MODELLING OF SPRAYS USING COMPUTATIONAL FLUID DYNAMICS CODES

Transcription

1 MODELLING OF SPRAYS USING COMPUTATIONAL FLUID DYNAMICS CODES Sergei Sazhin Sir Harry Ricaro Laboratory, School of Environment an Technology, University of Brighton, Brighton BN 4GJ, UK, Receive January 9; accepte???? 9 Abstract: Recently evelope approaches to roplet an spray moelling in computational flui ynamics (CFD) coes are reviewe. It is emphasize that the moelling of spray primary break-up nees to take into account the transient nature of sprays, while the moelling of roplet heating an evaporation nees to take into account a number of factors, incluing recirculation an finite thermal conuctivity of liqui, semi-transparency of roplets an the kinetic effects. This moelling nees to take place in realistic three-imensional enclosures, which makes it essential to fin a compromise between the accuracy of the moels an their CPU efficiency. Keywors: Sprays, Droplets, Computational Flui Dynamics Coes, Break-up, Heating, Evaporation 1. Introuction The importance of the accurate multi-imensional moelling of roplets an sprays in various engineering an environmental applications is well recognise (e.g. [1]-[4]). In the most general case, the moels nee to take into account a number of complicate flui ynamics, heat/mass transfer an combustion processes in realistic threeimensional configurations. This inevitably leas to the application of rather simplistic moels escribing iniviual processes. These inclue simplistic chemical moels, roplet break-up moels, which o not take into account transient effects, an roplet heating moels, which isregar the effects of internal temperature graient, an so on. The aim of this paper is to summarise the results of recent stuies into the necessity for, an feasibility of, relaxing some of the simplifying assumptions in the moelling of roplet break-up, heating an evaporation in engineering computational flui ynamics (CFD) coes. Some of the new moels have been implemente into the customise HU ISSN Akaémiai Kiaó, Buapest

2 MODELLING OF SPRAYS version of the KIVA computational flui ynamics (CFD) coe. The implementation of these moels into other CFD coes seems to be a straightforwar task. The moelling of the gas phase in CFD coes is generally base on the Eulerian approach when the analysis is focuse on the solution of iscretise equations of conservation of mass, momentum an energy for specific computational cells [5, 6]. An alternative approach is base on the solution of the vorticity equation [7]-[1], but this is rarely use in practical engineering applications. As for spray moelling, two approaches have been evelope an use: the Eulerian approach, when liqui is treate as a continuous meium or as the secon phase of the multiphase flow (e.g. [11]), an the Lagrangian approach when the analysis is focuse on tracing the trajectories of iniviual roplets or groups of roplets (roplet parcels) on a sub-gri scale (e.g. [1]). The secon approach is generally more popular than the Eulerian one [13]. It is implemente in an open-source non-commercial computer coe KIVA, wiely use for evelopment an valiation of spray moels [14, 15]. The Lagrangian approach to spray moelling has a number of well-known limitations, incluing the gri epenency of the results for ense sprays. These limitations an the ways in which they can be overcome were iscusse in [16]. One of the most important elements of Lagrangian spray moels is their focus on jet an roplet break-up. The nature of the break-up process epens on a spray region. Primary break-up takes place near the nozzle exit. Here isintegration of the liqui jet occurs. In the far-fiel spray, where the liqui phase is isperse in the gas, the seconary break-up of large roplets into smaller ones takes place. In many practical applications, unifie moels have been use for both primary an seconary break-up. In these moels, the jet is approximate by a chain of roplets, with initial iameters equal to the iameter of the nozzle, or slightly less than this iameter if the effects of cavitation are taken into account [17]. For implementation into CFD coes, esigne for computation of three-imensional sprays, the so-calle TAB (Taylor Analogy Break-up) an WAVE moels of break-up an their moifications are commonly applie [18]. An important limitation of the above-mentione break-up moels relates to the fact that they are base on an unrealistic assumption about single-size roplets create after the break-up. More realistic stochastic moels escribe the break-up in terms of the evolution of the roplet istribution function over time ([19]-[3]). Another limitation of the break-up moels use in CFD coes stems from an assumption about quasi-steay-state flow conitions, although in many practical applications spray injection is an essentially transient process. As for the moelling of heating an evaporation of iniviual roplets, only the simplest moels have been implemente into most available CFD coes. These moels are base on the assumption that the thermal conuctivity of roplets is infinitely high so that there are no temperature graients insie them. In evaporation moels it has been assume that vapour in the vicinity of a roplet s surface is always saturate, an so the problem of roplet evaporation reuces to the problem of vapour iffusion to the ambient gas (these are the hyroynamic moels) [4]. In what follows, attempts to overcome these limitations of the moel in CFD coes will be escribe.

