22

CHAPTER 2 LITERATURE REVIEW

2.1 Introduction

The simple droplet combustion model allows prediction of droplet

burning rate, flame radius, droplet surface and flame temperatures and fuel

vapour concentration at the droplet surface.

There are other complicating features of real burning droplets that are

ignored in the simplified theory [15]. In simplified model, all properties are

treated as constants. In reality, most properties possess strong temperature/

pressure and composition dependence. The variable property issue was treated

by some researchers who came up with a reference state for evaluation of

properties with respect to temperature and species mass fraction. This issue is

also important from the point of view of correct formulation of conservation

equations when ambient temperature and/or pressure exceeds the

thermodynamic critical point of the evaporating liquid i.e. supercritical

droplet evaporation/vaporisation which is an important subject for modelling

burning droplets in diesel and rocket engines.

Another important aspect which is ignored is the transient nature of

droplet combustion which leads to two important effects, namely fuel vapour

accumulation (between droplet surface and flame) and droplet heating.

Inclusion of vapour accumulation in the model leads to a variable flame to

droplet radius ratio which is in agreement with experimental observations

unlike the large constant flame to droplet radius ratio predicted by simple

models. Incorporation of this effect also results in capturing the flame

movement.

More sophisticated models of droplet burning include droplet heating

effects that take into account the time varying temperature field within the

liquid droplet. Proper treatment of the liquid phase is also important in the

evaporation and combustion of multicomponent fuel droplets. Another issue is

that of internal circulation within the droplet, occurring due to the shear

between the droplet surface and ambient gas.

23

Other important investigations include effects of ambient pressure

(subcritical and supercritical), ambient temperature, ambient atmosphere

composition, fuel types (single and multicomponent fuels including biodiesels),

droplet sizes and convection effects on evaporation and combustion

characteristics of fuel droplets. In addition, there are effects of radiation heat

transport, chemical kinetics, interaction between neighbouring droplets,

droplet-turbulence interaction.

Further, relatively less amount of literature is available concerning

emission data around a burning fuel droplet. Hence it is important to develop

an algorithm that can quantify effects of pure fuels, biodiesels, different fuel

blends and variables like ambient temperature on droplet emission

characteristics.

2.2 Brief Historical Background

Research on atomisation began as early as the 1920s, droplet behaviour in

the 1930s, interfacial exchange processes were examined in the 1940s, droplet

ignition and combustion phenomena were addressed in the 1950s [1]. From

the earlier part of the previous century to the 1970s, dilute sprays were

perceived as a collection of isolated droplets.

Early works of Godsave [16] and Spalding [17] are of historical

importance in the field of fuel droplet burning. Other noteworthy contributions

are the works of Isoda and Kumagai [18], Strahle [19], Waldman [20], Kotake

and Okazaki [21], Hubbard et al. [10], Law and Sirignano [22] and Law and

Law [23] to name a few.

Extensive review articles are provided by Alan Williams [24], Faeth

[2,25], C.K.Law [5], Sirignano [26], Law and Faeth [8], Givler and Abraham

[12], Chiu [1] and Sazhin [27] .

There are also excellent text books by Kanury [28], Chigier [29],

F.A.Williams [30], Lefebvre [14], Turns [15], Glassman [31], Borman and

Ragland [3], Sirignano [4] and Kuo [13] which present the subject with

various degrees of emphasis and diversification in broad segments of

research topics. We start from a simplified, spherically symmetric, droplet

combustion model.

24

2.3 Quasi-Steady, Unsteady and QS-Transient Evaporation and Combustion of Single Droplets

Pioneering work with regards Single Droplet Combustion was the

Classical approach given by Godsave [16] and Spalding [17] in the early

1950s. It treats an isolated, spherically symmetric, single component liquid

fuel droplet with a definite boiling point (no need for solving the liquid phase

energy equation) combusting in an infinite, quiescent (non convective),

subcritical ambient atmosphere Fig 2.1(a). All droplet processes are diffusion

controlled. The assumption of unity Lewis number permits the use of simple

Schvab-Zeldovich energy equation.

Here the gas phase properties are treated as constant. This model has since

been termed the 2d law− model because it predicts the square of droplet

diameter decreases linearly with time [5]. The 2d law− model is valid for

both pure droplet evaporation (absence of flame) and combustion situations. A

quasi-steady assumption is utilised in Spalding’s analysis which means that at

any instant of time the process can be described as if it were in steady state.

This assumption eliminates the need to deal with partial differential equations

in the gas phase.

The justification of this assumption is based on the relatively slow

regression rate of the fuel droplet as compared to gas phase transport

processes. Using this assumption, one can imagine that the droplet evaporates

so slowly that it can be replaced by a porous sphere of a fixed diameter and the

mass evaporation rate of fuel from its surface into the warm surroundings is

equal to the feeding rate of the liquid fuel from a small feeding tube to the

porous sphere. The relaxation of 2d law− assumption leads to the unsteady

droplet combustion model depicted in Fig 2.1 (b).

In droplet burning, there are several important questions to be considered

for example how long is the droplet lifetime? Which parameters govern the

rate of consumption of a fuel droplet? how does the burning rate of droplet

depend on the heat of reaction and flame temperature? what are the effects of

relative motion between the droplet and surrounding gas? what is the effect of

buoyancy? is there an internal recirculation within the droplet? what is the

effect of ambient pressure on droplet lifetime and other parameters?

25

Fuel

fT T, Products Liquid Droplet O2 Infinitely Thin Reaction Zone

g pg gk c Dρ= Fig 2.1(a) Classical d2-law model

(Godsave and Spalding, early 1950s)

Heat Conducted from the Flame to the Droplet Surface Utilised for Surface Evaporation cm vm

Conducted into Droplet Interior

Droplet Heating Vapour Accumulation

Fig 2.1(b) Relaxation of d2-law assumption

(Law, 1976 and 1980)

26

Three parameters are generally evaluated to answer some of the above

questions, namely, the mass burning rate, the flame front location and the

flame temperature. The most important parameter for engineering design is of

course mass evaporation rate/mass burning rate of the droplet, since it depicts

the rate of heat release in a combustor. It also permits the evaluation of

Evaporation Coefficient evk which is most readily measured experimentally

[13]. Moreover, evaporation rates have become more important in view of the

trend towards “alternative fuels”.

Estimations of evaporation rates and burning times of fuel drops are

simple if appropriate value of evk , the evaporation constant is known. Even if

no experimental data are available, it is still possible to calculate evk for any

given value of drop surface temperature. Under certain conditions, such as the

combustion of fuel drops in high temperature flames, or in high temperature

ambient gas atmosphere, it is often sufficiently accurate to equate sT to the

boiling point temperature of the fuel bT , then the droplet evaporation process

becomes “steady state” in nature. Although, the steady state assumption may

not be true for multicomponent fuels, nevertheless, for most light distillate fuel

oils, it is a convenient assumption [32].

Chin and Lefebvre [32] calculated values of evk for several fuels of

interest for the gas turbine, namely aviation gasoline, JP4, JP5, diesel oil

(DF2) and n-heptane in quiescent atmosphere. The calculated values of evk

were used to examine the influence of ambient gas pressure and temperature

on drop evaporation rates, and it was felt that the results obtained for single

droplets had direct relevance to combustion of fuel sprays. From the results it

was observed that evk increased markedly with an increase in ambient

temperature values at a given ambient the pressure, dependence of evk was

stronger at high pressures than at low pressures. For high ambient

temperatures, pressure dependence was positive, while for low ambient

temperatures it was negative.

27

It should be noted that variation of evk with ambient pressure and temperature for

convective situation can be readily incorporated by using empirical relations which

are functions of gas phase Reynolds and Prandtl numbers [2].

An isolated droplet combustion study under microgravity conditions serves as

an ideal platform in providing a basis for enhancing the existing understanding of

burning process. Microgravity condition is necessary not only for spherico symmetric

droplet combustion in quiescent atmosphere, but also for the resulting one

dimensional solution approach of combustion. Kumagai et al. [7,33,6] successfully

performed the first droplet combustion experiments in microgravity conditions to

validate the 2d law− . They showed that droplet vaporisation rate was constant over

time which is one of the most important feature of 2d law− .

A number of modelling studies under microgravity conditions have been

reported where researchers have derived the classical 2d law− , together with

expressions for burning constant bk , flame to droplet diameter ratio /F D , flame

temperature fT and transfer number TB given as equations (1.1-1.4), respectively of

the preceding chapter. The quasi-steady character of spherico symmetric combustion

has been extensively studied, analytically as well as numerically [34,35,36,30,37,38].

