This article was downloaded by: [Brown University]On: 03 May 2013, At: 01:54Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Aerosol Science and TechnologyPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/uast20

Computational Models for Simulating MulticomponentAerosol Evaporation in the Upper Respiratory AirwaysP. Worth Longest a & Clement Kleinstreuer ba Department of Mechanical Engineering, Virginia Commonwealth University, Richmond,Virginia, USAb Department of Mechanical and Aerospace Engineering, North Carolina State University,Raleigh, North Carolina, USAPublished online: 17 Aug 2010.

To cite this article: P. Worth Longest & Clement Kleinstreuer (2005): Computational Models for Simulating MulticomponentAerosol Evaporation in the Upper Respiratory Airways, Aerosol Science and Technology, 39:2, 124-138

To link to this article: http://dx.doi.org/10.1080/027868290908786

PLEASE SCROLL DOWN FOR ARTICLE

Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form toanyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses shouldbe independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims,proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly inconnection with or arising out of the use of this material.

Aerosol Science and Technology, 39:124–138, 2005Copyright c© American Association for Aerosol ResearchISSN: 0278-6826 print / 1521-7388 onlineDOI: 10.1080/027868290908786

Computational Models for Simulating MulticomponentAerosol Evaporation in the Upper Respiratory Airways

P. Worth Longest1 and Clement Kleinstreuer2

1Department of Mechanical Engineering, Virginia Commonwealth University, Richmond,Virginia, USA2Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh,North Carolina, USA

An effective model for predicting multicomponent aerosolevaporation in the upper respiratory system that is capable ofestimating the vaporization of individual components is neededfor accurate dosimetry and toxicology analyses. In this study, theperformance of evaporation models for multicomponent dropletsover a range of volatilities is evaluated based on comparisons toavailable experimental results for conditions similar to aerosolsin the upper respiratory tract. Models considered include asemiempirical correlation approach as well as resolved-volumecomputational simulations of single and multicomponent aerosolevaporations to test the effects of variable gas-phase properties,surface blowing velocity, and internal droplet temperature gra-dients. Of the parameters assessed, concentration-dependent gas-phase specific heat had the largest effect on evaporation andshould be taken into consideration for respiratory aerosols thatcontain high volatility species, such as n-heptane, at significantconcentrations. For heavier droplet components or conditions be-low body temperatures, semiempirical estimates were shown to beappropriate for respiratory aerosol conditions. In order to reducethe number of equations and properties required for complex mix-tures, a resolved-volume evaporation model was used to identifya twelve-component surrogate representation of potentially toxicJP-8 fuel based on comparisons to experimentally reported dropletevaporation data. Due to the relatively slow evaporation rate ofJP-8 aerosols, results indicate that a semiempirical evaporation

Received 15 March 2004; accepted 12 November 2004.This effort was sponsored by the Air Force Office of Scientific

Research, Air Force Material Command, USAF, under graft numberF49620-01-1-0492 (Dr. Walt Kozumbo, Program Manager). The U.S.Government is authorized to reproduce and distribute reprints for gov-ernmental purposes notwithstanding any copyright notation thereon.Use of the software package CFX4 from ANSYS, Inc. (Canonsburg,PA, USA) and access to the SGI Origin 2400 at the North CarolinaSupercomputing Center (Research Triangle Park, NC) are gratefullyacknowledged.

Address correspondence to Dr. P. Worth Longest, Departmentof Mechanical Engineering, Virginia Commonwealth University, 601West Main Street, P.O. Box 843015, Richmond, VA 23284-3015,USA. E-mail: pwlongest@vcu.edu

model in conjunction with the identified surrogate mixture providea computationally efficient method for computing droplet evapo-ration that can track individual toxic markers. However, semiem-pirical methodologies are in need of further development to effec-tively compute the evaporation of other higher volatility aerosolsfor which variable gas-phase specific heat does play a significantrole.

INTRODUCTIONReliable dose assessments of multicomponent aerosols in the

respiratory tract require accurate analyses of droplet heat andmass transfer characteristics. Applications in which multicom-ponent liquid respiratory aerosol evaporation is of significanceinclude the inhalation of potentially toxic pollutants (Brodayand Georgopoulos 2001; Zhang et al. 2004), as well as thegeneration and delivery of inhaled therapeutics (Finlay 2001;Edwards and Dunbar 2002). In either case, the dose receiveddepends on the characteristics of the inhaled liquid aerosol,biophysical transport (ICRP 1994), and the surface conditionsat the local site of aerosol deposition or vapor absorption,e.g., mucus or surfactant covering and thickness, metabolicreactions, etc. (Miller et al. 1985; Cohen Hubal et al. 1996;Overton et al. 2001; Kimbell et al. 2001). Evaporation of res-piratory aerosols affects the diameter and density of the par-ticle, thereby significantly affecting transport including depo-sition (Martonen et al. 1982; Stapleton et al. 1994; Broday andGeorgopoulos 2001; Martonen and Schroeter 2003; Zhang et al.2004). Moreover, individual chemical species of multicompo-nent aerosols are often of interest due to particular toxic ortherapeutic effects, making it necessary to differentiate amongchemical components. Hence, accurate dose assessments re-quire an effective model for the evaporation of multicompo-nent respiratory aerosols that is capable of predicting the forma-tion of individual vapors as well as determining the remainingdroplet characteristics and liquid mass fractions.

