International Journal of Heat and Mass Transfer 79 (2014) 704–712

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Heating and evaporation of a two-component droplet: Hydrodynamicand kinetic models

http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.08.0260017-9310/Crown Copyright � 2014 Published by Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +44 (0)1273642677; fax: +44 (0)1273642330.E-mail address: S.Sazhin@brighton.ac.uk (S.S. Sazhin).

S.S. Sazhin a,⇑, I.N. Shishkova b,a, M. Al Qubeissi a

a Sir Harry Ricardo Laboratories, Centre for Automotive Engineering, School of Computing, Engineering and Mathematics, Faculty of Science and Engineering,University of Brighton, Brighton BN2 4GJ, UKb Low Temperature Department, Moscow Power Engineering Institute, Krasnokazarmennaya 14, Moscow 111250, Russia

a r t i c l e i n f o a b s t r a c t

Article history:Received 26 June 2014Received in revised form 8 August 2014Accepted 8 August 2014

Keywords:Boltzmann equationDiesel fuel dropletn-Dodecanep-DipropylbenzeneHeat/mass transferMixture

A previously developed kinetic model for two-component vapour and background gas (air) is applied tothe analysis of droplet heating and evaporation in Diesel engine-like conditions. The model used in theanalysis is based on the introduction of the kinetic region in the immediate vicinity of the dropletsand the hydrodynamic region. The presence of two components in the vapour, finite thermal conductivityand finite species diffusivity in droplets are taken into account. It is pointed out that for parameters whichare typical of Diesel engine-like conditions, the heat flux in the kinetic region is a linear function of thetemperature at the outer boundary of this region, but is almost independent of the density of the com-ponents at this boundary. Mass fluxes of both components in the kinetic region are shown to decreasealmost linearly with increasing vapour density at the outer boundary of this region, but are almost inde-pendent of the temperature drop in the kinetic region. The model is tested for the analysis of heating andevaporation of a droplet with initial radius and temperature equal to 5 lm and 300 K, respectively,immersed into gas with temperatures 1000 K and 700 K for several mixtures of n-dodecane and p-dipro-pylbenzene. It is pointed out that an increase in the mass fraction of p-dipropylbenzene and kineticeffects lead to an increase in the predicted droplet evaporation time. The kinetic effects are shown toincrease with increasing gas temperature and molar fraction of p-dipropylbenzene.

Crown Copyright � 2014 Published by Elsevier Ltd. All rights reserved.

1. Introduction

An interest in modelling droplet heating and evaporation hasbeen stimulated by various engineering, environmental and phar-maceutical applications [1,2]. In most engineering applications,including automotive ones, the modelling of these processes hasbeen based on the hydrodynamic approximation, although the lim-itations of this approximation, even in the case when these pro-cesses take place at high pressures, are well known (see [3–5]). Ina number of studies, including [6–9], the evaporation of n-dodecaneC12H26 (an approximation for Diesel fuel) was studied and a newmodel was developed based on a combination of the kinetic andhydrodynamic approaches. In the immediate vicinity of dropletsurfaces (up to about one hundred molecular mean free paths),the vapour and ambient gas dynamics were studied based on theBoltzmann equation (kinetic region), while at larger distances theanalysis was based on the hydrodynamic equations (hydrodynamic

region). Mass, momentum and energy fluxes were conserved at theinterface between these regions. The authors of [7–9] consideredthe problem of n-dodecane evaporation into air and developed anew numerical algorithm for the solution of a system of twoBoltzmann equations for n-dodecane and air, taking into accountelastic collisions between: n-dodecane molecules; between airmolecules; and between n-dodecane and air molecules. A newapproach to taking into account the effects of inelastic collisionswas developed in [10] and applied to the problem of n-dodecanedroplet heating and evaporation in [11].

One of the important limitations of the approaches described in[6–11] is that they were based on the assumption that Diesel fuelcan be approximated by n-dodecane. A more detailed analysis ofthe composition of Diesel fuel showed that it includes hundredsof various hydrocarbon components [12–14]. It is obviously notpossible to take into account the contributions of all these compo-nents in the kinetic modelling. At the same time, one can see thatthese components can be subdivided into two main groups:alkanes and aromatics [14]. The assumption that n-dodecane canapproximate alkanes is widely used (see [12,15,16]), while aromat-ics could be approximated by p-dipropylbenzene [15]. In this case

Nomenclature

BM Spalding mass transfer numberBT Spalding heat transfer numberc specific heat capacityD binary diffusion coefficientf molecular velocity distribution functionE relative error defined by Eq. (18)h convection heat transfer coefficientj mass fluxJ collision integralk thermal conductivity‘ mean free pathL latent heat of evaporationLe Lewis numberm massM molar massNat the total number of atoms in a moleculeNu Nusselt numberp pressurePr Prandtl numberq heat fluxQL power spent on droplet heatingr radius-vectorR distance from the centre of the dropletRvðaÞ gas constant referring to n-dodecane (air)Rd droplet radiusSc Schmidt numberT temperaturev velocityX molar fractionY mass fraction