Most of these models were reported taking into account the temperature dependence

of transport properties, kinetic effects and transport mechanisms. Botros et al. [36]

considered the effect of fuel vapour accumulation during the initial transient period,

they retained time derivatives in the gas phase equations and showed that the fuel

vaporised during this transient portion is a significant fraction of the original liquid

fuel. The liquid phase was considered quasi-steady, with droplet at its boiling point

temperature.

Puri and Libbi [37] proposed a numerical model for steady state droplet

combustion with a proper description of the gas phase transport mechanism. Model

predictions for vaporisation rate and flame location showed a good agreement with

experimental data. Fachini [38] presented an analytical, steady state, droplet

combustion model with considerations of temperature dependence of transport

coefficients and non-unity Lewis number. However their results were not in good

agreement with the experimental results.

28

Based on several experimental results [33,6,39-41], it was found that

predictions of 2d law− for /F D ratio are not in accordance with the experimental

observations. Experiments have shown that /F D ratio continues to increase, while

the vaporisation rates follow a steady state behaviour shortly after the ignition period.

However, a better explanation of pure liquid droplet combustion can be given by

considering unsteady effects as well. Theoretical studies regarding the unsteadiness of

the droplet combustion has been described in detail elsewhere [42-45]. Contribution

of Aggarwal et al. [46] and review papers of Law [5], Faeth [25] and Sazhin [27] are

also noteworthy.

Droplet models available in the literature are basically liquid phase models

which can be coupled with either a quasi-steady or transient gas phase depending

upon one’s choice. Some authors have preferred a fully quasi-steady droplet

combustion model for both liquid and gas phases, while some researchers persist with

the fully transient droplet combustion model. There are also studies which retain the

quasi-steady liquid phase with a non steady gas phase in their droplet combustion

model.

King [47] used a numerical analysis procedure to relax the quasi-steady gas

phase assumption while maintaining other assumptions of the QS model for a

spherically symmetric droplet combustion under microgravity. Results indicated that

the gas behaviour was nearly quasi-steady at the droplet surface but deviated

significantly from quasi-steady behaviour at the flame location resulting in predictions

in qualitative agreement with experimental observations (a non constant /F D ratio).

In addition, this modelling approach was also applied to the analysis of combustion of

liquid heptane fuel injected through a porous sphere at a rate equal to the rate at which

it evaporates (no surface accumulation of liquid). These calculations also indicate

rapid approach to quasi-steady behaviour immediately adjacent to the porous sphere

(leading to nearly steady fuel vaporisation rate) but a much slower approach to quasi-

steady behaviour in the flame region.

Ulzama and Specht [48] developed an analytical, spherico symmetric

combustion model of an isolated n-heptane droplet in microgravity, taking into

account both the quasi-steady and transient behaviour of droplet combustion. The

model included an alternative approach in describing the droplet combustion as a

process where the diffusion of fuel vapour residing inside the region between the

29

droplet surface and flame interface experiences quasi-steadines while the diffusion of

oxidiser inside the region between the flame interface and the ambient surroundings

experiences unsteadiness. The modelling approach especially focussed on predicting

the variations of droplet and flame diameters with burning time, the effect of

vaporisation enthalpy on burning behaviour, the average burning rates and the effect

of change in ambient oxidiser concentration on flame structure. On comparison of the

modelling results with experimental data, it was observed that the simplified quasi-

steady transient approach towards droplet combustion yielded behaviour similar to the

classical droplet theory.

2.4 Fuel Droplet in a Convective Stream

In practical applications, droplets in a spray will be moving at some relative

velocity to the surroundings. The Reynolds number gRe based on relative velocity

and gas properties can be of the order of 100. Boundary layer present due to

convection surrounding the droplet enhances heat and mass transport rates over the

values for the spherically symmetric droplet. Further, shear force on the liquid surface

causes an internal circulation that enhances the heating of the liquid. As a result

vaporisation rate increases with increasing Reynolds number.

The general approach adopted in dealing with droplet vaporisation / combustion

in forced convective situations has been to model the drop as a spherically symmetric

flow field and then correct the results with an empirical correlation for convection [2].

The basis for this approach can be seen if we consider heat transfer to a sphere. For

constant fluid properties in the absence of convection, the heat transfer coefficient for

the sphere is /hD λ = 2 (λ is the thermal conductivity). The effect of forced or

natural convection is then treated with an additive correction which is a function of

Grashof or Reynolds number, as well as parameters describing potential differences

for heat, mass and momentum transport such as Prandtl and Schmidt numbers. In a

comprehensive theoretical and experimental study, Frossling [49] showed that the

effect of convection on heat and mass transfer rates could be accommodated by a

correction factor. For the case where heat transfer rates are controlling, the correction

factor is: 0.5 0.331 0.276 g gRe Pr+ , (where gas phase Reynolds number gRe is based on

droplet diameter). Another convective heat transfer correlation is due to Ranz and

Marshall [50], given as: 0.5 0.332 0.6g g gNu Re Pr= + , ( gNu is the gas phase Nusselt

30

number, gRe is based on droplet diameter) which corresponds to a correction factor of

0.5 0.331 0.3 g gRe Pr+ [14].

The droplet evaporation rate evm and droplet evaporation lifetime evt for forced

convection are given respectively as [14]:

0.5 0.332 ln(1 )(1 0.3 )gev T g g

pg

m D B Re PrCλ

π= + + (2.1)

( )( )2

00.5 0.338 ln 1 1 0.3

l

g pg T g gev

DtC B Re Pr

ρλ

=+ +

(2.2)

Where gRe is based on droplet diameter, D and 0D are instantaneous and original

droplet diameters respectively. For the case of convective droplet combustion, droplet

mass burning rate fm and combustion lifetime dt can be determined from equations

(2.1) and (2.2) respectively, using transfer number TB for combustion.

Ranz and Marshall correlation that corrects the spherically symmetric

vaporisation rate ssm⋅

is as follows [4]:

0.33 0.5[1 0.3 (2 ) ]ss g gm m Pr Re⋅ ⋅

= + (2.3)

Here gRe is based upon droplet radius and the correlation is based upon certain quasi-

steady, constant diameter, porous sphere experiments.

Faeth [2] has analysed the available data on convective effects and proposed a

correlation for gNu that approaches the correct limiting values at low and high

Reynolds numbers ( gRe < 1800 ).

0.5 0.33

1.33 0.5

0.5552

[1 1.232 /( )]g g

gg g

Re PrNu

Re Pr= +

+ (2.4)

This correlation yields the following correction factor [14] to account for the

augmentation of evaporation due to forced convection : 0.5 0.33

1.33 0.5

0.2761

[1 1.232 /( )]g g

g g

Re PrRe Pr

++

gRe is based on droplet diameter.

Another expression for determining mass burning rate for steady state combustion

incorporating convective effects is given by [15] as:

31

( ),2 ln 1

g pgg l c bf

pg

Nu r h C T Tm

LCπλ ν ∞

⎡ ⎤⋅ Δ + −= +⎢ ⎥

⎢ ⎥⎣ ⎦ (2.5)

For gNu = 2, the above equation reduces to an expression for a spherically symmetric

case.

The spherically symmetric quasi-steady and transient droplet combustion

models can be directly applied to the situation in which there is no relative motion

between the droplet and the ambient gas ( 0gRe = ) or in which a correction factor

based on the Reynolds number can be applied to account for convective heat transfer

from the gas to the liquid [4]. The focus has been on the models suitable for

implementation in computational fluid dynamics (CFD) codes like KIVA,

PHOENICS, FLUENT, VECTIS, STAR CD. The structure of these codes can vary

substantially. However, basic approaches to droplet and spray modelling used in them

are rather similar [27]. This allows the linking of models with many of these codes.

The Constant droplet temperature model (which yields the 2d law− ) allows the

reduction of the dimension of the system via complete elimination of equation for

droplet temperature. This appears to be particularly attractive for the analytical studies

of droplet evaporation and thermal ignition of fuel vapour/air mixture. More detailed

models have not been used and are not expected to be used in CFD codes in the

foreseeable future due to their complexity. These models can be used for validation of

more basic models of droplet vaporisation or for in depth understanding of the

underlying physical processes [4,51]. The advantage of modelling with regard to

industrial combustion systems such as diesel engine NOX emissions is discussed

elswhere [52].

An important aspect of convective droplet combustion is the influences of

convective flow and initial droplet diameter on isolated droplet burning rate. In a

convective flow at a velocity sV , the burning constant bk of an isolated fuel droplet

was demonstrated to adhere to the correlation of [53-55], given as: 0.5

0 (1 . )b gk k C Re= + , where 0k is the burning constant for the case without

convection, C is a correlation constant, and gRe is the gas phase Reynolds number

estimated by 0 /g g s gRe V dρ μ= ( gρ : gas density, gμ : gas viscosity, 0d : initial droplet

diameter). The value of 0k is determined by burning a droplet of same fuel in a

32

microgravity, quiescent ambience at the ambient temperature equaling to the flowing

gas temperature.