124

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

1:54

03

May

201

3

MULTICOMPONENT AEROSOL EVAPORATION 125

The effects of water condensation and evaporation (hygro-scopicity) on aerosols in the respiratory tract have been con-sidered in a number of studies as reviewed by Morrow (1986)and Hiller (1991). More recently, Finlay and Stapleton (1995)showed that two-way coupling between aerosol droplets and thecontinuous phase, due to water evaporation and condensation,significantly affects respiratory deposition. Due to high relativehumidity, studies that have considered the effects of evapora-tion and condensation on droplet transport and deposition in therespiratory tract have predominately focused on water vapor.For these conditions, simplifying assumptions often includespatially and temporally constant droplet temperature profiles,constant gas and liquid properties, and no effect from surfaceblowing. However, the evaporation and condensation of multi-ple component droplets and compounds with higher volatilitiesthan water may violate the assumptions typically made for res-piratory aerosol hygroscopicity.

Respiratory aerosols are characterized by low-to-moderateparticle Reynolds numbers and relatively mild variations intemperature. In contrast to combustion applications (Landis andMills 1974; Law 1982; Aggarwal et al. 1984), evaporation ofmulticomponent droplets for a range of volatilities under con-ditions consistent with the respiratory tract has been consid-ered in relatively few studies. For moderate Reynolds num-ber conditions (Rep < 130) and mild temperature variations,Chen et al. (1997) and Runge et al. (1998) both consideredthe evaporation of single and multicomponent fuel droplets.Chen et al. (1997) showed that a semiempirical rapid mixingmodel (RMM), which assumes spatially constant droplet tem-peratures and concentrations, performed well for the moderateevaporation rates considered. The model employed by Chen etal. (1997) was based on the solution of ordinary differentialequations (ODEs) for droplet heat and mass conservation withempirical Nusselt and Sherwood number correlations used toaccount for gas-phase transport. Similarly, Runge et al. (1998)showed that a semiempirical ODE-based solution could be usedto match their experimental results; however, relations for theproduct of gas-phase density and diffusivity were derived em-pirically for each fuel mixture considered.

For low temperature evaporation of multicomponent fuelblends, a homogeneous mixture model has been developedbased the process of batch distillation (Gauthier et al. 1996) andapplied to respiratory aerosol conditions (Zhang et al. 2004).This model assumes rapid mixing of the components and imple-ments composite properties for the fuel mixture of interest thatare calculated based on the method of Bardon and Rao (1991)and that depend on the fraction of mass evaporated (Catorieet al. 1999; Benaissa et al. 2002). As such, the fuel can beconsidered a single-component or homogeneous mixture, andonly one conservation of mass equation is necessary to predictdroplet evaporation. However, a constant effective diffusivitymust be empirically determined for the mixture, and the modelis not designed to differentiate among chemical compounds, asmay be of interest in dose assessments.

For cases in which empirical correction factors are not avail-able to describe the gas-phase transport, direct numerical simu-lations may be conducted. These resolved-volume techniquescan directly account for the effects of droplet arrays, shearin the underlying fluid media, and variable thermo-physicalproperties of the gas-phase. Due to the small diameter of res-piratory aerosols, surface tension is much greater than otherforces on the droplet such that a spherically symmetric ge-ometry may be assumed (Sadhal et al. 1997). For combustionapplications, several researchers have numerically simulatedthe evaporation characteristics of spherical droplets includingvariable thermo-physical properties, transient heating, inter-nal liquid circulation, and droplet rotation (Xin and Megaridis1996; Kleinstreuer et al. 1993; Chiang and Kleinstreuer 1992a,b). For instance, Chiang and Kleinstreuer (1992a, b) consid-ered collinear interacting and vaporizing single-component fueldroplets in a heated air stream, under high pressure, and forparticle Reynolds numbers of approximately Rep = 100. As-suming a spherical interface, gas- and liquid-phase flows werecoupled and fully resolved. In a similar study, Kleinstreuer et al.(1993) found that for single-component hydrocarbon fuels theLewis number is greater than unity and variable during the ini-tial phase of evaporation.

