Advanced models of fuel droplet heating and evaporation

Transcription

1 Progress in Energy an Combustion Science 32 (2006) Avance moels of fuel roplet heating an evaporation Sergei S. Sazhin * School of Engineering, Faculty of Science an Engineering, The University of Brighton, Cockroft Builing, Lewes Roa, Brighton BN2 4GJ, UK Receive 18 April 2005; accepte 2 November 2005 Available online 6 January 2006 Abstract Recent evelopments in moelling the heating an evaporation of fuel roplets are reviewe, an unsolve problems are ientifie. It is note that moelling transient roplet heating using steay-state correlations for the convective heat transfer coefficient can be misleaing. At the initial stage of heating stationary roplets, the well known steay-state result NuZ2 leas to uner preiction of the rate of heating, while at the final stage the same result leas to over preiction. The numerical analysis of roplet heating using the effective thermal conuctivity moel can be base on the analytical solution of the heat conuction equation insie the roplet. This approach was shown to have clear avantages compare with the approach base on the numerical solution of the same equation both from the point of view of accuracy an computer efficiency. When highly accurate calculations are not require, but CPU time economy is essential then the effect of finite thermal conuctivity an re-circulation in roplets can be taken into account using the so calle parabolic moel. For practical applications in computation flui ynamics (CFD) coes the simplifie moel for raiative heating, escribing the average roplet absorption efficiency factor, appears to be the most useful both from the point of view of accuracy an CPU efficiency. Moels escribing the effects of multi-component roplets nee to be consiere when moelling realistic fuel roplet heating an evaporation. However, most of these moels are still rather complicate, which limits their wie application in CFD coes. The Distillation Curve Moel for multi-component roplets seems to be a reasonable compromise between accuracy an CPU efficiency. The systems of equations escribing roplet heating an evaporation an autoignition of fuel vapour/air mixture in iniviual computational cells are stiff. Establishing hierarchy between these equations, an separate analysis of the equations for fast an slow variables may be a constructive way forwar in analysing these systems. q 2005 Elsevier Lt. All rights reserve. Keywors: Droplets; Fuel; Heating; Evaporation; Convection; Raiation Contents 1. Introuction Heating of non-evaporating roplets Convective heating Stagnant roplets Moving roplets Raiative heating Basic equations an approximations Mie theory * Tel.: C ; fax: C aress: s.sazhin@brighton.ac.uk /$ - see front matter q 2005 Elsevier Lt. All rights reserve. oi: /j.pecs

2 S.S. Sazhin / Progress in Energy an Combustion Science 32 (2006) Integral absorption of raiation in roplets Geometric optics analysis Droplet evaporation Empirical correlations Hyroynamic moels Classical moel Abramzon an Sirignano moel Yao, Abel-Khalik an Ghiaasiaan moel Multi-component roplets Kinetic moels Molecular ynamics simulations Evaporation an autoignition Couple solutions Concluing remarks Acknowlegements Appenix A. Physical properties of fuels Appenix B. Physical properties of tetraecane Appenix C. Physical properties of n-heptane Appenix D. Physical properties of n-oecane Appenix E. Physical properties of iesel fuel References Introuction The problem of moelling roplet heating an evaporation is not a new one. A iscussion of the moels evelope prior to the early fifties is provie in [1,2]. A number of wiely known monographs an review papers have been publishe since then, incluing [3 17]. Various aspects of this problem have been covere in numerous review articles, incluing those publishe in this journal [18 21]. In all these monographs an review articles, however, this problem was iscusse as an integral part of a wier problem of roplet an spray ynamics. This inevitably limite the epth an the breath of coverage of the subject. Also, most of the relevant monographs an reviews were publishe more than 5 years ago, an thus o not inclue the most recent evelopments in this area. In contrast to the articles reference above, this review will focus on the relatively narrow problem of roplet heating an evaporation. Although the application of the moels will be mainly illustrate through examples referring to fuel roplets, most of them coul be easily generalise to any liqui roplets if require. Only subcritical heating an evaporation will be consiere. Near-critical an supercritical roplet heating an evaporation was covere in the relatively recent reviews publishe in this journal [22, 23], an in [24]. Analysis of the interaction between roplets, collisions, coalescence, atomization, oscillations (incluing instabilities of evaporating roplets) an size istribution will also be beyon the scope of this review, although all these processes inirectly influence the processes consiere (see [25 40]). Neither will the problem of heating an evaporation of roplets on heate surfaces be consiere (see [37,41]). Although the phenomena consiere in this review can be an integral part of the more general process of spray combustion, the etaile analysis of the latter will also be beyon the scope of this work (see [42 45]). Although the problem of raiative heating of roplets is closely linke with the problem of scattering of raiation, the formal moelling of the two processes can be separate. The moels of the latter process were reviewe in [46] (see also [47]), an their analysis will be beyon the scope of this paper. Soret an Dufour effects will be ignore. Soret effect escribes the flow of matter cause by a temperature graient (thermal iffusion), while Dufour effect escribes the flow of heat cause by concentration graients. The two effects occur simultaneously. Both effects are believe to be small in most cases although sometimes their contribution may be significant (see [48 52]). In most moels of roplet evaporation it is assume that the ambient gas is ieal. This assumption becomes questionable when the pressures are high enough, as observe in internal combustion engines. The main approaches to taking into account