In quiescent ambiences, the burning rate of an isolated fuel droplet varies with

the initial droplet diameter 0d due to the closeness of droplet and the flame and the

subsequent strong action of flame heat in balance with its loss to ambience, on

burning. Suppressing the influence of 0d on burning rate was recognised [56] through

burning the droplet in a forced convective flow that sweeps the flame off the droplet

to weaken the action of flame heat on burning. The 0d dependent 0k , however

affected the correlation 0.50 (1 . )b gk k C Re= + for the burning rate in convective flows.

Different correlation constants C were acquired when using different 0d and its

corresponding 0k to fit the correlation. In hot conditions (633 K ), 0k was bigger for

larger 0d , causing a smaller constant C when taking a larger 0d for the correlation.

Against this, 0k was lower for larger 0d in room temperature ambiences, which

resulted in a mutual compensation of the effects from 0k and 0d on C such that C

was basically independent of the values of 0d and 0k . Besides, it was also observed

[56] that C was larger for gas flow with a higher temperature, revealing an increase

in the effect of gRe on burning with raising the gas temperature.

Droplet vaporisation and combustion experiments have mostly been conducted

with the droplet suspended on a support fibre [50,57-64] to avoid the experimental

difficulties for freely falling droplets, such as in obtaining high resolution droplet

images or in maintaining fixed test conditions. These experiments were under flow

conditions of either forced convection [50,57-58] or natural convection [50,58-62], at

atmospheric [50,57-59,61-64] or elevated pressures [59-63], at normal gravity [50,57-

59] or microgravity [60-61,63]. The fibre orientation was either horizontal [61,63] or

vertical [50,57-60,64]. The fibre material included quartz [59,61-64], glass [50,58,60]

or metals [65].

Yang and Wong [66] investigated the effect of heat conduction through the

support fibre on a droplet evaporating in a weak convective flow. In their experiment,

a droplet of n-heptane or n-hexadecane with an initial diameter of 700 or 1000 mμ

was suspended at the tip of a horizontal or vertical quartz fiber ( diameter 50, 150 or

33

300 mμ ) to evaporate in an upward hot gas flow (at 490 or 750 K ). A simple one

dimensional model of transient conduction was formulated in combination with

evaporation of the droplet. The calculations agreed well with the experiments. In

general, heat conduction through the fiber enhances evaporation, with a stronger

effect for a lower gas temperature and a thicker fiber. However the total heat inputs

were attenuated when the fiber diameter was 300 mμ . Orientation of the fiber was

unimportant. Also, the evaporation rate was enhanced in an oxygen containing gas

flow, due to the additional heating from oxidation around the droplet.

Avedisian and Bae [67] carried out experimental study to account for the

effect of helium/nitrogen concentration and initial droplet diameter on nonane droplet

combustion in an environment (with minimal convection) that promoted spherical

droplet flames. The oxygen concentration was fixed while the inert was varied

between nitrogen and helium. A range of initial droplet diameters 0D were examined

in each ambient gas: 0.4 mm< 0D < 0.8 mm; and an oxidising ambiance consisting of

30% oxygen (fixed) and 70% inert (fixed), with the inert in turn composed of

mixtures of nitrogen and helium concentrations of 0, 25, 50, 75 and 100% N2. The

experiments were carried out at normal atmospheric pressure in a cold ambiance

(room temperature) under low gravity to minimise the influence of convection and

promote spherical droplet flames. For burning within a helium inert (0 % N2), the

droplet flames were entirely blue and there was no influence of initial droplet

diameter on the local burning constant bk . With increasing dilution with nitrogen,

droplet flames showed significant yellow luminosity indicating the presence of soot

and the individual burning histories showed bk reducing with increasing 0D . The

evolution of droplet diameter ( )D t was nonlinear for a given 0D in the presence of

either helium or nitrogen inerts indicating that soot formation has little to do with

nonlinear burning. A correlation was presented in the form of: 2(1 ) ' 2(1 )

0 0( / ) 1 /D D k t Dε ε+ += − , where, 'k was defined as effective burning constant, ε

was a correction factor to the classical 2d law− , and n =3, 3/2 and 1 for burning of

solid fuels, for strong convection and for burning in turbulent flow respectively.

Several investigators have studied droplet combustion, employing methanol

as the fuel. Methanol offers several advantages such as simple chemical composition,

34

non-sooting behaviour, and a well understood oxidation mechanism. However, the

complication in using methanol droplets is the absorption of water into the droplet due

to the high polarity of both water and methanol molecules, which causes them to mix

rapidly. Water initially present in the environment and/or produced through

combustion condenses onto the droplet surface. The water at the droplet surface may

either vaporise or be transported to the droplet interior via mass diffusion, which may

be enhanced by liquid phase circulation. The condensation and transport of water

results in a continuous increase in the water content on the droplet surface and within

the droplet interior. At a particular point in time, the water content increases to a level

where the vaporising methanol cannot sustain the combustion reaction, leading to

flame extinction. Thus water absorption grossly affects the burning characteristics of

methanol droplets.

Methanol droplet combustion studies have either employed unsupported

droplets [68-71] or droplets suspended from thin fibers [72-73]. Both suspended and

unsupported droplet experiments are often conducted in a quiescent microgravity

environment [68-69,71-73] in order to approach spherically symmetric conditions and

to have a better understanding of the fundamental phenomena associated with droplet

burning. However, there are many difficulties in attaining spherically symmetric

conditions. Disturbances from various sources such as droplet drift velocity (of the

order of 1-4 mm/s), deployment of the droplet, effect of the fiber used to suspend the

droplet [66] and ignition of the droplet are present. These disturbances cause a non

uniform distribution of chemical species and temperature around the droplet surface,

resulting in deviation from spherically symmetric conditions.

Main objectives of the above experimental studies [68-73] were the

characterisation of the extinction diameter, evaporation constant and lifetime of

methanol droplets. Many of these experiments were carried out in drop towers [68-

69,71,73] to generate microgravity conditions. The use of drop tower restricts the total

available combustion time. The limited combustion time is often insufficient to

observe extinction when the environment is composed of air [69,71]. To overcome

this difficulty, oxygen mixed with inert gases other than nitrogen has been used in

some drop tower experiments [69] to shorten the combustion time and allow the

observation of droplet extinction. Experimental studies that varied the initial water

content in the ambient environment [70] and in the liquid droplet [71-72] have

35

addressed the influence of these factors in the absorption of water by methanol

droplets. It has been shown experimentally that the absorption of water causes a

methanol droplet to extinguish at a diameter larger than that of a hydrocarbon droplet

of the same initial diameter [68,72,74] and that for methanol droplets, the extinction

diameter varies approximately linearly with the initial droplet diameter [72].

Authors such as Marchese et al. [71], Shaw [75], Zhang et al. [76], and

Marchese and Dryer [77] have conducted analytical and numerical studies of

spherically symmetric methanol droplet combustion. The asymptotic analysis of

Zhang et al. [76] and the numerical results of Marchese and Dryer [77] both show

that considering only diffusion transport within the methanol droplet leads to the

under prediction of the extinction diameter when compared to experimental results.

Investigators have concluded that internal mixing is responsible for additional water

within the droplets which leads to much larger extinction diameters than those

predicted through one dimensional models. A small relative velocity between the

droplet and the surrounding gas phase, however, could be one of the disturbances that

could trigger surface tension effects which can lead to intense internal mixing.

Axisymmetric mumerical models have also been developed and reported in

the literature. Aharon and Shaw [78], in their numerical investigation of bicomponent

droplets using an axisymmetric droplet evaporation model, have concluded that the

thermal Marangoni effect (surface tension gradient due to temperature gradient) has a

stabilising effect and the solutal Marangoni effect (surface tension gradient due to

composition gradient) has a destabilising effect. Recently, Dwyer et al. [79-81]

numerically investigated surface tension effects for both vaporising and combusting

methanol droplets with their axisymmetric model. Their results showed that variations

in surface tension due to composition and temperature can have strong effects on the

flow patterns in the liquid droplet. Shih and Megaridis [82] considered the thermal

Marangoni effects on droplet evaporation in a convective environment. They

concluded that surface tension gradients due to spatial variation of temperature along

the interface had a profound impact on the droplet dynamic behaviour. Gogos and

coworkers [83-84] and Raghavan et al. [85] developed axisymmetric droplet

combustion models and predicted quantities such as mass burning rates, flame shapes,

extinction velocities and extinction diameters.

