Effect of Ambient Pressure and Droplet Size on Droplet ...€¦ · pressure during fuel injection...

Effect of Ambient Pressure and Droplet Size onDroplet Vaporisation and Combustion

Shah Shahood Alam*, Ahtisham A. Nizami, Tariq AzizAbstract

In the present work, two cases of liquid droplet vaporisation and combustion were investigated with respect tohigh pressure and droplet size effects. Supercritical droplet vaporisation models for n-heptane-N2 and LOX-H2systems were developed with considerations of high pressure liquid-vapour equilibrium, real gas effects andabsorption of ambient gas in liquid layer at the droplet surface. A generalised code was developed usingNewton-Raphson method. Appropriate relations were used for determining pressure dependent thermophysicaland transport properties. It was observed that for n-heptane-nitrogen system, the critical mixing temperatureincreased with an increase in ambient pressure in the subcritical regime while decreased in the supercriticalregime. Also, the solubility of nitrogen in liquid n-heptane increased with an increase in temperature in thisregime. Further, at a given pressure, mass fraction of n-heptane vapour at the droplet surface increased withdroplet temperature, while for a fixed droplet temperature, vapour mass fraction decreased with an increase inambient pressure. For LOX-H2 system, the critical mixing temperature increased progressively with pressure(unlike n-heptane-N2 system) while hydrogen solubility in liquid oxygen increased with increasing temperatureat supercritical pressures. Further, for a 100 m liquid oxygen droplet vaporising in hydrogen environment at1000 ,K evaporation lifetime for the droplet decreased for spherically symmetric case (gas phase Reynoldsnumber 0gRe ), as ambient pressure was increased from subcritical to supercritical value and furtherreduction in evaporation lifetime was noticed with an increase in gRe . Also, for fixed values of ambienttemperature and gRe , maximum evaporation rate increased as ambient pressure was increased from 5.04 bar(reduced pressure rP = 0.1) to 126 bar ( rP = 2.5).Effect of initial droplet diameter was explored by developing a transient gas phase combustion model. It wasobserved that droplet burning was an unsteady event. Effect of droplet size on combustion characteristicsshowed that at a given pressure, droplet lifetime and burning rate increased. Further, for a wide range of dropletsize (20 m to 5000 m ), the flame diameter first increased, reached a maxima and then reduced to a minimumvalue at the end of droplet lifetime, so was the case with flame standoff distance.

Keywords: High pressure droplet vaporisation model, effects of supercritical vaporisation, unsteady gas phasemodel, numerical technique, effects of droplet size, different fuels, spray combustion.

*Corresponding author. E-mail:[email protected]. Address: Pollution andCombustion Engineering Lab. Department of MechanicalEngineering, Aligarh Muslim University, Aligarh-202002,U.P. India.

International Journal of Scientific & Engineering Research, Volume 5, Issue 10, October-2014 ISSN 2229-5518 1090

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1 INTRODUCTION1.1 Droplet Evaporation and Combustion at HighPressures

Droplet evaporation and combustion isprevalent in the spray combustion of liquidhydrocarbon fuels occuring in the combustionchambers of gas turbine and diesel engines, oil firedfurnaces and liquid rockets and serves as thefundamental step for understanding the complexnature of spray burning.

Spherically symmetric assumption furtherleads to a simple geometry and one dimensionalapproach to the problem. It can be achieved inmicrogravity droplet combustion experiments. Insome applications like diesel engines, liquid rocketsand gas turbine engines, fuel droplets are subjectedto temperatures and pressures beyond their criticalpoint during the combustion process. This gives riseto a number of interesting phenomena at the criticalpoint. Of particular importance in the study are thechanges in the specific heat ( )pc of the fuel, whichgoes to infinity, and the latent heat of vaporisationwhich goes to zero at the critical point. Unresolvedand controversial topics of interest includeprediction of phase equilibrium at high andsupercritical pressure including the choice of aproper equation of state. Influence of 2 -d lawbehaviour at supercritical conditions, and thereduced surface tension.

High pressure and supercritical ambientconditions have a considerable influence on themechanisms controlling engine behaviour andperformance. Most of these effects are related todroplet behaviour. When liquid is injected into acombustion chamber that is filled with a gas atsupercritical thermodynamic conditions, all aspectsof the combustion process from atomisation tochemical reaction can be expected to departsignificantly from the better known subcriticalpatterns. Studies in the past have investigated howand to what extent supercritical conditions mayaffect various aspects of the combustion of anisolated droplet in a quiescent environment.

In major industrial and space applicationssuch as high output combustors for aircraft engines,

liquid fueled rocket engines and diesel engines,manufacturers are looking for high pressurecombustion systems for improving engineefficiency and power density. In fact, in recentyears, diesel engine manufacturers have striven toincrease the ambient chamber density and therebypressure during fuel injection for achieving bettermixing and increased rates of combustion. Theoperating pressure often exceeds the criticalpressure of the liquid fuel. The droplets in the liquidfuel spray ignite and burn in the gaseous medium attemperatures and pressures above thethermodynamic critical state of the fuel.

The study of droplet behaviour in highpressure environment presents a scientificallychallenging problem. The actual combustionprocess is characterised by the supercriticalcombustion of relatively dense sprays in highlyconvective environment. However, most studiesconsidered decoupled problems in order to isolate alimited set of issues. Consequently, most resultswere derived in the case of an isolated dropletvaporising (no reactions) in a quiescentenvironment. Convective effects, influence ofneighbouring droplets, detailed chemical kinetics orproduct condensation have received less and morerecent attention [1].