3 164 S.S. Sazhin / Progress in Energy an Combustion Science 32 (2006) Nomenclature a coefficient introuce in Eq. (73) (m Kb ) a l liqui fuel absorption coefficient (1/m) a w, b w, c w constants introuce in Eq. (34) A pre-exponential factor (1/s) A v, B v functions introuce in Eqs. (40) an (41) a 0,1,2 coefficients introuce in Eq. (74) (m Kb,1/ (K m b ), 1/(K 2 m b )) b coefficient introuce in Eq. (73) b 0,1,2 coefficients introuce in Eq. (74) (1, 1/K, 1/K 2 ) B branching agent B f parameter introuce in Eq. (87) B M Spaling mass number B T Spaling heat transfer number B l Planck function (W/(m 2 mm)) c specific heat capacity (J/(kg K)) c k, k functions introuce in Eqs. (62) (64) C f fuel vapour molar concentration (kmol/m 3 ) C 1,2 coefficients in the Planck function (W mm 3 /m 2, mm K) C g1, g2 coefficients introuce in Eq. (79) f iameter of fuel molecules (m) D binary iffusion coefficient (m 2 /s) E activation energy (J) E r,q,f components of wave electric fiel (N/C) f molecular istribution function f c function introuce in Eq. (35) f m relative contribution of components (see Eq. (137)) F force (N) F T,M correction factors: T / T0 ; M / M0 Fo Fourier number: tk g =R 2 g 0 (R) function efine by Eq. (78) (W/(m 2 mm)) h convection heat transfer coefficient (W/(m 2 K)) h m mass transfer coefficient (m/s) h 0 (hr /k l )K1 H e enthalpy (J) I property of the component (see Eq. (137)) I l spectral intensity of thermal raiation in a given irection (W/(m 2 mm)) Il 0 spectral intensity of thermal raiation integrate over all angles (W/(m 2 mm)) Il ext spectral intensity of external raiation in a given irection (W/(m 2 mm)) I 0ðextÞ l spectral intensity of external raiation integrate over all angles (W/(m 2 mm)) j mass flux (kg/(m 2 )) k thermal conuctivity (W/(m K)) k B Boltzmann constant (J/K) Kn Knusen number l coll characteristic mean free path of molecules (m) l K thickness of the Knusen layer (m) L specific heat of evaporation (J/kg) Le Lewis number: k g /(c pg r total ) m mass (kg) m i mass of iniviual molecules (kg) m l complex inex of refraction: n l Kik l M molar mass (kg/kmol) n inex of refraction (oes not epen on l) n l inex of refraction (epens on l) n N A Avogaro number (1/kmol) Nu Nusselt number _q heat flux (W/m 2 ) Q intermeiate agent Q a efficiency factor of absorption Q f specific combustion energy (J/kg) p(r) raiative power ensity (see Eq. (77)) (W/ m 3 ) p n coefficients introuce in formula (58) (K/s) p l (R) spectral istribution of raiative power ensity (W/(m 3 mm)) ~p v p v /p amb P(R) raiative term in Eq. (57) (K/s) P ch chemical power per unit volume release in the gas phase (W/m 3 ) P total total amount of raiation absorbe in a roplet (K/s) P 1 k associate Legenre polynomials Pe Peclet number Pr Prantl number R istance from the roplet centre (m) R cut parameter introuce in Eq. (151) (m) R g gas constant (J/(kg K)) R i positions of iniviual molecules (m) R ij istance between molecules (m) R m the value of R ij when VZK3 ij (m) R ref reflection coefficient R u universal gas constant (J/(kmol K)) R* raical R * R /n (m) RH hyrocarbon fuel Re Reynols number S function introuce in Eq. (65) Sc Schmit number Sh Sherwoo number

4 S.S. Sazhin / Progress in Energy an Combustion Science 32 (2006) t time (s) T temperature (K) ~T 0 ðrþ parameter introuce in Eq. (21) (K) u flui velocity (m/s) U value of the net velocity of the mixture (m/s) v molecular velocities (m/s) kv n (r)k parameter introuce in Eq. (21) V Lenar Jones 12-6 potential (J) V shift parameter introuce in Eq. (151) (J) w l normalise absorbe spectral power ensity of raiation We Weber number x position in space (m) x l size parameter: 2pR /l X, Y n- an m-imensional vectors Y relative concentration z parameter introuce in Eq. (104) Greek symbols a, b, g parameters introuce in Eq. (138) a v (Re) parameters introuce in Eq. (49) b c coefficients introuce in Eq. (33) b m evaporation or conensation coefficient b v parameter introuce in Eq. (53) g c parameter introuce in Eq. (6) g parameter introuce in Eq. (85) T,M film thickness (m) t time step use for calculation of roplet parameters (s) Dt global time step (s) e emissivity 3 small positive parameter 3 i, 3 j parameters introuce in Eq. (149) (J) 3 ij minimal energy of interaction between molecules (J) 3 m species evaporation rate z parameter introuce in Eq. (24) z k Riccati Bessel functions q angle relative to the velocity of unperturbe flow or the angle of wave propagation q R raiative temperature (K) Q Heavisie unit step function k thermal iffusivity (m 2 /s) k R k l =ðc l r l R 2 Þ (1/s) k l inex of absorption l wavelength (m or mm) l m 3.4 mm l n eigen values obtaine from the solution of Eq. (22) l v m l /m g l 1,2 spectral range of absorbe raiation (mm) L 0 function introuce in Eq. (73) m ynamic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi viscosity (kg/(m s)) m c 1Kð1=n 2 Þ m 0 (t) (ht g (t)r /k l )(K) m q cos q mq 0 parameter pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi introuce in Eq. (76) m * 1KðR =RÞ 2 n kinematic viscosity (m 2 /s) x parameter introuce in formula (86) p k function introuce in Eqs. (62) (63) r ensity (kg/m 3 ) r b moifie ensity efine by Eq. (147) (kg/ m 3 ) r l 2pR/l (rc) 12 r g c pg/ (r l c l ) s Stefan Boltzmann constant (W/(m 2 K 4 )) s i, s j parameters introuce in Eq. (150) (m) s ij zero energy separation between molecules (m) s s interfacial surface tension (N/m) t el time elay before the start of autoignition (s) t k function introuce in Eqs. (62) (64) t l a l R t 0 a l R f asymuthal angle measure from the plane of electric fiel oscillations F(u) function introuce in Eqs. (5) an (6) c k eff /k l (see Eq. (54)) c l,m molar fraction of the mth species in the liqui c t parameter efine by Eq. (17) j k Riccati Bessel functions U Y parameter introuce in Eq. (118) Subscripts b boiling abs absorbe amb ambient c centre or convection coll collision cr critical roplet iff iffusive r rift eff effective ext external f film surrouning roplets or fuel g gas ins incient

