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    A quasi-discrete model for heating and evaporation of complexmulticomponent hydrocarbon fuel droplets

    S.S. Sazhin , A.E. Elwardany, E.M. Sazhina, M.R. HeikalSir Harry Ricardo Laboratories, Centre for Automotive Engineering, School of Computing, Engineering and Mathematics, Faculty of Science and Engineering, University of Brighton,Brighton BN2 4GJ, UK

    a r t i c l e i n f o

    Article history:Received 15 March 2011Received in revised form 4 May 2011Available online 14 June 2011

    Keywords:Multicomponent dropletsHeatingEvaporationDiesel fuelsModelling

    a b s t r a c t

    A quasi-discrete model for heating and evaporation of complex multicomponent hydrocarbon fuel drop-lets is suggested and tested in Diesel engine-like conditions. The model is based on the assumption thatproperties of components are weak functions of the number of carbon atoms in the components ( n). Thecomponents with relatively close n are replaced by the quasi-components with properties calculated asaverage properties of the a priori dened groups of actual components. Thus the analysis of heating andevaporation of droplets consisting of many components is replaced by the analysis of heating and evap-oration of droplets consisting of relatively few quasi-components. In contrast to previously suggestedapproaches to modelling the heating and evaporation of droplets consisting of many components, theeffects of temperature gradient and quasi-component diffusion inside droplets are taken into account.The model is applied to Diesel fuel droplets, approximated as a mixture of 21 components C nH2n+2 for5 6 n 6 25, which correspond to a maximum of 20 quasi-components with average properties forn = n j and n = n j+1, where j varies from 5 to 24. It is pointed out that droplet surface temperatures andradii, predicted by a rigorous model taking into account the effect of all 20 quasi-components, are veryclose to those predicted by the model, using just ve quasi-components. Errors due to the assumptionsthat the droplet thermal conductivity and species diffusivities are innitely large cannot be ignored in the

    general case.Crown Copyright 2011 Published by Elsevier Ltd. All rights reserved.

    1. Introduction

    The models of multicomponent droplet heating and evapora-tion could be subdivided into two main groups: those based onthe analysis of individual components (Discrete Component Mod-els (DCM)) [19] , applicable in the case when a small number of components needs to be taken into account, and those based onthe probabilistic analysis of a large number of components (e.g.Continuous Thermodynamics approach [1017] and the Distilla-tion Curve Model [18] ). In the second family of models a numberof additional simplifying assumptions were used, including theassumption that species inside droplets mix innitely quickly ordo not mix at all.

    A model containing features of both these groups of models hasbeen suggested in [1921] . In [19] the mixtures of multicomponent(Diesel fuel, gasoline, biodiesel) and mono-component substanceswere studied based on the combination of the Continuous Thermo-dynamics approach and the Discrete Component Model. As in thecase of the classical Continuous Thermodynamics approach, it

    was assumed that the mixing processes inside droplets are in-nitely fast both for species and temperature. The analysis of [20,21] was based on the application of the Quadrature Method of Moments (QMoM), originally developed in [22] . This methodallowsoneto use2 or 3 pseudo-components foreach group of componentsinstead of dozens of real components for the whole mixture. Thenormal boiling point of each pseudo-component was allowed tochange during the vaporization process. This approach was shownto be particularly useful if the condensation process needs to bemodelled alongside the evaporation. As in the case of the conven-tional Continuous Thermodynamics approach, it was assumed thatthe droplets are well mixed. As follows from our analysis of heatingand evaporation of bi-component droplets [8], this assumption ap-pears to be questionable.

    The authors of [23] applied the Discrete Component Model forstudying the mixtures of substances containing many components(Diesel fuel, and gasoline) by approximating these fuels with rela-tively small numbers of physical species (six species for Diesel fueland seven species for gasoline). They took into account the effectsof temperature gradient inside droplets, but assumed that the masstransfer processes inside droplets are innitely fast. As alreadymentioned, this assumption is far from obvious.

    0017-9310/$ - see front matter Crown Copyright 2011 Published by Elsevier Ltd. All rights reserved.doi: 10.1016/j.ijheatmasstransfer.2011.05.012

    Corresponding author.E-mail address: [email protected] (S.S. Sazhin).