Greek symbolsa parameter defined by Eq. (6)b evaporation coefficient

dRd thickness of the kinetic region�i evaporation rate of individual speciesh angular coordinatej liquid diffusivityq densityr effective diameter of molecules/ angular coordinate

Subscriptsa airat atoma; b a ¼ a;n; p; b ¼ a;n; pd dropleteff effectiveg gash hydrodynamici componentsk kineticl liquidn n-dodecanep constant pressure or p-dipropylbenzener reference or reflectedRd outer boundary of the kinetic regions surfacetotal totalv fuel vapour0 initial1 ambient

Superscriptsi components0 after the collision� normalised

Fig. 1. Liquid, kinetic and hydrodynamic regions near the surface of the droplet. Ts

is the droplet surface temperature, qs ðn;pÞ are the n-dodecane (n) and p-dipropyl-benzene (p) vapour densities in the immediate vicinity of the droplet surface, TRd

and qRd ðn;pÞ are the temperature and the n-dodecane (n) and p-dipropylbenzene (p)vapour densities at the outer boundary of the kinetic region.

S.S. Sazhin et al. / International Journal of Heat and Mass Transfer 79 (2014) 704–712 705

it was suggested that a more accurate approximation of Diesel fuel,compared with the one based on its approximation by n-dodecane,could be its approximation by a mixture of n-dodecane and p-dipropylbenzene. Mass fractions of n-dodecane in this mixturecould vary from 0.8 to 0.7 [15,12].

A new kinetic algorithm for modelling of a three component(two components, approximating Diesel fuel, and air, approxi-mated by nitrogen) mixture was developed in [17]. Binary colli-sions between molecules were taken into account assuming thatthese collisions are elastic or inelastic. The functionality testingof the algorithm was performed using a simple test problem ofheat and mass transfer in a mixture of n-dodecane, p-dipropylben-zene and nitrogen between two infinite parallel walls. It was con-cluded that the predictions of the new kinetic algorithm arequalitatively consistent with the predictions of more basic modelsof the phenomena for which it was tested.

The aim of this paper is to investigate the kinetic effects onheating and evaporation of two-component droplets, approximat-ing Diesel fuel, assuming that this approximation is a mixture ofn-dodecane and p-dipropylbenzene. The numerical algorithmdeveloped in [17] will be used in our analysis.

The mathematical model, used in the analysis, is discussed in Sec-tion 2. In Section 3 this model is applied to the analysis of heatingand evaporation of a two-component (mixture of n-dodecane andp-dipropylbenzene) droplet. The results are compared with thosebased on the approximation of Diesel fuel droplets by n-dodecanedroplets. The main results of the paper are summarised in Section 4.

2. Mathematical models

As in [6–11], two regions above the surface of an evaporatingfuel droplet are considered: the kinetic and hydrodynamic regions.As in [11], we take into account that thermal conductivity of theliquid phase is finite, and identify the third region called the liquidphase region. All three regions are schematically shown in Fig. 1. Ts

and qs ðn;pÞ refer to the surface temperature and vapour density (for

706 S.S. Sazhin et al. / International Journal of Heat and Mass Transfer 79 (2014) 704–712

n-dodecane (n) and p-dipropylbenzene (p)) in the immediatevicinity of the droplet surface; TRd and qRd ðn;pÞ refer to the sameparameters but at the interface between the kinetic and thehydrodynamic regions. dRd is the thickness of the kinetic region.In contrast to [11], we take into account the diffusion of speciesin the liquid phase and the presence of up to 3 components inthe kinetic region. The conventional hydrodynamic analysis isapplied in the liquid and hydrodynamic regions, while vapourand air dynamics in the kinetic region are described by theBoltzmann equations. The hydrodynamic and kinetic models usedin our analysis are described below.

2.1. Hydrodynamic model

The hydrodynamic model for the liquid and gas phases, used inour analysis, is essentially the same as described in [14]. In whatfollows the most essential features of this model are brieflysummarised.

A droplet is assumed to be spherical, stationary (the generalisa-tion of the model to a moving droplet is described in [14]) and theprocess of its heating is described by the following transient heatconduction equation for the temperature T � Tðt;RÞ in the liquidphase:

@T@t¼ j

@2T

@R2 þ2R@T@R

!; ð1Þ

where j ¼ kl=ðclqlÞ is the liquid thermal diffusivity, kl; cl, and ql arethe liquid thermal conductivity, specific heat capacity, and densityrespectively, R is the distance from the centre of the droplet, t istime.

