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The effect of internal diffusion on an evaporating bio-oildroplet e The chemistry free case

James Brett a,*, Andrew Ooi a, Julio Soria b

aDepartment of Mechanical Engineering, University of Melbourne, Australiab Laboratory for Turbulence Research in Aerospace and Combustion, Department of Mechanical and Aerospace Engineering,

Monash University, Australia

a r t i c l e i n f o

Article history:

Received 13 March 2009

Received in revised form

2 March 2010

Accepted 9 March 2010

Available online 1 April 2010

Keywords:

Bio-oil

Evaporation

Simulation

Fast pyrolysis oil

Simulated mixtures

Numerical analysis

* Corresponding author. Tel.: þ61 383440024,E-mail addresses: brettjd@unimelb.edu.a

0961-9534/$ e see front matter ª 2010 Elsevdoi:10.1016/j.biombioe.2010.03.006

a b s t r a c t

The chemical diversity of the components in bio-oil has a significant effect on its evapo-

ration. The low boiling point compounds, such as simple acids and water, evaporate away

from the surface at a faster rate than the internal diffusion. As a result a significant

proportion still remains in the droplet core late in the evaporation process. As the droplet

temperature increases, these trapped chemicals, in the centre of the droplet, can reach

a sufficiently high temperature to cause them to vapourise resulting in rapid expansions,

which can fracture the droplet. In this study a numerical model of a stationary, spherically

symmetric evaporating bio-oil droplet, in a hot ambient atmosphere, is used to investigate

the effect of the rate of diffusion on the evaporation process.

ª 2010 Elsevier Ltd. All rights reserved.

1. Introduction be difficult to obtain. For this reason the model used in this

An important stage of the combustion of liquid fuels is the

evaporation of fuel droplets. This evaporation can occur either

prior to or during combustion. An excellent overview of the

evaporation of droplets can be found in the book by [1].

Evaporation models can be divided into two categories,

hydrodynamic and kinetic. Hydrodynamic models assume

that the fuel vapour is saturated at the droplet surface, while

in kinetic models a fraction of vapour molecules striking the

liquid surface are reabsorbed into the liquid. A comparison of

the hydrodynamic and kinetic models for diesel fuel was

conducted by [2], showing that the models produce slightly

different results. However the kinetic model requires an

accurately determined evaporation parameter, bm, which can

þ61 409537064 (mobile);u (J. Brett), a.ooi@unimelbier Ltd. All rights reserve

paper follows the hydrodynamic approach. The first model of

an evaporating droplet, the d2 law, is a hydrostatic model, and

can be expressed in the form published by [3],

�dd0

�2

¼ 1� bt; (1)

where d is the droplet diameter at time t, d0 is the diameter at

t¼0andb isaparameterdependentuponthe initialdroplet size,

temperature, liquidandgasproperties. The d2 law isbasedupon

the assumption of a quasi-steady evaporating droplet, where

the droplet temperature is in equilibrium with its surrounding

and is constant with respect to both space and time.

The d2 law is often used in simple evaporation models and

it is not appropriate for systems where transient effects are

fax: þ61 397478784..edu.au (A. Ooi), julio.soria@eng.monash.edu.au (J. Soria).d.

b i om a s s a n d b i o e n e r g y 3 4 ( 2 0 1 0 ) 1 1 3 4e1 1 4 0 1135

significant, especially those where the droplet contains

a diverse mix of components. In these cases a more detailed

model which accounts for the change in temperature of the

droplet with time is required.

When a single species droplet is evaporating, the evapo-

ration rate is determined by the temperature and the rate at

which the fuel vapour diffuses away from the surface. For

fuels with more than one fuel species the rate can also be

affected by the rate at which fuel diffuses from inside the

droplet, to the surface, where it evaporates. If any species

evaporates at a greater rate than the molecules are replen-

ished by diffusion and/or mixing, significant internal

concentration gradients occur.

Bio-oil, created by the pyrolysis of bio-mass, typically

consists of a wide range of chemical components, from water

and light weight acid components, which boil at relatively low

temperatures, through to heavy lignin molecules with much

higher boiling points. The vast differences in boiling point

results in the low boiling point components evaporating away

from the surface relatively quickly, and can lead to a build up

of light weight components like water in the centre of the

droplet. As the droplet heats up the water in the core can

vapourise, leading to explosive boiling which can fragment

the droplet. This complexity makes the understanding of

evaporating bio-oil droplets important for its use in engines.