Because high pressure tests are expensiveand sometimes dangerous, considerable effort hasbeen devoted to the development of accuratemodels capable of portraying the physics of dropevolution at high pressures. Predictions from suchmodels, for example the validity of the 2 -d law athigh pressures, would enable a considerablesimplification in the incorporation of drop modelsin the complex Computational Fluid Dynamics(CFD) codes [2].

Earlier research in this area includedSpalding [3], who theoretically considered highpressure combustion by approximating the dropletvapour as an instantaneous point source of fuel. Thecombustion process was represented by a flamesurface approximation, that is a diffusion flame withan infinitely thin reaction zone, constant propertieswere assumed, and convection was neglected. Thisanalysis was modified by Rosner [4] to account for



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the finite dimensions of the puff of gas. Theinfluences of convection, density variation andfinite rate chemical kinetics on supercriticalcombustion were studied by Brzustowski [5].Manrique and Borman [6] found that effect ofthermodynamic non idealities, property variationsand high pressure corrections for phase equilibriumcould influence the vaporisation mechanismssignificantly.

Lazar and Faeth [7] and Canada and Faeth[8] conducted a series of experimental andtheoretical studies on droplet combustion ofhydrocarbon fuels in both stagnant and forcedconvective environments, with special attentionfocussed on high pressure phenomena of phaseequilibrium. The effects of forced convection in thegas phase were treated by conventionalmultiplicative corrections.

Rosner and Chang [9] examined the effectsof transient processes, natural convection, and theconditions under which a droplet may be driven toits critical point. Kadota and Hiroyasu [10]conducted an experimental study of combustion ofsuspended fuel droplets of n-heptane, n-decane, n-dodecane, n-hexadecane, iso-octane and light oildrops at reduced pressures as large as 1.5, and even2.7 for oil under the influence of natural convection.

For all fuels, the final droplet temperaturewas nearly equal to its critical temperature andindependent of ambient pressure in supercriticalconditions. The combustion lifetime, defined as thetime from the appearance of flame to itsdisappearance, displayed an abrupt reduction with

rP upto the critical point after which the reductionwas only gradual, whereas the burning constantincreased with an increase in pressure throughoutsubcritical and supercritical conditions.

Effects of ambient pressure and temperatureon commercial multicomponent fuels like aviationgasoline, JP5 and diesel oil (DF2) were investigatedby Chin and Lefebvre [11]. Their results suggestedthat evaporation constant evk values were enhancedas ambient pressure and temperature was increased.

Other research groups like Hsieh et al. [12],Curtis and Farell [13], Jia and Gogos [14] andDelplanque and Sirignano [15] employed numerical

techniques to simulate high pressure dropletvaporisation and combustion with considerablesuccess. Among them, Delplanque and Sirignano[15] developed an elaborate numerical model toinvestigate spherically symmetric, transientvaporisation of a liquid oxygen (LOX) droplet inquiescent, gaseous hydrogen at moderate and highpressures. Computations were performed for purevaporisation of a 50 m LOX droplet initially at100 K at a reduced temperature rT =9.70, andreduced pressures of rP =2.0, 3.0 and 4.0.

They suggested that when film theory isused to model droplet combustion, the filmsurrounding the droplet can be assumed to be quasi-steady, since the characteristic time for heatdiffusion through the film is typically two orders ofmagnitude smaller than the droplet lifetime. Theyadvocated the use of Redlich-Kwong equation ofstate used by spray combustion community for itssimplicity and accuracy for computing gas phaseequilibrium mole fractions at high pressures. Theyfurther suggested that for a simplified dropletvaporisation model to be used in spray codes atsupercritical conditions, it can be assumed thatdissolved hydrogen remains confined in a thin layerat the droplet surface.

The effect of non equilibrium phasetransition on droplet behaviour was furtheraddressed by Harstad and Bellan [16-19] usingKeizer’s fluctuation theory. Umemura and Shimada[20] reported a numerical investigation ofspherically symmetric droplet vaporisation undersupercritical conditions. They identified thetransition from subcritical to supercritical state interms of a binary diffusion coefficient, which wassuitably modified so that it became zero as thedoplet surface reaches the critical mixing state.

Zhu and Aggarwal [21] carried outnumerical investigation of supercritical vaporisationphenomena for n-heptane-N2 system by consideringtransient, spherically symmetric conservationequations for both gas and liquid phases, pressuredependent thermophysical properties and detailedtreatment of liquid-vapour phase equilibriumemploying different equations of state. Yang [22]also analysed numerically a fully transient model



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for LOX-H2 system employing complex Benedict-Webb-Rubin EOS.

In another microgravity combustion studyby Vieille et al.[23], high pressure droplet burningcharacteristics of five fuels (methanol, ethanol, n-hexane, n-heptane and n-octane) were investigatedunder normal and reduced gravity conditions usingsuspended droplet technique employing initialdroplet diameters of about 1.5mm. An importantresult was that the 2 -d law holds even under veryhigh pressure and allows the estimation of anaverage droplet burning rate. The experimentalresults for all fuels showed that the droplet burninglifetime decreases strongly with increasing pressurein the subcritical regime.

Stengele et al.[24] conducted experimentaland theoretical study, where the evaporation of freefalling, non interacting, single and bicomponentdroplets in a stagnant high pressure gas wasinvestigated at different temperatures. Due to therelative velocity between the falling droplet and thestagnant gas, convective effects were incorporatedthrough experimental correlations. Theexperimental results were compared with numericalcalculations based on the conduction limit anddiffusion limit model. The effect of ambient pressure on theevaporation of a droplet and a spray of n-heptanewas investigated by Kim and Sung [25] using amodel for evaporation at high pressure. Their modelconsidered phase equilibrium using the fugacities ofliquid and gas phases for real gas behaviour and itsimportance on the calculation of the evaporation ofthe droplet or spray at high pressures wasdemonstrated. For the evaporation of single droplet,droplet lifetime increased with pressure at a lowambient temperature (453 K ) but decreased at hightemperatures. The evaporation of a spray wasenhanced by increasing the ambient pressure andthe effect was more dominant at higher ambienttemperatures.