Eq. (1) is to be solved for 0 6 R < Rd, where Rd is the dropletradius, with the following initial condition Tðt ¼ 0Þ ¼ Td0ðRÞ andthe boundary condition (assuming that the effects of evaporationcan be ignored):

qs � hðTg � TsÞ ¼ kl@T@R

����R¼Rd�0

; ð2Þ

where Ts ¼ TsðtÞ is the droplet’s surface temperature, Tg ¼ TgðtÞ isthe ambient gas temperature, h is the convection heat transfer coef-ficient, linked with the Nusselt number Nu via equationNu ¼ 2Rdh=kg ; kg is the gas thermal conductivity. We assume thatfuel vapour is dilute and kg is equal to the thermal conductivity ofambient air. Remembering the physical background to the problem,we are interested only in the solution which is continuously differ-entiable twice in the whole domain. This implies that T should bebounded for 0 6 R < Rd.

Our analysis is based on the analytical solution to Eq. (1), sub-ject to the above-mentioned boundary and initial conditions,which was implemented into a numerical code [14]. To take intoaccount the effect of droplet evaporation in the analytical solution,gas temperature was replaced with the effective temperaturedefined as: Teff ¼ Tg þ qlL _Rd

h , where L is the latent heat of evapora-tion, the value of _Rd is taken from the previous time step and esti-mated based on Eq. (8). Rd is assumed constant in the analyticalsolutions, but is updated at the end of the time step Dt to take intoaccount the effects of swelling and evaporation (the effects ofnon-constant Rd during the time step are discussed in our previouspapers, the results of which are summarised in [5]). The value ofNu is estimated as:

Nu ¼ 2lnð1þ BTÞ

BT; ð3Þ

where BT ¼ cpv ðTg�TsÞLeff

is the Spalding heat transfer number,Leff ¼ Lþ Q L

_md¼P

i�iLi þ QLPi

_mi;QL is the power spent on droplet

heating, cpv is the specific heat capacity of fuel vapour (estimatedfor the mixture of vapour species), �i ¼ �iðtÞ are the evaporationrates of species i; _mi ¼ �i _md; _md is the droplet evaporation rate(see Eq. (8); _md ¼

Pi

_mi). i ¼ n; p in our analysis, where n refers ton-dodecane while p refers to p-dipropylbenzene.

Equations for mass fractions of liquid species Yli � Yliðt;RÞ arepresented in the following form:

@Yli

@t¼ Dl

@2Yli

@R2 þ2R@Yli

@R

!; ð4Þ

where Dl is the liquid diffusivity, assumed to be the same for all spe-cies and all species concentrations.

Eq. (4) was solved analytically with the following boundarycondition:

að�i � YlisÞ ¼ �Dl@Yli

@R

����R¼Rd�0

; ð5Þ

and the initial condition Yliðt ¼ 0Þ ¼ Yli0ðRÞ, where Ylis ¼ YlisðtÞ areliquid components’ mass fractions at the droplet’s surface,

a ¼ j _mdj4pqlR

2d

: ð6Þ

Assuming that species concentrations in the ambient gas areequal to zero, the values of �i were found from the followingrelationship:

�i ¼YvisP

i¼n;pYvis; ð7Þ

where the subscript v indicates the vapour phase. The conditiona ¼ const can always be guaranteed for sufficiently small timesteps.

The Wilke–Chang approximation was used for the estimate ofDl assuming that the average Lennard–Jones length of moleculesis equal to that of n-dodecane (7.12 Å; see Table 1).

The droplet’s evaporation rate is estimated from the followingequation:

_md ¼ �4pRdDvqtotal lnð1þ BMÞ; ð8Þ

where Dv is the binary diffusion coefficient of vapour in air, BM isthe Spalding mass transfer number defined as:

BM ¼qvs � qv1

qgs¼ Yvs � Yv1

1� Yvs; ð9Þ

Yv is the vapour mass fraction. BT and BM are linked by equation

BT ¼ ð1þ BMÞu � 1,where u ¼ cpvcpa

� �1Le ; Le ¼ kg=ðcpaqtotalDvÞ ¼ Scd=

Prd is the Lewis number; it was assumed that Yv1 ¼ 0. Note thatthe derivation of Eq. (8) is essentially based on the assumption thatqtotal remains the same at all distances from the droplet surface.

We assume that Diesel fuel vapour diffuses from the surface ofthe droplet, without changing its composition, with the averagediffusion coefficient assumed to be equal to that of n-dodecanein air [18]

Dv ¼ 5:27� 10�6 T300

� �1:583

p�1; ð10Þ

where Dv is in m2/s, T is temperature in K, p is in bar. Note that thereis a typo in Eq. (39) in [14]; this typo does not affect the results ofthe paper as the correct formula was used in the numerical code.

All liquid properties are calculated for the average temperatureinside droplets. All gas properties are calculated for the referencetemperature Tr ¼ ð2=3ÞTs þ ð1=3ÞTg , where Ts and Tg are dropletsurface and ambient gas temperatures, respectively. Enthalpy of

Table 1Chemical formulae, molar masses and molecular diameters of n-dodecane, p-dipropylbenzene and nitrogen, used in our analysis.

Component Chemical formula Molar mass Molecular diameter

n-dodecane C12H26 170.3 kg/kmol 7.12 Åp-dipropylbenzene C12H18 162.27 kg/kmol 6.73 Ånitrogen N2 28.97 kg/kmol 3.617 Å

S.S. Sazhin et al. / International Journal of Heat and Mass Transfer 79 (2014) 704–712 707

evaporation and saturated vapour pressure are estimated at thesurface temperature Ts.