36

Literature strongly suggests that during microgravity droplet combustion

experiments, sources of disturbances such as droplet drift velocity, droplet

deployment, droplet support fibers and droplet ignition cause deviation from intended

spherical symmetry. As a result concentration and temperature gradients along the

droplet surface have been present in the microgravity experiments. These gradients

enable surface tension forces to come into play. The surface tension causes rapid and

complex circulatory flow patterns within the liquid phase, which enhance the amount

of water absorbed by the droplet.

Raghavan et al. [86] did numerical investigation of methanol droplet

combustion in a zero gravity, low pressure, low temperature environment.

Simulations were carried out using a predictive, transient, axisymmetric model, which

included droplet heating, liquid phase circulation and water absorption. A low

Reynolds mumber ( 0.01gRe = ) was used to impose a weak gas phase convective

flow, introducing a deviation from spherical symmetry. The resulting weak liquid

phase circulation was greatly enhanced due to surface tension effects, which created a

complex, time varying, multicellular flow pattern within the liquid droplet. The

complex flow pattern, which results in nearly perfect mixing caused increased water

absorption within the droplet, leading to larger extinction diameters. It was shown that

for combustion of a 0.43 mm droplet in a nearly quiescent environment ( 0.01gRe = )

composed of dry air, the extinction diameter was 0.11 mm when surface tension

effects were included, and 0.054 mm when surface tension effects were neglected.

Experimental work available in the literature for a 0.43 mm droplet reported

extinction diameters in the range of 0.16 to 0.19 mm. Results for combustion for a

nearly quiescent environment ( 0.01gRe = ) with varying initial droplet diameters

(0.16 to 1.72 mm) showed that including the effect of surface tension resulted in

approximately linear variation of the extinction diameter with the initial droplet

diameter, which was in agreement with theoretical predictions and experimental

measurements.

37

2.5 Multicomponent Droplet Vaporisation and Combustion

A brief introduction to multicomponent droplet vaporisation / combustion was

given in the preceding chapter. In general, chemically reacting flows appear in many

branches of engineering and science including chemical reactor design, combustion

and air pollution. The mathematical modelling of chemically reacting flows has been

an extremely challenging area due to several reasons. First, the mathematical

equations governing simultaneous chemistry, convection, and diffusion are in general

represented by a large set of partial differential equations (PDEs). These PDEs are

difficult to solve even numerically because of the high degree of numerical stiffness

and strong coupling among individual chemical species and temperature. Secondly,

the detailed mechanisms of kinetics and transport properties and

thermochemical/physical property data have not been sufficiently defined. Thus the

model results are susceptible to a large degree of uncertainty.

Previous analyses have depended heavily on numerous assumptions to simplify

physical/chemical processes and to derive simple mathematical formulations for

predicting objectives of interests in a narrow range of conditions. Frequently

employed assumptions have included infinitely fast chemical reaction or simplified

chemistry, unity Lewis number, constant physical/chemical properties, and quasi-

steady state approximations. Mathematical modelling using these assumptions has

provided important qualitative features of phenomena [43].

A major portion of the energy produced in the world today comes from burning

liquid hydrocarbon fuel in the form of droplets. A review of the existing literature

reveals that although evaporation and combustion of single component fuel droplet

has been studied over the years, multicomponent droplet studies constitute only a

relatively small fraction of the available literature. The impetus for continued research

in this field comes from the search for alternative sources of fuel like vegetable oils

and better ways of fuel utilisation in the face of increasing demand and dwindling oil

reserves. Also of major concern are the problems of combustion related pollution and

the use of combustion for disposal of hazardous wastes. In the process, fuels

developed are blends of several components. The use of blended fuels offer several

advantages like desired combustion and emission characteristics. When combustion

products dissolve back into an initially pure fuel, they can change it into a

multicomponent fuel, for example in the case of methanol and ethanol combustion,

38

where water vapour produced during combustion is absorbed by the alcohol droplet.

Another example of combustion products dissolving back into a pure liquid occurs in

supercritical droplet combustion.

Recent concern over the wide specification nature of synthetic and derived

fuels has generated much interest in the vaporisation and combustion of

multicomponent liquid fuel droplets which frequently determine the bulk combustion

characteristics of sprays in many forms of liquid fueled combustors. Compared with

the conventional petroleum fuels, these fuel blends have more complex compositions

and also wider and higher boiling point ranges. It is then obvious that not only the fuel

vaporisation process, but also strongly kinetically dependent gas phase combustion

phenomena such as ignition, extinction and pollution formation will depend

sensitively on composition of the liquid fuel and how its vaporisation is modelled

[23].

Mass diffusion in the liquid phase is very slow as compared with heat

diffusion in the liquid and extremely slow compared with momentum, heat or mass

diffusion in the gas phase or momentum diffusion in the liquid. A multicomponent

droplet therefore exhibits a significantly different vaporisation behaviour compared

with that of a pure fuel droplet. These differences have been attributed to transient

liquid mass transport in the droplet interior, volatility differential between the

constituent fuels, phase equilibrium at the droplet surface, and thermo transport

properties that are functions of mixture compositions, temperature and pressure [87].

Here, different components vaporise at different rates, creating concentration

gradients in the liquid phase and causing liquid phase mass diffusion. The theory

requires coupled solutions of liquid phase species continuity equations,

multicomponent phase-equilibrium relations, and gas phase multicomponent energy

and species continuity equations [4]. One of the earlier multicomponent droplet

combustion model of Law [88] considered pure vaporisation and combustion of a

spherically symmetric multicomponent droplet in a stagnant, unbounded atmosphere.

The droplet temperature and concentration were assumed to be spatially uniform but

temporally varying. This model is referred to in literature as infinite diffusivity model

or batch distillation model. The justification of this model was that in presence of

internal circulation within the droplet, a thorough mixing was achieved. The result of

this model indicated that at first, early in the droplet lifetime, the more volatile

39

substance will vaporise from the droplet surface leaving only the less volatile material

that vaporises more slowly. More volatile material still exists in the droplet interior

and tends to diffuse towards the surface because of the concentration gradients created

by prior vaporisation. This diffusion is balanced by the counter diffusion of the less

volatile fuel component towards the droplet interior and as a result of this process,

different components posses different vaporisation rates which can vary significantly

during the lifetime.

In the absence of any internal circulation, the infinite diffusivity model is

inappropriate. For such situations, Landis and Mills [89] carried out numerical

analyses using finite difference method to solve the coupled heat and mass transfer

problem for vaporising, spherically symmetric, miscible bicomponent droplets. This

model has been referred to in the literature as diffusion limit model. Pentane, hexane

and heptane were respectively mixed with octane. Their results indicated that batch

distillation or infinite diffusivity model are highly inaccurate in predicting the results

for vaporisation rates of the individual components. The agreement between the more

exact diffusion controlled model and the over simplified models appears to be slightly

better at 600 K ambient temperature than at 2300 K , but is still quite poor. Landis and

Mills also showed that disruptive boiling or microexplosions are also possible since

for certain regions of droplet interior, the equilibrium vapour pressure of the more

volatile component can exceed the ambient pressure. This is an important

phenomenon that may be important in multicomponent or emulsified spray

combustion. Landis and Mills further suggested that internal circulation would

decrease the differences between the batch distillation model and the diffusion

controlled model because effectively the diffusivity is increased.

Law, C.K [90] generalised the formulation of Landis and Mills to the case of

multicomponent droplets by developing a spherically symmetric, multicomponent

droplet vaporisation and combustion model. The model assumed equilibrium

vaporisation at the droplet surface, flame sheet combustion, and heat and mass

transport processes to be transient-diffusive in the liquid phase, and quasi-steady,

convective-diffusive in the gas phase. It was felt that since these processes are

governed by heat conduction type equations with a moving boundary, that is the

regressing droplet surface, they are only amenable to numerical solutions. The

numerical effects are further complicated by the stiffness of the coupled governing

40

equations because of the much faster thermal diffusion rate compared with mass

diffusion rate.

Results on the vaporisation of a binary droplet showed that owing to the

significant resistance to diffusion in the liquid phase, the vaporisation process

approximately consists of an initial transient regime, an intermediate diffusion limited

regime which is almost quasi-steady and a final volatility limited regime.

Feeling a need for an analytical solution, Law and Law [23] derived a

simplified approximate analytical solution for quasi-steady, spherically symmetric,

liquid phase mass diffusion controlled vaporisation and combustion of

multicomponent fuel droplets in case where liquid phase species diffusion is slow

compared to droplet surface regression rate. An ideal solution behaviour was

assumed. A unique feature of the diffusion dominated droplet vaporisation

mechanism is the possible attainment of approximately steady state temperature and

concentration profiles within the droplet, which then leads to a steady state

vaporisation rate. Based on this concept, Law and Law [23] formulated a 2d law−

model for multicomponent droplet vaporisation and combustion. It was noted that, the

mass flux fraction or the fractional vaporisation rate mε was propotional to the initial

liquid phase mass fraction of that species prior to vaporisation.