From the literature review, it was felt thatthere was still a need of developing a high pressuredroplet vaporisation model with considerations ofhigh pressure liquid-vapour equilibrium, real gaseffects, absorption of ambient gas in a thin layer at

the droplet surface and pressure dependentproperties which should be simple and realistic tobe incorporated in spray analysis and tested fordifferent systems. So that, the validity of 2 -d law ,surface temperature behaviour, solubility of ambientgas in the liquid and effects of convection couldthen be quantified.

1.2 Effect of Initial Droplet Diameter on DropletCombustion Characteristics

Apart from ambient variables, initial dropletdiameter plays an important role on dropletcombustion characteristics. For example, thelifetime of the largest droplet in a spray determinesthe minimum time the droplet must be allowed toreside inside the combustion chamber.

Kumagai and co-workers [26,27,28] werepioneers in conducting spherically symmetricdroplet combustion experiments in microgravityconditions through drop towers, capturing the flamemovement and further showed that /F D ratio variesthroughout the droplet burning history.

Waldman [29] and Ulzama and Specht [30]used analytical procedure whereas Puri and Libby[31] and King [32] employed numerical techniquesin developing spherically symmetric dropletcombustion models. The results of these authorswere however mainly confined to the observationsthat unlike quasi-steady case, flame is not stationaryand flame to droplet diameter ratio increasesthroughout the droplet burning period. The presentwork has tried to quantify the effect of initialdroplet diameter on important combustioncharacteristics through a droplet combustion model.2 PROBLEM FORMULATION2.1 Development of Droplet Vaporisation Model atHigh Pressure

Combustion chambers of diesel engines, gasturbines and liquid rockets operate at supercriticalconditions. To determine evaporation rates of liquidfuel sprays (made up of discrete droplets) at highpressures, a prior thermodynamic analysis is a must.Important assumptions employed in the present highpressure model are:

Droplet shape remains spherical, viscousdissipation effects are neglected, ambient gas getsdissolved in the droplet surface layer only, gas



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phase behaves in a quasi-steady manner, radiationheat transport is neglected and Soret and Dufoureffects are not considered.At high pressures, vapour-liquid equilibria for eachcomponent can be expressed as:

( ) ( )1 1( ) ( )

( ) ( )

v l

v l

v l

f fT TP P

(1)

where, the superscripts (v) and (l) stand for vapourand liquid respectively.

if is the fugacity for component or species “ i ” andcan be integrated through the following relation

, ,ln [( ) ] lnj

i uu T V uV

i in

f R TPR T dV R T Zx P n V

(2)The above equation suggests that if can be

determined by the properties of the constituentcomponents, the concentrations in both phases andthe temperature and pressure of the system. Tocompute the integral of the above equation, a realgas equation of state such as Redlich-Kwongequation of state was chosen. The cubic equationsand their constants are provided in (Table 1). Theycan be expressed in general form by the relation[33]:

2 2uR T aP

v b v ubv b

uR is Universal gas constantv is molar volumeT is absolute temperature ( )K

is Pitzer’s acentric factorEquation (2) is then integrated. After simplificationwe get equation (3) which is given as:

1

2ln ln ln 1 ln ln 1

N

j ijji i

i i

x ab bA Bf x P Z Z Bb B a b Z

This equation is valid for both vapour and liquidphases. The dimensionless parameters A and B aredefined as:

2 2 andu u

aP bPA BR T R T

(4)

Redlich-Kwong equation can be written in theform of compressibility factor as:

3 2 2 0Z Z A B B Z AB (5) The mixture compressibility factor “ Z ” can bedetermined using Cardan’s method, where highestvalue of Z corresponds to vapour phase and thesmallest value to liquid phase.Rewriting equation (3) for components 1 and 2respectively, we have,

1 11 1 1 11 2 12

2ln ln ln 1 ln ln 1b bA Bf x P Z Z B x a x ab B a b Z

(6)2 2

2 2 1 21 2 22

2ln ln ln 1 ln ln 1b bA Bf x P Z Z B x a x ab B a b Z

(7) The parameters a and b in the cubic equationcan be expressed in terms of composition and purecomponents parameters

2 2 2

1 1 1,i j ij i i

i j ia x x a b x b (8)

The constants aij and bi are essentially dependent

on the critical properties of each species. Inaddition, proper mixing rules have to be employedto determine various mixing parameters fordifferent mixtures.

The mixture constants a and b andsubsequently dimensionless parameters A and B canbe evaluated by substituting the assumed values ofthe mole fractions of the two components. Nowsince equation (3) can be applied to both vapour andliquid phases, if we have a two component, twophase system in equilibrium at a given temperatureand pressure then there are four mole fractions

1 2 1 2, , andv v l lx x x x as unknowns for the phaseequilibrium system. These unknowns have to besolved from four separate equations.