    International Journal of Heat and Mass Transfer 54 (2011) 43254332

    Contents lists available at ScienceDirect

    International Journal of Heat and Mass Transfer

    j ou r na l ho m e pa ge : www.e l s e v i e r. c om / l oc a t e / i j hm t

    http://dx.doi.org/10.1016/j.ijheatmasstransfer.2011.05.012mailto:[email protected]://dx.doi.org/10.1016/j.ijheatmasstransfer.2011.05.012http://www.sciencedirect.com/science/journal/00179310http://www.elsevier.com/locate/ijhmthttp://www.elsevier.com/locate/ijhmthttp://www.sciencedirect.com/science/journal/00179310http://dx.doi.org/10.1016/j.ijheatmasstransfer.2011.05.012mailto:[email protected]://dx.doi.org/10.1016/j.ijheatmasstransfer.2011.05.012
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    In this paper a new method of modelling heating and evapora-tion of multi-component droplets, suitable for the case when alarge number of components is present in the droplets, is sug-gested. As in [20,21] , this method is based on the introduction of pseudo-components, but these pseudo-components are introducedin a way which differs from the one described in [20,21] . In con-trast to the previously suggested models, designed for large num-bers of components, the new model takes into account thediffusion of liquid species and thermal diffusion as in the classical

    Discrete Component Models.It is possible to draw a parallel between our approach to mod-

    elling the heating and evaporation of multi-component dropletsand the modelling of absorption of thermal radiation in moleculargases, using the weighted-sum-of-gray-gases method [24,25] . In thismethod the medium is assumed to consist of different fractions of grey gases with different (but grey) absorption coefcients. Theaccuracy of this method turned out to be sufcient for most prac-tical engineering applications. At the same time its application ismuch more CPU efcient compared with the rigorous approachwhen all or most of the molecular absorption bands are accountedfor.

    The new model, based on the introduction of the concept of quasi-components, is described in Section 2. The thermo-physical

    properties of quasi-components are summarised in Section 3. Theresults of application of the new model to Diesel fuel dropletsare presented and discussed in Section 4. The main results of thepaper are summarised in Section 5.

    2. Model

    The Continuous Thermodynamics approach is based on theintroduction of the distribution function f m(I ) such that:

    Z I 2

    I 1 f mI dI 1; 1

    where I is the property of the component (usually taken as the mo-

    lar mass M ), f m characterises the relative contribution of the compo-nents having this property in the vicinity of I , I 1 and I 2 are limiting

    values of this property. For most practically important fuels f m(I )can be approximated by relatively simple functions. For examplethe following function was considered in [1017] :

    f mI I c

    a 1

    ba Ca exp

    I cb ; 2

    where C (a ) is the Gamma function, a and b are parameters thatdetermine the shape of the distribution, c determines the original

    shift. In the case of this choice of f m , we need to take I 1 = c andI 2 = 1 . The authors of [26] considered a more general function pre-sented as the weighted sum of the functions (2) (double-Gamma-PDF), but this approach is not widely accepted in the engineeringcommunity to the best of our knowledge.

    In most practically important cases the approximation forrealistic multicomponent fuel in the form (2) is valid only in thelimited range of I : I 1 > c and I 2 < 1 . In this case distribution (2)needs to be replaced by the following distribution:

    f mI C m I c

    a 1

    ba Ca exp

    I cb ; 3

    where constant C m is dened from Condition (1) as

    C m Z I 2

    I 1I c

    a 1

    ba Ca exp

    I cb dI " #

    1

    : 4Although molar mass is almost universally used to describe the

    property I , this choice is certainly far from being a unique one. Forexample, in [21] , this parameter was associated with the normalboiling points of individual components. Remembering that mostpractically important hydrocarbon fuels consist mainly of mole-cules of the type C nH2n+2 , where n P 1 in the general case orn P 5 for liquid fuels, it is more practical to write the distributionfunction f m as a function of the carbon number n rather than M [15] . These two parameters are linked by the following equation:

    M 14 n 2; 5where M is measured in kg/kmole.