Their solution allows direct evaluation of all combustion properties of interest,

including the liquid phase composition profiles, once the droplet surface temperature

is determined iteratively. Therefore utilisation of the their multicomponent 2d law−

is almost as simple as the classical pure component 2d law− .

They stated that since the droplet concentration profiles closely follow the

temperature variations at the droplet surface, they attain their steady state values

whenever the droplet surface has reached its steady state temperature. In particular

when there is only little droplet heating, which is likely to be the practical situation of

interest because only a sufficient hot droplet is ignitable, the droplet concentrations

will attain their steady state profiles fairly early in the droplet lifetime.

The fuel vaporisation process is crucial in determining the bulk combustion

characteristics of a spray combustor. As a fuel droplet travels through the combustor,

the more volatile fuel vaporises earlier while the less volatile fuel vaporises later. This

implies that certain regions will be relatively rich in the more volatile fuel vapours

41

while other regions are relatively rich in the less volatile fuel. This nonuniform

distribution of fuel is believed to have considerable influence on the characteristics of

ignition, flame stability and pollution formation. Therefore an accurate prediction of

composition of fuel vapour evaporating at the droplet surface is important for spray

combustion design analysis. In order to make study of transient vaporisation in sprays

feasible, simplified analysis is very useful because it reduces the required

computational effort. With the more complex model, computer cost would become

prohibitive in the analysis of spray with many droplets [91].

Tong and Sirignano [91] analysed the problem of transient vaporisation of a

multicomponent droplet in a hot convective environment. The model accounted for

the liquid phase internal circulation and quasi-steady, axisymmetric gas phase

convection. Essentially it was called the simplified vortex model for the liquid phase

(which is basically a diffusion limit model with axisymmetry rather than spherical

symmetry) and a simplified, quasi-steady, axisymmetric convective model for the gas

phase. The objective of the study were (i) to develop an algorithm for multicomponent

droplet vaporisation simple enough to be feasibly incorporated into a complete spray

combustion analysis and yet be accountable for important physics, (ii) comparison of

the developed model with existing models, and (iii) to compare the different models

with the available experimental data. Although many studies have focussed on the

spherically symmetric vaporisation, the practical problem of droplet vaporisation in

spray involves a convective situation, in which there is a relative gas-droplet velocity.

Lerner et al. [92] conducted experiments for measuring overall vaporisation

rates, droplet composition and droplet trajectories for free, isolated, bicomponent

paraffin droplets subject to large relative gas-droplet velocities. The experimental

results for bicomponent fuel droplets of heptane and dodecane vaporising at

atmospheric pressure were used for comparison with other theoretical models of Tong

and Sirignano [91].

Lara-Urbaneja and Sirignano [93] developed a more exact model for the

problem of convective, transient vaporisation of multicomponent droplets. According

to Tong and Sirignano, their simplified vortex model [91] required significantly less

computing time than the more detailed model [93]. The reduction in computing time

makes it feasible to incorporate the model into spray calculation [46].

42

Shaw, B.D [75] investigated spherically symmetric combustion of miscible

droplets for the case where liquid phase species transport was slow relative to droplet

surface regression rates. Attention was focussed on later periods of combustion,

following decay of initial transients, when droplet species profiles change slowly

relative to droplet size changes and 2d law− combustion closely holds. Spherical

combustion of heptane-dodecane droplet was considered at one atmosphere and

300 K . Asymptotic analysis was employed. The gas phase was assumed to remain

quasi-steady. Properties were not calculated as a function of temperature. A

concentration boundary layer where species profile changed sharply in the radial

coordinate was shown to be present at the droplet surface.

Mawid and Aggarwal [94] numerically analysed transient combustion of a

spherically symmetric 50-50 by mass heptane-decane liquid fuel droplet. The

unsteady effects caused by the liquid and gas phase processes were considered.

Temporal variations of the liquid mass fraction of heptane at the droplet

surface and the droplet surface temperature suggested that for lLe = 10, a small initial

period exists during which the mass fraction of heptane decreases rapidly because of

the preferential vaporisation of the more volatile species. This initial period was

followed by an intermediate regime of steady continuous decrease of the heptane

surface mass fraction. During this regime most of the heat arriving at the droplet is

utilised to heat up rather than vaporise the droplet, and hence the vaporisation rate is

low.

This allows more time for the heptane species to be transported from within the

droplet to the droplet surface where it vaporises. However, as the droplet surface

temperature increases, approaching the wet bulb temperature of the heptane

component, the heptane vaporisation accelerates, whereas vaporisation of the less

volatile species “decane” remains fairly slow because the droplet surface temperature

is still low relative to its boiling point.

An important aspect of multicomponent droplet combustion is the combustion

of chlorinated hydrocarbons, dealing with the effects of chlorination and blending.

Direct incineration is a promising technology for the disposal of hazardous wastes

with the potential of complete detoxification. Many hazardous wastes are chlorinated

hydrocarbons (CHCS) which are incineration resistant. This distinguishing property is

caused by two factors. The first is their low heat of combustion resulting from the

43

substitution of the hydrogen atoms by the chlorine atoms in the CHC molecule, and

the further reactions of these chlorine atoms with the remaining hydrogen atoms. The

second factor is the retardation of the rate of the crucial H+O2 reaction in the

hydrocarbon oxidation scheme because of scavenging of hydrogen radicals. It is

therefore reasonable to accept that hazardous waste incinerators could experience

difficulties with flame holding and thereby increased sensitivity to flame outs.

A comprehensive experimental investigation has been conducted [95] to

quantify the combustion characteristics of pure CHCS as well as their mixtures with

regular hydrocarbon fuels, with the specific interest of enhancing the incinerability of

CHCS through judicious blending with hydrocarbon fuels. The general result showed

that relative to normal alkanes, monochlorinated alkanes burn almost equally rapidly.

However, heavily chlorinated alkane such as 1,1,2,2,-tetrachloroethane (TECA)

exhibited the same vaporisation rate in either oxidising or inert environments.

Mixtures of TECA and various alkanes were studied to determine the role of

volatility differentials in the burning TECA. it was seen that addition of a more

volatile component such as heptane results in a slower vaporisation rate while the

opposite holds for the addition of less volatile components such as dodecane and

hexadodecane.

This interesting result can be understood by recognising that when the

hydrocarbon additive is more volatile, a substantial portion of it tends to be

preferentially vaporised in the early part of the droplet lifetime, thereby leaving the

droplet concentrated with the incineration resistant TECA and minimising the

beneficial effect of its addition. However, if the additive is less volatile, TECA is

preferentially vaporised and thereby enhances the droplet vaporisation rate. The study

demonstrated that incineration of a heavily chlorinated hydrocarbon can be promoted

through the addition of a small quantity of a less volatile regular hydrocarbon fuel,

and emphasises the importance of developing rational blending strategies in the

incineration of hazardous wastes.

Another important aspect of multicomponent droplet burning is vaporisation

of alcohols with respect to the water vapour condensation phenomenon. Recent

concern over environmental issues has spurred new interest in alternate fuels

including alcohol based fuels. Because of the relatively large values of their latent

heats of vaporisation as compared to those of the conventional hydrocarbon based

44

fuels, their exists a significant concern over their vaporisation efficiency and

consequently the heterogeneity and uniformity of the fuel/air mixture for combustion.

Some previous studies on the light alcohols, namely methanol and ethanol, have

however suggested that the droplet vaporisation rate can be substantially enhanced

through condensation of water vapour from the environment.

That is because the saturation temperatures of ethanol and methanol are lower

than that of water and because they are also completely water miscible, water vapour

from a humid environment could condense onto and subsequently dissolve into the

relatively cool alcohol droplet. The condensation heat release could be used by

alcohol for its own vaporisation, thereby facilitating its evaporation rate. The concept

is interesting in that it indicates the potential of capitalising on the environment

humidity, in addition to its thermal energy to effect and enhance droplet vaporisation.

This concept has been experimentally substantiated [96] for the slow vaporisation of

suspended alcohol droplets in room temperature humid environments. The existence

of water condensation has also been investigated for rapidly vaporising droplets

which undergo either pure vaporisation or combustion in a hot ambience [70].

For methanol droplets undergoing vaporisation in moist and dry environments,

it was seen that vaporisation in dry environment was well described by the 2d law− ,

however, substantial deviation from the 2d law− behaviour was observed for

vaporisation in the wet environment due to water condensation and dissolution into

the droplet.

Dee and Shaw [97] carried out experimental and theoretical investigations for

propanol/glycerol mixtures. These mixtures have physical properties that are useful

for scientific studies. Propanol and glycerol have widely different boiling points

(370 K and 563 K at 1 atm respectively), which lead to the sudden flame contractions.