3 MODELLING OF SPRAYS 3. Break-up of transient sprays In orer to account for the effect of transient injection on break-up, the analysis was focuse on the WAVE moel, the parameters of which, controlling the rate of spray isintegration, have been moifie [4]. The original WAVE break-up moel is base on the analysis of the Kelvin-Helmholtz instability of a liqui jet. This instability leas to stripping of chil roplets from the liqui core. The core is approximate by parent roplet parcels injecte from the nozzle. The raii of the roplets in these parcels continuously ecrease uring the break-up process, as escribe by the following equation: R t R R eq, (1) bu where bu is the characteristic break-up time, an R eq is the raius of roplets in equilibrium (stable roplets): Bo, Bo R.33 R (3 R U / ) eq () min, Bo R.33 (3R / 4) B.61 is the moel constant, an are the wave length an the frequency of o the fastest growing isturbance on the surface of a liqui jet moving in an invisci gas. These are functions of Reynols, Weber an Ohnesorge numbers [5]. The break-up is estimate as: time bu R bu 3.76 B 1. (3) The break-up constant B 1 was taken to be equal to 1 [5] base on the results of measurements of quasi-steay-state Diesel spray penetration at relatively low injection rates. Later, it was shown that the break-up constant B 1 can vary wiely epening on the type of injector. Patterson an Reitz [6] evelope this moel further in orer to account for the effect of the Rayleigh-Taylor (RT) instability of roplets. When the wave length corresponing to the maximal increment of this instability (4) RT 3 a l is less than the iameter of a roplet, the bag break-up of the roplet is expecte to take place. In this expression, is the surface tension, an 3 gu a C is the 4 D l D eceleration of the roplet ue to the rag force. For the Rayleigh-Taylor type of break- R

4 MODELLING OF SPRAYS 4 up, R was calculate as eq Req C RT RT, where RT =.5 is the moel constant [6]. The break-up time was estimate as: 3 3 bu, RT. (5) a al The Kelvin-Helmholtz instability is mainly responsible for the primary break-up, while the Rayleigh-Taylor mechanism prevails at the seconary atomisation stage. In both cases the roplets conserve momenta uring the break-up process. In the moifie version of the WAVE moel, the ecrease in (the frequency of the fastest growing isturbance on the liqui surface) with increasing injection acceleration was taken into account, while it was assume that the wavelength of critical instability is not affecte by the transient nature of the flow. This ecrease is accounte for by introucing the following relation for B 1: where a c a B 1 B1, st 1 c1, (6) D U inj Re is the acceleration parameter taking into account the effect of U t inj flow acceleration; c 1 an c are ajustable constants. In the steay-state limit, a is zero an B 1 B1. Following [5], it was assume that, st B 1, st 1. A rigi boy concept to escribe the propagation of isturbances along the liqui core was introuce. It was assume that parcels constituting the liqui core experience no rag from the gas an move as a rigi jet at a spee equal to the instantaneous injection U U. The stripping of the jet ue to the evelopment of the Kelvin- velocity jet inj Helmholtz instability, however, is allowe. This leas to a continuous ecrease in the iameter D of parcels in the spray core escribe by Equation (1). In computations, the length of the liqui core has been limite by the conition D. 5 Dinj, similar to the criterion applie by Reitz [5] ( D inj is the initial jet iameter at the nozzle exit). The introuction of the critical iameter of the liqui core allows us to istinguish between the primary an the seconary break-up regions in the moel. The results of calculations of spray tip penetration, using the conventional TAB an WAVE (with B 1 =1), moifie WAVE an stochastic (suggeste in [19]) moels, an the corresponing experimental ata for a highly transient spray, were compare [4]. As follows from this comparison, the conventional WAVE moel, TAB an stochastic moels significantly uner-preict the penetration at the initial stage of this process. At the same time the moifie version of the WAVE moel, escribe above, gave much better agreement between the preictions of the moel an experimental ata, as expecte, remembering the highly transient nature of the spray uner consieration [7]. It is recommene that this moel is implemente into other CFD coes use for the analysis of transient sprays. C