Studies dealing with respiratory aerosols typically focus onparticle diameters less than 10 µm. This is based on the ef-fective filtering mechanism of the lung as well as supportingevidence that only a small percentage of particles greater than10 µm reach the alveolar or pulmonary region during moderatebreathing conditions. However, studies have shown that parti-cles on the order of 100 µm are frequently inhaled through themouth and nose, i.e., an inhalation efficiency of approximately50% (ACGIH 1997; Hinds 1999). Nearly all of the inhaled par-ticles in this size range are deposited in the nasal and oral cav-ities or the pharynx (Stahlhofen et al. 1989). Analysis of toxicand therapeutic doses within these extrathoracic regions re-quires accurate assessments of aerosol transport and deposition,which are largely influenced by evaporation. Moreover, largerparticles deliver a significantly greater amount of the toxic ortherapeutic compound. For instance, deposition of a 100 µmparticle provides 8,000 times more dose than a 5 µm particleon a volumetric basis. While clearance is typically rapid in theextrathoracic airways, absorption through the mucus lining intothe blood may occur depending on the local metabolic char-acteristics. Desorption resulting in vapor formation may alsooccur after the liquid droplet deposits, resulting in gas-phasetransport to other regions of the lung. In this study, droplet di-ameters ranging from 5 to 100 µm within the upper respiratorytract will be considered. Particle diameters less than 5 µm havebeen excluded to avoid the influence of noncontinuum effectson heat and mass transfer.

An effective model for predicting multicomponent aerosolevaporation in the respiratory system that is capable of differ-entiating among individual components for a range of volatil-ities and that does not require an empirically derived effective

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

1:54

03

May

201

3

126 P. W. LONGEST AND C. KLEINSTREUER

gaseous diffusivity is needed for accurate dosimetry estimates.Considering the broad range of inhalable particles and the sig-nificance of deposition in the upper airways, models appropri-ate for dosimetry applications should be valid up to aerosoldiameters on the order of 100 µm and for low-to-moderateparticle Reynolds numbers, i.e., 0 < Rep ≤ 30. In this study,the performance of droplet evaporation models for multicom-ponent respiratory aerosols over a range of volatilities is eval-uated based on comparisons to available experimental resultsfor similar conditions. Models considered include a semiem-pirical ODE-based approximation as well as resolved-volumesimulations of single and multicomponent aerosol evaporationrates to test the effects of variable gas-phase properties, surfaceblowing velocity, and droplet temperature gradients. Sherwoodand Nusselt number correlations for respiratory aerosol heatand mass transfer under uniform flow conditions have alsobeen evaluated. The primary disadvantage of the heterogeneousevaporation models considered is the complexity introduced bythe physiochemical constants required for each species presentin multicomponent mixtures. In order to reduce the number ofrequired properties while maintaining the capability to track in-dividual components of interest, the appropriate multicompo-nent evaporation models can be used to define surrogate blendsof complex mixtures. To illustrate this concept, jet propellant-8 (JP-8) aviation fuel has been selected as a potential toxi-cant due to its wide use and chemical complexity. JP-8 is akerosene-based fuel consisting of over 200 hydrocarbons span-ning a wide range of volatilities that is exclusively used by theU.S. Air Force and may present a prevalent exposure hazard tomaintenance personnel (ATSDR 1998; Bakshi and Henderson1998). An appropriate droplet evaporation model has been usedto identify a twelve-component representative or surrogate mix-ture for JP-8 fuel based on comparisons to the experimentallyreported droplet evaporation data of Runge et al. (1998). The se-lected evaporation models and surrogate mixture can be used tosimplify computational toxicokinetic analyses in which track-ing specific chemical markers, e.g., benzene, contained withinmulticomponent aerosols is imperative. The evaporation mod-els evaluated are capable of accounting for the effects of rar-efied flow and relative humidity; however, these factors havenot been directly considered in this study.

METHODS

Evaporation Models and AssumptionsTo model the evaporation of multicomponent aerosols in

the upper respiratory tract, semiempirical and resolved-volumetreatments of the gas-phase have been employed. As with theclassic d2 law (Spalding 1953), the semiempirical approachimplements Nusselt and Sherwood number correlations to ac-count for heat and mass transfer at the droplet surface. Theother approach considered in this study directly simulates (orfully resolves) the gas-phase field around the droplet, includ-ing heat and mass transfer. Due to relatively slow evaporationrates of droplets that are consistent with respiratory aerosols, itis hypothesized that both approaches may employ rapid mix-ing assumptions for the liquid phase (Chen et al. 1997). Thatis, infinite conduction and diffusion may be assumed within thedroplet such that liquid temperature and concentration are spa-tially constant but vary with time, i.e., the lumped capacitancemethod (Sirignano 1999). To test the validity of the rapid mix-ing model assumption for respiratory aerosols, resolved-volumesimulations have also been conduced with a diffusion-limitedmodel in which droplet temperatures are assumed to vary tem-porally and radially.

The semiempirical approach requires solution of a coupledset of ordinary differential equations (ODEs) for heat and massconservation, which can be numerically computed very effi-ciently. However, this ODE-based approach may not be validfor conditions that violate the assumptions of the underlyingempirical approximations. For instance, empirical correlationsmay not be valid across a range of chemical volatilities andmay be inaccurate for the low-to-moderate Reynolds numberconditions of respiratory aerosols. Furthermore, thermophysi-cal properties of the gas-phase are assumed spatially constant,which may not effectively account for concentration depen-dence near the droplet surface.