5 166 S.S. Sazhin / Progress in Energy an Combustion Science 32 (2006) l lg m p r R R s sv t liqui from liqui to gas type of species in the liqui phase constant pressure raial component raiation outsie the Knusen layer surface saturate fuel vapour time epenent v fuel vapour q component in q irection f component in f irection K0 inner sie of the roplet surface C0 outer sie of the roplet surface N infinitely far from the roplet surface Superscripts average 00 absolute value per unit area of the roplet surface real gas effects have been iscusse in [53 57]. The analysis of these effects, however, will be beyon the scope of this review. This review is intene to be both an introuction to the problem an a comprehensive escription of its current status. Most of the review is planne to be a self-sufficient text. On some occasions, however, the reaer will be referre to the original papers, without etaile escription of the moels. Experimental results will be iscusse only when they are essential for unerstaning or valiation of the moels. The focus will be on the moels suitable or potentially suitable for implementation in computational flui ynamics (CFD) coes. These are the public omain (e.g. KIVA) or commercial (e.g. PHOENICS, FLUENT, VECTIS, STAR CD) coes. The structures of these coes can vary substantially. However, basic approaches to roplet an spray moelling use in them are rather similar. This will allow us to link the moels, escribe in this review, with any of these coes, without making any specific references. Following [13] the moels of roplet heating can be subivie into the following groups in orer of ascening complexity: (1) moels base on the assumption that the roplet surface temperature is uniform an oes not change with time; (2) moels base on the assumption that there is no temperature graient insie roplets (infinite thermal conuctivity of liqui); (3) moels taking into account finite liqui thermal conuctivity, but not the re-circulation insie roplets (conuction limit); (4) moels taking into account both finite liqui thermal conuctivity an the re-circulation insie roplets via the introuction of a correction factor to the liqui thermal conuctivity (effective conuctivity moels); (5) moels escribing the re-circulation insie roplets in terms of vortex ynamics (vortex moels); (6) moels base on the full solution of the Navier Stokes equation. The first group allows the reuction of the imension of the system via the complete elimination of the equation for roplet temperature. This appears to be particularly attractive for the analytical stuies of roplet evaporation an thermal ignition of fuel vapour/air mixture (see e.g. [58 62]). This group of moels, however, appears to be too simplistic for application in most CFD coes. The groups (5) an (6) have not been use an are not expecte to be use in these coes in the foreseeable future ue to their complexity. These moels are wiely use for valiation of more basic moels of roplet heating, or for in-epth unerstaning of the unerlying physical processes (see, e.g. [13,63 66]). The main focus of this review will be on moels (2) (4), as these are the ones which are actually use in CFD coes, or their incorporation in them is feasible. The review consists of two main parts. First, moels of non-evaporating roplets will be reviewe (Section 2). Then evaporation moels will be iscusse (Section 3). The main results of the review will be summarise in Section Heating of non-evaporating roplets Someone wishing to moel heating of non-evaporating roplets woul be require to take into account a number of important processes. These inclue the eformation of roplets in the air stream an the inhomogeneity of the temperature istribution at the roplet surface. Rigorous solution of this general problem, however, woul have been not only ifficult, but also woul have rather limite practical applications. Inee, in realistic engineering applications

6 moelling of simultaneous heating of a large number of roplets woul be require. Moreover, this moelling woul have to be performe alongsie gas ynamics, turbulence an chemical moelling. This leas to the situation where the parameters of gas aroun roplets may be estimate with substantial errors, which are often ifficult to control. This is why the focus has to be on fining a reasonable compromise between accuracy an computer efficiency of the moels, rather than on the accuracy of the moels alone. The most commonly use assumption is that the roplet retains its spherical form even in the process of its movement. This assumption will be mae in this review as well. The generalisations of the moels to roplets of arbitrary shapes were iscusse in a number of books an articles (see [2,65,67]). These generalisations coul be applie to simple moels of roplet heating, but it is not obvious how they can be applie to the more sophisticate moels iscusse below. Another simplification wiely use in roplet heating moels is the assumption that the temperature over the whole roplet surface is the same (although it can vary with time). This assumption effectively allows the separation of the analysis of heat transfer in gaseous an liqui phases. It is expecte to be a goo approximation in the case of a stationary or very fast moving roplet, when the isotherms almost coincie with streamlines [63]. The errors introuce by this assumption in intermeiate conitions are generally assume to be acceptable. In what follows the moels for convective an raiative heating of roplets will be consiere separately (Sections 2.1 an 2.2) Convective heating Stagnant roplets In the case of stagnant roplets, there is no bulk motion of gas relative to the roplets, an the problem of their heating by the ambient gas reuces to a conuction problem. The heat conuction equation can be solve separately in the roplet an the gas, an the solutions are matche at the roplet surface. Assuming the spherical symmetry of the problem, its mathematical formulation is base on the solution of the equation vt vt Z k where v2 T vr 2 C 2 R vt vr S.S. Sazhin / Progress in Energy an Combustion Science 32 (2006) ; (1) ( k Z k l Z k l =ðc l r l Þ when R%R k g Z k g =ðc pg r g Þ when R!R%N; (2) k l(g), k l(g), c l(pg), an r l(g) are the liqui (gas) thermal iffusivity, thermal conuctivity, specific heat capacity, an ensity, respectively, R is the istance from the centre of the sphere, t is time, subscripts l an g refer to liqui an gas, respectively. This equation nees to be solve subject to the following initial an bounary conitions: ( Tj tz0 Z T 0ðRÞ when R%R (3) T g0 ðrþ when R!R%N; Tj RZR K0 Z Tj RZR C0; vt vt k l vr RZR K0 Z k g vr Tj RZN Z T gn : RZR C0; (4) Assuming that T 0 (R)ZT 0 Zconst 1, an T g0 (R)Z T gn Zconst 2, Cooper [68] has solve this problem analytically, using the Laplace transform metho. His solution can be presente as rffiffiffiffiffi TðR%R Þ Z T gn C 2k l pk g ðn! 0 k g k l ðt gn KT 0 Þ R R ufðuþexp Ku 2 Fo k l sin u R ; k g R TðRRR Þ Z T gn C 2k rffiffiffiffiffi l k g ðt pk g k gn KT 0 Þ R l R ðn u! u FðuÞexp Ku2 Fo k l where FðuÞ Z Fo Z tk g =R 2 0 k g (5) rffiffiffiffiffi k g! cosðg c uþsin u C sin g k c l! k l ðu cos uksin uþ Csin u ; ð6þ k g ðu cos uksin uþ h i u 2 sin 2 u C k 2 ; g k l k l k g ðu cos uksin uþ Csin u ðfourier numberþ

