Modeling of liquid propellant combustion chamber

Chemical Engineering Journal xxx (2012) xxx–xxx

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Chemical Engineering Journal

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Modeling of liquid propellant combustion chamber

Ajinkya V. Pandit a, Ashok Kumar b, G. Srinivasa Rao b, C. Kedarnath b, R. Srihari b, Vivek V. Ranade a,⇑a Chemical Engineering & Process Development Division, National Chemical Laboratory, Pune 411 008, Indiab Thermal Propulsion Division, Naval Science & Technology Laboratory, Vishakhapattanam, India

h i g h l i g h t s

" A simple model is presented for the simulation of liquid propellant spray combustion." The model was implemented and verified by comparing model predictions with the experiments. The model predictions agreed reasonably with the

experiments." The model correctly captures the observed linear relationship between chamber pressure and liquid propellant flow rate." The model is shown to be useful to screen liquid propellant fuels with or without pre-mixed water." The model will be useful in design applications.

a r t i c l e i n f o

Article history:Available online xxxx

Keywords:Liquid propellantSingle droplet decompositionSpray combustion

1385-8947/$ - see front matter � 2012 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.cej.2012.06.033

⇑ Corresponding author.E-mail address: [email protected] (V.V. Ranade)

Please cite this article in press as: A.V. Pandit ej.cej.2012.06.033

a b s t r a c t

Liquid propellants are fuels which do not require external supply of oxygen for combustion. These fuelsare widely used for propulsion applications in oxygen rare environments (underwater, space). We hadpreviously presented a model for solid propellant decomposition in a combustion chamber. In this studywe extend the model for the case of combined solid and liquid propellant combustion. Sub-models relat-ing to solid propellant combustion and outlet flow-rate in the present study are borrowed from the pre-vious study. The combustion chamber is assumed to be a ‘well-mixed reactor’. A model is developed tocharacterize the liquid propellant spray assuming the spray is a collection of individually burning drop-lets. A single droplet decomposition model was developed and used along-with the spray model to solvefor the liquid propellant burning rate. Appropriate numerical methods and computer programs weredeveloped to solve the model equations. The model predictions were compared with the experimentaldata obtained in earlier studies. Appropriate methodology for calibrating the model parameters wasdeveloped and its usefulness was demonstrated by comparing the simulated pressure profiles with theexperimental pressure profiles. The presented model and results will be useful for selecting appropriateliquid propellants for propulsion systems and for the development of a system level model of propulsionsystems.

� 2012 Elsevier B.V. All rights reserved.

1. Introduction

Solid propellants and liquid propellant do not require an exter-nal source of oxygen to release energy. They have an ability to rap-idly release hot gaseous products. This ability is harnessed in anumber of ways in propulsion applications, typically in oxygenrare environments. Solid and liquid propellants are known to beused in chemical rockets, space shuttles as also in underwater mis-siles (torpedoes). Solid propellants are also used in airbag inflatorsused to protect car passengers from accidents. The solid propellantcan be used alone as a fuel or it can be used as an igniter for liquid

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.

t al., Modeling of liquid propell

propellants. Typically in propulsion applications, a solid propellantbrick is placed inside a closed combustion chamber which has anexit port for the gaseous products. The brick is ignited by meansof an electrical or mechanical shock. The solid propellant exother-mically decomposes leading to rapid evolution of extremely hotgases. This process significantly builds up the pressure in the com-bustion chamber (�100 bar). Gases escape through an exit portdue to the built up pressure. These gaseous products are used toprovide thrust (as in the case of rockets) or alternatively to drivea turbine. The solid propellant also can be used as an igniter for li-quid propellants. Once the solid propellant builds up the pressureand temperature of the combustion chamber, the liquid propellantis injected. The role of the solid propellant here is to realize condi-tions suitable for a continued self-sustaining decomposition of theliquid propellant.

ant combustion chamber, Chem. Eng. J. (2012), http://dx.doi.org/10.1016/

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Nomenclature

_mevap rate of vaporization of liquid propellant_mburn rate of decomposition of solid propellant_mo rate of outlet flow of decomposition products_s rate of regression of solid propellant surfaceAb area of burninga burning law coefficientn burning law exponentCp specific heat capacityV volumeT temperatureP pressureh enthalpyq densityDHR heat of decompositionk heat of vaporizationm massYm;s mass fraction of species ‘m’ in the decomposition prod-

ucts of solid propellantYm,f mass fraction of species ‘m’ in the decomposition prod-

ucts of liquid propellant

Ym mass fraction of a species ‘m’mwi molecular weight of species ‘i’L length of solid propellant brickDi inner diameter of solid propellant brickDo (uninhibited) outer diameter of solid propellant brickheff,ch effective heat transfer coefficient for heat transfer from

decomposition products to combustion chamberheff,sp effective heat transfer coefficient for heat transfer from

decomposition products to solid propellant_QHX rate of heat loss to the heat exchanger

Subscriptsg decomposition productssp solid propellantch combustion chamberliq liquid propellantinlet inlet

2 A.V. Pandit et al. / Chemical Engineering Journal xxx (2012) xxx–xxx

Design and development of such propulsion systems often re-quire extensive experimental testing. It is essential to complementsuch an experimental program by mathematical modeling forenhancing its effectiveness. In our previous study [1], we devel-oped a model for solid propellant combustion. The model wasimplemented and then validated by comparing with experimentaldata. In this work, we have extended the solid propellant model forthe case of combined solid and liquid propellant decomposition.The actual processes taking place during the decomposition of so-lid and liquid propellants are quite complex and span a wide rangeof spatio-temporal scales. It is however essential to develop a sim-ple system level model to facilitate interpretation of experimentaldata and provide guidelines for system design. The model for thecombustion chamber is the heart of such a system level model.

The liquid propellant is injected into the combustion chamber viaan atomizer. The atomizer breaks up the liquid jet into a distributionof droplets of various sizes. These droplets may further undergoatomization into smaller droplets (secondary atomization). In thepresence of a turbulent flow field, the droplets may collide with eachother or impinge on the combustion chamber walls. In a combustionchamber, the droplets are exposed to high temperatures and under-go rapid vaporization and combustion. These processes themselvesare inherently complex. After combustion, the hot gaseous productsescape rapidly through an exit port and are used to provide thrust.For developing a model for liquid propellant decomposition, weneed to consider all these complex processes and interactions takingplace with a sufficient level of detail.

Liquid spray has been a topic of enormous interest as it findsapplications over a wide spectrum. The formation of a liquid sprayfrom a jet of liquid is a complex process involving primary and sec-ondary breakup of the liquid jet, interaction with the surroundingflow field and such. Hence, modeling such a complicated phenome-non is challenging. A lot of research has been done to characterizesuch sprays. It is important from an engineering point of view to ob-tain the droplet size distribution of a given system. Many functionslike the Rosin–Rammler, Nukiya–Tanasawa and the log–normalfunctions are used to represent this droplet size distributionmathematically [2]. For such functions, it is sufficient to find certaincharacteristic parameters like the mean droplet diameter or theSauter mean diameter to obtain the complete size distribution ofthe droplets. Correlations are available to estimate the values of

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the characteristic parameters for particular systems [3]. Hence, forsuch systems the droplet size distribution can be obtained. Theoret-ical prediction of droplet size distribution is still difficult because ofthe level of complexity inherent in the phenomenon. Some theories,however, have been developed to attempt this task [4].

