Fang WANG,Jie YAO,Shaofeng YANG,Rui LIU,Jie JIN

Aero-engine Numerical Simulation Research Center,School of Energy and Power Engineering,Beihang University,Beijing 100083,China

Co-Innovation Center for Advanced Aero-Engine,Beijing 100083,China

A new stationary droplet evaporation model and its validation

Fang WANG*,Jie YAO,Shaofeng YANG,Rui LIU,Jie JIN

Aero-engine Numerical Simulation Research Center,School of Energy and Power Engineering,Beihang University,Beijing 100083,China

Co-Innovation Center for Advanced Aero-Engine,Beijing 100083,China

The liquid droplet evaporation character is important for not only combustion chamber design process but also high-accuracy spray combustion simulation.In this paper,the suspended droplets’evaporation character was measured in a quiescent high-temperature environment by micro high-speed camera system.The gasoline and kerosene experimental results are consistent with the reference data.Methanol,common kerosene and aviation kerosene droplet evaporation characteristics,as well as their evaporation rate changing with temperature,were obtained.The evaporation rate experimental data were compared with the prediction result of Ranz-Marshall boiling temperature model(RMB),Ranz-Marshall low-temperature model(RML),drift flux model(DFM),mass analogy model(MAM),and stagnant film model(SFM).The disparity between the experimental data and the model prediction results was mainly caused by the neglect of the natural convection effect,which was never introduced into the droplet evaporation concept.A new droplet evaporation model with consideration of natural convection buoyancy force effect was proposed in this paper.Under the experimental conditions in this paper,the calculation results of the new droplet evaporation model were agreed with the experimental data for kerosene,methanol and other fuels,with less than 20%relative deviations.The relative deviations between the new evaporation model predictions for kerosene and the experimental data from the references were within 10%.

©2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-NDlicense(http://creativecommons.org/licenses/by-nc-nd/4.0/).

1.Introduction

Liquid fuels are advantageous because of their high calorific value per volume,convenient transportation,and lower pollution emission than many gaseous and solid fuels.At present and in the near future,liquid fuels are and will be widely used in power machinery.However,the combustion of liquid fuels is a complex process,so that investigating it through experiment is difficult and expensive.The numerical simulation can be a powerful tool to study the detailed interaction relationship and flow structures in liquid fuel spray combustion.An accurate two-phase model set,particularly the droplet evaporation model,is crucial for spray combustion simulation in the engine combustor design process because it is highly significant to the flame structure,fuel burning rate and droplet spatial distribution.

During liquid fuel combustion,the droplets evaporate first,and then the gas phase mixture burns,with the spray burning rate positively correlating with the droplet evaporation rate.The liquid fuel is atomized before burning in engineering machinery to ensure the highest combustion efficiency possible in the combustor.The characteristics of single droplet evaporation are thus related to combustion chamber performance.In addition,the evaporation characteristics of liquid fuel droplet are the foundation in understanding many complicated combustion issues,such as combustion theory,combustor design,pollution and emission control,and fire safety.Therefore,the study of liquid fuel evaporation characteristics is theoretically and practically significant.

Droplet evaporation is a phenomenon associated with the processes of heat transfer,mass transfer,and gas flow.Many theoretical studies on the evaporation of a single droplet have been conducted.Godsave1and Spalding2proposedd2law,which is mainly applicable to the evaporation of a singlecomponent droplet in a high-temperature quiescent environment.Law and Sirignano3,4presented the rapid mixing model based on classicald2law,wheredis droplet diameter.This model suggests that the droplet temperature is spatially uniform but temporally varying.Sirignano5found that the internal circulation of a liquid droplet does not significantly reduce even if the vortex strength inside the droplet reaches a high level.Then Sirignano renamed the rapid mixing model as the infinite conduction model,which was more accurate in concept than the former.The finite heat conduction model was proposed by Prakash and Sirignano.6,7They presented that the internal circulation of a liquid droplet is unremarkable in static condition,and thermal diffusion dominates the heat transfer inside the droplet.Harstad8stated that the temperature inside a droplet is not uniform,and a temperature gradient in space exists;they also proposed a nonequilibrium model.Zhou9found that the intensive evaporation of a droplet in a forced convection environment does not have the characteristics of the boundary layer;consequently,he presented thick exchange layer theory.Zhou9proposed stagnant film theory to deal with the evaporation and combustion of droplets in a forced convection environment.