The mixtures are treated as ideal (Raoult’s law is assumed to bevalid). In this case, partial pressures of individual components areestimated as:

pvi ¼ Xlsi psatðiÞ; ð11Þ

where Xlsi are the molar fractions of liquid components at the sur-face of the droplet, psatðiÞ are saturated vapour pressures of the ithcomponent (i ¼ n; p) assuming that this is the only componentpresent in the droplet.

The following approximations for psatðiÞ are used in our analysis:

psatðnÞ ¼ exp 8:1948�7:8099 300=Tsð Þ�9:0098 300=Tsð Þ2h i

bar ð12Þ

for n-dodecane, and

log10 0:001� psatðpÞðnÞ� �

¼ AðnÞ � BðnÞT þ CðnÞ ; ð13Þ

where

AðnÞ ¼ 0:0007 n2 � 0:0064 nþ 6:0715;BðnÞ ¼ 51:811 nþ 1049:1;

CðnÞ ¼ 0:1215 n2 � 9:6892 nþ 11:161;

for p-dipropylbenzene. Pressure in Formula (13) is in Pa.Note that none of the above expressions for psatðiÞ can be consid-

ered reliable at temperatures close to or above critical tempera-tures. Heating of the droplets above these temperatures,sometimes predicted by the model at the very final stage of dropletevaporation, does not describe accurately the physical backgroundof the processes at this stage. The contribution of the processes atthis stage of droplet heating and evaporation to the overall dropletheating and evaporation, however, is expected to be small. To mit-igate this behaviour of droplet surface temperature, pressures pre-dicted by Formula (12) and (13) were artificially increased whenthe temperatures exceeded the corresponding critical tempera-tures. This affected the very final stage of droplet evaporation(when their mass becomes less than about 1% of the initial mass)and produced negligible effects on the overall process of heatingand evaporation.

The mass flux of components evaporating from the droplet sur-face is estimated as:

jvi ¼�i _mdj j4pR2

d

: ð14Þ

2.2. Kinetic model

The kinetic model for a three component mixture, described in[17], is used in our analysis. We consider a mixture of air (approx-imated by nitrogen), n-dodecane and p-dipropylbenzene (exactlythe same analysis is applicable to any other mixture). The evolu-tion of the molecular velocity distribution functions of these threecomponents, f a � f aðr; t;vÞ; f n � f nðr; t;vÞ f p � f pðr; t;vÞ, is con-trolled by the following system of Boltzmann equations:

@f a@t þ va

@f a@r ¼ Jaa þ Jan þ Jap

@f n@t þ vn

@f n@r ¼ Jna þ Jnn þ Jnp

@f p

@t þ vp@f p

@r ¼ Jpa þ Jpn þ Jpp

9>>=>>;; ð15Þ

where Jab (a ¼ a;n;p; b ¼ a;n;p) are collision integrals, taking intoaccount the contribution of the collisions between molecules; thefollowing explicit expressions for the collision integrals Jab are used[7]:

Jab ¼r2

ab

2

Z þ1

�1dv1

Z p

0sin hdh

Z 2p

0d/ f 0af 0b1 � f af b1

� �va � vb1

�� ��;ð16Þ

where rab ¼ ðra þ rbÞ=2;ra and rb are the corresponding effectivediameters of molecules of air ‘a’, n-dodecane ‘n’ and p-dipropylben-zene ‘p’, h and / are angular coordinates of molecules b relative tomolecules a, superscript 0 indicates the velocities and the distribu-tion functions after collisions, subscript 1 indicates that the functionf a is modified under the influence of collisions with molecules ofthe type b. The first integral on the right hand side of (16) is calcu-lated in the three dimensional velocity space. Expression (16) is thesame as that used in [7], where the contribution of only 2 compo-nents in the kinetic region was taken into account. This is justifiedby the fact that triple collisions are ignored.

All collision integrals Jab are calculated taking into account thecontribution of internal degrees of freedom (inelastic collisions)as described in [10]. A degree of freedom is defined as a parametercorresponding to each independent variable necessary to describethe energy of a molecule [19]. A monoatomic molecule has threedegrees of freedom corresponding to its translational energies inx; y and z directions. Polyatomic molecules have additional degreesof freedom corresponding to their rotational and vibrationalmotions (see [20] for the analysis of degrees of freedom of CO2

molecules). The total number of degrees of freedom in any mole-cule is equal to 3Nat where Nat is the total number of atoms in amolecule [21]. It is assumed that air (approximated as nitrogen)has 2 internal degrees of freedom, while both hydrocarbons (n-dodecane and p-dipropylbenzene) have 20 internal degrees of free-dom each. As shown in [10,11], taking into account larger numbersof internal degrees of freedom does not affect the results.