Droplet heating and vaporization has also been a subject ofinterest which also presents a lot of complexities. In this field re-search was done mainly along four areas of basic phenomenon,droplet slip and internal circulation, transient heating of droplets,multi-component fuel vaporization and combustion of dropletsprays [5]. Several models of varying complexity have been devel-oped for the case combustion of droplet sprays. Rosner et al. [6]have assumed that the combustion chamber is a well-mixed reac-tor due to the highly turbulent nature of the system, thus removingall spatial complexities. Several models using the ‘well-mixed’assumption have also been previously used [7]. On the other handdetailed computational fluid dynamic models have also been suc-cessfully implemented to model spray combustion [8,9]. In thisstudy, we wish to develop a simple model and hence, have as-sumed that the combustion chamber is ‘well-mixed’.

For the case of single droplet combustion, theories have beendeveloped for diffusion controlled (as for fuel–oxidizer mixture)or vaporization controlled (as in our case) processes [10]. The tran-sient heating of the droplets have also been considered in formu-lating some of these models. Several authors have publishedresearch relating to the combustion of specifically liquid monopro-pellant droplets. The basic mechanism assumed for liquid propel-lant droplet decomposition is that due to the heat received to thedroplet surface through conduction, vaporization of the liquiddroplet takes place. The fuel vapors then stream outwards fromthe droplet until they reach the flame front where they spontane-ously decompose to evolve hot gaseous products [11]. Work hasbeen done in various aspects with regards to improving this model.Faeth [12] analyzed the effect of the motion of the droplet on theburning rate. In calculating the burning rate from the model pre-sented above, many researchers assume a quasi-steady statehypothesis, i.e. the flame front and the steady state temperatureprofile are obtained immediately after the diameter changes. How-ever, this assumption is not strictly valid and a variety of transienteffects are observed while the flame zone develops which influencethe burning rate [13]. Most models proposed including the model

lant combustion chamber, Chem. Eng. J. (2012), http://dx.doi.org/10.1016/



A.V. Pandit et al. / Chemical Engineering Journal xxx (2012) xxx–xxx 3

above assume a Lewis number of unity [11,14]. However, even thisassumption is not reasonable under certain circumstances. In thepresent study, we have developed a model similar to the modelpresented by Jain [11], however, we have made some simplifyingassumptions like a constant flame front temperature which areappropriate for the system considered in this work.

The decomposition of a spray of liquid propellant is an extre-mely complicated process and we will attempt to briefly summa-rize these complexities here. The liquid propellant is firstinjected into the combustion chamber through an atomizer. Theliquid propellant jet is broken by the atomizer into an ensembleof droplets of various sizes. Once they enter the combustion cham-ber, the droplets may be further broken down due to the occur-rence of secondary atomization. These droplets may also impingeon the walls as well as impinge on each other. The droplets enterthe extremely turbulent atmosphere of gaseous products and theystart heating up. The droplets while heating up simultaneously un-dergo vaporization. The vaporized liquid propellant then streamsoutwards from the droplet surface towards the flame front. Forthe case of burning of a spray, a common flame front may beformed for a collection of droplets which in turns affects theamount of heat being conducted inside towards the droplets andthus the vaporization rate. The combustion products then flow intothe swirling reservoir of gaseous products eventually to exit thecombustion chamber through an exit port. The system is thusinherently extremely complex. However, keeping in mind theobjective of developing a system level model in mind we havemade certain approximations without compromising the essentialaspects of the system.

For the purposes of the present study, the complexities of multi-component fuel droplet vaporization are approximated by assum-ing a ‘lumped’ single component model to be representative of theliquid propellant droplet. We have assumed that the spray is anensemble of individually combusting droplets. The spray is repre-sented by a Rosin–Rammler type of droplet size distribution. Cor-relations to obtain the distribution parameters are not knownand hence, will be obtained while calibrating the model. A suitable(to considered system) single droplet model was developed in thisstudy. The transient effects of droplet heating have been consid-ered although it is assumed that no vaporization occurs duringheat up. A quasi-steady state assumption was made to calculatethe burning rate in the single droplet model. The liquid propellantmodel is implemented along-with the solid propellant model. Themodel was calibrated using the experimental values and its useful-ness is demonstrated by comparing model predictions with theexperiments. Our purpose is to develop simple model for the

Fig. 1. Illustration of the combustion chamber.

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combustion chamber which is vital to a system level model of pro-pulsion systems. Such a simple to implement yet detailed systemlevel model would be very useful for providing design guidelinesin propulsion applications.

2. Mathematical model

A typical combustion chamber with an exit port is considered inthis work. The schematic of the considered combustion chamberand typical operation of such a combustion chamber is discussedin our previous paper [1] and is shown in Fig. 1. The solid propel-lant used in this study is in the shape of a hollow cylindrical brick.The brick is placed inside a cylindrical combustion chamber. Solidpropellant burns rapidly and the combustion gases travel throughthe internal combustion port and exit the combustion chamber viathe exit port. The combustion gases are produced at a much fasterrate than the rate of exit from the combustion chamber. These re-sults in a very high pressure build up (�100 bar). At such highpressures, the compressibility effects of the gas become important.Furthermore, the velocity of the exiting gas cannot exceed thevelocity of sound irrespective of the high pressure differential. Thisphenomenon is known as ‘choking flow’ and is expected to occur inour system. The combustion chamber is cooled externally by acooling coil, although the cooling is not significant compared tothe rate of evolution of heat. Liquid propellant is injected to thecombustion chamber via atomizer before all the solid propellantis exhausted to realize sustained burning conditions. At the timeof liquid propellant injection, the conditions within the chamberare conducive to rapid decomposition of the liquid propellant.The atomizer breaks the jet of liquid into a distribution of dropletsof various sizes. Each of the droplets vaporizes and then the fuelvapors decompose to release energy. The decomposition productsescape through an outlet nozzle into the atmosphere. As in the caseof solid propellant decomposition, there is a high pressure build upin the combustion chamber due to which compressibility effects ofthe gaseous products and the phenomenon of ‘choking flow’ be-come important. In our previous study [1] we formulated a modelfor decomposition of solid propellant in the combustion chamber.The model was implemented and validated against experimentaldata. In the present study we have extended the previous modelto simulate combined solid and liquid propellant decomposition.

2.1. Overall balance equations

In the present study, the model equations presented in our pre-vious work [1] are modified to include the liquid propellant spraydecomposition. We have assumed that the combustion chamber isa ‘well-mixed’ reactor. Hence, spatial variation of properties withinthe combustion chamber will not be considered. The combustiongases are assumed to follow the ideal gas law. Heat, mass and spe-cies balances for our system are presented below.

Mass balance:

dðqgðVg � VspÞÞd

¼ _mevap þ _mburn � _mo ð1Þ

Heat balance:

dðqgðVg � VspÞhgÞdt

¼ _mevapðhliq;inlet � DHR;liq � kliqÞ

þ _mburnð�DHR;spÞ � _mohg � _Q sp � _Q ch ð2Þ

Species balance:

dðqgðVg � VspÞYm;gÞdt

¼ _mburnYm;s þ _mevapYm;f � _moYm;g ð3Þ





where the species balance equation is written for each ‘m’. After re-arranging the above equations, we obtain the following set of differ-ential equations:

qgðVg � VspÞdYm;g

dt¼ _mburnðYm;s � Ym;gÞ þ _mevapðYm;f � Ym;gÞ ð4Þ

qgðVg � VspÞCpgðTÞdTg

dt¼ _mevapðhliq;inlet � DHR;liq � kliq � hgÞ

þ _mburnð�DHR;sp � hgÞ � _Q sp � _Qch ð5Þ

By differentiating the ideal gas equation with respect to time,we obtain:

ðVg � VspÞdPg

dt¼ RTg

dng

dtþ ngR

dTg

dt� Pg _mburn

qspð6Þ

where

dnch

dt¼ _mevap

Xm

Ym;f =ðmwÞm þ _mburn

Xm

Ym;s=ðmwÞm

� _mo

Xm

Ym;g=ðmwÞm ð7Þ

which is obtained by a simple mole balance. Here the volume of thegas is taken to be equal to the volume of the combustion chamber.