7 168 rffiffiffiffiffi k g c Z l k g RKR R u: As expecte, this solution preicts iffusion of heat from gas to roplets, if T gn OT 0. As a result, roplet an gas temperatures approach each other in any finite omain. Initially, the heat flux from gas to roplets preicte by the equation vt _q ZKk g (7) vr RZR C0 is infinitely large. It approaches zero at t/n. This solution was originally applie to the problem of heating a stationary liqui soium sphere in UO 2 atmosphere [68]. However, the applicability of this solution to more general problems, involving the time variations of gas temperature ue to external factors an evaporation effects, is questionable. The only practical approach to solve this general problem is currently base on the application of computational flui ynamics (CFD) coes. In this case one woul nee to take into account the istribution of temperature insie roplets at the beginning of each time step an the finite size of computational cells. These effects were taken into account in the solution suggeste in [69,70]. However, these papers i not iscuss how practical it is to implement the solution into a CFD coe. Practically all the available an, known to me, CFD coes are base on separate solutions for gas an liqui phases, followe by their coupling [19,28,44,71]. Hence, some kin of separation of the solutions for gas an liqui phases woul be essential to make them compatible with these coes. The require separation between the solutions coul be achieve base on the comparison between the thermal iffusivities of gas an liqui. Let us consier typical values of parameters for iesel fuel spray roplets an assume that these roplets have initial temperature 300 K an are injecte into a gas at temperature 800 K an pressure 30 atm [61]: r l Z 600 kg=m 3 ; k l Z 0:145 W=ðm KÞ; c l Z 2830 J=ðkg KÞ; r g Z 23:8 kg=m 3 ; k g Z 0:061 W=ðmKÞ; S.S. Sazhin / Progress in Energy an Combustion Science 32 (2006) c pg Z 1120 J=ðkg KÞ: A more etaile iscussion of fuel properties is given in Appenix A. For these values of parameters we obtain k l Z8.53!10 K8 m 2 /s an k g Z2.28!10 K6 m 2 /s. This allows us to assume: k l /k g : (8) This conition tells us that gas respons much more quickly to changes in the thermal environment than liqui. As a zeroth approximation we can ignore the changes in liqui temperature altogether, an assume that the roplet surface temperature remains constant in time. This immeiately allows us to ecouple the solution of Eq. (1) from the trivial solution of this equation for the liqui phase (T(R%R )Zconst). The former solution can be presente as TðROR Þ Z T gn C R R ðt 0KT gn Þ 1Kerf where erfðxþ Z p 2 ffiffiffi p ð x 0 expðkt 2 Þt: RKR p 2 ffiffiffiffiffiffi ; k g t (9) In the limit RZR, Eq. (9) gives TZT 0. In the limit t/0, but RsR, this equation gives TZT gn. Having substitute (9) into (7) we obtain the following equation for the heat flux from gas to roplets [72]: _q Z k gðt 0 KT gn Þ 1 C R pffiffiffiffiffiffiffiffiffi : (10) R pk g t The same expression follows from the analysis reporte later in [73,74], who were apparently not aware of the original paper [72]. Moreover, this expression might have been erive even earlier, as in 1971 it was referre to in [75] as the well known conuction solution without giving any references. For t[t hr 2 =ðpk g Þ Eq. (10) can be further simplifie an rewritten as: j _qj Z hðt gn KT 0 Þ; (11) where h is the convection heat transfer coefficient efine as h Z k g : (12) R Remembering that the convection heat transfer is commonly escribe by the Nusselt number NuZ2R h/ k g, Eq. (12) is equivalent to the statement that NuZ2. Solution (11) coul be obtaine irectly from Eq. (1) if the time erivative of temperature is ignore (steaystate solution). It gives us the well known Newton s law for heating of stationary roplets so long as the bounary layer aroun roplets has ha enough time to evelop. Note that for the values of parameters mentione above t Z3.5 ms. That means that except at

8 S.S. Sazhin / Progress in Energy an Combustion Science 32 (2006) the very start of roplet heating, this process can be base on Eqs. (11) an (12). These equations are wiely use in CFD coes. Comparing Eqs. (10) (12), it can be seen that Newton s law (Eqs. (11) an (12)) can be use to escribe the transient process iscusse above, if the gas thermal conuctivity k g is replace by the time epenent gas thermal conuctivity k g(t) efine as [72,74] k gðtþ Z k g ð1 Cz t Þ; (13) where rffiffiffiffiffiffiffiffiffiffiffi c pg r g z t Z R : (14) pk g t This is applicable only at the very start of roplet heating (at the start of calculations when a roplet is injecte into the gas). Unless abrupt changes in gas temperature occur, one may assume that the bounary layer aroun the roplet has ha enough time to ajust to varying gas temperature. This woul justify the application of Newton s law in its original formulation (Eqs. (11) an (12)). Note that in the limit t/n solution (9) is simplifie to DT ht KT gn Z R R ðt 0KT gn Þ; (15) where DT inicates the local changes in gas temperature after the bounary layer aroun the roplet has been forme. The change of gas enthalpy, ue to the presence of the roplet, in this case can be obtaine as: DH e Z ðn R r g c pg DT4pR 2 R calculate using Eq. (16). In practice DH e has never been calculate from Eq. (16), to the best of my knowlege. Instea, the amount of heat gaine or lost by the roplet uring a certain perio of time Dt has been calculate base on the values of Nu. This gives reasonable results provie that the assumption that the roplet surface temperature remains constant in time is vali. Cooper s [68] solution also allows us to present the heat flux in the form (11), by replacing h with h t Zc t hzc t k g =R, where K k l k g c t Z Ð N 0 ðu cos uksin uþfðuþexp Ku 2 Fo k l k g u Ð ; N 0 sin ufðuþexp Ku 2 Fo k l k g u (17) F(u) is the same as in Eqs. (5) an (6). The plot of c t versus Fo for the same values of parameters as use earlier is presente in Fig. 1. As follows from this figure, c t is large for small Fo, asin the case of the solution with constant roplet surface temperature. In contrast to the latter solution, however, this correction to h approaches not 1, but 0att/N (or Fo/N). These small values of c t are observe at the final stage of roplet heating when the ifference between the ambient gas temperature an roplet surface temperature becomes close to zero. As mentione earlier, at this stage the applicability of Eqs. (5) an (6) becomes questionable. To summarise this part of the review, we may say that simple Eqs. (11) an (12) seem to be the most Z 4pR ðt 0 KT gn Þr g c pg ð N R R R ZN: (16) Thus the establishment of the require bounary layer of a single roplet leas to an infinitely large change in the enthalpy of the gas. It seems that Toes [72] was the first to raw attention to this fact. One can see, however, that if the value of DT was calculate for any tsn from Eq. (9) then DH e woul have remaine finite. This appears to be ue to the fact that at any t!n the secon term in the square brackets in p Eq. (9) cannot be ignore at RKR O2 ffiffiffiffiffiffi k g t. This means that Eq. (15) is not vali at these raii, an DH e cannot be Fig. 1. Plot of c t, efine by Eq. (17), versus Fo.