As in [6,8,9,11], the effects of the curvature of the droplet sur-face are ignored. This is justified by the fact that the thickness ofthe kinetic region is very small; in our case it is assumed equalto 10 mean free paths for n-dodecane molecules in saturatedn-dodecane vapour at temperature equal to 600 K (‘).

Eqs. (15) are solved subject to the boundary conditions at theinterface between the kinetic and liquid regions and at the inter-face between the kinetic and hydrodynamic regions (see Fig. 1).The first boundary condition for both components of the vapourcan be presented as:

f v i ðoutÞ ¼ bif vis þ ð1� biÞf vir ; ð17Þ

where f vis is the distribution function of molecules leaving theliquid surface assuming that bi ¼ 1; f vir is the distribution functionof reflected molecules. Both f vis and f vir are assumed to be isotropic

Fig. 2. The plots of normalised heat flux ~qk versus ~qRd for eT Rd ¼ 1:05 and eT Rd ¼ 1:10for an 80% n-dodecane and 20% p-dipropylbenzene mixture at droplet surfacetemperature 600 K.

Fig. 3. The plots of mass fluxes, predicted by the kinetic model, of n-dodecane ~jk ðnÞ

and p-dipropylbenzene ~jk ðpÞ versus eT Rd for ~qRd ¼ 0:7 under the same conditions as inFig. 2.

708 S.S. Sazhin et al. / International Journal of Heat and Mass Transfer 79 (2014) 704–712

Maxwellian. The temperature for f v is is assumed to be equal to Ts,while the temperature for f v ir is assumed to be equal to TRd. Thisis justified by the fact that the thickness of the kinetic region issmall and the gas temperature just above the droplet surface isclose to TRd [3]. At the boundary between the kinetic and hydrody-namic regions the distribution function of vapour components andair molecules entering the kinetic region is assumed to be Maxwell-ian, controlled by qRd for both components and TRd.

The contributions of both mass and heat transfer in the kineticregion are taken into account following the approach described in[11]. In [11] we took into account the dependence of the evapora-tion coefficient for n-dodecane on temperature as inferred fromour molecular dynamics analysis [22]. Since no such analysis wasperformed for p-dipropylbenzene, we believe that it would bemore consistent to assume that in both cases the evaporationcoefficients for both components are equal to 1 at this stage.

The kinetic calculations were performed for pure n-dodecanedroplets, and the following n-dodecane and p-dipropylbenzenemolar fractions: 80% n-dodecane and 20% p-dipropylbenzene;70% n-dodecane and 30% p-dipropylbenzene.

Chemical formulae, molar masses and molecular diameters ofthese vapour components and nitrogen (approximating air), usedin our analysis, are given in Table 1.

As in [11], the first step in the solution of Eqs. (15) is to performan investigation of mass and heat transfer processes in the kineticregion for a set of values of qRd (for both vapour components) andTRd. We consider the problem of heating and evaporation of drop-lets in a hot gas (Diesel engine-like conditions) and these parame-ters are assumed to be in the ranges: qRd < qs and TRd > Ts. Duringthe droplet heating process, the temperature increases away fromthe droplet; the evaporation process is possible when the vapourdensity decreases away from the droplet surface. For the chosenvalues of qRd and TRd, the solution to the Boltzmann equation(15) in the kinetic region allows us to calculate the normalisedmass and heat fluxes at the outer boundary of this region:

~jk ðn;pÞ ¼ jk ðn;pÞ= q0

ffiffiffiffiffiffiffiffiffiffiffiRvT0

p� �; ~qk ¼ qk= p0

ffiffiffiffiffiffiffiffiffiffiffiRvT0

p� �;

where Rv is the gas constant referring to n-dodecane vapour, T0 isthe reference temperature chosen equal to 600 K, p0 and q0 arethe saturated n-dodecane vapour pressure and density correspond-ing to T0;q0 is calculated from the ideal gas law, subscript k standsfor kinetic.

In [11] it was shown that for the case of heating and evapora-tion of n-dodecane droplets the values of ~qk are almost indepen-dent of ~qRd � qRd=qs in a certain range of ~qRd and the values of ~jk

are almost do independent of eT Rd � TRd=Ts in a certain range ofeT Rd relevant to the conditions typical for Diesel engines. In whatfollows, it is demonstrated that this property is observed for thecase of heating and evaporation of two-component (a mixture ofn-dodecane and of p-dipropylbenzene) droplets.

Let us assume that molar fractions of n-dodecane andp-dipropylbenzene in the droplet are 80% and 20% respectively,droplet surface temperature is equal to 600 K, and~qRd ðnÞ ¼ ~qRd ðpÞ � ~qRd. The plots of ~qk versus ~qRd for eT Rd ¼ 1:05and 1.1 are shown in Fig. 2. As one can see in this figure, the plotsfor these values of eT Rd are the lines which are almost perfectlyparallel to the ~qRd axis. A similar conclusion was reached in thegeneral case when ~qRd ðnÞ – ~qRd ðpÞ. The same result was obtainedfor the 70% n-dodecane and 30% p-dipropylbenzene mixture. Thisallows us to ignore the dependence of ~qk on ~qRd in agreementwith the similar result obtained in [11].