Rao and Lefebvre10examined kerosene droplet evaporation characteristics under different air conditions,and a new empirical formula for liquid fuel evaporation rate was developed from the experimental data.Stengele et al.11performed an evaporation experiment offree-falling N-pentane and N-nonane droplets;droplet evaporation law was obtained under the experimental conditions.The experimental results agreed well with the calculation results of the finite conductivity model.Wu et al.12conducted experiments on droplet evaporation in a turbulent environment and found two different evaporation regions.Ghassemi et al.13hung heptane and a 16 alkyl mixture droplet with quartz wire statically to investigate liquid droplet evaporation in a normal-gravity environment.Okamoto et al.14examined the evaporation and diffusion characteristics of kerosene and gasoline mixture through experiments.Irfan15,16,Mishra17,Hyemin18,and other researchers conducted many evaporation experiments on single-component and mixed-liquid droplets,including kerosene,ethanol,and heptane.They determined the relationship between droplet evaporation rate and temperature,as well as the micro explosion phenomenon of kerosene droplet in the evaporation process.The evaporation of gasoline and ethanol was studied by Liu19and Hallett20et al.Khan21and Irfan22et al.also investigated kerosene evaporation through experiments.The single droplet evaporation process in microgravity was studied experimentally by Chauveau et al.23,and the entire evaporation process was found to conform tod2law.Ma et al.24investigated droplet evaporation in a hightemperature gas flow environment.They found that temperature and flow velocity affect the stability of droplet evaporation.They also found that a critical temperature exists,at which the initial heating time of the droplet in the entire droplet evaporation time reaches the maximum.

In this paper,the evaporation phenomenon of liquid fuel droplets in a quiescent high-temperature environment through an electric heating suspended droplet experimental system is investigated firstly.Secondly,the prediction results of the existing popular evaporation models are compared with experimental data.Finally,thick exchange layer theory and a new droplet evaporation model are studied and applied to prediction of the droplet evaporation rate versus ambient temperature.

2.Experiment

2.1.Experimental device

The sketch map of experimental device used in this paper is shown in Fig.1.It consists offive parts:the heating control system,temperature control system,experimental chamber,high-speed camera,and data acquisition system.The apparatus can provide a stable heating source,suspend droplet stably,and obtain images and digital signals synchronously.

2.2.Experimental method

The experimental devices were connected as shown in Fig.1.The heating furnace was heated to the test temperature with the temperature controller.The high-speed camera(MONO,SN1400183,Olympus Company,UK)and data collector(NI USB-6008,National Instruments Company,US)were prepared for the image and temperature data collection simultaneously.After the preparation process,the droplet was hung on the thermocouple wire with a microliter syringe.The droplet was pushed promptly and stably into the furnace along the slide rail.Then,the image and temperature data were collected with the high-speed camera and LabVIEW program synchronously.Instantaneous variations in droplet size during the evaporation process were recorded on a computer in the form of pictures.The diameter of the droplet was obtained from the pictures through Photoshop software and MATLAB program.

In the photo,the 0.08 mm thermocouple wire diameter was used as the reference.It was considerably smaller than the droplet diameter,and it was measured using a micrometer with 0.001 mm precision.The initial droplet diameter was obtained from the comparison between the droplet size and the thermocouple wire size.Fig.2 displays one of the droplet images.It clearly shows that the droplet is an ellipsoid in shape and has a smooth surface.The droplet image was delineated with Photoshop software to obtain the droplet shape profile,and MATLAB software was then used to measure the diameters.

The droplet shape is an ellipsoid because of the gravity effect.Its major axis length isdv,and its minor axis length isdh.The droplet volume is

The volume of a sphere whose diameter isdis obtained as follows:

LetVs=Ve,and then

wheredis the equivalent diameter of the droplet.

The experimental results of this study were compared with those from the literature to test the accuracy of the experimental method in this paper.Fig.3 shows the experimental result(EXP)and the literature data,tis the time andd0is the initial droplet diameter.The results of the experiment were generally consistent with those of Liu and Avedisian.19

Experimental data from the reference Ref.19 at 773 K and the experimental results in this paper(EXP)at 773 K,932 K and 1073 K are shown.Staggered scales for different temperatures are shown.

2.3.Experimental parameters

Table 1 shows the ambient temperaturesTfor the methanol droplets are 473 K,580 K,687 K,777 K,885 K,983 K and 1087 K.The ambient temperatures for the kerosene droplets are 777 K,878 K and 982 K,whereas the ambient temperatures for the aviation kerosene droplets are 778 K,880 K and 983 K.

The initial droplet temperature was equal to the room temperature of approximately 298 K.The heating furnace was precisely calibrated with a measurement accuracy of 1 K.