5 MODELLING OF SPRAYS 5 3. Moelling of roplet heating an evaporation Although the heating an evaporation of roplets is a complex process, involving a close interaction between the liqui an gas phases (e.g. [8]), the moelling of these processes in CFD coes is commonly base on the separate analysis of the two phases. 3.1 Liqui phase moels The liqui phase moels actually use in CFD coes, or the ones which can potentially be use, are the ITC (infinite thermal conuctivity) an ETC (effective thermal conuctivity) moels. The ITC moels are base on the energy balance equation of the roplet as a whole. The solution to this equation can be presente as [1, 4]: 3ht T T s Tg ( Ts Tg )exp, (7) cl l R where T s an T g are the initial roplet temperature an ambient gas temperature respectively, c l an l are liqui specific heat capacity an ensity respectively. Droplet temperature T oes not epen on the istance from the roplet centre R in this case. Assuming that the process is spherically symmetrical, the roplet transient heating in finite liqui thermal conuctivity moels is escribe by the following equation [4]: T Kl T R P( t, R), (8) t R R R where K = k l /(c l ρ l ) is the liqui thermal iffusivity, k l is the liqui thermal l conuctivity, assume to be constant, T is specifie at the initial moment of time as T t= =T (R), P takes into account the raiative heating of a roplet. The bounary conition at R= follows from the problem symmetry T / R. It was assume that the roplet is heate by convection from the surrouning gas, an coole ue to evaporation. The general analytical solution of Equation (8), taking into account the changes in roplet raius ue to evaporation, woul be a ifficult task. This coul be consierably simplifie if we take into account that this solution is use in the numerical analysis when the time step is small. In this case we can assume that the roplet raius is constant, but the effect of evaporation can be taken into account by replacing T g with the effective temperature [4]: R T T L h, (9) eff g l / t where L is the latent heat of evaporation, effects of swelling are ignore at this stage, h is the convection heat transfer coefficient. The solution to (8) for h=const can be presente as [4]: R

6 MODELLING OF SPRAYS 6 T T eff R R n1 sin n () qn exp( t) n n Vn 1 sin n ( ) exp n( t ) n Vn nr R pn nr sin 1 exp( nt)]sin R R n1 n R where ht eff ( t) R, (11) k l 1 1 q n TVn ( R) ( R / R ), (1) V n T is the roplet initial temperature istribution, T RT / ( R) R, κ = k l /(c l ρ l R ), λ n are solutions to the equation:, (13) n cos n h sin n hr Vn.5(1 sin n / n k h = V n l 1, ), sin( R / R ), (14) n 1 1 RP( t, R) pn V ( ) ( / ) n R R R Vn, (15) R P(R) takes into account the raiative heating of roplets. The variations of all parameters with temperature an time were accounte for when analytical solution (1) was incorporate into a numerical coe. The finite liqui thermal conuctivity moel coul be generalise to take into account the internal recirculation insie roplets. This coul be achieve by replacing the thermal conuctivity of liqui k l with the so-calle effective thermal conuctivity k eff = χ k l, where the coefficient χ varies from about 1 (at roplet Peclet number < 1) to.7 (at roplet Peclet number >5) [9]. This is known as the effective thermal conuctivity (ETC) moel. Various moels for P(R), taking into account the variation of thermal raiation absorption insie roplets were suggeste (see [4]). It transpire that this function has only minor effects on the heating of roplets, which can be ignore in most practical applications. Ignoring the epenence of the istribution of thermal raiation absorption on R, we can present the expression for P(R) as [4]: (1)