Therefore, equation (3) can be written forliquid and vapour phases for component “1”(equation 6). Then since at equilibrium,

1 1 ,l vf f we get a relationship between 1vx and

1 .lx In the same manner, using equation (7) for

component “2”, we get a relationship between 2vx

and 2lx from the fact that 2 2

l vf f at equilibrium,



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further, in a mixture, 1 2 1l lx x and( ) ( )

1 2 1v vx x , hence the four unknowns can bedetermined. The solution procedure requires evaluation oftwo lengthy, non linear equations which need to besolved iteratively. In the present work, Newton-Raphson method has been used. The solution isobtained in the form of a diagram representingvapour-liquid phase equilibrium compositions for aparticular system at various pressures. For a spherically symmetric Oxygen fueldroplet vaporising in a stagnant high pressure H2environment in a (LOX-H2 system), mass fractionof the fuel vapour (oxygen) at the droplet surface isgiven as:

2 2

v v FO O

t

Wy xW

(9)

where,

2 2 2 2(total molecular weight) v v

t O O H HW x W x W

Here, vx (mole fraction of vapour phase) of boththe components are determined from the computerprogramme. The same equation can be used for n-heptane-N2 system.The heat transfer number TB for subcriticalpressures is given as:

pg bT

pl b lo

C T TB

L C T T (10)

where, loT is the initial droplet temperature,HV or L is the latent heat of vaporisation and

for subcritical pressures given as:

2 2

2

1

0.38

1

11

rO O

r

THV HV

T (11)

where, 1HV is at normal boiling point ; r b cT T T .Equation (11) can also be used for n-heptane-N2system.For supercritical pressures, TB for LOX-H2 systemcan be taken as [34]:

o

cT

c l

T TBT T

(12)

where, cT is the critical mixing temperature.

Mass evaporation rate evm for spherical droplet is

2 ln(1 )mix

mixgev T

pg

m D BC

(13)

Droplet evaporation lifetime, evt is20

8 ln 1mix

mix mix

lev

g pg T

Dt

C B (14)

Instantaneous droplet radius can be determinedfrom

ln 1mix

mix mix

glT

l pg l

dr Bdt C r

(15)

Proper determination of thermophysicaland transport properties are essential specially formulticomponent, high pressure, high temperaturestudies since simulation results depend heavily uponthe correct evaluation of properties. For the presentwork, it was observed that in general the error lieswithin 10% of the experimental values.2.2 Development of Spherically Symmetric,Unsteady Droplet Combustion Model for the GasPhase

Important assumptions invoked in the developmentof spherically symmetric gas phase dropletcombustion model of the present study

Spherical liquid fuel droplet is made up ofsingle chemical species and is assumed to be at itsboiling point temperature surrounded by aspherically symmetric flame, in a quiescent, infiniteoxidising medium with phase equilibrium at theliquid-vapour interface expressed by the Clausius-Clapeyron equation.

Droplet processes are diffusion controlled(ordinary diffusion is considered, thermal andpressure diffusion effects are neglected). Fuel andoxidiser react instantaneously in stoichiometricproportions at the flame. Chemical kinetics isinfinitely fast resulting in flame being representedas an infinitesimally thin sheet.

Ambient pressure is subcritical and uniform.Conduction is the only mode of heat transport,radiation heat transfer is neglected. Soret andDufour effects are absent.



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Thermo physical and transport properties areevaluated as a function of pressure, temperature andcomposition. Ideal gas behaviour is assumed.Enthalpy ‘ h ’ is a function of temperature only. Theproduct of density and diffusivity is taken asconstant. Gas phase Lewis number gLe is assumedas unity.

The overall mass conservation and speciesconservation equations are given respectively as:

22

1 0rr vt r r

(16)

22

1 0r

Y Yr v Y Dt r rr

(17)

Where;t is the instantaneous timer is the radial distance from the droplet center

is the density

rv is the radial velocity of the fuel vapourD is the mass diffusivityY is the mass fraction of the speciesequations (16) and (17) are combined to give speciesconcentration or species diffusion equation for thegas phase as follows

2

2

2 gg r

DY Y Y YD vt r r r r

(18)

gD is the gas phase mass diffusivityThe relation for energy conservation can be writtenin the following form

22

1 . 0r p

h Tr v h D Ct r r r

(19)

The energy or heat diffusion equation for the gasphase, equation (20) can be derived with the help ofoverall mass conservation equation (16) andequation (19), as:

2

2

2 gg r

T T T Tvt r r r r

(20)

T is the temperature, g is gas phase thermaldiffusivity. Neglecting radial velocity of fuel vapour

rv for the present model, equations (18) and (20)

reduce to a set of linear, second order partialdifferential equations (eqns 21 and 22).

2

2

2 gg

DY Y YDt r r r

(21)

2

2

2 gg

T T Tt r r r

(22)

The solution of these equations provide thespecies concentration profiles (fuel vapour andoxidiser) and temperature profile for the inflameand post flame zones.

The boundary and initial conditions based on thiscombustion model are as follows

, , ; , 0, 0f f o f F fat r r T T Y Y

, ; , 0.232,oat r r T T Y

, 0; , , 1.0lo b F Sat t r r T T Y

1/ 2where 1 / for 0 ,l lo d dr r t t t t

is the moving boundary condition coming out fromthe 2 -d law (since droplet is at its boiling pointtemperature). Here, fT and T are temperatures atthe flame and ambient atmosphere respectively.

,F SY and ,F fY are fuel mass fractions respectivelyat the droplet surface and flame. ,oY and ,o fY areoxidiser concentrations in the ambience and at theflame respectively, t is the instantaneous time, dtis the combustion lifetime of the droplet, lor is theoriginal or initial droplet radius and lr is theinstantaneous droplet radius at time “ t ”.