The theoretical research showed that the variation of droplet diameterdin the droplet evaporation process with timetis

Table 1 Experimental conditions offuel droplets.

whereKis the evaporation constant.In this study,the evaporation rateCvof the droplet is defined as the time derivative of the droplet’s squared diameter:

Kcan be considered as the average value ofCvwhen the droplet was evaporating stably.

3.Experimental results and discussion

The experimental data of the droplet evaporation process at a set of representative environmental temperatures are analyzed,and the variations of the normalized squared diameter(d/d0)2value with time are obtained(Fig.4).

Fig.4 shows that droplet’s dimensionless diameter changes faster along with the increase of ambient temperature.Therefore,environment temperature considerably affects the evaporation rate.Fig.4(a)depicts that the dimensionless squared diameter decreases linearly with time.However,Fig.4(b)and(c)shows thatd2and timetare not strictly linearly relative.That is because common kerosene and aviation keroseneare both multicomponent mixtures,and their droplet evaporation law is more complicated than that of elementary substance,such as methanol droplet.Fig.4(b)and(c)shows that the droplet evaporation process generally has two stages.The first is heating stage.In this stage,droplet is heated and expands.After the heating stage,droplet enters the stable evaporation stage,the droplet temperature tends to stabilize,and droplet evaporation approximately follows the classicald2law in this stage.

The evaporation rates of methanol,common kerosene and aviation kerosene at different temperatures are given in Fig.5.

Fig.5 illustrates that the evaporation rate increases with the ambient temperatureT:the higher the ambient temperature is,the faster the evaporation rate increases.The methanol evaporation rate reaches a large value when the temperature is higher than 1000 K(Fig.5(a)).Fig.5(d)depicts that when the temperature is higher than 800 K,the kerosene and aviation kerosene evaporation rates are greater than those of methanol.In addition,the methanol and kerosene evaporation rates are similar when the ambient temperature is approximately 770 K.However,kerosene evaporates considerably faster than methanol when the temperature reaches 900 K because of the different characteristics of the fuels.This observation shows that for different fuels,the effect of ambient temperature on the evaporation rate is different,and droplet fuel type properties also affect the evaporation rate.

Scholars have proposed many theoretical models to describe droplet evaporation.For example,Godsave1and Spalding25proposed classical droplet evaporation model,and it is universally used in most commercial software,such as FLUENT software. Ranz-Mashall boiling modelis applicable when ambient temperature is above the boiling temperature of the droplet fuel,and Ranz-Mashall low temperature model2+0.6Re1/2Sc1/3,Scis applicable for the ambient temperature lower than the boiling temperature of the droplet fuel,whererwis droplet radius,Shis Sherwood number,Scis Schmidt number,DABis diffusion coefficient of two material,Reis Reynolds number,ρ∞is density offar field gas,ρlis density of liquid,cpis constant pressure specific heat,T∞andTware temperatures of circumstance and surface of liquid respectively,qeis evaporation heat of liquid,μ is dynamic coefficient of viscosity,ρ is density.Zhou9proposed the stagnant film theory.Stagnant film theory supposed the droplet to be relatively static without thermal radiation and thermal dissociation.The evaporation processes were considered to be stationary,and the evaporation rate can be expressed aswhereis average thermal conductivity. Crowe et al.26proposed mass analogy model(Cv=3πrwρ∞ShDAB(Yi,s-Yi,∞)/ρl).Mashaye27proposed drift flux model (Cvv,s=3πrwρ∞ShDABBm/ρl),whereYis mass fraction in the mixture of gas,Bmis transfer number of mass.

The evaporation rates of methanol and kerosene are calculated with these models,and the results are compared with the experimental data of this study(Fig.6)Experimental data(EXP);model predictions:Ranz–Marshall boiling temperature model(RMB),Ranz–Marshalllow-temperature model(RML),mass analogy model(MAM),drift flux model(DFM),and stagnant film theory model(SFT).Fuel properties adopted in this calculation was the mass weighted mean value at room temperature of approximately 298 K.Because both common kerosene and aviation kerosene are mixture consisting of some kinds of components,their properties are complex,changing with ambient temperature.

Fig.6(a)illustrates that the mass analogy model can predict the variation trend of methanol evaporation rate in the temperature range of 500–700 K.However,a deviation exists between the mass analogy model simulation results and the experimental data,and the mass analogy model cannot predict the trend of methanol evaporation rate anymore when the temperature is higher than 700 K.It also can be seen from Fig.6(a)that some errors exist between the results of the experiment and the calculations of the models:RMB,RML,DFM,and SFT.Fig.6(b)shows that the existing models fail to simulate the variation of kerosene droplet evaporation rate accurately.The reason is that these existing models neglect the effect of natural convection in the environment.The next section introduces the natural convection in thick exchange layer theory.