To assess the implicit limitations of the ODE-based semiem-pirical model applied to respiratory aerosols, resolved-volumesimulations have been employed. As indicated in Table 1, theresolved-volume models considered are of increasing complex-ity to allow for the isolation and evaluation of a number of crit-ical assumptions. The first three resolved-volume approaches

Table 1Description of computational models implemented for droplet evaporation

Conditions: Solution of Spatially variable gas Internal dropletModel the flow field specific heat Surface blowing temperature gradient

ODE Xa

RMM1 XRMM2 X XRMM3 X X XDLM X X X X

aNot included in ODE solution shown.

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

1:54

03

May

201

3

MULTICOMPONENT AEROSOL EVAPORATION 127

include rapid mixing assumptions for the liquid phase, denotedas RMM1 through RMM3. Specifically, the effects of calcu-lating the gas-phase flow field (RMM1), spatially variable gas-phase specific heat (RMM2), and surface blowing (RMM3) areevaluated. The fourth resolved-volume model accounts for ra-dial droplet temperature gradients that may arise for rapid evap-oration conditions, i.e., a diffusion-limited model (DLM). Inreality, spatial variations arise for both temperature and con-centration fields within the droplet; however, the interdiffusionof more than two chemical species at similar concentrations isdifficult to predict (Bird et al. 1960). As such, this study will fo-cus on the effects of temperature gradients and use comparisonsto empirical droplet evaporation data to assess the assumptionof infinite mass diffusion within the droplet.

In addition to the RMM and DLM approximations, a numberof other simplifying assumptions can be applied for respiratoryaerosol conditions. The relatively small droplet diameters of in-terest and low-to-moderate particle Reynolds numbers resultin dominate surface tension forces, allowing for the assump-tion of spherically symmetric droplets with no internal vortexmotion (Clift et al. 1978; Sadhal et al. 1997). Constant sur-face temperature and concentration values have been assumedbased on expected rapid mixing within the droplet and the re-sults of Chen et al. (1997). In addition, Niazmond et al. (1994)have shown that the thermal Marangoni effect induces signif-icant surface flows, arising from temperature-dependent vari-ations in surface tension, that aid in equalizing droplet sur-face temperatures and concentrations (Dwyer et al. 2000). Forthe gas-phase the assumptions of low mass fluxes and dilutechemical species, which simplify the mass transport equations,were made. Reynolds and Mach number conditions as well asmoderate temperature variations allow for the assumptions oflaminar incompressible flow. For the resolved-volume models,beginning with RMM2, the gas-phase specific heat is calcu-lated at each location in the flow field and is based on weightedcontributions from each mass fraction. All gas-phase propertieshave been calculated at the current film temperature.

Governing Equations and Numerical MethodsConservation of energy for an immersed droplet, indicated

by the subscript d, under rapid mixing model conditions can beexpressed as

dT

dtmdCpd = −

∫surf

qconvdA−m∑

s=1

∫surf

nsLsdA [1]

In the above equation md is the droplet mass, Cpd is the compos-ite liquid specific heat, qconv is the convective heat flux, ns is themass flux of each evaporating chemical species at the dropletsurface, and Ls is the latent specific heat of each liquid com-ponent. The integrals are performed over the droplet surfacearea, A. Conservation of mass for an immersed droplet based

on evaporating fluxes can be expressed as

d (md )

dt= −

m∑s=1

∫surf

nsdA [2a]

Implementing the chain rule, Equation (2a) is often rewritten interms of droplet radius (Aggarwal et al. 1984):

dr

dt= −

m∑s=1

(∫surf nsdA

Aρs

)[2b]

with r(t = 0) = Ro. For the semiempirical ODE-based model,surface-averaged heat and mass fluxes are assumed such thatthe surface integrals in Equations (1 and 2) may be written as

∫surf

qconvdA = q̄conv · A [3a]

and

m∑s=1

∫surf

nsdA =m∑

s=1

(n̄s · A) [3b]

The primary difference between the semiempirical andresolved-volume solutions is defined by the method used to cal-culate the surface heat and mass flux values. For the semiem-pirical solution, heat and mass fluxes can be determined fromappropriate correlations available for uniform flow. Based on56 observations with particle Reynolds numbers from 2 to 200and temperatures up to 220oC, Ranz and Marshall (1952) es-tablished the widely used correlation

Sh = 2. + 0.6Re1/2p Sc1/3 [4a]

Blowing effects are absent in this expression, which is at-tributed to the fact that the observations by Ranz and Marshall(1952) were mostly below 90oC (Renksizbulut et al. 1991). Atnonzero Reynolds numbers, experimental observations show astrong analogy between heat and mass transfer allowing for acorresponding Nusselt number correlation of

Nu = 2. + 0.6Re1/2p Pr1/3 [4b]

The analogy between heat and mass transfer expressions is notexact largely due to property variations in the gas-phase asso-ciated with moderate to high flux rates (Clift et al. 1978). Ina more recent analysis, Clift et al. (1978) correlated availabledata for Rep < 400 with the expressions:

Nu = 1 + (1 + Rep Pr)1/3 max[1, Re0.077

p

][5a]

and

Sh = 1 + (1 + RepSc)1/3 max[1, Re0.077

p

][5b]

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

1:54

03

May

201

3

128 P. W. LONGEST AND C. KLEINSTREUER

Once Nusselt and Sherwood numbers have been determined,the associated dimensional heat and mass transfer parameterscan be computed from

h = Nukg

d[6a]

and

hm,s = ShDs

d[6b]

where kg is the thermal conductivity of the gas and Ds is the bi-nary diffusion coefficient of species s with air. For the semiem-pirical ODE-based solution, these transport coefficients are em-ployed to evaluate the mean fluxes at the droplet surface, i.e.,

q̄conv = h(Td − T∞) [7]

and accounting for the convective effect of droplet evaporation(blowing velocity)

n̄s = ρghmωs,surf − ωs,∞

1 − ωs,surf[8a]

where ωs is the gas-phase mass fraction of each species andρg is the gas mixture density, assumed to be that of air. Surfacemass flux is often expressed in the following equivalent form

n̄s = ρghm ln(

1 − ωs,∞1 − ωs,surf

)[8b]

which arises from an ambient solution (Rep = 0) of the speciesconservation equation.

Provided evaporation-induced convective effects (blowingvelocity) can be neglected and assuming low evaporation rates,the surface mass flux reduces to

n̄s = ρghm(ωs,surf − ωs,∞) [8c]

For the resolved-volume approach, heat and mass fluxes atthe liquid-gas interface are determined from a solution of thesurrounding flow field. The gas-phase conservation equationsfor laminar incompressible flow on a moving mesh in strongconservation law format can be written (Hawkins and Wilkes1991)

Conservation of mass

ρg√g

∂

∂t(√

g) + ρg∂

∂x j

(u j − ∂ x̃ j

∂t

)= 0 [9a]

Conservation of momentum

ρg√g

∂

∂t(√

gui) + ρg∂

∂x j

[(u j − ∂ x̃ j

∂t

)ui

]= − ∂ p

∂xi+ µ

∂2ui

∂x2j

[9b]

Conservation of energy

ρgcpg√g

∂

∂t(√

gT ) + cpgρg∂

∂x j

[(u j − ∂ x̃ j

∂t

)T

]= kg

∂2T

∂x2j

[9c]

Conservation of dilute chemical species

ρg√g

∂

∂t(√

gωs) + ρg∂

∂x j

[(u j − ∂ x̃ j

∂t

)ωs

]= ρgDs

∂2ωs

∂x2j

[9d]

In these equations, x̃i represents the moving mesh locationand

√g is the metric tensor determinate of the transforma-

tion, i.e., the local computational control-volume size. As such,terms of the form 1√

g∂∂ t (ρgφ

√g) account for the gain or loss of

the variable ρgφ due to control-volume motion. For a stationarymesh,

√g = 1 and ∂ x̃ j

∂ t = 0 .In the resolved-volume approach, heat and mass fluxes are

computed at the gas–liquid interface using the gas-phase sur-face gradients, i.e.,

qconv = −kg∂T

∂n

∣∣∣∣surf

[10a]

and

ns = −ρgDs∂ωs,surf

∂n

(1 − ωs,surf)[10b]

These flux values are then applied to determine the area-averaged transient heat and mass transfer parameters,

h =∫

surf qconvdA

A(T − T∞)[11a]

and

hm,s =∫

surf nsdA

Aρg(ωs,surf − ωs,∞)/(1 − ωs,surf)[11b]

which are written in non-dimensional form as the Nusselt(Nu = h d/kg) and Sherwood (Sh = hm,s d/Ds) numbers. For lowmass flux cases in which blowing velocity may be neglected(i.e., RMM1), the term (1-ωs,surf ) is absent in Equations (10b)and (11b).

For the resolved-volume solution with rapid mixing modelassumptions (RMM1–RMM3), the area-averaged heat andmass fluxes may be substituted directly into Equation (1). Ifspherical droplet conduction is assumed, as with the DLM, con-servation of energy for the droplet is governed by

1

r2

∂

∂r

(kdr2 ∂T

∂r

)= ρdCpd

∂T

∂t[12]

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

1:54

03

May

201

3

MULTICOMPONENT AEROSOL EVAPORATION 129

The area-averaged surface flux values determined from theresolved-volume simulation are then used to establish theinterface boundary conditions, and Equation (12) is numericallysolved to establish the internal droplet temperature gradient atthe current time.

For both the semiempirical and resolved-volume solutions,assumptions of thermodynamic equilibrium at the droplet sur-face, ideal gas conditions, and Raoult’s law have been made,such that the gas-phase surface concentrations can be expressedas

ωs,surf = xsPsat,s (Td )

ρgRsTd[13]

In the above expression, xs is the liquid-phase mole fraction ofcomponent s, Rs is the gas constant, and Psat,s(Td) is the tem-perature dependent saturation pressure. To estimate the satura-tion pressure of each component on the surface, the Clausius-Clapeyron equation has been employed, i.e.,

Psat,s(Td ) = Po,s exp[

Ls

Rs

(1

To,s− 1

Td

)][14]

where Po,s and To,s represent saturation conditions at a refer-ence state.