Separate heat balance equations for the solid propellant brickand the combustion chamber were also formulated as:

For solid propellant,

dðmspCp;spTspÞdt

¼ _Q sp ¼ heff ;spðT � TspÞ ð8Þ

For combustion chamber,

dðmchCp;chTchÞdt

¼ _Q ch � _Q HX ¼ heff ;chðT � TchÞ � _Q HX ð9Þ

where _Qsp is the rate of heat transfer from the combustion productsto the solid propellant brick, _Qch is the rate of heat transfer from thecombustion products to the combustion chamber, and _QHX is the rateof heat transfer from the combustion chamber to the heat exchanger.

The values of Cp,sp and Cp,ch are assumed to be constant. The val-ues of heff,sp, heff,ch can be obtained from heat-transfer correlations.Due to the large coolant flow-rate it was observed experimentallythat the coolant exit temperature does not change significantly.Moreover, the magnitude of the rate of heat exchange is very smallin comparison with the magnitude of the rate of heat generation(�2%). Hence, in the present study we have assumed the exit tem-perature of the coolant to be constant. Hence,

_Q HX ¼ ð _mCPðTexit � TinletÞÞcoolant

2.2. Solid propellant model

The model to calculate solid propellant burning rate is ex-plained briefly, however it is not required to be implemented forthe purposes of the present study. It is sufficient to specify initialconditions which are favorable for sustained liquid propellantburning. It was shown in the previous work [1] that the shapeand size of the solid propellant brick influences the burning profilesignificantly. Hence, the shape of the solid propellant brick must betracked with time. We assumed that the solid propellant brick wasa cylindrical brick with a cylindrical bore (shown in Fig. 1), wherethe outer and side surfaces of the brick are inhibited. From theseconsiderations, equations can be written for the rate of change oflength and inner diameter of the brick as follows:

dLdt¼ �2_s ð10Þ


dDi

dt¼ þ2_s ð11Þ

where _s is the rate of regression of the solid propellant surface andis given by:

_s ¼ aPn ðSt: Robert’sLawÞ ð12Þ

The instantaneous rate of burning of the solid propellant isgiven by:

_mburn ¼ Abqsp _s ð13Þ

where

Ab ¼ pDiLþ pðD2o � D2

i Þ=2 ð14Þ

The volume of the solid propellant is given by:

Vsp ¼ pðD2o � D2

i ÞL=4 ð15Þ

2.3. Outlet flow model

In such high pressure systems, the compressibility effects of thegases become important. A phenomenon known as ‘choking flow’,i.e. where the flow stream of the exiting gas cannot exceed Machnumber of unity is expected in such systems. The procedure givenin Levenspiel [15] was used to calculate the outlet flow-rate for thissystem. The outlet flow is modeled as an ideal adiabatic flow of acompressible gas through a nozzle. However, this ‘ideal’ flow-mod-el needs to be corrected accounting for the chamber pressure andthe exit port configurations. A correction factor (u) is multipliedto the ideal flow-rate to obtain the actual flow rate. Through preli-minary calculations, it was found the system operates under chok-ing flow conditions for most of the period of operation. Underchoking flow conditions, the correction to the ideal model doesnot depend on the chamber pressure [15]. Hence, for this system,we assume that the correction factor depends only on the exit portconfiguration or specifically on the frictional head loss of the flowthrough the exit port. The frictional head loss is given as:

N ¼ 4f F Ld

ð16Þ

Here, N is the effective frictional head loss. fF is the fanning frictionfactor and an equivalent L/d ratio is calculated for the given geom-etry. fF can be calculated from the Moody charts [15]. fF depends onthe Reynolds number of the flow and the effective roughness of theexit port. However, at very high Reynolds numbers, as are obtainedin this system, fF is constant and independent of the Reynolds num-ber and depends only on the effective roughness of the pipe whichis constant for a given geometry. Hence, for this system we can as-sume that fF is constant. For a fixed geometry, the equivalent L/d ra-tio is constant. Hence, we can assume that the frictional head lossfor the system is constant under choking flow conditions. As thecorrection factor under choking flow conditions depends only uponthe frictional head loss [15], the correction factor is constant for thespecific system. The gas flux for the ideal compressible flow throughan adiabatic nozzle is given by:

_Gexit ¼ PgðMaÞ gcðmwÞkg

RTgY ðkgþ1Þ=ðkg�1Þ

" #1=2

ðkg=m2 sÞ ð17Þ

where

Y ¼ Pg

Pambient

� �ðkg�1Þ=kg

Ma ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðY � 1Þðkg � 1Þ

s

However, as Ma = 1, conditions for choking flow are obtained. Y iscalculated by taking Ma = 1. At low flow-rates the exit flow can bemodeled as a compressible or an incompressible flow. A comparison





between the two outlet flow models revealed that at low flow rates(Ma < 0.05), results obtained with compressible flow and incom-pressible flow models are almost the same. An incompressible flowmodel is therefore used at low flow rates (Ma < 0.05) for the sake ofcomputational convenience. The gas flux for the ideal incompress-ible flow through a nozzle is given by:

_Gexit ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2PgMWgðPg � PambientÞ

RTg

sðkg=m2 sÞ ð18Þ

The rate of outlet flow is thus given by:

_mexit ¼ U _GexitAorifice ðkg=sÞ ð19Þ

2.4. Liquid phase modeling

The liquid propellant spray is injected via an atomizer into thecombustion chamber after the solid propellant burning raises thepressure and temperature of the combustion chamber to condi-tions suitable for self-sustained liquid propellant burning. Theatomizer breaks the jet of liquid propellant into a distribution ofdroplets of various sizes which then decompose. We wish to calcu-late the total burning rate of the liquid propellant. For this, we con-sider the spray as a collection of individually decomposing isolateddroplets. Each of the droplets is assumed to first undergo a heatingup period due to heat received from the combustion chambergases, until it reaches the wet bulb temperature. After that, thedroplet starts vaporizing. The rate of decomposition of the fuel va-pors is assumed to be significantly greater than the rate of vapor-ization. Hence, the rate of combustion of the liquid propellantdroplet is equal to the vaporization rate. The total combustion rateis calculated by summing up the individual droplet vaporizationrates.

The spray resulting from the atomizer is modeled here as a col-lection of spherical droplets with a Rosin Rammler type of size (i.e.diameter) distribution. Here, in modeling the liquid phase, we haveassumed that the volume fraction of the liquid phase is negligiblein comparison with the volume of the combustion chamber. Theslip velocity of liquid droplets is assumed to be very small com-pared to the gas phase velocities and therefore the residence timeand internal age distributions of liquid droplets are assumed to besame as that for the gaseous phase in the combustion chamber. Tocalculate the total combustion rate, we first develop a single drop-let model to calculate the rate of vaporization of the droplet withthe ‘age’ of the droplet. The effects of heat-up and vaporizationare considered with the assumption that both the processes occurstrictly individually, the heat up preceding the vaporization. Thedroplet size distribution is broken down into specified number ofdiscrete size classes. The number of droplets in each size class,the age distribution of the droplets in the size class and the totalnumber of droplets are calculated. Then, along-with the equationsdeveloped for the single droplet model, this analysis was used tocalculate the total rate of vaporization.