Hallett and Clark [4] developed a numerical model for

a stationary spherically symmetric bio-oil droplet, in which

they represented the bio-oil composition with four contin-

uous distributions, representing the acid, aldehydeyde/

ketone, water and lignin fractions. They treated the liquid

phase as well mixed, so that its composition and temperature

was uniform, and included the breakdown of lignin to char

and gas. This paper uses a discrete representation of a bio-oil,

with the components and fractions chosen to replicate the oil

modelled by Hallet and Clark. A discrete representation was

modelled to enable us to later expand the model with internal

chemistry. In an actual bio-oil droplet, such chemistry will

include polymerization of the heavier compounds, which can

result in spatial variations of fuel properties, and the break-

down of components to form gasses, and char in combustion

cases. Unlike Hallet and Clark’s model, liquid phase diffusion

was modelled, to better replicate the dynamics of the evapo-

rating droplet. The oil represented by Hallet and Clark’s

distribution is not necessarily a good representation of many

bio-oils, it differs significantly in the compounds present from

those found inmany other bio-oil studies (for example [8e10]),

but it was adopted in this study to allow for comparisons.

2. Numerical model

The model developed for this research considers a spherically

symmetric droplet of radius R evaporating in a hot gas envi-

ronment. Three regions are considered, the interior of the

droplet, the surrounding gas phase, and the liquidegas

interface.

Inside the droplet (r < R), the temperature profile is gov-

erned by thermal conduction. For a sphere with internal

conduction, under the assumption of spherical symmetry, the

heat balance equation can be expressed as,

rlbCl

vTl

vt¼ 1

r2v

vr

�r2ll

vTl

vr

�; (2)

where rl is the density, bCl is the constant pressure specific

heat, ll is the thermal conductivity and Tl is the temperature of

the liquid.

The temperature at the outer surface, Ts, of the droplet

must match the vapour temperature, and to ensure spherical

symmetry and continuity, the center must have a zero

temperature gradient. These boundary conditions can be

written as,

Tl;s ¼ Ts ¼ Tg;s (3)

at r ¼ R and

vTl

vr¼ 0; (4)

at r ¼ 0, where the subscript l denotes liquid phase and g

denotes the gas/vapour phase.

In the region surrounding the droplet (r > R) the tempera-

ture profile is governed by both thermal conduction and

convection. Due to the assumption of spherical symmetry the

only velocity is the radial velocity caused by the expansion of

the fluid as it evaporates.

Applying the continuity equation to the gas phase leads to

r2vravt

þ vrgur2

vr¼ 0; (5)

where ra is the density of the gas phase, and u is the radial

velocity. The boundary conditions for vapour temperature are

a constant far field temperature at the outer limit of the

computational domain and Eq. (3) at the droplet surface.

With the assumption of constant pressure, the vapour

velocity at the surface can be calculated as

u ¼ rl_Rras

rgra; (6)

where ras is the density of the gas phase at the droplet surface

and rg is the density of the fuel vapour at the surface

conditions.

The temperature at the surface of the droplet is evalu-

ated by energy conservation. The surface of the gas and

liquid phases is required to be at equal temperatures,

denoted Ts. The energy conservation equation balances the

heat flow from the gas with the heat flowing into the

droplet interior and the energy used to vaporise the liquid

and is given by

lgR2dTg

dr

����s¼ llR

2dTl

dr

����sþ _RLrl: (7)

this can be discretised spatially using a finite difference

scheme and rearranged to solve for Ts.

The rate of evaporation, expressed as a change in radius

can be calculated from mass balance to be

_R ¼ rg

rl�1� Yfs

��vYf

vr

�s

(8)

where Yf is the molar concentration of the fuel vapour in the

surrounding atmosphere. This concentration is calculated

from diffusion using the following equation

Fig. 1 e Fuel distribution, as a function of molecular mass,

used by [4], reproduced with permission from Elsevier.

Table 1 e Chemical compounds modelled to createa discrete representation of the continuous bio-oildistribution of [4].