9 170 S.S. Sazhin / Progress in Energy an Combustion Science 32 (2006) useful for practical application in CFD coes. At the initial stage of heating, the corrections escribe by Eq. (13) can be introuce if require. The range of applicability of these equations for transient heating, however, has not been rigorously justifie (cf. Eq. (17)). The next stage of the analysis will be focuse on the processes in the liqui phase. When the first group of moels iscusse in Section 1 (T (R%R )Zconst) is use, then the problem of liqui roplet heating oes not occur. For the secon group of moels (no temperature graient insie roplets) the roplet temperature can be foun from the energy balance equation: 4 T 3 pr3 r l c l Z 4pR 2 t hðt gn KT Þ: (18) This equation merely inicates that all the heat supplie from gas to roplet is spent on raising the temperature of the roplet. It has a straightforwar solution T Z T gn CðT 0 KT gn Þexp K 3ht c l r l R 0 ; (19) where T (tz0)zt 0. Eq. (18) an its solution (19) are wiely use in various applications. In conjunction with the mass transfer Eq. (escribe in Section 3) Eq. (18) was use to etermine experimentally the heat transferre by convection to roplets [71]. Solution (19) is wiely use in most CFD coes. Sometimes this is justifie by the fact that liqui thermal conuctivity is much higher than that of gas. However, the main parameter which controls roplet transient heating is not its conuctivity, but its iffusivity. As shown earlier, in the case of iesel engine sprays the iffusivity for liqui is more than an orer of magnitue less than that for gas. This raises the question of whether the secon group of moels is applicable to moelling fuel roplets in these engines. The only reasonable way to answer this question is to consier the thir group of moels, which take into account the effect of finite liqui thermal conuctivity. The application of the thir group of moels can be base on the solution of Eq. (1) insie the roplet with the following bounary conitions vt hðt g KT s Þ Z k l (20) vr RZR K0; ((T/(R)j RZ0 Z0, an the initial conition T(tZ0)Z T 0 (R), where T s ZT s (t) is the roplet s surface temperature, T g ZT g (t) is the ambient gas temperature, the subscript N has been omitte. Assuming that hzconst, this solution can be presente as [76,77] ( TðR; tþ Z R X N q R n exp½kk R l 2 ntš nz1 K sin l n jjv n jj 2 m l 2 0 ð0þexp½kk R l 2 ntš n 9 K sin l ð t n m 0 ðtþ = jjv n jj 2 exp½kk l 2 n t R l 2 nðtktþšt ; 0 R!sin l n CT g ðtþ; ð21þ R where l n are solutions of the equation l cos l Ch 0 sin l Z 0; (22) jjv n jj 2 Z 1 2 q n Z 1 R jjv n jj 2 k l k R Z c l r l R 2 1Ksin 2l n 2l n ðr 0 R ~T 0 ðrþsin l n R Z C h 0 h 2 ; 0 Cl2 n ; m 0 ðtþ Z ht gðtþr k l : R; h 0 ZðhR =k l ÞK1, ~T 0 ðrþzrt 0 ðrþ=r. The solution of Eq. (22) gives a set of positive eigenvalues l n numbere in ascening orer (nz1,2,.). Proof of the convergence of the series in (21) is given in [77]. In the limit k l /N solution (21) reuces to solution (19) as expecte [78]. Solution (21) was generalise to the case of almost constant convection heat transfer coefficients [77]. In the case of the general time-epenent convection heat transfer coefficient, the solution of the ifferential equation is reuce to the solution of the Volterra integral equation of the secon kin [77]. However, the practical applicability of these solutions to CFD coes has turne out to be limite [79]. The benefits of taking into account the finite thermal conuctivity of fuel roplets in the CFD moelling of combustion processes in iesel engines was first emonstrate in [80]. These authors performe the calculations using KIVA II CFD coe with the conventional moel for roplet heating base on Eq. (19) (calle the Spaling moel), an with the moels taking into account the finite thermal conuctivity of roplets. Note that the latter effect in this paper was taken into account base on the irect numerical solution of Eq. (1), rather than on the