The plots of mass fluxes of n-dodecane and p-dipropylbenzenepredicted by the kinetic model (~jk ðnÞ and ~jk ðpÞ) versus eT Rd for~qRd ¼ 0:7 for both components under the same conditions as inFig. 2 are shown in Fig. 3. As one can see in this figure, the plots

for these values of ~qRd and in the range of eT Rd shown, are the lineswhich are almost parallel to the eT Rd axis. In contrast to the caseshown in Fig. 2, a very weak dependence of jk on eT Rd can beobserved, but this can be ignored in our analysis. A similar conclu-sion was reached in the general case when ~qRd ðnÞ – ~qRd ðpÞ. The sameresult was obtained for the 70% n-dodecane and 30% p-dipropyl-benzene mixture. This allows us to ignore the dependence of ~jk

on eT Rd. Thus the results shown in Figs. 2 and 3 allow us to decouplethe analysis of heat and mass fluxes in the kinetic region for ~qRd inthe range (0.7 – 1) and eT Rd in the range (1–1.1).

The values of eT Rd for two-component droplets were obtainedfollowing the same procedure as described in [11] for monocompo-nent droplets. This procedure is illustrated in Fig. 4 for an 80%n-dodecane and 20% p-dipropylbenzene droplet of radius 5 lm,surface temperature 600 K and gas temperature 1000 K. The valueof ~qRd was taken equal to 0.7. Recalling Fig. 2, we expect the resultsnot to depend on the actual values of ~qRd. The plots of the heat fluxpredicted by the kinetic model, ~qk, versus eT Rd, and the heat flux inthe hydrodynamic region predicted by the hydrodynamic model,~qh ¼ qh=ðp0

ffiffiffiffiffiffiffiffiffiffiffiRvT0p

Þ, versus eT Rd (horizontal line), for these valuesof parameters are shown in Fig. 4. The intersection between the

Fig. 4. The plots of the heat flux predicted by the kinetic model, ~qk, versus eT Rd andthe heat flux predicted by the hydrodynamic model, ~qh ¼ qh=ðp0

ffiffiffiffiffiffiffiffiffiffiffiRv T0p

Þ, versus eT Rd

(horizontal line) for ~qRd ¼ 0:7 and an 80% n-dodecane and 20% p-dipropylbenzenemixture droplet of radius 5 lm, surface temperature 600 K and gas temperature1000 K. The intersection between the ~qk and ~qh gives the value of eT Rd ¼ 1:022.

S.S. Sazhin et al. / International Journal of Heat and Mass Transfer 79 (2014) 704–712 709

horizontal and inclined lines gives the required value ofeT Rd ¼ 1:022.The plots of the mass flux predicted by the kinetic model, ~jk,

versus ~qRd for eT Rd ¼ 1:05 (as follows from the analysis based on

Fig. 3, the result is not expected to depend of eT Rd) and the mass flux

predicted by the hydrodynamic model, ~jh ¼ jh= q0

ffiffiffiffiffiffiffiffiffiffiffiRvT0p �

, for

n-dodecane (~jk ðnÞ and ~jh ðnÞ) and p-dipropylbenzene (~jk ðpÞ and ~jh ðpÞ)versus ~qRd (~qRd ðnÞ or ~qRd ðpÞ) are shown in Fig. 5. The mass fluxes pre-dicted by the hydrodynamic model are shown by the horizontallines. This figure is presented for the same parameters as inFig. 4. Following [11], it was assumed that qRd in Eq. (14) can bereplaced with qs. The intersections between the horizontal andinclined lines give the required values of ~qRd : ~qRdðnÞ ¼ 0:983 forn-dodecane and ~qRdðpÞ ¼ 0:987 for p-dipropylbenzene. Similar val-

ues of eT Rd and ~qRd were obtained for other values of Tg ; Ts and Rd

relevant for Diesel engine conditions (Tg ¼ 750 K, Tg ¼ 700 K; forvalues of Ts in the range 300 K to temperatures close to the critical

Fig. 5. The plots of the mass flux of the components predicted by the kinetic model,~jk, versus ~qRd for eT Rd ¼ 1:05 and the hydrodynamic model, ~jh ¼ jh=ðp0

ffiffiffiffiffiffiffiffiffiffiffiRv T0p

Þ, for n-dodecane (~jh ðnÞ) and p-dipropylbenzene (~jh ðpÞ) versus ~qRd (~qRd ðnÞ and ~qRd ðpÞ for thesame parameters as in Fig. 4). ~jh ðnÞ and ~jh ðpÞ are shown by horizontal lines. Theintersections between the horizontal and inclined lines give the required values of~qRd : ~qRdðnÞ ¼ 0:983 for n-dodecane and ~qRdðpÞ ¼ 0:987 for p-dipropylbenzene.

temperature, and the range of molar fractions of n-dodecane andp-dipropylbenzene predicted by hydrodynamic calculations). The

corresponding values of eT Rd and ~qRd were used for the analysis ofheating and evaporation of mono- and two-components dropletsin realistic Diesel engine-like conditions.