compound chemicalcomposition

molecularweight

[g mol�1]

initialvolumefraction

water H2O 18.02 0.267

propanal C3H6O 58.08 0.144

butanal C4H8O 72.11 0.109

pentanal C5H10O 86.13 0.021

phenol C6H6O 94.11 0.054

guaiacol C7H7O2 124.14 0.108

coniferyl alcohol C10H12O3 180.20 0.190

formic acid C2H2O2 46.03 0.042

acetic acid C2H4O2 60.05 0.037

propanoic acid C3H6O2 74.08 0.017

butanoic acid C4H8O2 88.11 0.011

b i om a s s an d b i o e n e r g y 3 4 ( 2 0 1 0 ) 1 1 3 4e1 1 4 01136

v�rgr

2Yf

�vt

þv�rgur

2Yf

�vr

�v�rDr2

vYf

vr

�vr

¼ 0; (9)

where D is the coefficient of diffusivity for the vapour in air. Yf

is assumed to asymptote to a constant far field ambient

condition (0 in this study). The concentration at the surface is

calculated using the ClausiuseClapeyron equation as,

Yf ¼ Py

P¼ Pref

Pe

�LM=R

�1

Tref� 1Ts

��; (10)

where L is the latent heat of vapourisation, M is the Molar

mass, R is the ideal gas constant P is the ambient pressure

and Pref and Tref are the vapour pressure and temperature

of the gas at a known reference state. At high pressures,

when the effects of fugacity become more important, the

ClausiuseClapeyron is inaccurate, as shown by [11]. However,

for the pressures considered in this study it is sufficiently

accurate.

The internal liquid diffusion can be calculated from

conservation of matter, using the MaxwelleStefan equations

for molecular flux [5],

ðJÞ ¼ � ct½B��1½G�ðVYÞ: (11)

where

ðJÞ ¼

0BB@

J1J2«Jn�1

1CCA ðVYÞ ¼

0BB@

VY1

VY2

«VYn�1

1CCA (12)

and the diffusivity matrix, [B], and thermodynamic factor, [G],

are squarematrices of size n�1.PYi is the spacial derivative of

Yi, the Molar concentration of fuel species i, in the direction of

the diffusion calculation, and ct is the Molar density of the

fluid (Mole per volume). The flux of the last species, Jn, is

calculated by balancing the system as follows,

Xni¼1

Ji ¼ 0: (13)

The thermodynamic factor, [G], incorporates molecular

interactions into the diffusion model. For this study these

interactions were assumed to be small, allowing [G] to be

approximated by the identity matrix. The elements of [B] are

given by,

Bii ¼ xi

Dinþ

Xn

k ¼ 1isk

xk

Dik; (14)

and

Bij ¼ �xi

�1Dij

� 1Din

�(15)

where i s j.

The values of the MaxwelleStefan coefficients of diffusion,

Dij, were found using the Vignes method (see [5] for details),

Dij ¼�Do

ij

�ðxjÞ�Do

ji

�ðxiÞ: (16)

The infinite dilution diffusion coefficient, Doij, is the diffusion

coefficient for an infinitely weak solution of species i in

species j. This diffusion coefficient can be estimated using an

empirical formulation. We selected the Wilke and Chang

formulation (see [5] for more details).

The equations detailed in this section were solved

numerically in a Cþþ program, using finite difference

methods and Runge-Kutta time stepping. All fluid properties,

except the liquid phase density, weremodelled as functions of

temperature using experimental data (taken from [6]) where

available. When experimental data as a function of tempera-

turewas not available empirical estimations (sourced from [7])

were used. The density of liquids was held constant, at the

value for 300 K, to avoid any expansion of the droplet as it

heated. This assumption was made to simplify the model,

although any expansion due to droplet heating in practice is

likely to be reasonably small as the change in radius is

proportional to the cubed root of any increase in volume.

To create a model of bio-oil similar to that used by [4], 11

chemical components were chosen, based on the represen-

tative chemicals listed in their paper. The volume fractions

were selected to approximate the continuous distributions

specified in their paper. Hallet and Clark’s distributions are

shown diagrammatically in Fig. 1. The chosen composition is

detailed in Table 1. The very heavy molecular weight

Fig. 2 e Evolution of radius versus time for n-hexane

droplets. Initial radii: solid line R0 [ 5mm, dashed line

R0 [ 10mm, dash-dot line R0 [ 15mm.

Fig. 4 e Non-dimensionalised radius squared versus time

for bio-oil droplets. Ambient Temperature: solid line

Tamb [ 773 K, dashed line Tamb [ 573 K.

b i om a s s a n d b i o e n e r g y 3 4 ( 2 0 1 0 ) 1 1 3 4e1 1 4 0 1137

molecules in the lignin component were not modelled accu-

rately, due to their chemical complexity and very slow evap-

oration rates, instead they were grouped in with the coniferyl

alcohol which evaporates very slowly due to its high boiling

point. The resultant fuel is similar but not identical in

composition to the fuel studied by Hallet and Clark. The most

notable difference is that it doesn’t have the extremely light or

extremely heavymolecules included in the continuousmodel.