10 S.S. Sazhin / Progress in Energy an Combustion Science 32 (2006) vali at the very beginning of the heating process when TZT c in most of the roplet. Substitution of Eq. (23) into the bounary conition at RZR (Eq. (20)) gives [81] T s KT c Z z 2 ðt gkt s Þ; (24) where zzðnu=2þðk g =k l Þ, Nu is the Nusselt number introuce earlier. Eq. (23) oes not satisfy Eq. (1) but it shoul satisfy the equation of thermal balance, which can be obtaine from integrating Eq. (1) along the raius R r l c l 3 where T t Z hðt gkt s Þ; (25) Fig. 2. In-cyliner pressure versus crank angle. Reprouce from [80] with permission of the authors an Elsevier. analytical solution of this Eq. (see solution (21)). The results of their calculations are shown in Fig. 2 as incyliner pressure versus crank angle (proportional to time). The results of measurements of this pressure are shown in the same figure. It follows that taking into account the effects of finite thermal conuctivity of roplets leas to better agreement with experimental ata, compare with the conventional moel. Although noboy seems to contest the benefits of taking into account the finite thermal conuctivity of fuel roplets in CFD moelling, the evelopers of CFD coes o not embrace this moelling opportunity too eagerly. The main reason for this is the aitional cost of CPU time involve. This has stimulate efforts to evelop a moel taking into account the effect of finite thermal conuctivity of roplets, but with minimal extra emans on CPU time. The iea of this new moel may have been prompte by looking at the character of the plots T(R) preicte by Eq. (21). Assuming that initially T(R) is constant, one can see that, except at the very start of heating, the shape of the curve T(R) looks very close to a parabola. This allows us to approximate T(R) as[81] R 2 TðR; tþ Z T c ðtþ C½T s ðtþkt c ðtþš ; (23) where T c an T s are the temperatures at the centre (RZ0) an on the surface (RZR ) of the roplet, respectively. Approximation (23) is obviously not R T Z 3 R 3 ðr 0 R 2 TðRÞR (26) is the average temperature of the roplet. Note that Eq. (25) is ientical to Eq. (18) escribing roplet heating in isothermal (infinite liqui conuctivity) approximation, if h is replace by h* foun from the moifie Nusselt number: Nu Z Nu T gkt s T g K T : (27) Having substitute Eq. (23) into Eq. (26) an remembering Eq. (24) we obtain: T Z 2T c C3T s Z T 5 s K0:2zðT g KT s Þ: (28) Substitution of Eq. (28) into Eq. (27) gives: Nu Z Nu 1 C0:2z : (29) On the other han, the combination of Eqs. (24) an (28) gives [81]: T s Z T C0:2zT g 1 C0:2z ; (30) T c Z ð1 C0:5zÞ T K0:3zT g : (31) 1 C0:2z The combination of Eqs. (29) (31) gives the full solution of the problem of convective roplet heating as preicte by Eq. (23). At first, Eq. (18) is solve with h replace by h* from Eq. (29). Then the values of T s an T c are obtaine from Eqs. (30) an (31). These two parameters are then substitute into Eq. (23) to give the raial istribution of temperature

11 172 S.S. Sazhin / Progress in Energy an Combustion Science 32 (2006) insie roplets. For practical engineering applications we are primarily intereste in the values of T s, which etermine the rate of evaporation an break-up of roplets. The moel base on Eq. (23) was calle the parabolic temperature profile moel [81]. As follows from the comparison of this moel with the numerical solution of Eq. (1), the preictions of this moel show goo accuracy at large times, but can iffer consierably from the numerical results for small times. The perio of time when the latter happens is usually reasonably short an can be ignore in practical calculations. Actual implementation of this moel into a customise version of the CFD coe VECTIS is escribe in [82]. The preictions of the moel were reasonably close to those preicte by Eq. (21), an its aitional CPU requirements are very small an coul be acceptable in most practical applications. The parabolic temperature profile moel was generalise to take into account the initial heating of roplets in [81], but the implementation of this generalise moel into CFD coes has not yet been investigate. Note that a moel similar to the parabolic temperature profile moel can be evelope more rigorously base on Eq. (21) if only the first term in this series is taken into account an the initial temperature insie roplets is assume to be constant. This approach was suggeste in [83]. In the limit t/n Eq. (18) of [83] is ientical with Eq. (28) in this paper. For t/0 the accuracy of Eq. (18) of [83] becomes questionable since in this case all terms in Eq. (21) become comparable. The analysis of Eq. (1) presente in this section was essentially base on the assumption that thermal conuctivities, specific heat capacities an ensities of gas an liqui are constant. This assumption is reasonable for CFD applications, where the calculations are performe over small time steps when the variations of these parameters can be ignore. They can be upate from one step to another. Attempts to take into account the temperature epenence of these parameters (i.e. to solve a non-linear heat conuction equation) starte more than 50 years ago an are still continuing (e.g. [84 90] Moving roplets The analysis of heat exchange between gas an stationary roplets was simplifie by the fact that this problem is essentially one-imensional in space. The complexity of the problem of heat transfer between gas an moving roplets lies in the fact that this problem is at least two-imensional in space. This leas to the nee to replace Eq. (1) by a more general equation for the temperatures of both liqui an gas an present it in the form [16] vt vt CV$ðuðt; xþtþ Z kv2 T; (32) where u(t,x) is the flui velocity, which epens on time t an position in space x in the general case. The Laplacian P 2 is three-imensional in the general case; its two-imensional approximation is almost universally use. In most practical applications, the time-epenence of u is ignore, even if the problem of transient heat transfer is consiere. In the general case, this equation is solve both in roplets an in the surrouning gas. Eq. (32) oes not contain the socalle history terms. These will be briefly iscusse later. Application of Eq. (32) instea of Eq. (1) leas to qualitative ifferences between the mechanisms of heat transfer in the cases of stationary an moving roplets. In the case of stationary roplets, the heat transfer takes place via conuction, an is escribe by the heat conuction equation in both liqui an gas. In the case of moving roplets, however, convection heat transfer takes place, which incorporates effects of bulk flui motion (avection) an iffusion effects (conuction). The moelling of heat transfer in this case nees to be base on the Navier Stokes equations for enthalpy an momentum. For stationary roplets the thickness of the bounary layer aroun roplets can be infinitely large. In the case of moving roplets, however, this thickness is always finite. All these ifferences between the heat transfer processes in the case of stationary an moving roplets require the evelopment of ifferent methos of analysis. While in the case of stationary roplets we starte with the general transient solution an ene up with the analysis of the limiting steay-state case, in the case of moving roplets, the starting point will be the simplest steay-state case Steay-state heating. In the case of steaystate roplet heating the first term in Eq. (32) can be assume equal to zero, an this leas to consierable simplification of the solution of this equation. In some practically important cases, however, this solution can be avoie altogether, an the analysis can be performe at a semi-qualitative level, backe by experimental observations. As mentione in Section 1, the first simplification of the problem is base on the assumption that there is no spatial graient along a roplet s surface. This