The location where the maximumtemperature fT T or the correspondingconcentrations , 0F fY and , 0o fY occur, wastaken as the flame radius fr . Instantaneous time “ t ”was obtained from the computer results whereasthe combustion lifetime “ dt ” was determined fromthe relationship coming out from the 2 -d law .Other parameters like instantaneous flame todroplet diameter ratio ( / )F D , flame standoff



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distance ( - )/2F D , dimensionless flamediameter 0( / )F D etc, were then calculated as afunction of timeSolution technique

Equations (21) and (22) were solvednumerically using the “finite difference technique”.The approach is simple, fairly accurate andnumerically efficient [35]. Here the mesh size inradial direction was chosen as h and in timedirection as k . Using finite differenceapproximations, equations (21) and (22) weredescritised employing three point central differenceexpressions for second and first space derivatives,and the time derivative was approximated by aforward difference approximation resulting in a twolevel, explicit scheme (eqn 23) which wasimplemented on a computer.

11 1 1 1 11 1 2 1n n n n

m m m m m mT p T T p T(23)

Here, 21( ) / , /m mmesh ratio k h p h r ,

m lor r mh , 0,1, 2.... . = ,ol NNh r r m

The solution scheme is stable as long as thestability condition 1 1/ 2 is satisfied.

3 RESULTS AND DISCUSSION3.1 Effect of Ambient Pressure on BurningConstant, Droplet Lifetime, Mass Burning Rate and

/F D Ratio

Burning constant The burning constant bk , a primary factor forcharacterising burning droplets has been verifiedexperimentally. The variation of burning constantwith increase in reduced pressure rP is shown inFig 1. The results of present study were comparedwith experimental results of Kadota and Hiroyasu[10] who conducted an experimental study ofcombustion of single suspended droplets insupercritical gaseous environments under theinfluence of natural convection. The trend of thetwo results is the same which suggests that bkincreases with an increase in ambient pressure.

The slight difference (approximately 15%)in the variation of bk with rP for both the studiesmay be due to the difference in the values ofthermophysical and transport properties for thepresent study which were subsequently used tocalculate combustion parameters such as TB , bkand dt given in the form of correlations. (In thepresent work, adiabatic flame temperature wascalculated first by Gülder’s method [36]. Variousproperties were then determined by using propertyestimation techniques [33] at an averagetemperature of bT and fT for the inner zone and fTand T for the outer zone). Another reason may bethe presence of natural convection in theexperimental investigation of [10] which resulted inslightly higher values of bk , and lower values of dt(Fig 2) at each rP .

The same trend was also observed in theresults of Nomura et al.[37] who consideredspherically symmetric n-heptane dropletevaporating in a high pressure (upto 20atmospheres) nitrogen ambience. Here, the dropletsize was varied between 0.6 to 0.8 mm and ambientnitrogen temperature varied between 400 K to800 K . Their results suggested that as ambientpressure was increased, corresponding values of evkwere increased at each pressure, as the temperaturewas raised from 400 K to 800 K .

Droplet lifetime Combustion lifetime (time fromestablishment of flame till the burnout of thedroplet) or simply droplet lifetime is a criticalparameter of importance in the design ofcombustion chambers. It is also related to otherdesign parameters such as spray injection velocity,combustion geometry etc. Results of the presentstudy were compared with those of Kadota andHiroyasu [10] in Fig 2. Here values of combustionlifetime of droplet dt are plotted against reducedpressure rP . The results of both studies indicate that

dt decreases with an increase in the value ofambient pressure P .



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This trend is also consistent with theinvestigations of other workers like Faeth et al.[38],who conducted supercritical droplet combustionexperiments in near zero gravity conditions in afreely falling apparatus with n-decane dropletshaving initial diameters in the range of 800 m to1000 m . The gas environment consisted of variousmixtures of nitrogen and oxygen initially at roomtemperature. The pressure range of the experimentswas 100-2000 psia. Similar trends in the variationof dt with P were observed by Kim and Sung [25]who modelled the evaporation of n-heptane dropletinjected into gaseous nitrogen at high pressures(upto 50 atmospheres). The ambient temperaturestaken were 600 K and 1200 K and initial dropletdiameter was 0.5 mm. Their results also showedthat at high ambient temperatures, evt decreasedwith increase in ambient pressure.

The same behaviour was also observed fromthe results of Jia and Gogos [39], who modelled theevaporation of spherically symmetric high pressureevaporation of n-hexane droplet in nitrogenenvironment. 0D was taken as 100 m , P variedbetween (1-100 atmospheres), T between (1000-2000 K ).Mass burning rate This result (Fig 3) can be interpreted using

2f l l bm r k (equation 24), i.e. if ambientpressure P is increased, then the cumulativeeffects of l and bk for a fixed droplet radius lrlead to an increase in the mass burning rate.

/F D ratio against reduced pressure at / dt t =0 Effect of ambient pressure on /F D ratio isshown in Fig 4. It was observed that as reducedpressure values increase, there was a correspondingincrease in /F D ratio at a particular time, / dt t =0.The reason is that as ambient pressure is increased,boiling point of fuel is increased which leads to anincrease in flame diameter F obtained fromcomputed results. Then as droplet radius lr isdecreasing continuously, the ratio /F D increases as

a function of rP . The same trend was observed inthe results of Kotake and Okazaki [40].