7 MODELLING OF SPRAYS 7 P 3 SB b1 4 ( R) ar Text lcl, (16) where we assume that the raiation temperature θ R is equal to the external temperature T ext, a an b are polynomials (quaratic functions in most cases) of θ R, σ SB is the Stefan- Boltzmann constant. For low sulphur ESSO AF 1313 Diesel fuel, it was foun that the best approximation for a an b in the ranges 5m R 5 m an 1 T ext 3 K is provie by the functions [4]: a= (T ext /1) +.8 (T ext /1). b= (T ext /1) (T ext /1). (17) Equation (16) is essentially an approximation of the preiction of the Mie solution for the absorption of thermal raiation by semi-transparent roplets. Note that in all CFD coes known to us the effects of the semi-transparency of roplets are not taken into account an the roplets are treate as grey opaque spheres (see [4] for a more etaile analysis of this problem). 3. Gas phase hyroynamic moels The values of h are controlle by the conitions in the gas phase. Various approximations for h are usually escribe in terms of the corresponing approximations for the Nusselt number Nu hr / k. Droplet heating, iscusse in the previous section, is accompanie by roplet evaporation, which is escribe by the following equation: R k g Sh, (18) t c R l pg where k g is the gas thermal conuctivity, c pg is the gas specific heat capacity at constant pressure, Sh is the Sherwoo number. The ifference between various gas moels is essentially escribe in terms of the approximations of Nusselt an Sherwoo numbers [3]. As shown in [3], the gas phase moel, originally suggeste in [9], preicts the evaporation time closest to the one base on the approximation of experimental ata. This gas phase moel was recommene for practical application in CFD coes. In most cases, the roplet evaporation time epens strongly on the choice of gas phase moel. The epenence of this time on the choice of liqui phase moel, however, is weak if the roplet break-up processes are not taken into account. On the other han, the epenence of the roplet surface temperature, at the initial stage of heating an evaporation, on the choice of gas phase moel is weak, while its epenence on the choice of liqui phase moel is strong [3]. Solution (1) with the raiation term in the form (16) was use in the new approach to the numerical moelling of roplet heating an evaporation by convection an raiation from the surrouning gas [4]. This solution was applie at the first time step, using the initial istribution of temperature insie the roplet. The results of the analytical solution over this time step were use as the initial conition for the secon g

8 MODELLING OF SPRAYS 8 time step, an so on. This approach was compare with those approaches base on the numerical solution of the iscretise heat conuction Equation (8), an those base on the assumption that there is no temperature graient insie the roplets (see Equation (7)) [4]. All of these approaches were applie to the numerical moelling of fuel roplet heating an evaporation in conitions relevant to Diesel engines, but without taking into account the effects of roplet break-up. The algorithm base on the analytical solution for constant h has been shown to be more effective (from the points of view of accuracy an CPU time requirement) than the approach base on the numerical solution of the iscretise heat conuction equation insie the roplet, an more accurate than the solution ignoring the temperature graient insie roplets [4]. This numerical algorithm was implemente into a zero-imensional coe, use for the moelling of coupling between roplets an gas, an the KIVA II CFD coe [4]. The importance of taking into account the effects of temperature graients insie roplets was confirme by irect temperature measurements insie roplets [31]. The valiity of the effective thermal conuctivity moel with the simplifie raiation term was justifie by irect comparison of the preiction of this moel an the rigorous moel, taking into account the recirculation insie, an the istribution of the raiation absorption insie, roplets [3, 33]. It was recommene that the effects of temperature graient insie roplets are taken into account in computational flui ynamics coes, using the moel escribe above. 3.3 Gas phase kinetic moels As mentione in the introuction, the currently use gas phase hyroynamics moels for roplet evaporation, incluing the one escribe in the previous section, implicitly assume that the rate of etachment of molecules of fuel is such that the concentration of fuel vapour at the roplet surface is maintaine at saturation level. The applicability of this assumption to the problem of moelling Diesel fuel roplet evaporation in realistic Diesel engines is not at first evient. In [34] a comparative analysis of hyroynamic an kinetic approaches to the problem of Diesel fuel roplet evaporation was presente, base on a previously evelope approximate kinetic moel. This moel was base on a number of assumptions the applicability of which to realistic (incluing Diesel engine) conitions is not immeiately obvious. For example the contribution of air in the vicinity of the roplet surfaces was ignore. A new numerical kinetic moel for roplet evaporation into a high-pressure backgroun gas, approximate by air, is escribe in [35]-[37]. As in [3], two regions above the surface of the evaporating roplet were consiere. These are the kinetic region, where the analysis is base on the Boltzmann equation, an the hyroynamic region. The contribution of air an heat flux in the kinetic region was taken into account. The evolution of the istribution function of air f f ( r, v, t) an fuel vapour f v f ( r, v, t) in the kinetic region is controlle by the corresponing Boltzmann v equations: a a