12 S.S. Sazhin / Progress in Energy an Combustion Science 32 (2006) allows us to separate the analysis of the gas an liqui phases. If the surface temperature of the roplet is fixe, then using the imensional analysis one can show that Nusselt number (Nu) can epen only on Reynols an Prantl numbers (Re an Pr). If ReZ0 (stagnant roplet) then NuZ2. The qualitative analysis of the problem for large Re, but in the laminar bounary layer region, shows that Nu is expecte to be proportional to Re 1/2 Pr 1/3 [2]. Thus the general formula for Nu can be written as Nu Z 2 Cb c Re 1=2 Pr 1=3 ; (33) where the coefficient b c cannot be erive from this simplifie analysis. It shoul be obtaine either from experimental observations or from more rigorous numerical analysis of the basic equations. The most wiely use value of this coefficient, supporte by experimental observations, is b c Z0.6 (see e.g. [15]). In a number of papers (see e.g. [13,63]) the value b c Z0.552 is use. Sometimes the power 1/3 of the Prantl number is replace by 0.4 [16]. Whitaker [91] rew attention to the fact that in the wake region, for large Re, Nu is expecte to be proportional to Re 2/3 Pr 1/3. Also, the ifference in gas ynamic viscosities in the vicinity of the roplet surface (m gs ) an in the bulk flow (m gn ) ha to be taken into account. Hence, he looke for the correlation for Nu in the form Nu Z 2 C a w Re 1=2 Cb w Re 2=3 Pr 1=3 m cw; gn (34) m gs where the constants a w, b w an c w ha to be calculate base on available experimental ata. The analysis of these ata le to the following result: a w Z0.4, b w Z 0.06 an c w Z0.25. Eq. (34) might be expecte to be more accurate than Eq. (33) (at least ue to the larger number of fitting constants), but it has rarely been use in practical applications. Note that the ratio m gn /m gs is expecte to be very close to unity. As notice in [6,63], Eq. (33) over-estimates the heat transfer rate at low Reynols numbers (Re%10). Also, this equation an Eq. (34) preict the physically incorrect super-sensitivity of the heat transfer rate to the small velocity fluctuations near ReZ0, since vnu/vre/n when Re/0 [63]. As an alternative to Eq. (33), the following correlation was recommene [6,63] Nu Z 1 Cð1 CRePrÞ 1=3 f c ðreþ; (35) where f c (Re)Z1 atre%1 an f c (Re)ZRe at 1! Re%400. Eq. (35) approximate the numerical results obtaine by a number of authors for 0.25!Pr!100 with an error less than 3% [63]. Analyses of the process of heating of a spherical boy in a flow of flui were often restricte to the case of small Re when the flow can be consiere as Stokesian. In this case, the raial an asymuthal components of flow velocity aroun a spherical soli boy can be foun from the following formulae [75,92]: 1 u r Zu C 3Re 2K 3R 16 R C R 3 cos q R K 3Re 2K 3R 64 R C R 2 K R 3 C R 4 R R R!ð3 cos 2 qk1þ COðRe 2 log ReÞ ; ð36þ u q Zu 0 K C 3Re 4K 3R 16 R K R 3 sin q R C 3Re 4K 3R 64 R C R 3 K2 R 4 cos q sin q R R COðRe 2 log ReÞ ; ð37þ where u 0 is the flow velocity unperturbe by the sphere, u r an u q are the raial (away from the centre of the sphere) an circumferential (angle q is estimate relative to the irection of the velocity in the unperturbe flow) flow velocity components. As follows from Eqs. (36) an (37) both components of flow velocity are zero at RZR as expecte (no slip conition). In the limit Re/0 Eqs. (36) an (37) can be simplifie to: u r Z u 0 1K 3R 2R C 1 R 3 cos q; (38) 2 R u q ZKu 0 1K 3R 4R K1 4 R 3 R sin q: (39) Note that the main avantages of Eqs. (36) an (37) compare with Eqs. (38) an (39) can be observe at R comparable with R. In the limit R/N, Eqs. (38) an (39) preict the unperturbe flow velocity. In the case of roplets in which re-circulation is allowe, Eqs. (38) an (39) can be generalise to [93] (p. 697), [73]: u r Z u 0 1K A vr R C2B R 3 v cos q; (40) R