3. Evaporation of a single component droplet

The evaporation of a single component droplet was modelled

to provide a comparison case to reveal the effects of multiple

Fig. 3 e Non-dimensionalised radius squared versus time

for n-hexane droplets. Initial radii: solid line R0 [ 5mm,

dashed line R0 [ 10mm, dash-dot line R0 [ 15mm.

components and diffusion. N-hexane droplets were selected

for this purpose. Droplet of three different initial diameters

(5,10 and 15mm) at an initial temperature of 300 K in air at an

ambient temperature of 325 K were simulated. The evolution

of the radii of these droplets versus time is shown in Fig. 2.

The non-dimensionalised radius squared, for all three of

the cases shown Fig. 2, is plotted in Fig. 3. All three cases tend

towards a straight line, after an initial heating period. This

behaviour is to be expected, because as the droplets approach

a steady temperature, they should also approach the linear d2

law. Similar simulations were also run for n-octane and water

droplets. The same behaviour was observed in all cases. This

provides a good indication of the expected evaporation rate

when the internal diffusion process does not limit the rate at

which liquid reaches the surface.

4. Evaporation of a bio-oil droplets

Two different bio-oil cases were selected, with values chosen

to replicate Hallet and Clark’s simulations and experimental

observations. The first case is a 1.7 mm diameter bio-oil

droplet, in an ambient atmosphere at 573 K, and the second

is a 1.6 mm droplet with a 773 K ambient temperature. In both

cases the droplet is initially at 300 K and the ambient pressure

is 100 kPa. The non-dimensionalised radius squared against

time of both cases is shown in Fig. 4. In both cases the heavier

components of the fuel prevent the droplet from evaporating

completely. Initially a similar trend to that shown in Fig. 3

occurs, with the rate of surface regression increasing and

tending towards a constant, until it suddenly decreases to

a reduced rate. This occurs when the evaporation rate of one

or more of the fuel series is limited by the rate of diffusion

inside the droplet.

The remaining fraction of the initial droplet mass for the

573 K case is plotted in Fig. 5, against the results from Hallet

and Clark’s simulation. Their experimental observations are

shown under the time axis. Both models agree well in the

Fig. 5 e Plot of remaining mass fraction versus time for an

evaporating 1.7 mm bio-oil droplet at 573 K (solid line)

compared to the simulation by [4] (dashed line).

Observations from the experimental study by [4] under the

same conditions are shown below the plot, note that in the

experiments, the droplet became solid after approximately

40 s. Data from [4] reproduced with permission from

Elsevier.

Fig. 7 e Plot of remaining mass fraction versus time for an

evaporating 1.6 mm bio-oil droplet at 773 K (solid line)

compared to the simulation by [4] (dashed line). The dash-

dot line is the char prediction from [4]. Observations from

the experimental study by [4] under the same conditions

are shown below the plot. Data from [4] reproduced with

permission from Elsevier.

b i om a s s an d b i o e n e r g y 3 4 ( 2 0 1 0 ) 1 1 3 4e1 1 4 01138

early stages of evaporation, there are visible differences, but

this is to be expected due to slight differences in the fuel

models. As time progresses themodels deviatemore, with the

diffusing model evaporating significantly slower. From t ¼ 27s

onwards the evaporation rate has dropped so low that it

appears negligible compared to the earlier behaviour. The

model also appears to agree qualitatively with the experi-

mental observations, exhibiting a rapid evaporation period

initially, before settling down.

Fig. 6 shows the volume of three of the eleven chemical

components in the droplet expressed as a fraction of the initial

Fig. 6 e Plot of volume fractions versus time for selected

chemicals in a evaporating bio-oil droplet at 573 K Legend:

(solid line) propanol, (dashed line) coniferyl alcohol, (dash-

dot line) water.

droplet volume. These chemicals all show different evapo-

rating behaviour. The coniferyl alcohol has a very high boiling

point, andexhibits analmostnegligible amountof evaporation,

with the amount staying almost constant. The propanol

initially evaporates at a rapid rate, before slowing down

suddenly at t z 15s. This coincides with the time when the

concentrationofpropanolat thesurface is reducedtozero,after

which the rate at which the propanol evaporates is limited by

acombinationof, therateatwhich itdiffuses to thesurface,and

the rate at which the surface regresses towards the more

propanol rich core. Thewater component shows a fairly steady

rate of evaporation, without the point of inflection at which it

becomes limitedby liquiddiffusion.This isa resultof thehigher

initial concentration, which means it will take significantly

longer before it becomes diffusion limited.

The remaining dropletmass against time for the 773 K case

is shown in Fig. 7. There is a significant difference between the

two models earlier in the lifetime compared to the 573 K case.