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

Pr = P / Pc

Kb

(mm

2/s

)

Pr

Figure 1: Effect of ambient pressure P onburning constant kb for n - heptane ( D0 = 2000

microns, T = 298 K, Yo , = 0.232 )

Present ModelExperiment of Kadota and Hiroyasu [10]

0.0 0.3 0.5 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

t d(s

)

Pr

Present ModelExperiment of Kadotaand Hiroyasu [10]

Figure 2: Ambient pressure effect on droplet lifetime

0 0.2 0.4 0.6 0.8 10

1

2

3

mf(m

g/s

)

PrFigure 3 : Mass burning rate



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0 0.25 0.5 0.75 10

6

12

18

24

t / td = 0

F/D

Pr

Figure 4: F / D ratio

3.2 Mole Fraction of n-Heptane Predicted byRedlich-kwong Equation of State for n-Heptane-Nitrogen System in Equilibrium at Sub andSupercritical Pressures

The variation of n-heptane mole fractions inliquid and vapour phases with temperature, whenambient pressure P is 13.7 bar (reduced pressure;

rP =0.5) is shown in Fig 5. This variation is alsoobtained for P =24.66 bar ( rP = 0.9) and forsupercritical pressure P =68.5 bar ( rP =2.5) shownin Fig 6. The critical temperature and pressure fornitrogen are; 126.2 ,cT K 33.9 barcP and for n-heptane; 540.3cT K , 27.4 barcP respectively.The critical mixing point obtained by employingRedlich-Kwong equation of state in the presentwork is the highest point of the curve whereenthalpy of vaporisation goes to zero and there is nowell defined interfacial boundary separating theliquid phase (n-heptane) from ambient gas(nitrogen).

It was observed that in the sub-criticalregime the critical mixing temperature increasesfrom 480K for rP =0.5 to 530.1K for rP =0.9 andthen in supercritical region ( rP =2.5) decreases to520K. This trend regarding the behaviour of criticalmixing temperature is in good agreement with thesimulation results of Zhu and Aggarwal [21], whodeveloped a comprehensive, transient supercriticaldroplet vaporisation model by numerically solvingconservation equations in liquid and gas phasesusing pressure dependent variable thermophysical

properties and detailed treatment of liquid-vapourphase equilibrium employing different equations ofstate as compared to the simpler approach adoptedin the present study. The small difference may bedue to the methods employed for the evaluation ofproperties. Another important observation is that in thesubcritical regime, the amount of nitrogen dissolvedin the liquid fuel (n-heptane) is very small at 300K,decreasing progressively with increasingtemperature and reducing to zero at the criticalmixing point. In supercritical pressure regime ( rP =2.5), shown in Fig 6, it was observed that thesolubility of nitrogen in the liquid n-heptanebecame progressively higher as the temperature wasincreased and becomes equal to about 0.3 at thecritical mixing state. The amount of nitrogen dissolved in n-heptaneincreased as the pressure was increased further,above rP =2.5. Therefore, at appreciably highpressures, the amount of nitrogen dissolved in n-heptane may be quite significant and may modifythe liquid properties and subsequently thevaporisation behaviour. Hence it is felt thatsupercritical modelling studies should include liquidphase solubility of gases.

0 0.25 0.5 0.75 1300

350

400

450

500

Tem

pera

ture

(K)

Figure 5 : Vapour-liquid phase equilibrium compositionspredicted by Redlich-Kwong EOS for

n-heptane-nitrogen system

n-Heptane - Nitrogen System

P=13.7 bar

Model of Zhuand Aggarwal [21]

Vapour

Liquid

Pr = 0.5( )

Present Model

Mole fraction of n-heptane



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0 0.25 0.5 0.75 1300

350

400

450

500 n-Heptane -N2 System

Pr = 0.9

Pr = 2.5

Figure 6: Vapour-liquid phase equilibrium compositionsat subcritical and supercritical pressures

Mole fraction of n-heptane

Tem

pera

ture

(K)

3.3 Variation of n-Heptane Vapour Mass Fractionat the Droplet Surface with Droplet Temperature

Figure 7 shows the variation of n-heptanevapour mass fraction

7 16

v

C Hy with droplet surface

temperature at three different pressures. The molefractions of n-heptane vapour

7 16

v

C Hx were computed

from the computer programme at each temperatureand the corresponding vapour mass fractions

7 16

v

C Hy

were determined using equation 9. For eachpressure, mass fraction increased with an increase indroplet temperature and at a given temperature,mass fraction decreased with increasing pressure.The maximum values being 1.0 for subcritical and0.9 for supercritical pressures respectively. Also atsupercritical pressure of 68.5 bar, the mass fractionvariation was small about 0.42 till 500K, butincreased suddenly at a slight increase intemperature to a value of about 0.9 at 520K. Areason for this difference in behaviour may be dueto the absorption of ambient nitrogen in liquidheptane droplet that could modify the dropletvaporisation characteristics.

300 400 5000

0.4

0.8

1.2

Figure 7: n-Heptane vapour mass fraction variation withdroplet temperature at different pressures

Droplet temperature (K)

yv C7H

16

n-Heptane - Nitrogen System13.7 bar24.66 bar68.5 bar

3.4 Mole Fraction of Oxygen Predicted by Redlich-Kwong Equation of State for LOX–H2 System inEquilibrium at Different Pressures

Figure 8 suggests that the critical mixingtemperature increases progressively with anincrease in pressure from 110 K at rP =0.1 to132 K at rP =0.4 and 138.0 K at rP =2.5 which istrue for both the present results as well as those ofYang [22] who employed a detailed, fully transientnumerical model with the complex eight constantBenedict-Webb-Rubin EOS, as compared to thepresent study which is simpler and accurate. It isfurther observed that in subcritical regime ( rP =0.1to rP =0.4), the amount of hydrogen dissolved inliquid oxygen was very small, decreasingprogressively with increasing temperature andreducing to zero at the critical mixing temperature.However, at the supercritical pressure rP =2.5,hydrogen solubility in liquid oxygen increasedsharply with increasing temperature, beingapproximately equal to 0.4 for the present model.