13 174 S.S. Sazhin / Progress in Energy an Combustion Science 32 (2006) u q ZKu 0 where 1K A vr 2R KB v R 3 R sin q; (41) A v Z 3l v C1 2ðl v C1Þ ; B v Z l v 4ðl v C1Þ ; l v Zm l /m g is the ratio of ynamic viscosities (note that this efinition of l v is ifferent from the one use in [73,93] but consistent with the efinition of the parameter use later in this review). In the case when l v ZN (soli boy) Eqs. (40) an (41) refer to the case when RRR, an they reuce to Eqs. (38) an (39). Similar equations for the volume insie the roplets coul be erive from Eq. ( b) of [93]. These escribe the well known Hill s spherical vortex [93]. A more in-epth analysis of re-circulation within a flui sphere is given in [94]. If the effects of viscosity are ignore then for both soli an liqui moving spheres, the components of velocity can be presente in particularly simple forms vali for RRR (see [93], p. 562): u r Z u 0 1K R 3 cos q; (42) R u q ZKu 0 1 C 1 R 3 sin q: (43) 2 R The values of the components of flui velocity preicte by the above-mentione systems of equations are substitute into Eq. (32). All these systems of equations have actually been use in the analyses of steay-state an transient heating of roplets or, more generally, spherical boies. These will be reviewe below. Eqs. (38) an (39) were use by [95] for the asymptotic analysis of the process of heating of a spherical boy in a flow of flui. They assume that both Reynols number an Peclet number (PehRe$Pr) are small, an use the technique originally evelope in [92], which le to the erivation of Eqs. (36) an (37). Their final formula was presente in the form: Nu Z2 1C 1 4 Pe C 1 8 Pe2 ln Pe C0:01702 Pe 2 C 1 32 Pe3 ln Pe COðPe 3 Þ : ð44þ As follows from this equation, Nu/2 when Pe/0. Although this equation was erive uner the assumption that Re/1 an Pe!1, it can still be applie for Re an Pe in the range from 0 to 0.7 [16]. Sometimes Re is efine base on R an not 2R (as in our case). This le to a ifferent form of Eq. (44) [96]. Using Eqs. (36) an (37) but assuming that Pe/N, an alternative expression for Nu was erive [95]: NuZ0:991Pe 1=3 1C 1 16 ReC Re2 lnreco Re 2 : (45) The generalisation of the analysis by [95] for the case when Re!1 an Pe!1 by taking extra terms in the velocity fiel (that is replacing Eqs. (38) an (39) with Eqs. (36) an (37)), was reporte in [96]. The generalisations of this equation to the case of boies of arbitrary shape were reporte in [97 99]. Base on the numerical stuy of the transient heat transfer from a sphere at high Reynols an Peclet numbers, the following steay-state correlation was suggeste [100]: Nu Z 0:922 CPe 1=3 C0:1Re 1=3 Pe 1=3 : (46) If ReZ0 then Eq. (46) reuces to the solution by Acrivos [99]. Also the preiction of this equation is close to that of Eq. (45) at large Pe, but relatively small Re. In [101] a mathematical moel was evelope to escribe the heat transfer process when a melting sphere is immerse in a moving flui. Base on this moel, the following correlation for Nu was obtaine Nu Z 2 C0:47Re 0:5 Pr 0:36 ; (47) where 0:003%Pr%10; 10 2 %Re%5!10 4 : This moel was valiate with various experimental results involving metals an water. Nu preicte by Eq. (47) is reasonably close to Nu preicte by Eq. (33). None of Eqs. (44) (47) take into account recirculation insie the roplets. The latter coul be accounte for base on asymptotic or rigorous numerical analysis of couple flui ynamics an heat transfer equations. Levich [3] (see p. 408) was perhaps the first to provie an asymptotic solution for small Re, but large Pe in the form (see also [16]): sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 1 Nu Z Pe: (48) 3p 1 Cl v As this formula is vali for very large Pr, its practical applicability to the analysis of roplet heating in a gas is expecte to be limite.

14 S.S. Sazhin / Progress in Energy an Combustion Science 32 (2006) The earlier stuies in this irection were reviewe by Feng an Michaelies [102]. Also, in this paper a comprehensive numerical analysis of heat an mass transfer coefficients of viscous moving spheres (roplets) was reporte. Their analysis was base on the conventional assumptions mentione in Section 1. Namely, it was assume that roplets retain their spherical forms an there are no temperature graients along the roplet surfaces. In what follows their main results will be summarise. One of the aims of [102] was to investigate the epenence of Nu on Re, PehRe$Pr, l v Zm l /m g an r l /r g, where m l(g) an r l(g) are ynamic viscosities an ensities of liqui (gas). As follows from their analysis, the epenence of Nu on r l /r g was negligibly small for r l /r g in the range (bubbles an roplets), an the range of Pe from 1 to 500, ReZ10 an l v Z1. It was not expecte to be important for other ranges an values of parameters. Hence, the analysis was focuse on the stuy of the epenence of Nu on Re, Pe an l v. The results are presente in a table, covering the range of Re from Re/0 to 500, the range of Pe from 1 to 1000, an of l v from 0 to N (soli). Base on the information presente in this table the following working correlations were evelope [102]. At small but finite Re (0!Re!1) an 10%Pe%1000, the following expression for Nu was erive [102] Nuðl v ; Pe; ReÞ Z 0:651 Pe 1=2 C 0:991l v Pe 1=3 1 C0:95l v 1 Cl v ½1 Ca v ðreþš C 1:65ð1Ka vðreþþ C l v ; 1 C0:95l v 1 Cl v (49) where a v ðreþ Z 0:61Re Re C21 C0:032: Note that in the limit Re/0 an l v /N, Nu given by expression (49) is not reuce to 2, as preicte for stationary roplets. In this limit Pr/N to satisfy the conition that PeO10. At 1%Re%500 an 10%Pe%1000, the following expressions for Nu were suggeste [102] Nuðl v ; Pe; ReÞ Z 2Kl v Nuð0; Pe; ReÞ 2 C 4l v Nuð2; Pe; ReÞ ð50þ 6 Cl v for 0%l v %2, an Nuðl v ; Pe; ReÞ Z 4 2 Cl v Nuð2; Pe; ReÞ C l vk2 NuðN; Pe; ReÞ l v C2 ð51þ for 2%l v %N, where Nuð0; Pe; ReÞ Z 0:651Pe 1=2 1:032 C 0:61Re Re C21 C 1:60 C 0:61Re ; Re C21 Nuð2; Pe; ReÞ Z 0:64Pe 0:43 ½1 C0:233Re 0:287 Š C1:41K0:15Re 0:287 ; NuðN; Pe; ReÞ Z 0:852Pe 1=3 ½1 C0:233Re 0:287 Š C1:3K0:182Re 0:355 : These equations coul be potentially incorporate into any CFD coe an use in engineering applications Transient heating. The complexity of the problem of transient heating of moving roplets lies in the fact that both variations in temperature an flow velocity nee to be accounte for in the general case. This is something that is performe in most commercial CFD coes. This analysis, however, is usually case epenent an the results might have limite general practical applications. At the same time, in many applications the characteristic times of variation of flow parameters are much longer than the characteristic times of roplet heating. This allows us to consier the problem of roplet heating assuming that the flow is fixe. This assumption is almost universally use in the analysis of this problem. In the case of small Re, we can further assume that flow is Stokesian, with the velocities escribe by Eqs. (36) an (37), or their simplifie versions (Eqs. (38) an (39)). Any effects of roplet oscillations on the heating process are generally ignore. This coul be justifie when the roplet Weber number WeZ2u 0 R r l /s s (where s s is the interfacial surface tension) is less than 4 [103]. This result was obtaine in [103] base on experimental stuies of falling roplets at Re in the range Most approaches to investigation of the problem of transient heating of moving roplets performe so far can be presente in two groups. The first approach is base on the assumption that the temperature of the roplet surface remains constant throughout the whole heating process. The secon approach allows for