Statements for the conservation of energy and mass—Equations (1) and (2)—represent a coupled set of ordinarydifferential equations that can be solved for droplet tempera-ture and radius over time. With the semiempirical approach,heat and mass transfer parameters are readily available for aspecified particle Reynolds number such that multiple-pointpredictor-corrector algorithms can easily be used to solve thecoupled ODE set. As such, a fourth-order Runge Kutta algo-rithm (Press et al. 1992) has been implemented for the ODE-based semiempirical solution to compute droplet flux values,temperature, radius, and mass over time for single and multi-component droplets. Variable time-step and error control rou-tines were included to ensure efficient and accurate solutions.The error control algorithm ensures that differences among so-lutions calculated using variable time increments �t and 2 ×�t/2 are negligible at each step (Press et al. 1992; Longest et al.2004).

Solution of the droplet heat and mass transport equationswith the resolved-volume approach is complicated by the nu-merical evaluation of the surrounding flow field. The computa-tional fluid dynamics program CFX 4.4 (ANSYS, Inc.), whichis a finite-volume code that allows for multiblock structuredmeshes, was implemented to solve the flow field conservationequations. Mesh deformation was included to account for theeffects of droplet evaporation. To improve computational ac-curacy on the scales of interest, a cgs (centimeters, grams, andseconds) system of units and double precision calculations wereemployed. A dimensionally based code was selected to avoidcomplications associated with geometry motion occurring due

to droplet evaporation. All flow field equations were discretizedto be at least second-order accurate in space. The third-orderQUICK scheme was employed for the advection terms in orderto better resolve the upstream diffusional effects that arise forthe lower Reynolds number flows of interest. The use of movinggrids to resolve droplet evaporation limits temporal discretiza-tion to first-order accuracy, i.e., a fully implicit backward Eulerscheme. However, time-step size and not method order deter-mine solution accuracy, such that an acceptable solution can beachieved for a sufficiently small time-step.

Once the gas-phase heat, mass, and flow fields were de-termined for a time-step, effects on the particle are evaluated.Convective and mass fluxes are computed at the droplet surfaceusing Equations (10a) and (10b). Surface gradients of tempera-ture and mass fraction were computed to be second-order accu-rate on the deformable nonuniform grid surrounding the droplet(Tannehill et al. 1997). Given the use of a first-order method intime for the flow field equations, a first-order Euler approxi-mation was also used to determine the droplet temperature andcomponent mass at each solution step. Total droplet mass andliquid mass-fraction can then be directly computed. Due to nu-merical accuracy, it is most effective to determine next the newdroplet volume, composite density, and finally the diameter andmixture specific heat. Equations (11a) and (11b) are then usedto calculate the area-averaged heat and mass transfer parametersat the current time-step. Based on the new spherical diameter,the mesh resolving the flow field around the droplet is adjusted.The solution is then advanced by setting the new droplet tem-perature and mass fraction at the spherical surface and repeatingthe flow field solution.

The solution of the parabolic expression for droplet conduc-tion, Equation (12), was computed using a Crank-Nicholsonscheme. Singularity at the droplet center was addressed by onlyconsidering the second derivative spatial term at this location(Tannehill et al. 1997). To minimize numerical diffusion, spatialdiscretization and time-steps were selected to satisfy the condi-tion α �t/�r2 = 0.5, where α is the composite liquid thermaldiffusivity.

Properties that must be set for each of the chemical speciescontained in a droplet include liquid density (ρ l), molecularweight (Ml), liquid mass fraction (ωl), gas constant (Rs), latentheat of vaporization (Ls), liquid specific heat (Cp,l), referencesaturation pressure and temperature (Psat,s(Td)), and the binarygas diffusivity constant of each component with air (Ds). Theinitial droplet diameter and temperature are also explicitly set.For a specified initial set of liquid mass fractions and a diameterbased on spherical volume (V), the liquid mixture density of thedroplet (ρd) is determined by solving a coupled equation set ofthe form

ωlρdV = ρlVs, [15]

where Vs is the volume of each component. For the gas-phase,the local concentration-dependent specific heat depends on the

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

1:54

03

May

201

3

130 P. W. LONGEST AND C. KLEINSTREUER

mass fractions of air and the n dilute chemical species presentin vapor form, i.e.,

Cpg = Cp,airωair +n∑

s=1

Cp,sωs [16]

To resolve diffusion on the micron-size spatial scales of thedroplets, a very small time-step is needed. Furthermore, the useof moving grids and the fully implicit backward Euler schemeseverely limit the allowable time-step size. Using numerical ex-perimentation, it was found that initial resolution of the devel-oping gas-phase temperature and mass concentration profiles,along with accurate prediction of droplet temperature changes,required a time-step on the order of �t = 2 × 10−4 s forthe resolved-volume approach. Considering that droplet evap-oration often requires minutes to several hours, the associatednumber of time-steps is unreasonable. However, it was foundthat once pseudosteady temperature and concentration fieldsdeveloped, time-steps could be increased significantly as fur-ther evaporation occurs. The upper limit for time-step size afterpseudosteady concentration fields have developed was found tobe on the order of �t ≈ 0.1 s.