0 0.2 0.4 0.6 0.8 180

90

100

110

120

130

140O2 - H2 System

Figure 8 : Vapour-liquid phase equilibrium compositionsfor O2/H2 system at sub and supercritical pressures

Oxygen mole fraction

Pr = 0.1Pr = 0.4

Pr = 2.5

Tem

pera

ture

(K) Present Model

Model of Yang [22]



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3.5 Oxygen Vapour Mass Fraction Variation at theDroplet Surface with Droplet Temperature

Figure 9 shows the variation of oxygenvapour mass fraction

2

voy with droplet surface

temperature at different pressures. At each pressure,vapour mole fractions of oxygen

2

vox were

determined from the computed results as a functionof temperature. The corresponding oxygen massfraction

2

voy were then calculated using equation 9.

For each pressure, mass fraction increasedwith droplet temperature. Further at a fixed droplettemperature, mass fraction decreased as pressurewas increased. Also, unlike n-heptane-N2 system,the initial mass fraction present at the dropletsurface was 0.8 which decreased progressively withincrease in pressure to a minimum value. For thesupercritical pressure of 126 bar, there was a steeprise in the value of mass fraction with dropletsurface temperature as compared to. rP =0.1 and 0.4cases. It was also noted that the maximum value ofmass fraction was 0.959 for 126 bar whereas forsubcritical pressures, maximum

2

voy value was 1.0,

emphasising the change in vaporisingcharacteristics at very high pressures.

90 110 130 1500

0.4

0.8

1.2

Figure 9: Oxygen vapour mass fraction variation withdroplet temperature at various pressures

Oxygen - Hydrogen System

5.04 bar20.16 bar126 bar

yv O2

Droplet temperature (K)

3.6 Square of Dimensionless Droplet Diameter2

0( / )D D Versus Time

The plot of 20( / )D D (also called square of

dimensionless surface area) with time for a hundredmicrometer diameter oxygen droplet vaporising inhydrogen environment at a fixed ambient

temperature of 1000 K and 100gRe is shown inFig 10. The procedure for getting the variation ofinstantaneous droplet radius lr and subsequently

20( / )D D with time was achieved by calculating

high pressure specific heatmixpgC and thermal

conductivitymixg of the gaseous mixture at T =

1000 K . Lee-Kesler method [33] was used todetermine

mixpgC at high pressure and temperaturetogether with Lee-Kesler mixing rules. Lowpressure gas thermal conductivity was calculatedfirst for pure components using Stiel and Thodosrelation and for low pressure mixtures usingWassiljewa Equation, then using mixing rulessuggested by Yorizane et al.[33], high pressureinverse gaseous mixture thermal conductivity wasdetermined by Roy and Thodos formula andfinally

mixg at high pressure was obtained usingappropriate relation with respect to reduced mixturedensity

mr.

For high pressure liquid mixture densitiy

mixl , Hankinson and Thomson method wasemployed at low pressures, with high pressurecorrection provided by the modified Hankinson-Brobst-Thomson relation. Boiling point bT andlatent heat of vaporisation L were determined usinga particular EOS and equation 11 respectively.Specific heat of liquid mixture at high pressure

lmixpC was evaluated from Rowlinson method. Theheat transfer number TB was calculated usingequation 10 and droplet evaporation lifetime evtfollowed from equation 14. Finally the relationshipbetween instantaneous droplet radius lr with timewas obtained by integrating equation 15. Forsupercritical pressure of 126 bar, TB was calculatedusing equation 12.

For this forced convection case, theexpression for droplet evaporation lifetime wasmodified [41] by incorporating the term (1+0.3 0.5

gRe 0.33gPr ). grP was taken as unity since

quasi-steady gas phase. Equation 15 was used for



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getting the variation of lr with time and

subsequently 20( / )D D with time for three different

pressures corresponding to rP = 0.1, 0.4, 2.5. Thisresult represents a plot of 2

0( / )D D versus time forpractical systems where gRe is of the order of 100[1]. The results suggests that as ambient pressureis increased for a fixed value of ambienttemperature, the evaporation lifetime of the dropletdecreases and is in general agreements withexperimental and modelling results [34,10,22].Further, it was observed that as gRe is decreased,the evaporation lifetime increased in the samemanner with respect to the pressure. Also theevaporation follows the 2 -d law .

0 2 4 6 8 100

0.5

1

(D/D

0)2

t (ms)Figure 10: Square of dimensionless droplet diameter versus

time for Reg = 100

Pr = 0.1Pr = 0.4Pr = 2.5

O2-H2 System

3.7 Droplet Evaporation Rate Versus Time atDifferent Ambient Pressures for a Fixed Value ofReynolds Number and Ambient Temperature

A plot of evm with time for a 100 mdiameter oxygen droplet vaporising in hydrogenatmosphere at rP =0.1, 0.4 and 2.5 and T =1000 Kis depicted in Fig 11. It is observed that theevaporation rate was highest in the beginning andgradually reduced to a minimum value at the end ofthe droplet lifetime for each pressure. The

evm versus time plot were quite similar for rP =0.1and 0.4, but there was a sudden increase inevaporation rate for the supercritical pressurecorresponding to rP = 2.5, as a result the dropletgets consumed in about 5.5 ms. However, as gRe is

reduced, evm reduces thereby increasing the dropletlifetime.3.8 Droplet Diameter Effect on Lifetime andBurning Rate For investigating this effect, the droplet sizewas varied from 20 to 2000 microns. From Figs 12-13, it was observed that for a particular droplet

0 2 4 6 8 100

100

200

mev

(g

/s)

t (ms)Figure 11: Vaporisation rate at different times for Reg = 100

Pr = 0.1Pr = 0.4Pr = 2.5

diameter, an increase in pressure decreased thedroplet lifetime, whereas burning rate wasenhanced. Also, lifetime and burning rate increasedas droplet size was increased at a given pressure.