Flame contractions are caused by rapid droplet heating, which occurs as the mass

fraction of the low volatility component (glycerol) near the droplet surface approaches

unity. The onset time for flame contraction can be used to estimate effective liquid

species diffusivities (diffusivity enhanced by convection) of mixture droplets.

Experiments on combustion of propanol/glycerol mixture droplets of 1 mm

diameter were performed in atmospheric conditions in reduced gravity. Experiments

showed flame contractions, and data on burning rates and onset times for flame

contraction allowed effective species diffusivities to be estimated. Wei and Shaw [98]

45

conducted experiments on combustion characteristics of Hydroxylammonium Nitrate

(HAN, chemical formula: NH3OHNO3), which is a major constituent in a class of

liquid monopropellants that have many attractive characteristics and display

phenomena that differ significantly from those displayed by other liquid

monopropellants. HAN based propellants have attracted attention as liquid gun

propellants and for spacecraft propulsion applications and are composed primarily of

HAN, water and a fuel species. For example methanol, glycine and

triethanolammonium nitrate (TEAN) have been investigated as fuel species in HAN

based monopropellants. Spacecraft thrusters generally operate in low gravity

environments, providing a pragmatic motivation for studying combustion

characteristics of HAN-methanol-water droplets in reduced gravity. These

combustion characteristics include ignition, extinction, burning rates, aerosol

formation, and droplet bubbling, which are also of interest from scientific standpoint.

Lowering the gravitational level also reduces the influence of buoyancy, which is

desirable so that comparisons with simplified theory that neglect buoyant flows may

be made.

Only a few previous studies of HAN based monopropellant droplet

combustion have been reported in the literature. Most of these studies were perform in

normal gravity [99]. In addition, most previous studies have not investigated HAN-

methanol-water mixtures, but rather mixtures of HAN, water and TEAN. In practical

combustor, that does not have an ambient gas oxidiser (e.g. air), HAN might first

react in the condensed phase, which would release reactive species such as N2O, NO,

NO2, and HNO3. Then, as oxidisers, these species will react with methanol in the gas

phase to release most of the energy of combustion.

Wei and Shaw [98] conducted reduced gravity experiments using 1 mm

diameter droplets burning in air at about 25°C and with pressures from 0.1 to 1 MPa.

Initial droplet compositions varied from zero (initially pure methanol droplets) to a

stoichiometric mixture of 69.4% HAN, 15.2% water, and 15.45 methanol by mass.

Results indicated that increasing the pressure increased burning rates, delayed

extinction and promoted easier ignition of droplets. Decreasing the initial mass

fraction of methanol reduced burning rates, increased the difficulty of ignition and

promoted gas phase flame extinction. Internal bubbling was observed at certain initial

droplet compositions. Aerosol formation was observed for higher HAN loadings at

46

elevated pressures after the visible gas phase flame had extinguished, which may be

indicative of condensed phase HAN reactions.

An important and interesting phenomenon accompanying multicomponent

droplet combustion is microexplosion. Microexplosion (fragmentation of liquid

droplets due to violent internal vaporisation) has potential in improving engine

performance since it can be used to promote the atomisation of heavy fuels by adding

certain amounts of light fuels [100]. Although experimental studies related to

microexplosion have been performed in past years, numerical studies on

microexplosion are rather limited. There were several studies attempting to relate the

occurrence of microexplosion to the superheat degree of the droplet. Modelling of the

break up process associated with microexplosion are very rare, and the determination

of the secondary droplets (such as droplet size and velocity) is adhoc.

The occurrence of microexplosion is caused by the finite speed of mass

diffusion within the droplet. Lighter components inside the droplet cannot emerge to

the surface sufficiently fast to compensate its faster vaporisation rate than the rest of

components, and thus the mass fractions of the light components inside the droplet is

larger than that at the droplet surface. As a consequence, even though the droplet

surface does not pass the boiling point state, the temperature in some region within

the droplet is likely to be higher than the local boiling point. When the temperature is

high enough to support the nucleation, one or two bubbles are generated inside the

droplet [101]. Their subsequent rapid growth results in a violent explosion of the

droplet.

Zeng and Lee [100] presented a numerical model of microexplosion for

multicomponent droplets. The first part of the model addressed the mass and

temperature distribution inside the droplet and the bubble growth within the droplet.

The bubble generation is described by a homogeneous nucleation theory, and the

subsequent bubble growth leads to the final explosion i.e. break up. The second part

of the model determined when and how the break up process proceeded. Unlike

adhoc/empirical approaches reported in the literature, the size and velocity of sibling

droplets (secondary droplets) were determined by a linear instability analysis. After

validated against available experimental data for bubble growth and homogeneous

nucleation, the developed model was first used to study the effects of various

parameters on the onset of microexplosion. It was found that optimum composition

47

and high ambient pressure favour microexplosion, however, extremely high pressures

suppress microexplosion because the volatility differential decreases.The vaporisation

behaviour of an oxygenate diesel blend was analysed at the end. It was found that

microexplosion was possible under typical diesel engine environments for this type of

fuel. Occurrence of microexplosion shortens the droplet lifetime, and this effect is

stronger for droplets with larger sizes or a near 50/50 composition.

2.6 Droplet Evaporation and Combustion in High Pressure Environment

As briefly discussed before, for obtaining higher thermal efficiency, operating

pressures in combustion chambers of liquid fueled internal combustion engines

including gas turbines, diesel engines and rocket motors have been largely increased.

In fact, in recent years, diesel engine manufacturers have striven to increase the

ambient chamber density and thereby pressure during fuel injection for achieving

better mixing and increased rates of combustion. The operating pressure often exceeds

the critical pressure of the liquid fuel. The droplets in the liquid fuel spray ignite and

burn in the gaseous medium at temperatures and pressures above the thermodynamic

critical state of the fuel. Because high pressure tests are expensive and sometimes

dangerous, considerable effort has been devoted to the development of accurate

models capable of portraying the physics of drop evolution at high pressures.

Predictions from such models, for example the validity of the 2d law− at high

pressures, would enable a considerable simplification in the incorporation of drop

models in the complex Computational Fluid Dynamics (CFD) codes [102].

Effects of ambient pressure and temperature on commercial multicomponent

fuels like aviation gasoline, JP5 and diesel oil (DF2) were investigated by Chin and

Lefebvre [32]. Results suggested that evaporation constant evk values were enhanced

as ambient pressure and temperature was increased.

The study of droplet behaviour in high pressure environment presents a

scientifically challenging problem. The actual combustion process is characterised by

the supercritical combustion of relatively dense sprays in highly convective

environment. However, most studies considered decoupled problems in order to

isolate a limited set of issues. Consequently, most results were derived in the case of

an isolated droplet vaporising (no reactions) in a quiescent environment. Convective

effects, influence of neighbouring droplets, detailed chemical kinetics or product

condensation have received less and more recent attention [4].

48

Low pressure droplet models are generally not valid at high pressure conditions.

For example, the gas phase non idealities and the liquid phase solubility of gases are

negligible at low pressures, but become essential considerations at high pressures.

Consequently, a single component liquid fuel droplet would assume a

multicomponent behaviour, and liquid mass transport in the droplet interior would

become an important process. Secondly, as the droplet surface approaches the

transcritical state, the latent heat reduces to zero, and the gas and liquid densities

become equal at the droplet surface. The transient effects in the gas phase would then

become as important as those in the liquid phase, since the characteristic times for

transport processes in the two phases become comparable. In addition, the liquid and

gas phase thermophysical properties become pressure dependent. Also, under

convective conditions, the droplet distortion and breakup become important

processes, as the surface tension is greatly diminished and approaches zero at the

critical point [103].

Extensive reviews of supercritical droplet vaporisation and burning have been

conducted by Givler and Abraham [12], Sirignano [4], Bellan [104] and Kuo [13].

Spalding [105] theoretically considered high pressure combustion by approximating

the droplet vapour as an instantaneous point source of fuel. The combustion process

was represented by a flame surface approximation, that is a diffusion flame with an

infinitely thin reaction zone, constant properties were assumed, and convection was

neglected. This analysis was modified by Rosner [106] to account for the finite

dimensions of the puff of gas. The influences of convection, density variation and

finite rate chemical kinetics on supercritical combustion were studied by Brzustowski

[107]. Manrique and Borman [108] found that effect of thermodynamic non idealities,

property variations and high pressure corrections for phase equilibrium could

influence the vaporisation mechanisms significantly.