4.A new evaporation model

In this paper,the natural convection effect is introduced into the droplet evaporation model for the first time.The thick exchange layer theory presented by Zhou9in 1961 for droplet evaporation in a high-temperature and forced convection environment is closer to the experimental observation.Therefore,the new droplet evaporation model combined the thick exchange layer analysis method with the natural convection effect.

This section explains the new thick exchange layer theory in consideration of the natural convection buoyancy factor to theoretically analyze droplet evaporation in a hightemperature stationary air environment.

The fundamental assumptions of the new thick exchange layer theory in consideration of the natural convection buoyancy factor are as follows:

(1)The flow around a droplet is symmetric;

(2)Natural convection is considered,whereas the effects of radiation are ignored;

(3)No circulation exists inside a droplet;

(4)The flow field near a droplet has no characteristics of the boundary layer,and the heat transfer and mass exchange layer thicknesses are considerably larger than the droplet radius;

(5)The radial flow diffusion and heat flux are more significant than those in the circumferential direction.

The axisymmetric two-dimensional laminar flow equations of droplet evaporation in the spherical coordinate system are as follows.

Continuity equation:

whereris radial position,vris radial velocity,vθis circumferential velocity and θ is coordinate direction.According to the principle of Archimedes,the buoyancy generated by a temperature difference isFb= ρ∞gΔV,whereFbis buoyancy force,ρ∞the density offluid at temperatureT∞,gthe gravitational acceleration,and ΔVthe volume difference of a fluid element massmdue to temperature difference:

where β is the coefficient of expansion,V∞the volume offluid element.Therefore,the buoyancy of a fluid element per unit of mass is

Thus,the momentum equation is

The diffusion equation is

The energy equation is

The boundary conditions are as follows:

where subscribe ‘w” means the position at droplet radius,λ is thermal conductivity,Dis diffusion coefficient and φ is mass flow.

This saturation condition is a general expression,i.e.this condition varies according to different kinds offuels.qeis the evaporation heat,Lis the latent heat of evaporation,cplis the constant pressure specific heat of liquid,Tlis the average temperature of liquid:

The following dimensionless variables are obtained with the concept offinite thickness:

Defineur=vr/U,uθ=vθ/U,(whereUis the characteristic velocity,and defined asU= μ/ρd.)

whereRais Rayleigh number,subscriptTandDrepresent the transportation of heat and mass,hDandhTare thickness of mass exchange layer and heat exchange layer.The whippletree over letters means nondimensionalization to the quantities.

Then,the continuity equation,diffusion equation,and energy equation can be transformed into

The boundary conditionsare transformed into the following:

When ζ=1,

Saturation condition

According to the generalized Reynolds similarity,no chemical reaction exists in space,Fw=ϑw,whenLe=1,namelycp(T∞-Tw)/qe=YFw/(1-YFw).Leis Lewis number,YFwis mass fraction offuel at liquid surface.Then with continuity Eq.(17)and boundary conditions Eqs.(20)–(23),diffusion Eq.(18)and energy Eq.(19)are rewritten as

The approximate solution of Karmen–Pohlhausem is imitated to deal with the pressure gradient boundary layer flow,and the flow around fuel droplets is assumed to be between Stokes flow and inviscid flow.Driven by buoyancy,the general expression of sphere flow function with a nonspherically symmetric evaporating source under natural convection is

The following are solved and obtained:

whereB0,B1,B2,C0,C1,C2are coefficient offunctions.

Boundary conditions Eqs.(20)–(23)are used to obtain the following:

The definite integral is calculated as follows:

wherecis integration constant.

According to Eq.(35),

where δ=A0cis an experimentally determined constant.Thus,

The dimensionless thickness of the diffusion layer is then obtained:

The dimensionless heat transfer layer thickness is

Then,

Therefore,the droplet surface average heat transfer and mass exchange formulas are

wherePris Prandtl number,NuDandNuTare average Nusselt number of mass and heat transfer.

The relationship between evaporation constantKand the dimensionless heat transfer Nusselt number is

5.Prediction results and discussion

The new droplet evaporation model was applied to the droplet of kerosene and other fuels.The comparison of the kerosene evaporation rate between the new model prediction and the experimental data in this paper and other references is shown in Fig.7,Experimental data from Refs.16–28(Irfan).Experimental data from Ref.29(Ghassemi).Experimental data from this study(experiment).