    Nomenclature

    A, B, C parameters introduced in Eq. (12)C m constant dened by Eq. (7)D diffusion coefcient f m distribution functionI property of a componentI 1,2 limiting values of I kB Boltzmann constantL latent heat of evaporationM molar massn number of carbon atomsN f number of quasi-components p pressureRu universal gas constantT temperatureV parameter dened by Eq. (23) X molar fractionY mass fraction

    Greek symbolsa , b , c parameters of f m (see Eq. (2))ev ,a LennardJones energyl dynamic viscosity

    r LennardJones lengthu associated parameter of solventX D parameter used in Eq. (18)

    Subscriptsa airb boilingc critical f nali speciesl liquidm mixtures surfacev vapour0 initial

    Superscriptssat saturation0 dilute solute normalised

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    3.3. Critical and boiling temperatures

    Using data provided in [28] , the dependence of critical and boil-ing temperatures on n is approximated by the following equations:

    T cn ac bc n c c n2 dc n3 ; 16T bn ab bbn c bn2 dc n3 ; 17where the coefcients are presented in Table 1 .

    The plots of T c(n) and T b(n) are shown in Fig. 2 alongside the val-ues of these parameters for individual n as reported in [28] . As fol-lows from this gure, Approximations (16) and (17) are reasonablyaccurate and can be used in our model. Having replaced n in Eqs.(16) and (17) with n j we obtain the required values of T c and T bfor all quasi-components.

    3.4. Density, heat capacity, viscosity and thermal conductivity

    Following [15] the dependence of these thermodynamic param-eters and transport coefcients for liquid and vapour phases on nwill be ignored, and it will be assumed that they are equal to thoseof n-dodecane (see Appendix A4 of [29,30] ). The temperaturedependence of these coefcients is taken into account. This ap-proach is consistent with the one used in [15] .

    Using different values of these coefcients for n in the range 525 would affect the actual values of droplet temperature and ra-dius, but we do not anticipate any effect of this choice on the mainconclusion of the paper regarding the choice of the number of qua-si-components. Also, it might be possible to consider these proper-ties for individual quasi-components, as we have done for vapourpressure and latent heat of evaporation. This requires furtherinvestigation which is beyond the scope of this paper.

    3.5. Diffusion coefcient for gas

    In the case of one-component droplets, the value of the diffu-sion coefcient of vapour in air Dva can be estimated from the Wil-ke and Lee formula [28] :

    Dv a 3:03 0:98 =M 1=2v a h i10 7T 3=2 pM 1=2v a r 2v a X D ; 18

    where Dva is in m 2/s, T is temperature in K,

    M v a 21=M v 1=M a1 ;

    M v and M a are molar masses of vapour and air respectively, p ispressure in bar, r va = (r v + r a)/2, r v and r a are characteristicLennardJones lengths for vapour and air respectively, measuredin Angstrom (A), X D is the function of the normalised temperatureT = kBT /eva given by Eq. (B6) in [29] , ev a

    ffiffiffiffiffiffiffiffiffi ev eap ; ev and ea are char-

    acteristic LennardJones energies for vapour and air respectively, kBis the Boltzmann constant.

    Assuming that air is the dominant component in the air/fuel va-pour mixture, the same formula will be used for multicomponentdroplets with M v dened as

    M v X jN f j1

    M n j X v j.X jN f j1

    X v j; 19

    where the additional subscript v indicates that X j refers to the va-pour phase. Parameters r v and X D are assumed to be equal to thoseof n-dodecane [29] , since no reliable information referring to thedependence of these parameters on n is available to the best of our knowledge.

    3.6. Diffusion coefcient for liquid

    Following [8] , the diffusion coefcient of component j relativeto all other components can be estimated as

    D jm X jD0mj X mD

    0 jm; 20

    where m refers to the mixture of all other components, D0 jm and D0mj

    are diffusivities of dilute solute j in solvent m , and dilute solute m insolvent j respectively, both are in m 2/s. Note that there are typos inthe corresponding expressions for D jm given in [8,9] .

    As in [8], among various approximations for D0 jm and D0mj we

    have chosen the Wilke-Chang approximation given by the follow-ing formula [31] :

    D0 AB 7:4 10 12 ffiffiffiffiffiffiffiffiffiu M Bp T l BV 0:6 A ; 21

    where M B is the molar mass of solvent B, kg/kmol, T temperature inK, l B dynamic viscosity of solvent B, cP (1 cP = 10 3 kg m 1 s 1), V Ais the molar volume of solute A at its normal boiling temperature,cm 3/mol (this can be recalculated from density if required, takinginto account the molar mass of the substance), u is the associatedparameter of solvent B (following the recommendation by [31] itis assumed equal to 1).