Following the above procedure, a set of values of ~qRd and eT Rd

were obtained for a number of specific pairs of values of dropletsurface temperatures and radii, predicted by the hydrodynamicmodel. Once the values of ~qRd were obtained, then for kinetic mod-elling the values of qs in Eq. (8) were replaced by qRd. For the inter-mediate values of these parameters the values of ~qRd and eT Rd wereinterpolated. The thickness of the kinetic region is assumed to beinfinitely small. Since qRd < qs, then the value of BM predicted bythe kinetic model (BM; k) is always less than the value of BM pre-dicted by the hydrodynamic model (BM; h). Hence, we can expectthat the evaporation rate predicted by the kinetic model is alwaysless than the one predicted by the hydrodynamic model for thesame droplet surface temperature.

The decrease in the values of BM predicted by the kinetic modelis expected to lead to a corresponding decrease in the values of BT

and ultimately the values of the convection heat transfer coeffi-cient h. This will lead to a decrease in the values of Teff . On theother hand, slowing down of the evaporation process, predictedby the kinetic model, will lead to a decrease in j _Rdj, and ultimatelyan increase in Teff . The balance between these two processes willlead to either a decrease or an increase in the predicted dropletsurface temperatures.

Note that we did not take into account the effect of changes inthe droplet evaporation rates due to the changes in droplet surfacetemperatures predicted by the kinetic model, as in this case wewould need to use the corrected values of jvi for the mass fluxespredicted by the hydrodynamic model. This would lead to new val-ues of ~qRd and the new values of droplet surface temperature etc.The investigation of this effect is beyond the scope of this paper.

In the next section, the results of modelling of specific dropletsare demonstrated.

3. Results

The results of calculation of the radii and surface temperaturesof a droplet with initial radius and temperature equal to 5 lm and300 K, respectively, immersed into gas with temperature 1000 Kare shown in Fig. 6. Three types of droplets have been considered:pure n-dodecane, a mixture of 80% n-dodecane and 20%p-dipropylbenzene, and a mixture of 70% n-dodecane and 30%p-dipropylbenzene (the contributions of components refer to theirmolar fractions; see Table 2). Results of both hydrodynamic andkinetic calculations are presented. In kinetic calculations both heatand mass transfer in the kinetic region and the effects of inelasticcollisions are taken into account. The evaporation coefficient isassumed equal to 1 for both components.

Since the values of qRd were estimated as perturbations of qs

predicted by the hydrodynamic model, the calculations of thedroplet radii predicted by the kinetic model had to be terminatedbefore the evaporation time predicted by the hydrodynamic model(which is always less than the evaporation time predicted by thekinetic model). Then the values of the droplet radii predicted bythe kinetic model were extrapolated until the complete evapora-tion of the droplet.

As one can see from Fig. 6, both the addition of p-dipropylbenzeneand kinetic effects lead to an increase in the evaporation time ofdroplets. To estimate the effect of droplet composition on the kineticeffects, the following error function is introduced:

E � teðkÞ � teðhÞ

teðkÞ� 100%; ð18Þ

Fig. 6. The plots of Rd and Ts versus time, as predicted by the kinetic and hydrodynamic models for droplets with initial radii and temperature equal to 5 lm and 300 K,immersed into gas with temperature 1000 K; pure n-dodecane and mixtures of n-dodecane and p-dipropylbenzene have been considered; the link between the modellingconditions and the plots is shown in Table 2. A refers to the zoomed part of the figure for droplet surface temperatures; B refers to the zoomed part of the figure for dropletradii.

Table 2The link between the modelling conditions and the plots shown in Figs. 6 and 7.

Plots Model Molar fraction of n-dodecane (%)

Molar fraction of p-dipropylbenzene (%)

1 Kinetic 100 02 Kinetic 80 203 Kinetic 70 304 Hydrodynamic 100 05 Hydrodynamic 80 206 Hydrodynamic 70 30

710 S.S. Sazhin et al. / International Journal of Heat and Mass Transfer 79 (2014) 704–712

where teðk ðhÞÞ is the evaporation time predicted by the kinetic(hydrodynamic) models. E is estimated as a percentage. Note thatteðkÞ > teðhÞ. The values of E for the three mixtures presented inFig. 6 are the following: 100% n-dodecane – E ¼ 1:61%, 80%n-dodecane – E ¼ 1:70%, 70% n-dodecane – E ¼ 2:5%. Thus, the val-ues of E increase with increasing p-dipropylbenzene contribution.In all cases, these values remain less than 3%, and they need to betaken into account only in the case of very accurate modelling ofthis process.