In reality the neighboring droplets will influence the burning ofindividual droplets. Droplet–droplet interactions may also influ-ence flame characteristics. However, modeling such a complexphenomenon is extremely difficult and computation intensive.Keeping the overall objective of developing a system level modelwithout excessive burden on computational resource in mind, wehave not modeled influence of neighboring droplets (and drop-let–droplet interactions) explicitly in the present work. The devel-oped model was identified using the experimental data and thusthe parameters obtained from this exercise indirectly account forreal situation. The developed modeling framework may be ex-tended to include more sophisticated droplet burning models.


The modeling of all such aspects of liquid propellant combustionis discussed in the following section.

2.4.1. Single droplet modelThe physical picture assumed to calculate the rate of combus-

tion is as follows: fuel vapors stream outwards after vaporizingfrom the surface of a spherical droplet till they reach the ‘flamefront’. At the flame front the fuel vapors are assumed to igniteand undergo complete combustion instantly (see Fig. 1 of [11]).At the droplet surface, we have assumed that the heat requiredto vaporize the fuel is made available purely by the mechanismof conduction. The assumption was made after comparing the heattransfer coefficients for the three modes of heat transfer. The effec-tive heat transfer coefficients for conduction and convection wereestimated using the Ranz–Marshall correlation [16] for heat trans-fer to a spherical droplet. The effective radiative transfer coefficientwas obtained from Stefans law. The coefficients were calculated forconditions pertaining to our system for a single droplet. It wasfound that for the range of droplet diameter values pertaining toour system, the contribution through the conductive mode of heattransfer was much greater than either of the convective or radia-tive modes. Moreover, as the droplet diameter decreases further,during vaporization of the droplet, the heat transfer becomes evenmore conduction dominated. Hence, for the present model formu-lation, the effects of radiation and convection are assumed to benegligible compared to the contribution of conduction. Thisassumption has been made by several other convectional singledroplet burning models [10].

Energy balance equations for the regions ‘‘Droplet Surface toFlame Front’’ and ‘‘Outside Flame front’’ assuming only conductivemode of transfer were formulated. A steady state was assumedwhile calculating the rate of vaporization. Fixed value boundaryconditions are used to solve the equations in both the regions.The temperatures at the droplet surface (Ts), flame front (Tf) andat infinity (T1) are assumed to be known. The temperature varia-tion as a function of the radial distance from droplet center was ob-tained and is given as follows:

TðrÞ ¼ ðTf � T1Þeð�ZT _mF=rÞ þ Tf eð�ZT _mF=rsÞ � Tseð�ZT _mF=rf Þ

eð�ZT _mF=rsÞ � eð�ZT _mF=rf Þð20Þ

for rs < r < rf

TðrÞ ¼ ðTf � T1Þeð�ZT _mF=rÞ þ T1eð�ZT _mF=rf Þ � Tf

eð�ZT _mF=rf Þ � 1ð21Þ

for r > rf

where _mF is the vaporization rate of the liquid propellant.And

ZT ¼Cp;g

4pkgð22Þ

In the present analysis, we have assumed that the specific heat ofthe gas (Cp,g) and the thermal conductivity of the gas (kg) are con-stant. We now form two energy balances, one at the droplet surfaceand second at the flame front. At the droplet surface, the heat re-ceived through conduction is completely utilized to vaporize the li-quid fuel. This can be written mathematically as;

4pr2s ð _qcondÞrs

¼ _mFk ð23Þ

At the flame front, the heat generated due to the combustion ofthe liquid fuel vapors is transferred towards the inner region (to-wards droplet surface) and outside. This can again be written as;

4pr2f ð _qcondÞinwards

rfþ 4pr2

f ð _qcondÞoutwardsrf

¼ _mFð�DHcÞ ð24Þ

where,





ð _QcondÞr ¼ �kgdTdr

� �r

ð25Þ

We now define two dimensionless numbers for the sake ofconvenience:

B1 ¼kþ Cp;gðT1 � TsÞ � ð�DHcÞkþ Cp;gðTf � TsÞ � ð�DHcÞ

ð26Þ

B2 ¼kþ Cp;gðTf � TsÞ

kð27Þ

From the above equations, and after re-arranging the equations, weobtain the following relationship using the numbers B1 and B2:

_mF ¼rs

ZTlnðB1B2Þ ð28Þ

Here, to calculate the value of B1 and B2, we have assumed that thevalues for Tf (flame front temperature) and Ts (droplet surface tem-perature) are known. By mathematical considerations, we knowthat,

B1B2 > 0

Hence, B1 and B2 > 0 or, B1 and B2 < 0.As B2 is always positive, we infer that B1 must be positive. Hence

we obtain the condition that both the numerator and the denom-inator of B1 may either be negative or positive, simultaneously.Hence we obtain that the following conditions must necessarilyhold:

Tf and T1 >ðð�DHcÞ � kÞ

Cp;gþ Ts or; ð29Þ

Tf and T1 <ðð�DHcÞ � kÞ

Cp;gþ Ts ð30Þ

As first condition cannot hold for all values of T1, the second condi-tion will be used for further analysis. Here, we define another vari-able, Tcrit, the upper bound value for Tf and T1. From the secondcondition we can see that,

Tcrit ¼ðð�DHcÞ � kÞ

Cp;gþ Ts: ð31Þ

Here, the value of T1 is known and we make the approximation thatTs � TWB. TWB is the wet bulb temperature of the liquid propellant.

At this point Tf remains as the only unknown parameter. Here tocalculate Tf, we introduce the concept of a flame speed. The flamespeed (typically measured in cm/s or mm/s) is the measured rateof expansion of the flame front in a combustion reaction. The flamespeed is the property of the liquid propellant and can be obtained atdifferent conditions through experiments. Hence, for the case of acombusting droplet, the flame front will stabilize at a radius wherethe rate of vaporization of the liquid propellant equals the total gasflux exiting from the flame front assuming the gases exit at a veloc-ity equal to flame speed. Assuming the value of the flame speed wecan thus write the following mass balance at the flame front:

_mF ¼ 4pr2FqVf ð32Þ

where Vf is the flame speed of the liquid propellant in that specifiedatmosphere. Using this equation in conjunction with the heat bal-ance equations, we can calculate the value of Tf and thus the rateof vaporization. However, such an approach requires the solutionof an implicit equation to obtain the value of Tf. Hence, it is desir-able to simplify this model. Combustion rates and droplet lifetimeswere calculated versus droplet radius within the range of operationof our system using both the constant flame speed assumption aswell as the constant Tf assumption. The droplet lifetime versusthe droplet initial diameter profiles calculated using both of these


assumptions showed a similar dependence of droplet lifetime onthe initial diameter. As the dependence of droplet lifetimes on thedroplet initial diameter is similar using both the assumptions, inthis work we have used specified value of flame front temperature,Tf (which was obtained by calibrating the simulated results with theexperimental data as discussed later).

Now, we move onto model the spray of the liquid propellant tocalculate the total rate of combustion. The combustion rate ob-tained for the single droplet model above will be used in the spraymodel. The spray is modeled as a collection of droplets having asize distribution which can be represented by a Rosin–Rammlerfunction. The slip velocity of liquid droplets is assumed to be verysmall compared to the gas phase velocities and therefore the resi-dence time and internal age distributions of liquid droplets are as-sumed to be same as that for the gaseous phase in the combustionchamber. With these assumptions, we formulate the liquid propel-lant model.

2.4.2. Total number of droplets and number droplets in each size classLet the droplet diameter size distribution function be denoted

by f(x), where x is the diameter. f(x) is a Rosin Rammler distributionfunction and is given by;

f ðxÞ ¼ kk

xk

� �k�1e�ðx=kÞ

k ð33Þ

k is the spread parameter of the distribution and k is the spreadparameter of the distribution.