    This model is further simplied, assuming that D jm is the samefor all species and estimated as

    D jm Dl 7:4 10 12 ffiffiffiffiffiffiffiM v p T l lV 0:6v ; 22

    where M v is dened by (19) , but for the liquid phase, l l is the dy-namic viscosity for liquid Diesel fuel, V v is dened as

    V v r v =1:183 ; 23

    where the LennardJones length r v in A is taken from Appendix B of [29] .

    300

    400

    500

    600

    700

    800

    5 10 15 20 25

    Tcr (K)check_TcrTb (K)check Tb

    T ( K )

    values of T capproximation for T cvalues of T bapproximation for T b

    n

    Fig. 2. Plots of T c and T b, and their Approximations (16) and (17) , versus n .

    Table 1

    Coefcient ac bc c c dc Value 242.3059898052 55.9186659144 2.1883720897 0.0353374481Coefcient ab bb c b dbValue 118.3723701848 44.9138126355 1.4047483216 0.0201382787

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    3.7. Parameters of the distribution function

    Our new model will be applied to the analysis of heating andevaporation of Diesel fuel droplets. Following [15] the molar frac-tions of components of this fuel are described by Eq. (9) witha = 18.5, b = 10, c = 0, n0 = 5 and n f = 25. Having substituted thesevalues into expression (7) , we obtain C m = 14.0081.

    The plot of f m (n) versus n is shown in Fig. 3. As one can see fromthis plot, the maximal value of f m(n) is reached at n = 12.4. This va-lue is slightly different from the value of n j 12 : 5644 for n j 1 = n0and n j = n f , as predicted by Eq. (8). Both these values are reasonablyclose to n = 12, referring to n-dodecane, which is commonly con-sidered as a close approximation of Diesel fuel.

    3.8. Liquid and gas phase models

    As in our previous paper [8], we used the Effective Thermal Con-ductivity (ETC) and Effective Diffusivity (ED) models for the liquidphase, and the model suggested in [32] for the gas phase. When-ever appropriate, the results will be compared with the predictionof the Innite Thermal Conductivity (ITC) and Innite Diffusivity

    (ID) models.These models are described in detail in our previous paper [8] .In contrast to most previous studies, our analysis is based on theincorporation of the analytical solutions to the heat transfer andspecies diffusion equations inside droplets into a numerical code,rather than on the numerical solutions of these equations. Theapplicability of the ETC model to the analysis of droplet heatingand evaporation has been demonstrated in [32] for the simplestcase when the effects of thermal radiation and the dependence of transport coefcients on temperature are ignored and in [33,27]in the general case when both these affects are taken into account.The applicability of the ED model has been investigated in [34] .

    4. Results

    To illustrate the efciency of the model, described above, weuse the same values of parameters as in [29] . Namely, we assumethat the initial droplet temperature is equal to 300 K, and is homo-geneous throughout its volume. Gas temperature is assumed to beequal to 880 K and gas pressure is assumed to be equal to 3 MPa.The initial composition of droplets is described by distributionfunction (6) .

    The plots of droplet surface temperature T s versus time for theinitial droplet radius equal to 10 l m and velocity 1 m/s are shownin Fig. 4. The droplet velocity is assumed to be constant during thewhole process. The calculations were performed for the case of

    N f = 1 (one quasi-component droplet) and N f = 20 (20 quasi-components droplet), using the ETC/ED and ITC/ID models.

    As one can see from this gure, the evaporation times and T s,especially at the nal stages of droplet heating and evaporation,predicted by the ETC/ED models, using one and twenty quasi-components are noticeably different. The model, using twentyquasi-components predicts higher surface temperatures and

    longer evaporation time compared with the model using onequasi-component. This can be related to the fact that at the nalstages of droplet evaporation the species with large n becomethe dominant, as will be demonstrated later. These species evapo-rate more slowly than the species with lower n and have higherwet bulb temperatures.

    Also, there are noticeable differences in predictions of the ETC/ED and ITC/ID models, using twenty quasi-components, especiallyin the case of surface temperature at the initial stages of dropletheating and evaporation. The accurate prediction of this tempera-ture is particularly important for prediction of the auto-ignitiontiming in Diesel engines [30] . This questions the reliability of themodels for heating and evaporation of multicomponent droplets,based on the ITC/ID approximations. As mentioned in the Introduc-tion, these models are almost universally used for modelling theseprocesses, especially when a large number of components are in-volved in the analysis.