Fig.7 illustrates that the prediction result of the new droplet evaporation model,which added the natural convection buoyancy into the thick exchange layer model,is consistent with the experimental data from both references and this paper.The maximum relative error between the new model’s calculation and Irfan’s experimental data is less than 7%,and the relative error is no more than 10%between the new model’s calculation and Ghassemi’s experimental data.Fig.7 also shows that the percentage error between the new model and the experimental data from this paper is no more than 20%.The new droplet evaporation model with consideration of the natural convection can accurately predict kerosene droplet evaporation in a wider temperature range.

Fig.8 presents the new droplet evaporation model prediction for methanol droplet(δ=8×10-7)and the experimental data in this paper.The prediction results and the experimental data show good agreement.

Figs.7 and 8 show that the prediction results of the new droplet evaporation model for the evaporation rate of the kerosene and methanol droplets are consistent with the experimental data in the high-temperature environment condition.Natural convection around the droplet affects droplet evaporation in these kinds of conditions.Therefore,the new model that considers the natural convection factor in thick exchange layer theory can successfully predict the droplet evaporation rate.

The comparison between the new droplet evaporation model and the existing popular evaporation model described in the third section in this paper is shown in Fig.9.

Fig.9(a)indicates that the new thick exchange layer model well predicts the methanol droplet evaporation rate.The calculation result is consistent with the experimental data in the temperature range of 450–1000 K.The mass analogy model can predict the trend of the methanol droplet evaporation rate in the temperature range of 500–700 K.However,the mass analogy model results significantly vary from the experimental data.It also can be observed from Fig.9(a)that some errors exist between the calculation of other evaporation models and the experimental results.

In Fig.9(b),the new droplet evaporation model well predicts the kerosene droplet evaporation rate,with the calculation consistent with the experimental data in the temperature range of 450–750 K.The calculations of the evaporation models,namely the Ranz–Marshall boiling evaporation model,Ranz–Marshall low-temperature evaporation model,mass analogy model,drift flux model,and the stagnant film theory model,significantly deviate from the experimental result.

Fig.9 shows the comparison between the prediction results of the different models and the experimental data.The new droplet evaporation model that considers natural convection can predict the droplet evaporation rate more accurately than the classical models,such as the Ranz–Marshall boiling evaporation model.

Furthermore,the droplet evaporation rate value of gasoline,diesel fuel,and n-heptane was calculated by the new droplet evaporation model and tested by experimental data in this paper.The model calculation results are close to the experimental data,which are shown in Fig.10.

Fig.10 shows that the result of the calculation is consistent with the experimental data.The new droplet evaporation model can accurately predict the trend of the evaporation rate of different kinds offuel droplet.

Figs.7,8 and 10 illustrate that some discrepancies still exist between model predictions and experimental data.The neglect of radiation in the new model is an important factor causing the discrepancies.In addition,the physical properties of the liquid droplets are source of error.The physical and chemical properties of the gas mixture around the droplet surface change broadly in a high-temperature environment and thus cause errors as well.

6.Conclusions

In this paper,the single droplet evaporation rate in a hightemperature environment was studied with experiments,and the evaporation characteristics were examined with theoretical analyses.A new droplet evaporation model was proposed.The following conclusions were obtained:

(1)The evaporation characteristics of liquid fuel generally agree withd2law,and the evaporation rate increases with the increase of ambient temperature.

(2)The calculations of the Ranz–Marshall boiling evaporation model and other models,which do not consider natural convection,deviate significantly from the experimental results.

(3)The calculation results of the new droplet evaporation model are consistent with the experimental data for kerosene and methanol,and the relative deviations from the experimental results are less than 20%.The relative deviations between the new droplet evaporation model predictions for kerosene and the experimental results from the references are within 10%.

(4)The new droplet evaporation model that considers natural convection can predict droplet evaporation more accurately than the classical models,such as the Ranz-Marshall boiling evaporation model.

(5)In future research,the model proposed in this paper will be tested and developed further in consideration of the radiation and evaporation sequence of different components in the multicomponent fuel.

Acknowledgements

This study was supported by the National Natural Science Foundation of China(No.51106006).

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11 May 2016;revised 19 October 2016;accepted 26 December 2016

Available online 1 July 2017

*Corresponding author at:Aero-engine Numerical Simulation Research Center,School of Energy and Power Engineering,Beihang University,Beijing 100083,China.

E-mail address:fwang@buaa.edu.cn(F.WANG).

Peer review under responsibility of Editorial Committee of CJA.

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Droplet;

Evaporation model;

Evaporation rate;

Kerosene;

Natural convection;

Thick exchange layer theory