Through a simple mass balance, sum of the mass of all dropletsis equal to the inlet flow rate of liquid propellant. This can be writ-ten as;

_minlet ¼Z 1

0

pqliq

6x3 _Nf ðxÞdx ¼

pqliq

6_NZ 1

0x3f ðxÞdx ð34Þ

Hence,

_N ¼_minlet

pqliq

6

R10 x3f ðxÞdx

ð35Þ

f(x)dx is the fraction of droplets having diameter between x andx + dx, _N is the number of droplets entering the combustion cham-ber per unit time, and _minlet is the inlet mass flow rate.

The size distribution is broken down into discrete size classes(the number of classes being provided by the user). The minimumand the maximum diameters are chosen suitably. However, evenfor discrete size classes, we must preserve the identity;X

i

f ðdiÞDd ¼ 1 ð36Þ

For this, we can adopt a normalization method, where the drop-let size distribution function is divided by the following value toobtain the required droplet size distribution;

M ¼Z d max

d minf ðxÞdx ð37Þ

We also need to choose a suitable value for the mean diameter for aparticular size class. If the mean function value of class ‘i’ be repre-sented by f(di) and the upper and lower diameter limits of the classbe di+ and di� respectively; then

f ðdiÞ ¼R diþ

di�f ðxÞdx

diþ � di�

The number of droplet entering the chamber of each size classcan then be written as;_Ni ¼ _Nf ðdiÞDd ð38Þ

where _Ni is the number of droplets of size class ‘i’ entering the com-bustion chamber per unit time, di is the representative diameter ofsize class ‘i’, and Dd is the width of each size class.





Now we wish to find the total evaporation rate. It is convenientto calculate the evaporation rate for the separate size classes as thedroplet–droplet interaction is assumed to be absent. The totalevaporation rate will be the sum of the individual contributions.

Let the number of droplets of size class ‘i’ residing at any instantin the combustion chamber be given by Ni. Now, by a balance onthe number of droplets, for a size class ‘i’;

Number of droplets in chamber ¼ Number of droplets entered

� Number of droplets exited

The number of droplets of size class ‘i’ entered until time ’t’ will begiven by _Nit.

Now to calculate the number of droplets those have exited thechamber:

The number of droplets that enter between ‘h’ and ‘h + dh’,where ‘h’ is a time instance such that 0 < h < t, is given by: _Nidh.

The fraction of droplets that exit the chamber between ‘h’ and tis given by:

R th Eðt � hÞdt.

Hence, the number of droplets exiting that had entered between‘h’ and ‘h + dh’ are: _Nidh

R th Eðt � hÞdt.

Hence, the total number of droplets that had exited at time ‘t’ isgiven by:

R t0

_NidhR t

h Eðt � hÞdt.Hence the balance can be re-written as:

Ni ¼ ð _NiÞt �Z t

0

_NidhZ t

hEðt � hÞdt ð39Þ

for a CSTR,

EðtÞ ¼ 1s

e�t=s ð40Þ

where,

s ¼ mean residence time ¼ VolumeGas phase volumetric flow rate

ð41Þ

Hence,

Ni ¼ _Nisð1� eð�t=sÞÞ ð42Þ

2.4.3. The rate of vaporization with the age of dropletHeat supplied to the droplet is utilized for droplet heat up and

droplet vaporization. It is assumed that the droplet reaches a tem-perature TWB (wet bulb) from the initial inlet temperature withoutany significant evaporation. In reality, the droplets may undergosimultaneous vaporization as well as heat up [10]. The magnitudeof heat exchange contributed towards heating up the spray to thewet bulb temperature is negligible compared to the heat ex-changes occurring in the system (combustion, vaporization). Theeffect of heat up of a droplet becomes important while calculatingthe droplet lifetimes. However, for the present study, it was foundthat the quantitative impact of not considering the vaporizationduring the heat up period is very small. Hence, for the sake of sim-plicity, we assumed that heat up and vaporization do not occursimultaneously. This approach is consistent with the overall objec-tive of developing a simple system level model for the consideredsystem. Moreover, sophisticated models addressing the differentcomplexities of the droplet burning can be easily incorporated intothe overall modeling framework presented in this study if such aneed arises.

Heat is given to the droplet by means of conduction and convec-tion. The contribution of radiation to the overall heat transfer wasfound to be negligible in comparison to the conductive and convec-tive contributions. For the current purposes, a ‘lumped’ heat trans-fer model will be used. Hence,

_Q ¼ heff AdðT � TdÞ ð43Þ


_Q is the number of droplets of size class ‘i’ entering the combustionchamber per unit time, heff is the effective heat transfer coefficient, Tis the chamber temperature, Td is the droplet temperature, and A isthe surface area of the droplet.

While evaluating the droplet heat transfer coefficient it shouldbe noted that, in reality, heat transfer to the droplet would be af-fected by the presence of neighboring droplets. The droplet heattransfer may be enhanced (if in presence of combusting droplets)or hampered as compared to heat transfer rates predicted for a sin-gle isolated droplet. The actual phenomena occurring inside thecombustor are quite complex. However, to obtain useful predic-tions, we need to simplify the model appropriately. It was foundthat the duration for the heat up is small in comparison to thevaporization time scales for isolated burning droplets. Hence, thecombustion profile would not be sensitive to variations in heattransfer coefficient within the range of operation of our system(confirmed later by simulations). Therefore the effective heattransfer coefficient (heff) values were estimated from correlationsfor heat transfer to isolated droplets. The effects of conductionand convection are lumped and represented by heff. In the presentstudy, the values for heff were estimated using the Ranz–Marshallcorrelation [16]. The area available for heat transfer and the valueof heff change with droplet diameter. Hence, the rate of heat trans-fer varies with the droplet diameter.

Consider a droplet of size class ‘i’. Let the initial diameter of thedroplet be do,i. During heat-up, no vaporization occurs and thediameter of the droplet does not change. We wish to calculatethe rate of vaporization of the droplet as a function of the age ofthe droplet and for a size class. Let ‘h’ denote the age of the particle.

All the heat is utilized into heating up the droplet. The droplettemperature is considered to be uniform (infinite conductance in-side the droplet). Hence the rate of change in temperature is gi-ven by:

dðmiCpTdÞdh

¼ _Q ð44Þ

mi is the mass of droplet of size class ‘i’ and CP is the specific heatcapacity of the liquid.

The heat up period continues till Td = TWB. Let the age at whichthe droplet of size class ‘i’ attains TWB be given by hWB,i. The vapor-ization process occurs only for age h greater than hWB,i. For thevaporization period the temperature of the droplet remains con-stant at the wet bulb temperature. All the heat transferred to thedroplet goes into vaporizing the droplet. Hence, the rate of vapor-ization is given by,

For h 6 hWB,

dmi

dh¼ 0 ð45Þ

For hcomp > h > hWB

_mF ¼rs

ZTlnðB1B2Þ ð46Þ

from Eq. (28).The liquid phase burning rate is calculated at every time step

while solving the overall balance equations. Hence, for a particulartime, the change in diameter of a droplet is calculated with its agein the combustion chamber assuming that the temperature is con-stant and equal to the chamber temperature at the present itera-tion. It is thus assumed here that the droplet residence time(�0.01 s) is much less than the time scale of temperature changes(�1 s). This assumption is reasonable for the case of decompositionof only liquid propellant. The assumption may not hold in the caseof combined liquid and solid propellant decomposition where theinitial rate of temperature change is extremely high. For the com-bined case the process is extremely complex and rapid making itdifficult to predict. Fortunately the period of such rapid rate of





change of temperature occurs over a very small period comparedto the overall time scales of the process. In the present work there-fore we have used the aforementioned assumption.

hcomp, is the time for complete vaporization. Obviously forh > hcomp, the rate of vaporization is zero as the droplet is com-pletely consumed. Using the above equations the mass of dropletcan be obtained as a function of the age, h. Now the quantities heff

and ‘A’ in the above expressions depend on the diameter of the par-ticle. Assuming the particle is spherical, the droplet diameter canbe calculated from the mass of the droplet, as:

mi ¼pd3q

6ð47Þ

2.4.4. Total vaporization rateThe total vaporization rate is calculated by summing the indi-

vidual contributions by vaporizing droplets of particular size clas-ses. Let the internal age distribution be given by I(h) where I(h)dhdenotes the fraction of droplets having age between h and h + dh.For a CSTR, I(h) is known and is given by,

IðhÞ ¼ 1s

eð�h=sÞ ð48Þ

The internal age distribution, I(h), is only valid for a steady state. Inthe case considered in the present work, the time scale of the pro-cess, (t � 10 s) is much larger than residence time (t� s). Hence,assumption of steady state internal age distribution exists is a rea-sonable assumption.

