas phase and pk is the partial pressure of the component k with molar mass I. In the case of a continuous distribution function, the mole fractions of the different components inside the liquid are described by a distribution function, which changes its shape during evaporation. Thus, the mole fractions at the surface also change, and in this case Raoult’s law is derived by integrating over the complete distribution function:

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The heat of evaporation can be described via an empirical relation. It is assumed that ideal gas behavior can be applied and that pressure effects can be neglected. According to Reid et al. [113] the heat of evaporation for a single component is given as

where pcrit is the critical pressure and Tcrit is the critical temperature. Tamin and Hallett [137] have modified this approach for multi-component fuel mixtures:

where ah = 2.07·10-7 and bh = 1.35·10-5 for units of h fg of J/kmol.

The thermal conductivity of a gaseous fuel component can be described as a

ficient between alkanes and air. This correlation assumes a linear relation between molar mass and diffusion coefficient. However, for low and very large molar masses, this function underpredicts the diffusivity, see Fig. 4.42. Fuller (see [113]) uses a non-linear approach,