    The plots of T s and Rd at time equal to 0.25 ms versus the num-ber of quasi-components N f , predicted by the ETC/ED and ITC/IDmodels, are shown in Fig. 5 for the same conditions as in Fig. 4.Symbols refer to those N f for which calculations were performed.As follows from this gure, for N f P 5 the predicted T s and Rd nolonger depend on N f . Hence, heating and evaporation of Diesel fueldroplets can be safely modelled using just 5 quasi-components.This number can even be reduced to 3 if errors less than about0.3% can be tolerated. The errors due to the ITC/ID approximationin this case are signicantly larger than those due to the choiceof a small number of quasi-components, especially for the surfacetemperature. These errors cannot be ignored in most engineeringapplications, and this questions the applicability of the modelsusing the ITC/ID approximation, including the widely used Contin-uous Thermodynamics models.

    Plots similar to those shownin Fig.5 but attime equal to1 msareshown in Fig. 6. As one can see from this gure, both droplet surface

    n

    f m ( n )

    Fig. 3. A plot of f m(n) versus n as predicted by Eq. (6) .

    Fig. 4. Plots of T s and Rd, predicted by three models, versus time. The initial dropletradius and temperature are assumed to be equal to 10 l m and 300 K respectively,the droplet velocity is assumed to be equal to 1 m/s and its changes during theheating and evaporation process are ignored, gas temperature is assumed equal to880 K. These are the models used for calculations: Effective Thermal Conductivity(ETC)/Effective Diffusivity (ED) model using one quasi-component (black solid),ETC/ED model using twenty quasi-components (grey solid), Innite Thermal

    Conductivity (ITC)/Innite Diffusivity (ID) model using twenty quasi-components(dashed).

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    temperatureand radius canbe wellpredicted if only5 quasi-compo-nents are used. This number can even be reduced to 3 if errors of about 0.5% can be tolerated. In contrast to the case shown in Fig. 5,the droplet surface temperatures predicted by the ETC/ED and ITC/ID models practically coincide, butthe difference in predicted drop-let radii is signicantly larger than in the case shown in Fig. 5.

    The closeness of the temperatures predicted by ETC/ED and ITC/ID models at the later stages of droplet heating and evaporationcan be related to the fact that at this stage the droplet temperaturebecomes almost homogeneous (see Fig. 9) and the effects of tem-perature gradient inside droplets can be ignored. Smaller dropletradii predicted by the ITC/ID model, compared with the ETC/EDmodel, can be related to lower temperatures at the initial stagesof droplet heating and evaporation predicted by the ITC/ID model

    compared with the ETC/ED model.Comparing Figs. 5 and 6 one can see that at early stages of drop-

    let heating and evaporation ( t = 0.25 ms), the predicted droplet ra-dius reduces slightly with the increase in the number of quasi-components used, while at a later stage ( t = 1 ms) the opposite ef-fect is observed. This could be related to the fact that at the earlystages, droplet evaporation is controlled by the most volatile qua-si-components, while at the later stages it is controlled by less vol-atile quasi-components. When the number of quasi-componentsincreases then the volatility of the most volatile component in-creases and that of the least volatile decreases.

    The conclusions drawn from Figs. 46 remain essentially thesame for droplets with initial radius equal to 25 l m (the plotsare not shown).

    To illustrate the time evolution of the distribution of mass frac-tions of species inside droplets, we consider the case shown in

    Fig. 4 for three quasi-components. The plots of Y i, where i = 1, 2,3, versus R/Rd for t = 0, 0.3 ms, 0.5 ms and 1 ms are shown inFig. 7. As one can see from this gure, the mass fraction of theheaviest component ( Y 3) is always increasing with time, especiallynear the droplet surface. At the same time, the mass fraction of thelightest component ( Y 1) decreases with time, and almost disap-pears at time 1 ms. The behaviour of the middle component ( Y 2)is more complex. Initially, it increases with time, especially nearthe droplet surface, similarly to Y 3. At later times ( t 1 ms), how-ever, it decreases with time, similarly to Y 1. These plots clearlyshow the signicance of the gradients of concentration of all com-ponents at all times except the initial moment of time. This illus-trates the limitations of the ID model, widely used in engineeringapplications.