The rate of vaporization of droplet as a function of age is known.Hence, the rate of vaporization of a size class ‘i’ is given by;

_mevap;i ¼Z 1

0Ni _miðhÞIðhÞdh ð49Þ

Thus,

d _mevap;i

dh¼ Ni _miðhÞIðhÞ ð50Þ

Ni is the number of droplets within the combustion chamber of sizeclass ‘i’

Now; _mevap ¼X

i

_mevap;i ð51Þ

Hence;_mevap

dh¼X

i

Ni _miðhÞIðhÞ ð52Þ

This completes our liquid propellant model formulation. Thedroplet first undergoes a heating up period followed by a vaporiza-tion period. Firstly, the heat up times for each of the size classesneed to be calculated by solving Eq. (44) and determining whenthe droplet temperature reaches the given wet bulb temperaturefor each size class. After this initial ‘heat-up time’, Eq. (52) is com-bined with Eqs. (45) and (46) formulated for each of the size clas-ses, to obtain system of differential equations which can be solvedsimultaneously to calculate the total vaporization rate.

The Eqs. (4)–(6), (8)–(11) form the set of ordinary differentialequations which characterize the considered our system. Theequations were found to be stiff during preliminary simulations.Hence, an implicit method is required to solve this system. TheSUNDIALS suite of packages (https://computation.llnl.gov/casc/sundials/), in particular the CVODE solver, was used in the presentwork. The method selected in this work was variable-order,variable-step multi step methods based on backward differenceformula and Newton type of iterations. Adequate care was takento ensure that selected solver settings and tolerances have no influ-ence on simulated results. The system of ordinary differentialequations presented to calculate the liquid propellant source term


need to be solved at each time step for both heat up and vaporiza-tion. For evaluating the liquid propellant source term, an opensource numerical platform (OpenFOAM) was used where the equa-tions were solved using the Runga–Kutta method. Adequate carewas taken to ensure that selected solver settings and toleranceshave no influence on simulated results.

3. Results and discussion

3.1. Overview of model parameters

In this section, we wish to identify the characteristic parametersof the liquid propellant model as also the minimum number ofparameters that characterize the model. In the previous work [1],we implemented the model for the case of combustion of only solidpropellant. The parameters required then were, the effective heattransfer coefficients from combustion products to the chamberwall and the solid propellant brick (heff,sp and heff,ch), the burningrate parameters (a and n), the outlet flow correction factor (U)and the heat of combustion. We reduced this set of parametersto just two parameters: ‘a’ and ‘U’ by showing their equivalencewith other parameters with regards to influencing the burning pro-file of the solid propellant. The values of ‘a’ and ‘U’ were obtainedby calibrating the model while the other parameters were assignedfixed representative values. For the present study, these parame-ters of solid propellants are used [1].

To integrate the system of ordinary differential equations men-tioned previously, we require values of several additional parame-ters to be known. These parameters are:

� The droplet mean size (d) and the spread (f) parameters for theRosin–Rammler distribution which is used to characterize theinlet liquid droplet size distribution.� The minimum and maximum droplet diameters.� The number of size classes that should be used to discretize the

droplet size distribution.� The wet bulb temperature of the liquid propellant.� The flame front temperature for an individually combusting

droplet.� The effective heat transfer coefficient for heat transfer between

the combustion products and a single droplet for the heat upperiod.� The composition of the combustion product gases.� The composition of the liquid propellant.� The thermal conductivity and the specific heat of the fuel vapors

(here it is assumed that these properties are equal for the com-bustion products gases and fuel vapors).� The heat of vaporization for the liquid propellant.� The heat of combustion of the liquid propellant.� The inlet Temperature of the liquid propellant.� The inlet flow rate of the liquid propellant.� The density of the liquid propellant.� Solid propellant configuration.

Values for several parameters out of these were known for ourparticular experimental configuration and are summarized inTable 1. The values for the other parameters are not known. Forthe sake of simplifying the single droplet combustion model, thethermal conductivity as well as the specific heat properties of thecombustion products was assumed to be constant and also takento be equal to those of the fuel vapors. Representative values suit-able over the range of operation of our system were assigned tothese parameters. The Rosin Rammler parameters (d and f) dependon numerous factors, chiefly on the nozzle characteristics, liquidpropellant transport properties and flow rate. These are typically


https://www.computation.llnl.gov/casc/sundials

https://www.computation.llnl.gov/casc/sundials



Table 1Values of Important parameters used to perform liquid propellant simulations.

Parameter Value Unit

Known liquid propellant parameters (provided byNSTL)

Heat of vaporization 0.31 MJ/kgHeat of decomposition 3.0 MJ/kgDensity 1231.0 kg/m3

Inlet temperature 300.0 K

Assumed liquid propellant parametersWet bulb temperature 400.0 KFlame front temperature 1700.0 KDroplet effective heat transfer coefficient 3000.0 W/

(m2 K)Specific heat of combustion products 2000.0 kJ/

(kg K)Thermal conductivity of combustion products 0.1 W/

(m K)Mean diameter of droplets in spray 1.20E�004 mSpread parameter of droplet size distribution 1.5 –Number of size classes 10 –Minimum diameter 1.00E�005 mMaximum diameter 3.50E�004 m

Solid propellant configurationBrick geometry (Do � Di � L) 67 � 35 � 85 mmBurning law coefficient 1.30E�006 –Burning law exponent 0.55 –Density 1610.0 kg/m3

Heat of decomposition 2.0 MJ/kg

System configurationChamber Volume 7.50E�004 m3

Outlet port exit area 2.83E�005 m2

Correction for outlet flow 0.805 –Effective heat transfer coefficient (combustion

products to chamber wall)1000.0 W/

(m2 K)Effective heat transfer coefficient (combustion

products to solid propellant brick)1000.0 W/

(m2 K)


obtained through experiments. The minimum and the maximumdroplet sizes can be fixed once the Rosin Rammler parametersare known.

The composition of the liquid and solid propellant which wereused to perform the experiments were known. These are however,

Fig. 2. Sensitivity of the mean diamete


not mentioned in this study due to confidentiality constraints. Themolecular weights and the specific heat functions were obtainedfrom this known data. The specific heats of the components as afunction of temperature were taken from the published literature.From the known composition and using an ideal gas assumption,the effective specific heat as a function of temperature was ob-tained. It was found that it could be fitted reasonably well by usinga linear function (though, no such restriction is required by themodel). Such a linear fit was obtained for the known range of com-position of combustion product gases for both the solid and liquidpropellant and was used in the present study.