It is suggested that this results from the increased rate of

evaporation with the elevated temperature, which leaves less

time for the diffusion process to occur, resulting in the lack of

light weight compounds limiting the evaporation sooner.

Hallet and Clark’s model included an empirical equation to

model the conversion of lignin to char at higher temperatures.

The model presented in this paper did not include chemical

reactions.

Selected chemical components are shown in Fig. 8. The

behaviour shownis thesameas thatof the573Kcase, shownin

Fig. 6. The increased temperature results in faster evaporation,

leaving less time for diffusion to occur. This can be observed

from the difference in the remaining volume fraction of the

propanol at the point which the rate of evaporation slows. At

Fig. 8 e Plot of volume fractions versus time for selected

chemicals in a evaporating bio-oil droplet at 773 K Legend:

(solid line) propanol, propanol, (dashed line) coniferyl

alcohol, (dash-dot line) water.

Fig. 10 e Plot of the remaining volume fraction versus time

for the Propanol component of an evaporating 1.6 mm

bio-oil droplet at 773 K (solid line) compared to 53 diffusion

(dashed line), and no diffusion (dash-dot line).

b i om a s s a n d b i o e n e r g y 3 4 ( 2 0 1 0 ) 1 1 3 4e1 1 4 0 1139

the higher temperature, the remaining fraction is approxi-

mately 25% higher than the 573 K case at the limiting point.

These results suggest that the rate of diffusion within the

droplet limits the evaporation rate. To test this hypothesis the

simulations were re-runwith the diffusion ratesmultiplied by

a factor of 5 to allow the effect of the diffusion rate to be

examined. This is plotted in Fig. 9, along with a zero diffusion

case for comparison. All these cases, including the zero

diffusion case feature some mixing effects in the outermost

grid point (1/79th of the droplet diameter for these simula-

tions). Thismixing is a result of the numerical model used and

the need to apply continuity equations over a volume rather

than a surface when updating the droplet components as

evaporation occurs.

From these results it can be deduced that removing the

diffusion from the code drastically reduces the evaporation

Fig. 9 e Plot of remaining mass fraction versus time for an

evaporating 1.6 mm bio-oil droplet at 773 K (solid line)

compared to 5 3 diffusion (dashed line), and no diffusion

(dash-dot line).

rate very early in the droplet lifetime. The increased rate of

diffusion does not visibly differ from the standard rate until

approximately the 4.5 s mark, when the standard rate curve

reaches the limiting rate point for one of the chemical

compounds and suddenly slows down its rate of evaporation.

The increased diffusion rate delays the onset of the low

boiling species being exhausted from the surface. This same

effect is seen at the 8 and 14 s marks. The overall result is that

the droplet with faster diffusion tends towards a significantly

lower remaining mass, and even then the rate of evaporation

is noticeably faster as the light compounds diffuse to the

surface faster.

The effect of varying the rate of diffusion is much more

visible in Fig. 10. This shows the amount of propanol (as

a fraction of the initial droplet volume) remaining for the three

different diffusion cases. The increased diffusion rate case

doesn’t show signs of being diffusion limited until the

remaining volume fraction is much lower than the standard

case. Also, the increased rate of diffusion nearly leads to the

complete evaporation of this portion of the droplet. This

suggests that with sufficient internal diffusion and mixing,

the phenomenon of explosive boiling, caused by low boiling

point components trapped in the droplet core, can be signifi-

cantly reduced.

5. Conclusion

Anumericalmodel to simulate the evaporation of a stationary,

spherically symmetric fuel droplet consisting of a mixture of

chemicals was developed. This model was tested against the

model used by [4]. The limited rate of diffusion within the

dropletwas shownnot to affect theearly stagesof evaporation,

but significantly slowed evaporation later in the process.

These results suggest that the presence of low boiling point

compounds near the centre of the droplet late in the evapo-

ration, is highly sensitive to the rate of internal diffusion (and

b i om a s s an d b i o e n e r g y 3 4 ( 2 0 1 0 ) 1 1 3 4e1 1 4 01140

mixing). Therefore, when bio-oils are vapourised, and burnt in

engines, increasing the rate of mixing will reduce the occur-

rence of explosive boiling. Possible methods of promoting

internal mixing in droplets that may be worth further inves-

tigation include preheating the fuel to enhance the rate of

diffusion, including additives in the fuel to reduce viscosity, or

designing spray nozzles and droplet sizes to increase internal

circulation.

Acknowledgement

The authorswould like to acknowledge the support of the ARC

for this research through grant DP0556098.

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