Manrique and Borman also considered solubility of the inert gas in liquid

phase in a numerical study of spherically symmetric liquid carbon dioxide droplets

undergoing quasi-steady vaporisation at 500-600 K and 70-120 atm employing

Redlich-Kwong EOS in addition to the effects of non ideal mixtures, variation of

transport properties and non ideality of the energy required for phase change. Their

calculations showed that the droplet vaporisation rate increased with increasing

pressures. However, the heat up of the droplet interior was neglected, the gas phase

49

was treated as quasi-steady and the solubility was examined in the limiting case where

absorption of nitrogen into the liquid phase was assumed to be confined to a thin layer

at the droplet surface.

Lazar and Faeth [109] and Canada and Faeth [110] conducted a series of

experimental and theoretical studies on droplet combustion of hydrocarbon fuels in

both stagnant and forced convective environments, with special attention focussed on

high pressure phenomena of phase equilibrium. The effects of forced convection in

the gas phase were treated by conventional multiplicative corrections.

Rosner and Chang [111] examined the effects of transient processes, natural

convection, and the conditions under which a droplet may be driven to its critical

point. Kadota and Hiroyasu [112] conducted an experimental study of combustion of

suspended fuel droplets of n-heptane, n-decane, n-dodecane, n-hexadecane, iso-octane

and light oil drops at reduced pressures as large as 1.5, and even 2.7 for oil under the

influence of natural convection. For all fuels, the final droplet temperature was nearly

equal to its critical temperature and independent of ambient pressure in supercritical

conditions. The combustion lifetime, defined as the time from the appearance of flame

to its disappearance, displayed an abrupt reduction with rP upto the critical point after

which the reduction was only gradual, whereas the burning constant increased with an

increase in pressure throughout subcritical and supercritical conditions.

Other research groups like Hsieh et al. [113], Curtis and Farell [114], Jia and

Gogos [115], Delplanque and Sirignano [116], Jiang and Chiang [117,118] and Daou

et al. [119] employed numerical techniques to simulate high pressure droplet

vaporisation and combustion with considerable success. Hsieh et al. [113] developed

a comprehensive model on high pressure droplet vaporisation considering real gas

effects as well as ambient gas solubility. Results were presented for an ambient

temperature of 2000 K and it was predicted that droplet vaporisation rate increases

progressively with pressure. Curtis and Farell [114] developed a high pressure model

that predicted droplet vaporisation rate, droplet temperature and the critical mixing

state. They found that anomalies in the transport properties of a fluid near its critical

mixture point are insignificant in droplet vaporisation under conditions similar to

those in a diesel engine.

Delplanque and Sirignano [116] developed an elaborate numerical model to

investigate spherically symmetric, transient vaporisation of a liquid oxygen (LOX)

50

droplet in quiescent, gaseous hydrogen at moderate and high pressures. Computations

were performed for pure vaporisation of a 50 mμ LOX droplet initially at 100 K at a

reduced temperature rT = 9.70, and reduced pressures of rP = 2.0, 3.0 and 4.0. They

suggested that when film theory is used to model droplet combustion, the film

surrounding the droplet can be assumed to be quasi-steady, since the characteristic

time for heat diffusion through the film is typically two orders of magnitude smaller

than the droplet lifetime. They advocated the use of Redlich-Kwong equation of state

used by spray combustion community for its simplicity and accuracy for computing

gas phase equilibrium mole fractions at high pressures. They further suggested that

for a simplified droplet vaporisation model to be used in spray codes at supercritical

conditions, it can be assumed that dissolved hydrogen remains confined in a thin layer

at the droplet surface.

Jiang and Chiang [117,118] devoted several investigations to the study of drop

interactions in a monodisperse cloud. Their single drop model included real gas

effects. The drop was assumed liquid and all properties inside the drop were

calculated accordingly, whereas the surrounding fluid was assumed to be a gas with

equivalent gas transport properties. Thermodynamic equilibrium prevailed at the

droplet surface and solubility effects were included. The model was exercised for

C2H5-N2 system with pressures as large as 6 MPa and temperatures as high as

1250 K .

All of these models, however, adopted certain basic assumptions and empirical

formulas for fluid properties extrapolated from low pressure cases. In order to remedy

the deficiencies of [113-119], a series of fundamental studies [120-127] were

conducted using the state of the art treatment of thermodynamic and transport

phenomena. The effect of non equilibrium phase transition on droplet behaviour was

further addressed by Harstad and Bellan [124-127] using Keizer’s fluctuation theory.

In addition, Umemura and Shimada [128-130] developed approximate analysis to

explain many intriguing characteristics of supercritical droplet vaporisation.

Umemura and Shimada [129] reported a numerical investigation of spherically

symmetric droplet vaporisation under supercritical conditions. They identified the

transition from subcritical to supercritical state in terms of a binary diffusion

coefficient, which was suitably modified so that it became zero as the doplet surface

reaches the critical mixing state.

51

The “supercritical vaporisation” mode refers to the condition when the droplet

surface is at the critical mixing state, i.e. when sT = cmT , throughout the droplet

lifetime. The term supercritical vaporisation is perhaps a misnomer since both the heat

of vaporisation and surface tension become zero at the critical mixing point, and there

is no distinct gas-liquid interface that defines vaporisation as in the subcritical mode.

The droplet or the dense fluid representing the droplet may also undergo considerable

deformation in the supercritical regime. Compared to the subcritical mode, there is

relatively little understanding of the supercritical vaporisation mode because of the

lack of experimental data reported for this mode. Consequently, there is a need for

viable models to describe surface regression rate in the supercritical mode [131]. One

possible approach is suggested by Zhu and Aggarwal [132] and Yang [133], who

describe supercritical vaporisation by the inward motion of the critical mixing surface.

The surface condition of a droplet under supercritical vaporisation is very different

from that of subcritical vaporisation as shown in Fig 2.2.

Zhu and Aggarwal [132] carried out numerical investigation of supercritical

vaporisation phenomena for n-heptane-N2 system by considering transient, spherically

symmetric conservation equations for both gas and liquid phases, pressure dependent

thermophysical properties and detailed treatment of liquid-vapour phase equilibrium

employing different equations of state. Yang [133] also analysed numerically a fully

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

The transcritical vaporisation implies that as the drop surface reaches the

critical mixing state, the interfacial processes change significantly. Both the heat of

vaporisation and surface tension go to zero, and the gas solubility into liquid becomes

significant. The subsequent droplet regression process is qualitatively different from

that in the subcritical state. Prior to attaining the critical mixing state, the droplet

surface or interface is well defined, characterised by sharp gradients in density and

52

other properties. For a pure fuel droplet, the subcritical vaporisation is reasonably well

characterised by the quasi-steady mass or heat transport in the gas phase and the

transient heat transport in the liquid phase. These two processes are incorporated in

the classical quasi-steady vaporisation model, which yields the droplet vaporisation

rate as: .

4 ln(1 )ev s Tp

m r BCλπ= + ; where ( ) /T p S vB C T T h∞= − Δ ;

hereλ and pC denote respectively the average thermal conductivity and specific heat of

the gas layer in the droplet vicinity, sr the droplet radius, vhΔ is the enthalpy of

vaporisation and T∞ and ST are the ambient and droplet surface temperatures

respectively. As the droplet surface reaches the critical mixing state, the classical

model would predict an exceedingly high or infinite vaporisation rate, since vhΔ

approaches zero and pC approaches infinity, making the transfer number go to

infinity. This constitutes one of the major difficulties in modelling the droplet

transcritical vaporisation behaviour [131].

Once the droplet surface reaches the critical mixing state, the thermodynamic

properties and the interface conditions change dramatically. In contrast to the

subcritical vaporisation, the supercritical vaporisation is not characterised by a distinct

interface. However, the interface or droplet surface may be defined by the critical

mixing state. Then the surface regression can be characterised by the heat transport

process whereby the droplet interior is heated and the critical mixing surface moves

continuously inward. Aggarwal et al.[131] carried out numerical investigation to

characterise the transcritical vaporisation of n-hexane fuel droplets in a supercritical

nitrogen environment. Simulations were based on the numerical solution of the time

dependent conservation equations for both liquid and gas phases. One of the

important result was that once the critical mixing state is reached, the subsequent

surface regression or supercritical vaporisation rate is given by the inward velocity of

the critical surface, which is determined by the gas phase thermal diffusivity and the

difference between the critical mixing temperature and liquid temperature in the

droplet interior.

Hongtao Zhang [134] developed a comprehensive numerical model to study

evaporation of a suspended n-heptane droplet in hot, high pressure, convective

53

nitrogen environments. The model included real gas effects, liquid phase internal

circulation, variable thermophysical properties, solubility of inert species into the

liquid phase and gas and liquid phase transients. Numerical predictions for the

suspended droplet within a zero gravity environment were in very good agreement

with the microgravity experimental data. Numerical results showed that at high

ambient pressure the droplet swells initially due to the heat up of the cold droplet, and 2d law− is not followed during the early stages of droplet evaporation. The

numerical results also show that the droplet lifetime decreases with increasing

ambient pressure or ambient temperature. The results further indicate that the

solubility of nitrogen cannot be neglected at higher ambient pressures, however it can

be neglected at low ambient pressures.