0 500 1000 1500 20000

1

2

3

4

5( Pr )

t d(s

)

D0 ( microns )Figure 12: Effect of variation of initial droplet diameter D0

on droplet lifetime td at a fixed value of Prfor n - heptane (T = 298 K, Yo, = 0.232 )

Reduced Pressure0.0370.2250.450.6750.9

0 500 1000 1500 20000

500

1000

1500

2000

2500

mf

(g

/s)

D0 ( microns )

( Pr )

Figure 13: Droplet mass burning rate

Reduced Pressure0.0370.2250.450.6750.9



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3.9 Droplet Size Effect on Other BurningCharacteristics

The present code was used for a wide range ofinitial or original droplet diameter 0D ranging from20 to 5000 microns covering three importantburning regimes namely homogeneous 0D 20microns, transition 30 0D 80 microns andheterogeneous 0D > 80 microns. It was observed from Figs (14–15) for a n-heptane droplet undergoing combustion in standardatmosphere that when the variation of flamediameter F (determined from computed results) wasplotted against dimensionless time / dt t , flamediameter first increased, reached a maxima andthen decreased towards the end of the dropletlifetime. This behaviour is consistent with thetransient burning of the droplet verified byexperiments. It was further noted that for 0D > 50microns, there was a gradual increase in value offlame diameters, being about 0.375 cm and 4 cm for500 and 5000 micron droplet respectively. For 0D

50 microns, the increase in flame diameters Fwith 0D was small. It was also observed that themaximum value of flame diameter F occured whenabout 25% of droplet burning was completed( / dt t 0.25 ).

From above results, it can be said that for aparticular fuel, as the droplet size increased, itssurface area increased correspondingly which gaverise to increased evaporation rate of fuel vapourfrom the droplet surface and hence bigger flamediameters.

+ + + + + + + + + + + + + + +0 0.25 0.5 0.75 1

0

2

4

6

+

D0 in microns 205025050020005000

Flam

edi

amet

er(c

m)

Dimensionless time ( t / td )Figure 14: Variation of flame diameter F with dimensionless time

t / td for different initial droplet sizes of n-heptane( P = 1 atm, T = 298 K, Yo , = 0.232 )

0 0.2 0.4 0.6 0.8 10

2

4

6

Flam

edi

amet

er(m

m)

( t / td )

n - Heptane D0 ( m )2050

100500

Figure 15: Initial droplet diameter effect

It could be inferred from Fig 16 that for a n-heptane droplet burning in standard conditions,there was a linear variation of 2

0( / )D D with time.It was further observed that as the droplet size wasincreased from 20 to 500 microns, there was acorresponding increase in droplet lifetime.

*******************

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

(D/D

0)2

Time ( ms )Figure 16: Square of dimensionless droplet diameter versus time

n - Heptane D0 ( m)

* 2050100300500

From Figure 17, it was observed that for a given

fuel, an increase in the droplet size led to an increase in

the flame standoff distance. It was also noted that the



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flame standoff distance first increased, reached a

maxima and then decreased, emphasising that droplet

combustion is an unsteady phenomenon as verified by

experiments.

4 CONCLUSIONSThe present model provides the necessary

information regarding droplet vaporisation at high

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

(F-D

)/2

(mm

)

t / td

n - Heptane D0 ( m)2050100500

Figure 17: Flame standoff distance variation with time

pressures which includes pressure effects onburning constant, burning rate and droplet lifetime(in non convective and convective environments),variation of critical mixing temperature, solubilityof ambient gas in the liquid, variation of vapourmass fraction with droplet temperature at variouspressures and vaporisation behaviour in hotconvective surrounding at different pressures. Thepredictions of the present study were in goodagreement when compared with the experimentaldata and theoretical results of other authors whohave used more complicated models.

Feeling the need to explore the effect ofinitial droplet size on burning, a sphericallysymmetric, transient gas phase droplet combustionmodel was evolved and the effects of initial dropletdiameter on important droplet burning parameterswere quantified. Further, the versatility code wastested by handling a wide range of droplet diametersranging from 20 to 5000 microns (0.02mm-5mm).The gas phase model of the present work can befurther investigated with respect to ambienttemperature, composition and different fuels.



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TABLE 1 Summary of Important Equations of State and their Constants.

Equation u b a

Van der Waals 0 08

u c

c

R TP

2 22764

u c

c

R TP

Redlich- Kwong 1 00.08664 u c

c

R TP

2 2.5

0.5

0.42748 u c

c

R TPT

Soave-Redlich-Kwong 1 0

0.08664 u c

c

R TP

2 2 20.5

2

0.42748 1 1

where, 0.48 1.574 0.176

u cr

c

R T f TP

f

Peng-Robinson 2 -10.07780 u c

c

R TP

2 2 20.5

2

0.42748 1 1

where, 0.37464 1.5423 0.26992

u cr

c

R T f TP

f

TABLE 2Effect of Ambient Pressure on , andb d fk t m for n-Heptane

( 0 2000D m )

Reduced Pressure/r cP P P

AmbientPressure

( )P atm

BurningConstant

2 ( / )bk mm s

Droplet Lifetime ( )dt s

Mass BurningRate

fm ( / )mg s

0.037 1.0 1.036 3.86 1.1130.225 6.075 1.47 2.72 1.1960.450 12.15 1.69 2.37 1.2580.675 18.225 2.15 1.86 1.3560.900 24.3 3.53 1.14 1.691

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