We will now proceed to identify the characteristic parametersof the liquid propellant model.

3.2. Sensitivity analysis

A sensitivity analysis was done to identify important parame-ters as well as the equivalence of certain parameters. In the exper-iments performed, the liquid propellant flow rate was varied andthe corresponding pressure profile was obtained. For a constantflow-rate, the chamber pressure will remain constant at an equilib-rium value. Hence, the chamber pressure versus flow rate plot is anatural choice to evaluate the effect that changing a particularparameter has on the liquid propellant combustion. The corre-sponding values obtained from different experiments were alsoplotted on the same graph to help maintain a perspective on therange of operation of our system. As the data obtained from theexperiments must be kept confidential, the units of pressure arescaled by an arbitrary factor.

Firstly, a certain set of typical values was chosen and a base casesimulation was performed. The effect of significantly changing thevalue of a single parameter on the Chamber Pressure versus Flowrate plot was studied for each of the unknown parameters. Theparameters considered for this analysis were:

� The droplet mean size (d) parameter for the Rosin Rammler dis-tribution which is used to characterize the inlet liquid dropletsize distribution.� The wet bulb temperature of the liquid propellant.� The flame front temperature.

r on liquid propellant combustion.





� The effective heat transfer coefficient for heat transfer (via con-duction and convection) between the combustion products anda single droplet.� The thermal conductivity and the specific heat of the combus-

tion products.

The spread parameter (f) for the Rosin Rammler distribution isexpected to influence the system similar to the droplet mean size(d) parameter and hence was not considered.

For characterizing the sensitivity of a particular parameter,influence of these parameters on a comparative chamber pressureversus flow rate plot within the range of operation of our systemwas investigated. A sample of results obtained by varying the drop-let mean diameter is shown in Fig. 2. From the figure, we can seethat the chamber pressure depends linearly on the liquid propel-lant flow-rate. The simulation profile can further be approximatedclosely by a line passing through the origin. This characteristic wasobserved for the rest of the parameters as well. The effect that eachof the parameters has on the simulation profile is to merely changethe slope of this chamber pressure versus flow rate line. Hence, theslope of this line is the only characteristic of the liquid propellantmodel. Hence, we can also conclude that all parameters are equiv-alent from the standpoint of influencing the combustion profile asthey all affect only the slope of the line.

3.3. Calibrating the model

The objective of the present study is to develop a simple systemlevel model for the burning of a liquid propellant spray. An appro-priate modeling framework was presented and implemented inthe present study to achieve this end. Such system level simula-tions will provide better insights on key trends as well as will pro-vide useful guidance for conducting experiments. Conductingexperiments for such system is very time consuming and expen-sive. The proposed modeling efforts were directed towardsenhancing effectiveness of the experimental program. For this pur-pose, the accurate simulation of pressure profiles (and correspond-ing thrust profiles) is adequate. The thrust estimation also requires

Fig. 3. Comparison of inlet liquid prop


estimation of temperature. However, in the present study, thetemperature data for the experiments could not be obtained (sev-eral attempts to measure temperature in the combustion chamberfailed due to extreme conditions prevailing there). However, it wasseen from simulation experiments (Fig. 10) that the temperatureprofiles do not change significantly for varying flow rates. There-fore in this work the model was calibrated using the pressure pro-files data.

Based on the discussion in the previous section, since all param-eters are equivalent, we can choose any one parameter as being thecharacteristic of the model and assign suitable representative val-ues to the other parameters for the purpose of describing theexperiments. The specific heat of the gas and the thermal conduc-tivity are physical properties and can be suitably estimated fromthe available literature over the range of operation pertaining toour system. The wet bulb temperature can be closely approxi-mated by the boiling point of the liquid propellant. However, asthe boiling point is not known accurately, we assign a suitable va-lue for the wet bulb temperature. The droplet heat transfer coeffi-cient for the heat up period was estimated from heat transfercorrelations (Ranz–Marshall correlation). The value of the flametemperature was taken such that it is near to but lesser than thecritical flame temperature (Eq. (30)).

The only parameter that remains is the mean droplet size diam-eter. The value of this parameter is difficult to estimate withoutperforming experiments. Hence, we obtained the value for thisparameter by calibrating the model. For calibration, the rest ofthe parameters are assigned the values mentioned in Table 1. Sim-ulations were carried out using the droplet mean diameter valuesas 90, 100, 110, 120, 130 and 140 lm. The R2 value was calculatedby comparing the simulation results with experiments (Fig. 2) foreach value of the mean droplet diameter. Through such an analysisit was found that for a mean droplet diameter of 120 lm, the sim-ulations best resembled the experiments and hence, that diameterwas chosen for further simulations.

The calibrated model was then used to simulate the transientpressure profiles for each of the experiments and furtherinvestigations.

ellant flow rate for Experiment 1.




Fig. 4. Comparison of pressure profile for Experiment 1.

Fig. 5. Comparison of inlet liquid propellant fuel flow rate for Experiment 2.


3.4. Model validation

The calibrated model is now used to simulate the combined li-quid and solid propellant decomposition for different experimentalconfigurations. The configuration of the solid propellant is notimportant for the present study and hence, one of the configura-tions used in our previous work [1] was chosen and used for sim-ulation. In all the experiments, the inlet flow rate is approximatedby a piece-wise linear function. Using the piece-wise linear inletflow rate function, the transient pressure profile was obtainedand compared with the pressure profile obtained from correspond-ing experiments. The other parameter values required to perform


the complete simulation are listed in Table 1. Comparison of thesimulations results with three different experiments is presentedhere.

The liquid propellant inlet flow rate profile (both for experi-mental and for simulations) is shown in Fig. 3. The correspondingpressure profile for Experiment No. 1 is shown in Fig. 4. It can beseen that the simulations under-predict the combustion chamberpressure initially (10–30 s). However, after this period, the simula-tions over predict the chamber pressure values. The flow rate andthe pressure profiles for Experiment No. 2 are shown in Figs. 5 and6. Even for this case, we can see that the experimental values areunder-predicted by the simulations. For the case of Experiment





Fig. 7. Comparison of inlet liquid propellant flow rate for Experiment 3.


No. 3 (Figs. 7 and 8), initially simulations under-predict the exper-imental values while at higher flow-rates, simulations over-predictthem in comparison with the experimental results.

The source for this difference can be explained by consideringthe procedure used for calibration. For calibration, we obtained abest match between the simulated profile and all the experimentalresults. As can be seen in Fig. 2, for a mean diameter of 120 lm, thesimulation results under predict the pressure values for Experi-ment No. 2, under predict at lower flow rates and over predict athigher flow rates the pressure values for Experiment No. 1 andsimilarly for Experiment No. 3. This explains the observationsmade earlier. In the present study, we have assumed that the meandroplet diameter remains constant with changing flow-rate. How-ever, this may not be the case in reality. The mean droplet diameter


influences the slope of the pressure versus flow rate plot as shownin Fig. 2. As seen in this figure the slope for a smaller mean dropletdiameter is higher than that for a larger mean diameter. This is ex-pected as the time for vaporization is proportional to the size of thedroplet. It is thus clear that if we were to assume that the meandroplet diameter increases with increasing flow rate, we may beable to explain the discrepancy between simulated and experimentresults as observed in Figs. 4 and 8. If the mean droplet diameterincreases with liquid flow rate, the simulated pressures obtainedwill be lesser than in comparison to those obtained assuming aconstant mean diameter (and therefore will agree better with theexperimental data). Conversely, at lower flow rates if the meandiameter is lower than for the case of constant mean diameter,the simulated pressures would be greater in comparison to those





Fig. 9. Effect of variation in the heat of decomposition of liquid propellant on chamber pressure.


obtained in the later case. Possible dependence of mean dropletdiameter on liquid flow rate will be investigated further in the sub-sequent work.