The same plots as in Fig. 6 but for gas temperature equal to 700 Kare shown in Fig. 7. Comparing Figs. 6 and 7 one can see that thedecrease in gas temperature from 1000 K to 700 K leads to more thandoubling of the evaporation time and reduction of the kinetic effectsfor all three mixtures. For 100% n-dodecane – E ¼ 0:22%, 80%n-dodecane – E ¼ 0:32%, 70% n-dodecane – E ¼ 0:58%. As in the caseof gas temperature equal to 1000 K, the values of E increase withincreasing p-dipropylbenzene contribution. These errors can besafely ignored in all feasible applications.

We can expect that the kinetic effects will become even morenoticeable at temperatures greater than 1000 K. In this case, how-ever, the droplet surface temperatures are expected to approachthe critical temperature well before its final evaporation. Ourmodel is expected to be less reliable as it is based on the assump-tion that droplet surface temperature is not close to the criticaltemperature.

The main problem with approximation of Diesel fuel by amixture of n-dodecane and p-dipropylbenzene is that the accu-racy of this approximation for modelling droplet heating andevaporation has not yet been carefully investigated for a widerange of available Diesel fuels. In the case of Diesel fuel

Fig. 7. The same as Fig. 6, but for the gas temperature 700 K.

S.S. Sazhin et al. / International Journal of Heat and Mass Transfer 79 (2014) 704–712 711

considered in [11], the evaporation time of droplets with initialradii 5 lm in gas at temperature 700 K, predicted by thisapproximation, turned out to be about one half of the evapora-tion time predicted for a Diesel fuel droplet in these conditions,although the prediction of this approximation is better than theone for a pure n-dodecane droplet. As shown in [14], a reason-ably accurate approximation for both the evaporation time andtime evolution of droplet surface temperature can be achievedwhen Diesel fuel is approximated by about 15 quasi-compo-nents/components. Of course, it is not possible to performkinetic modelling for such a mixture; the approximation of Die-sel fuel by a two-component mixture is bound to be crude,although nobody, to the best of our knowledge, has attemptedto investigate this problem in depth and found an optimaltwo-component approximation of Diesel fuel (apart from theapproximation used in this paper). We were able to show thatthe mixture 10% n-dodecane and 90% dodecylbenzene (C18H30)leads to the prediction of droplet evaporation times close tothe one predicted for a Diesel fuel droplet. To achieve this, how-ever, it was necessary to tolerate rather large differences in thepredicted surface temperatures for this approximation and forthe approximation of Diesel fuel considered in [14].

4. Conclusions

The previously developed kinetic model for two-componentdroplet heating and evaporation into a high pressure backgroundgas (air) has been applied to the analysis of n-dodecane and p-dipropylbenzene mixture droplet heating and evaporation in Die-sel engine-like conditions. As in our previous papers (e.g. [11]),the kinetic heating and evaporation model is based on the intro-duction of the kinetic region in the immediate vicinity of theheated and evaporating droplets, where the dynamics of moleculesare described in terms of the Boltzmann equations for vapour com-ponents and air, and the hydrodynamic region. In contrast to [11],the presence of two components in the vapour are taken intoaccount. The boundary conditions at the outer boundary of thekinetic region are introduced by matching the mass fluxes ofvapour components leaving the kinetic region and entering intothe surrounding hydrodynamic region. The effects of finite thermalconductivity inside the droplets and inelastic collisions in thekinetic region are taken into account as in [11]. In contrast to[11], we have also considered the effects of species diffusivityinside the droplets. The evaporation coefficient for both compo-nents is assumed equal to 1.

712 S.S. Sazhin et al. / International Journal of Heat and Mass Transfer 79 (2014) 704–712

It is pointed out that for the parameters typical for Dieselengine-like conditions, the heat flux in the kinetic region is a linearfunction of the vapour temperature at the outer boundary of thisregion, but is almost independent of the densities of vapour com-ponents at this boundary in a certain range of these densities, asin the case of monocomponent droplets (see [11]). The mass fluxesof both components in the kinetic region are shown to decreasealmost linearly as their densities at the outer boundary of thekinetic region increase, but are almost independent of the temper-atures at this boundary in a certain range of these temperatures, asin the case of monocomponent droplets. Using the matching condi-tions at the outer boundary of the kinetic region, the values of tem-perature and densities of both components at this boundary havebeen found.

The model is tested for the analysis of heating and evaporation ofdroplets with initial radii and temperature equal to 5 lm and 300 K,immersed into gas with temperatures 1000 K and 700 K for threedroplet compositions (in terms of molar fractions): pure n-dodecane,a mixture of 80% n-dodecane and 20% p-dipropylbenzene, and amixture of 70% n-dodecane and 30% p-dipropylbenzene. It is shownthat both the addition of p-dipropylbenzene and kinetic effects leadto an increase in the evaporation time of droplets. In all cases, thekinetic effects on the droplet evaporation times increase withincreasing p-dipropylbenzene contribution and gas temperature.

Acknowledgements

The authors are grateful to EPSRC (UK) (Project EP/J006793/1),INTERREG IVa (Project E3C3, Reference 4274) and the RussianFoundation for Basic Research (Grant 14-08-00467) for theirfinancial support of this project.

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