Kadota et al.[135] conducted experimental study of evaporation, autoignition

and combustion of octadecanol fuel droplet which solidifies at 331 K in supercritical

gaseous environments under microgravity conditions produced by using a parabolic

flight of aircraft. A fuel droplet suspended at the tip of a fine quartz fibre in the cold

section of the high pressure combustion chamber was subjected to a hot gas in an

electric furnace. A video camera recorded the behaviour of the fuel droplet as well as

the flame around the droplet. The experiments were carried out in quiescent gaseous

environments at low oxygen concentration to reduce soot produced in the flame and

to make it possible to observe the entire droplet burning process. Important results

indicated that droplet burning time (defined as the period of time between the onset of

auto ignition and the end of burning) showed minima at reduced pressure near unity.

The burning constant showed a peak at reduced ambient pressure near 1.5 and the

flame diameter increased almost linearly with the lapse of time at the early stage of

droplet combustion.

In another microgravity combustion study by Vieille et al.[136], high pressure

droplet burning characteristics of five fuels (methanol, ethanol, n-hexane, n-heptane

and n-octane) were investigated under normal and reduced gravity conditions. The

reduced gravity experiments were conducted using the parabolic flights of aircrafts. A

fully automated high pressure droplet vaporisation facility was developed for these

experiments. Rapid videography was used to determine the time histories of burning

droplets from which average droplet burning rates were determined. For all

experiments, suspended droplet technique was used.

54

Initial droplet diameters were about 1.5 mm. Subcritical and supercritical

burning regimes were explored. Droplet time histories were only determined for

weekly sooting fuels such as methanol. An important result was that the 2d law−

holds even under very high pressure and allows the estimation of an average droplet

burning rate. The experimental results for all fuels showed that the droplet burning

lifetime decreases strongly with increasing pressure in the subcritical regime. When

the pressure is increased above the critical pressure of the pure liquid, the droplet

burning lifetime remains constant on the average.

To account for gas phase unsteadiness and its influence on droplet vaporisation

in sub and supercritical environments, two comprehensive models of high pressure

droplet vaporisation, namely a transient model and another assuming gas phase quasi-

steadiness were presented by Zhu et al. [137]. The physical model considered initially

a subcritical state n-heptane droplet introduced into a stagnant N2 environment. A

spherically symmetric vaporisation process was assumed. Both models were first

compared with experimental data and then used to calculate vaporisation processes of

single droplets of different initial sizes for environmental conditions in which the

ambient pressure and temperature ranged from 1-150atm and 500-2000 K ,

respectively.

It was shown that strong gas phase unsteadiness existed during the early period

of the vaporisation process. The unsteadiness attained a maximum value in the gas

near the droplet surface and decreased quickly to a nearly steady value within a short

distance from the surface. With increasing ambient pressure, the unsteadiness

increased nearly linearly at low ambient temperatures and rapidly at high ambient

temperature. Gas phase unsteadiness also increased with increasing ambient

temperature and was effected even more strongly by temperature. Compared to the

transient model, the quasi-steady model predicted a smaller regression rate initially,

and a larger regression rate during the later period. Further, vaporisation process using

the QS model reached the critical mixing state earlier than the transient model.

Yang and Wong [138] undertook the study to solve the discrepancies between

theoretical and experimental results for microgravity droplet evaporation. Since all the

experiments for microgravity droplet evaporation have been conducted in a hot

furnace with the droplet suspended by a fibre.Yang and Wong proposed that

discrepancies result from the fact that current theoretical models ignored the

55

conduction into the droplet through the fiber and the liquid phase absorption of the

radiation from the furnace wall. For verification, they formulated a comprehensive

model which incorporated the effects of fiber conduction and liquid phase radiative

absorption. For droplet size variation and evaporation rate constant, a good agreement

was found between their calculations and experimental data of Nomura et al.[63].

Radiative absorption and fiber conduction were found to enhance the evaporation rate

significantly. At a low temperature of 470 K , the discrepancies were mainly due the

additional fiber conduction, while at a high temperature of 750 K , the liquid phase

radiative absorption became mainly responsible.

Another interesting numerical study was contributed by Zhu and Aggarwal

[103]. The simulation was based on time dependent conservation equations for liquid

and gas phases, pressure dependent variable thermophysical properties and a detailed

treatment of liquid phase equilibrium at the droplet surface. Three different equations

of state (EOS) namely Redlich-Kwong, Peng-Robinson and Soave-Redlich-Kwong

were employed to represent phase equilibrium at the droplet surface. In addition, two

different methods were used to determine liquid density. Results indicated that for the

phase equilibrium of n-heptane-nitrogen system, the RK-EOS predicted higher liquid

phase solubility of nitrogen, higher fuel vapour concentration, lower critical mixing

state temperature, lower enthalpy of vaporisation, higher droplet vaporisation rates

and lower droplet lifetimes compared to those predicted by PR and SRK EOS. A

detailed investigation of the transcritical droplet vaporisation phenomena indicated

that at low to moderate ambient temperatures, droplet lifetime first increases and then

decreases as the ambient pressure is increased. At high ambient temperatures, droplet

lifetime decreases monotonically with pressure.

Aggarwal and Mongia [87] developed a spherically symmetric,

multicomponent, high pressure droplet vaporisation model and suggested that it could

be used as a sub model in the currently employed spray codes for predicting gas

turbine combustor flows. Their study was motivated by the consideration that the drop

submodels that are currently employed in spray codes for gas turbine applications do

not adequately incorporate multicomponent fuels (although it has been recognised

that gas turbine fuels are multicomponent with a wide distillation curve).

In brief, for moderate and high power operation, a suitably selected single

component (50% boiling point) fuel can be used to represent the vaporisation

56

behaviour of a bicomponent fuel, provided one employs the diffusion limit or

effective diffusivity model. Simulation of a bicomponent fuel by a surrogate fuel

becomes increasingly better at higher pressures

Stengele et al.[139] conducted experimental and theoretical study, where the

evaporation of free falling, non interacting, single and bicomponent droplets in a

stagnant high pressure gas was investigated at different temperatures. Due to the

relative velocity between the falling droplet and the stagnant gas, convective effects

were incorporated through experimental correlations. The experimental results were

compared with numerical calculations based on the conduction limit and diffusion

limit model.

The effect of ambient pressure on the evaporation of a droplet and a spray of

n-heptane was investigated by Kim and Sung [140] using a model for evaporation at

high pressure. Their model considered phase equilibrium using the fugacities of liquid

and gas phases for real gas behaviour and its importance on the calculation of the

evaporation of the droplet or spray at high pressures was demonstrated. For the

evaporation of single droplet, droplet lifetime increased with pressure at a low

ambient temperature (453 K ) but decreased at high temperatures. The evaporation of

a spray was enhanced by increasing the ambient pressure and the effect was more

dominant at higher ambient temperatures.

2.7 Motivation for the Present Work

After going through the literature review, it is felt that there is a need for a

simple yet comprehensive gas phase model that can predict the flame behaviour and

important combustion parameters as a function of ambient pressure, temperature and

composition and could be readily extended to include effects of convection, droplet

size and fuels.

There is a lack of information regarding biodiesel fuel properties as well as

combustion data with respect to droplet burning.

A realistic transient multicomponent droplet evaporation/combustion model

should be evolved with consideration for the mixing of air and fuel vapour since that

is going to effect the vaporisation behaviour, also, effect of Lewis number must be

quantified on multicomponent vaporisation. Feasibility of the present multicomponent

model in spray analysis as compared with other existing models can then be

discussed.

57

A high pressure model with considerations of high pressure liquid-vapour

equilibrium, real gas effects, absorption of ambient gas in a thin layer at the droplet

surface and pressure dependent properties can be developed and tested for different

systems. Validity of 2d law− , surface temperature behaviour, solubility of ambient

gas in the liquid and effects of convection can then be quantified.

Emission data for spherically symmetric single and multicomponent fuel

droplets with respect to important species like CO, NO, CO2 and H2O can be obtained

and effect of temperature on concentration of these species quantified with a simple

approach, for the main objective of providing a general trend.

Correct estimation of properties is an essential part of any modelling study,

therefore attention has to be focussed on this aspect and methods for evaluating

properties should be provided separately in detail.

Once the above sub models are developed, they can be represented by

computer programmes that may require less CPU time and can be used for specific

conditions and also in spray analysis.