    Plots of Y si versus time of the same quasi-components as inFig. 7 are presented in Fig. 8. The results presented in this gure

    are consistent with those shown in Fig. 7. The values of Y 1monotonically decrease with time, while those of Y 3 monotoni-cally increase with time. The values of Y 2 initially increase withtime, but at later times they rapidly decrease with time. At timesclose to the moment when the droplet completely evaporates,only the quasi-component Y 3 remains. Since this quasi-compo-nent is the most slowly evaporating one, and has the highestwet bulb temperature, the model based on three quasi-compo-nents is expected to predict longer evaporation times and largerdroplet surface temperatures at the nal stages of droplet evap-oration, compared with the model using one quasi-component.This result can be generalised to the case when the number of quasi-components is greater than 3. It is consistent with resultsshown in Fig. 4.

    In Fig. 9 the time evolution of the distribution of temperatureinside droplets is shown for the same case as in Fig. 4. Two cases

    ETC, ED models

    ITC, ID models

    Time = 0.25 ms

    Number of quasi-components

    T s

    ( K )

    ETC, ED models

    ITC, ID models

    Time = 0.25 ms

    Number of quasi-components

    R d

    ( m

    )

    (a)

    (b)

    Fig. 5. Plots of T s (a) and Rd (b) versus the number of quasi-components N f for thesame conditions as in Fig. 4 at time 0.25 ms as predicted by the ETC/ED (squares)and ITC/ID (triangles) models.

    605

    610

    615

    620

    625

    0 5 10 15 20

    Ts-ETC-ED

    Ts-ITC_ID

    ETC, ED models

    ITC, ID models

    Time = 1 ms T s ( K )

    Number of quasi-components

    6.8

    7

    7.2

    7.4

    7.6

    0 5 10 15 20

    Rd-ETC-ED

    Rd-ITC_ID

    ETC, ED models

    ITC, ID models

    Time = 1 ms R d

    ( m

    )

    Number of quasi-components

    (a)

    (b)

    Fig. 6. The same as Fig. 5 but at time 1 ms.

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    are considered: one quasi-component and 20 quasi-components.As follows from this gure, in both cases, initially mainly the areaclose to the droplet surface is heated and a noticeable temperaturegradient near the droplet surface can be clearly seen. At later times,however, the temperature inside the droplet becomes more homo-geneous, which could justify the application of the ITC model. Inagreement with Fig. 4, the model using 20 components predictshigher temperatures compared with the model using one quasi-component at t = 1 ms.

    Note that apart from the number of quasi-components the CPUefciency of the model depends on a number of other parameters,including the number of terms in the series used in analytical solu-tions for temperature and species mass fractions, time step and thenumber of grid points along the droplet radius. This problem wasinvestigated in detail in our previous paper [9] .

    5. Conclusions

    A new approach to modelling the heating and evaporation of multicomponent droplets is suggested and tested for realistic Die-sel fuel droplets in engine-like conditions. The model is based uponthe assumption that properties of components vary relativelyslowly from one component to another and depend on a singleparameter. This parameter is chosen to be the number of carbonatoms in the components ( n). The components with relatively closen are replaced by quasi-components with properties calculated asaverage properties of the a priori dened groups of actual compo-nents. Thus the analysis of the heating and evaporation of droplets

    consisting of many components is replaced by the analysis of theheating and evaporation of droplets consisting of relatively fewquasi-components. In contrast to previously suggested approachesto modelling the heating and evaporation of droplets consisting of many components, the effects of temperature gradient and quasi-component diffusion inside droplets are taken into account.

    The model is applied to Diesel fuel droplets, approximated as amixture of 21 components C nH2n+2 for 5 6 n 6 25, which corre-spond to a maximum of 20 quasi-components with average prop-erties for n = n j and n = n j+1, where j varies from 5 to 24, in realisticengine-like conditions. It is pointed out that droplet surface tem-peratures and radii, predicted by a rigorous model taking into ac-count the effect of all 20 quasi-components, are almost the sameas those predicted by the model using ve quasi-components.

    Moreover, if errors less than about 1% can be tolerated, then thenumber of quasi-components used can be reduced to three. On

    Fig. 7. Plots of Y i versus R/Rd for three quasi-component droplets ( i = 1, 2, 3) at fourmoments of time as indicated near the curves. The same droplet and gas parametersas in Fig. 4 are used.

    Fig. 9. Plots of T s versus R/Rd for one quasi-component (solid) and twenty quasi-component (dashed) droplets at ve moments of time as indicated near the curves.The same droplet and gas parameters as in Fig. 4 are used.

    Fig. 8. Plots of Y si versus time of the same quasi-components as in Fig. 7.

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