The external flow system of an isolated droplet representsa large flow environment with a relatively small disturbancefield, especially for the higher Reynolds number flows consid-ered. As such, significant residual reductions of all equationsare required to ensure convergence. Numerical experimenta-tion indicates that convergence could be assumed, based on a1% relative local error criterion on all variables, once initialglobal residual values have been reduced by five to six ordersof magnitude. In addition, the rate of residual reduction wasalso observed and used to establish convergence for all equa-tions. While strict residual criteria had to be enforced for theexternal flow field compared to typical internal conditions, in-creased under-relaxation factors allowable for the low Reynoldsnumber solutions reduced the number of iterations required toreach convergence, thereby decreasing solution time.

Geometry and Boundary ConditionsTo assess the resolved-volume evaporation models consid-

ered, a system consisting of a two-dimensional spherical ax-isymmetric droplet in an infinite gas was employed (Figure 1).A closed system was formulated for numerical analysis by spec-ifying walls far from the droplet in a manner that does not inter-fere with heat and mass transfer characteristics. Uniform flowwas specified sufficiently far upstream from the droplet, repre-senting motion relative to the surrounding air, i.e., particle slip.Zero mass concentration and ambient temperature conditionswas specified on the lateral boundaries and upstream of thedroplet. While slip is allowed on the boundary walls, the dropletsurface is considered rigid due to high relative surface tensionforces, resulting in a no-slip boundary condition. Droplet sur-face temperature is assumed spatially constant and is specified

Figure 1. Axisymmetric two-dimensional geometry of aspherical droplet in a nearly infinite media (from Longest andKleinstreuer 2004 with permission).

according to initial and previous time-step conditions. Based ondroplet temperature, saturation pressure is determined for eachspecies from the Clausius-Clapeyron relation, Equation (14),and is used to calculate the constant gas-phase surface concen-tration, Equation (13).

Comparisons to Empirical DataRespiratory aerosols are characterized by a wide range of

particle diameters and Reynolds numbers. Particles inhaled intothe extrathoracic region may vary from the nanometer range toseveral hundred microns (ACGIH 1997; Hinds 1999). Respira-tory aerosols on the order of 5 µm have a momentum responsetime (τp = ρpd2

p/18 µ) of approximately 0.08 ms, indicatingnearly instantaneous acceleration to surrounding conditions andresulting in a particle Reynolds number of nearly zero. In con-trast, the momentum response time of a 100 µm liquid aerosolis on the order of the mean residence time within the upperairways through the trachea, i.e., approximately 30 ms. For aninhaled 100 µm droplet initially at rest, the particle Reynoldsnumber is approximately Rep = 30 in the mid-oral cavity regionand Rep = 10 in the orthopharynx. In addition, near-wall dragmay significantly increase particle Reynolds numbers for respi-ratory aerosols of all sizes (Dahneke 1974; Loth 2000; Longestet al. 2004).

In this study, particle diameters less than 5 µm are notconsidered to avoid the influence of noncontinuum effects onheat and mass transfer. Hence, droplet diameters ranging from5 to 100 µm within the upper respiratory tract with particleReynolds numbers ranging from 0 < Rep ≤ 30 are of interest.Empirical data satisfying these diameter and Reynolds numberspecifications was not found for evaporating multicomponentaerosols of variable volatility. However, evaporation model per-formance is expected to be similar for spherical droplets be-low the onset of external vortex shedding (Rep ≈ 130) and in-ternal liquid vortex formation (Clift et al. 1978). In addition,relatively low evaporation temperatures are necessary. Based onthese specifications, the empirical results of Runge et al. (1998),which are characterized by 30 ≤ Rep ≤ 127, d ≤ 600 µm,and T ≤ 300 K, have been selected as an acceptable match to

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

1:54

03

May

201

3

MULTICOMPONENT AEROSOL EVAPORATION 131

Figure 2. Representative computational flow field results: (a) Velocity vectors, contours of velocity magnitude, and (b) massfraction for an axisymmetric 480 cm n-heptane droplet far from the wall with T∞ = 298 K, u∞ = 100 cm/s ω∞ = 0, and Rep =30.2 (from Longest and Kleinstreuer 2004 with permission).

the respiratory aerosol conditions of interest. For the maximumparticle Reynolds number and diameter considered, the dropletWeber number is approximately 10−4, which indicates domi-nance of the surface tension over inertial forces, such that spher-ical droplets may be assumed.