9 MODELLING OF SPRAYS 9 fa t f r a v a Jaa Jav, (19) fv t where f r v v v Jva Jvv, () J (α=a,v; β=a,v) are collision integrals efine as ' ' J v1 sin ( 1 1) v v1, f f f f (1) ( ) /, an are the corresponing iameters of molecules, θ an are angular coorinates of molecules β relative to molecules α after the collision, superscript ' inicates the velocities an the istribution functions after collisions, subscript 1 inicates that molecules of type β collie with molecules of type α an as a result of this interaction the function f is moifie. When eriving these equations it was assume that molecules are rigi elastic spheres an boy forces acting on them are negligible. A moifie version of the previously evelope metho of irect numerical solution of these Boltzmann equations was use. Although the moel escribe in [35]-[37] shows consierable progress in the evelopment of kinetic moels for roplet evaporation, it still has a number of important limitations. Its preictions still rely heavily on the choice of the value of the evaporation coefficient. The ifficulties with the measurement or calculation of this coefficient are well known. In most cases it can be assume that the system is in a state of equilibrium, when the evaporation coefficient coincies with the conensation coefficient. The rigorous theoretical estimation of both coefficients woul require the application of molecular ynamics methos, the analysis of which is beyon the scope of this paper (see [38-4]). Also, it remains unclear how kinetic effects can possibly be taken into account in CFD coes. In the most recent evelopments the kinetic effects were escribe as perturbations of the preictions of the hyroynamic moels, using simple approximations [41, 4]. This approach is expecte to open the way to taking kinetic effects into account in CFD coes. The work in this irection has not yet been complete. 4. Conclusion Recently evelope moels for transient spray break-up, roplet heating an evaporation, suitable for implementation into computational flui ynamics (CFD) coes, are reviewe with a view to specific application to the moelling of the processes in Diesel engines. It is pointe out that the gas phase moel, taking into account the finite thickness of the thermal bounary layer aroun the roplet, in the form suggeste by Abramzon an Sirignano [9], preicts the evaporation time closest to the one base on the approximation of experimental ata. This gas phase moel is recommene for

10 MODELLING OF SPRAYS 1 practical application in CFD coes. In most cases, the roplet evaporation time epens strongly on the choice of the gas phase moel. The epenence of this time on the choice of the liqui phase moel, however, is weak if the roplet break-up processes are not taken into account. On the other han, the epenence of the roplet surface temperature, at the initial stage of heating an evaporation, on the choice of the gas phase moel is weak, while its epenence on the choice of the liqui phase moel is strong. The relatively small contribution of thermal raiation to roplet heating an evaporation allows us to escribe it using a simplifie moel, which takes into account their semi-transparency, but oes not consier the spatial variations of raiation absorption insie roplets. The algorithm base on the analytical solution for constant h is more effective (from the points of view of accuracy an CPU time requirement) than the approach base on the numerical solution of the iscretise heat conuction equation insie the roplet, an more accurate than the solution base on the assumption that the thermal conuctivity insie roplets is infinitely large. It is recommene that kinetic effects are taken into account when accurate analysis of Diesel fuel roplet evaporation is essential. They may be taken into account in CFD coes if the results of kinetic calculations are escribe as the perturbations to the preictions of the hyroynamic moels, using simple approximations. Acknowlegements The author is grateful to the Royal Society for the financial support of this project. References [1] Sirignano WA. Flui Dynamics an Transport of Droplets an Sprays, Cambrige, UK: Cambrige University Press, [] Michaelies EE. Particles, Bubbles & Drops. New Jersey: Worl Scientific, 6. [3] Jancskar I, Ivanyi A Phenomenological hysteresis moel for vapour-liqui phase transition, Pollack Perioica, Vol. 3,8, pp [4] Sazhin SS. Avance moels of fuel roplet heating an evaporation, Progress in Energy an Combustion Sci. Vol. 3, 6, pp [5] Versteeg HK, Melalasekera W. An Introuction to Computational Flui Dynamics, Longman, [6] Date AW. An Introuction to Computational Flui Dynamics, Cambrige University Press, 5. [7] Batchelor GK. An Introuction to Flui Dynamics, Cambrige University Press, [8] Saffman PG. Vortex Dynamics, Cambrige University Press [9] Lim T, Nickels T. Vortex rings. In Flui Vortices (e. S. I. Green) Kluwer, Dorrecht, [1] Kaplanski F, Sazhin SS, Fukumoto Y, Begg S, Heikal M. A generalise vortex ring moel, J Flui Mechanics, Vol. 6, 9, pp

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