Due to the complexity of the system considered, performingaccurate experimentation is difficult. As can be seen in Fig. 2, allother factors remaining constant, the experiments show large devi-ations in pressures even for the same flow rates. Obtaining morereliable experimental data will help in better calibrating the modeland thus improving the predictions of the model. The model is nev-ertheless successful in capturing the linear dependence of chamberpressure and flow rate and other key aspects (which are discussedin the following).


3.5. Model application

The proposed model provides a reasonably adequate agreementwith experimental results. The use of developed model for thecombustor for possible design application is illustrated here witha hypothetical example. In order to enhance the thrust and rangeof propulsion system, fuels of higher and higher energy contentare being developed. The high energy fuels however will have sig-nificant large adiabatic temperature rise and therefore may com-promise mechanical integrity of the combustion chamber. Hence,it is essential to develop a predictive tool to provide guidelinesfor safe and efficient design of the system while using such high





energy fuels. The model discussed earlier was used to carry outsome simulations to gain better understanding of the combustionof such fuels.

To study the effect of using different fuels, simulations werecarried out by varying the heat of decomposition. All other param-eters were assigned values same as the base case parameters.Pressure and temperature profiles were obtained for heat ofdecomposition values of 3, 4 and 5 MJ/kg. However, by changingthe heat of decomposition, the critical temperature defined in Eq.

Fig. 10. Effect of variation in the heat of decomposition of l

Fig. 11. Effect of variation of heat of decomposition


(31) also changes. Thus the flame front temperature for a singledroplet is expected to change. In the present study, we wish to iso-late the effect of heat of decomposition on the chamber pressureand temperature profiles. Hence, as the critical temperaturechanges, the flame front temperature was also appropriatelychanged. In the previous simulations, we have assumed that theflame front temperature is a constant parameter which needs tobe supplied. For the purposes of this study, we set the flame fronttemperature by specifying parameter Uf as follows;

iquid propellant on combustion products’ temperature.

and heat of vaporization on chamber pressure.




Fig. 12. Effect of variation of heat of decomposition and heat of vaporization on combustion products’ temperature.


Uf ¼Tf

Tcrit

The value of this parameter was obtained from the base case simu-lations and was used here.

Thus, three simulations were performed for heats of decompo-sition of 3, 4 and 5 MJ/kg. Comparative plots for pressure versusflow-rate and temperature versus flow-rate profiles obtained areshown in Figs. 9 and 10 respectively. Varying the heat of decompo-sition of the liquid propellant changes the slope of the pressureversus flow-rate line. As the heat of decomposition is increased,the slope of the line increases. The temperature versus flow-rateprofile shows that an increased heat of decomposition causes a risein the temperature of the combustion chamber for any given flow-rate. Both these effects are expected due to the associated increasein the release of energy. The present model can thus be used to ob-tain guidelines for selection of liquid propellant fuels as well as togauge the fuel performance.

As seen from the above simulations, the high energy fuels havea significantly large adiabatic temperature rise and therefore maycompromise mechanical integrity of the combustion chamber. Itis therefore essential to develop ways by which one can increasethe thrust without jeopardizing mechanical integrity of the com-bustion chamber. One of the commonly used strategies is pre-mix water with high energy liquid propellants. Such a water pre-mixed fuel will reduce adiabatic temperature rise and also helpfulin creating larger thrust. To take first steps in modeling such acomplex phenomenon, it is useful to understand the effect of theeffective heat of vaporization of the ‘modified’ liquid propellanton the combustion profile. The present model is a useful tool toprovide guidelines for experimentation along these lines as it pro-vides estimates of what temperature and pressure profiles may beexpected. To understand the effect that addition of water has on thecombustion profile of high energy fuels, two cases were considered.

It was observed that the addition of water has two effects on thecombustion profile. Firstly, the slope of the chamber pressure ver-sus the flow-rate line reduces, that is, the thrust per unit inlet fuelis reduced. However, the quantity of pure liquid propellant re-quired to produce this thrust also reduces. Secondly, the tempera-


ture of the combustion chamber is brought down significantly. Todemonstrate these effects, we carried out certain simulation exper-iments as follows. Simulations were done for a base liquid propel-lant (A) (DH = 3 MJ/kg and k = 0.3 MJ/kg) and the correspondingpressure and temperature profiles were obtained versus flow-rate(Figs. 11 and 12 respectively). Now we wish to compare the com-bustion profiles of water premixed liquid propellants with the basecase propellant. Consider another propellant (B) (DH = 9 MJ/kg andk = 0.3 MJ/kg). Let us assume that the effective fuel after premixingwith water contains 45% water. Hence the effective heat of vapor-ization is 1.2 MJ/kg and the effective heat of decomposition is 5 MJ/kg (assuming kwater = 2.25 MJ/kg). As we can see the pressure pro-file of this fuel is similar to the fuel before. Hence, the performanceof both these fuels would be the same. Also note that the temper-ature of this fuel is still higher than that of the base profile. This isexpected as the heat of decomposition is significantly higher thanfor the base case. However, were it not premixed with water, thetemperature profiles would have been significantly higher andthe operation of the chamber would not have been feasible.Consider another propellant (C) (DH = 7 MJ/kg and k = 0.3 MJ/kg).This fuel is also premixed such that the water content is 45%.The effective heat of vaporization for the fuel then becomes1.2 MJ/kg and the effective heat of decomposition becomes3.9 MJ/kg. As can be seen this fuel would deliver a poorer thrustthan the base fuel. However, the quantity of fuel required to deliversuch a thrust is significantly lesser. The temperature attained forthis case is nearly similar to that obtained for the base fuel makingthe operation of combustion chamber feasible.

Hence, we can see that premixing the fuel with water has signif-icant potential in improving the fuel efficiency while at the sametime bringing down the temperature of the combustion chamber.However, the limitation of this model for the case of waterpremixed liquid propellant decomposition is its inability to providecriteria for the extinction or quenching of the flame. Further workwould be to evolve such criteria and incorporate it into the modelto predict the critical mass fractions of water which can be addedto the fuel without quenching the reaction. Such experimentationis essential for improving the design efficiency of the system. The





model can thus be used effectively as a means to obtain guidelinesfor planning different experiments needed to test and to enhanceperformance of such propellant systems.

4. Conclusions

In this study, the model of solid propellant combustion (pre-sented in previous work [1]) was extended for the case of liquidpropellant combustion. The model was implemented as a numeri-cal code and calibrated using experimental results. Some of theconclusions of the analysis and work presented here are:

� The heat transfer during the droplet heat up and vaporization isdominated by conduction.� Differences in the predicted vaporization rate from the single

droplet using the constant flame temperature or constant flamespeed assumptions are not significant.� The pressure generated in combustion chamber is linearly pro-

portional to the flow rate of the liquid propellant.� Key system parameters like heat of decomposition, heat of

vaporization, droplet diameter and so on essentially changethe slope of generated pressure versus propellant flow rate line.This slope can be considered as a unique characteristic of thesystem.� It is sufficient to choose only one parameter for calibration

using the experimentally observed slope while assigning theother parameters suitable values.� The mean droplet diameter serves as a useful calibration

parameter to describe the experiments. It may be necessary torelate mean droplet diameter with flow rate of liquid propellantto improve the agreement between the simulated and experi-mental results.


The developed computational model can be used to draw usefulguidelines for design.

References

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[10] S. Turns, An Introduction to Combustion, second ed., McGraw-Hill Europe,2000.

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Modeling of liquid propellant combustion chamber

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