RESULTSRepresentative computational solutions of the flow field

equations for an n-heptane droplet (T∞ = 298 K and Rep =30.2) are shown in Figures 2a and b, including velocity vectors,contours of velocity magnitude, and mass fractions. Moderatemomentum diffusion due to viscous forces results in noticeableinfluences on the flow field upstream of the droplet and rapiddownstream recovery of a uniform velocity profile. Upstreammass diffusion is much less pronounced due to the relativelyhigh Schmidt number of n-heptane in air (Sc = 2.4) and thecontour levels selected.

Resolved-volume simulations were used to evaluate theavailable Sherwood and Nusselt number correlations that are

needed for the semiempirical ODE-based solution approach.Results are compared to the empirical relations of Ranz andMarshall (1952) and Clift et al. (1978) for respiratory aerosolconditions (0 ≤ Rep ≤ 30) in Figures 3a and b. Numerical es-timates have been computed using RMM1 due to the absenceof variable property effects and blowing velocity in the low-to-moderate flux correlations. Considering Sherwood numbervalues, the correlation of Clift et al. (1978) provides the bestfit to the numerical data. The Ranz and Marshall (1952) cor-relation marginally overpredicts the Sherwood values for parti-cle Reynolds numbers less than approximately 2. Retaining thewidely used form of the Ranz and Marshall (1952) correlation,a best fit to the numeric Sherwood data in the range of 0 ≤Rep ≤ 30 results in

Sh = 2.0 + 0.474Re0.6p Sc0.333 [17a]

This relation provides an improved representation of thedata (r2 = 0.99 and p-value ≤ 0.01), especially in the criticalrange of Rep ≈ 1. Considering Nusselt number values, the more

Figure 3. Comparison of two-dimensional simulation results for uniform flow to (a) Sherwood and (b) Nusselt number correla-tions of Clift et al. (1978) and Ranz and Marshall (1952).

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

1:54

03

May

201

3

132 P. W. LONGEST AND C. KLEINSTREUER

complex expression of Clift et al. (1978) provides the best fit tothe numerical results. Applying the heat and mass transfer anal-ogy, a Nusselt number relation comparable to Equation (17a),can be written

Nu = 2.0 + 0.474Re0.6p Pr0.333 [17b]

This expression results in improved Nusselt number predictionsin the range of Rep ≈ 1 compared to the Ranz and Marshall(1952) correlation. While Nusselt number estimates calculatedusing Equation (17b) are marginally higher than the simulatedvalues, the effect on the evaporation solution is not expected tobe large considering the relatively mild temperature variationsof interest. As such, Equations (17a) and (17b) have been usedin the semiempirical calculation of droplet evaporation basedon their simplicity and familiar form. The effects of implement-ing the more exact expressions of Clift et al. (1978) were alsoassessed, as discussed below.

Based on similarity to multicomponent respiratory aerosolconditions, the evaporation models considered were comparedto the experimental results of Runge et al. (1998) for single-and two-component droplets (Figures 4a–d). Considering a rel-

atively volatile droplet of n-heptane at near body temperatureconditions (T∞ = 298 K), the models considered predict evap-oration consistent with one of two rates (Figure 4a). The ODEand RMM1 solutions significantly overpredict droplet evapo-ration. Minor differences between these two models arise dueto variations in the empirical and computed Nusselt numbersshown in Figure 3b. In fact, implementing the expression ofClift et al. (1978) in the semiempirical ODE approximation re-sults in a solution that is practically indistinguishable from theRMM1 prediction. In contrast to the ODE and RMM1 solu-tions, the remaining models all predict the empirical dropletevaporation rate of n-heptane inline with experimental results(Figure 4a). The only difference between the RMM1 andRMM2 solutions is that the latter accounts for concentration-dependent gas-phase specific heat throughout the flow field. Assuch, it appears that an accurate estimate of variable specificheat is a significant factor, resulting in the improved predictionof the RMM2 approximation. Accounting for the blowing ve-locity (RMM3) only marginally increases the evaporation ratecompared to RMM2, largely due to a marginal increase in wet-bulb temperature. Inclusion of a radial droplet temperature gra-dient (DLM) in addition to the blowing velocity results in a

Figure 4. Computational estimates of normalized droplet surface area (d2/do2) over time compared to the experimental results

of Runge et al. (1998) for (a) high volatility n-heptane with T∞ = 298 K and Rep = 30.2; (b) high volatility n-heptane withT∞ = 272 K and Rep = 107.3; (c) low volatility n-decane with T∞ = 272 K and Rep = 94.1; and (d) multicomponent 50:50heptane-decane mixture with T∞ = 272 K and Rep = 107.0. Descriptions of the numerical models are given in Table 1.

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

1:54

03

May

201

3

MULTICOMPONENT AEROSOL EVAPORATION 133

reduction in the evaporation rate arising from a more accurateprediction of droplet surface cooling; however, this effect ap-pears negligible.

In comparison to Figure 4a, Figure 4b illustrates evapora-tion of a similar 570 µm n-heptane droplet in a lower tempera-ture environment, as may occur in the upper respiratory airwaysunder cold external conditions. Despite a significant increasein Reynolds number (107.3 versus 30.2), the lower tempera-ture environment results in a significant reduction in the dropletevaporation rate (Figure 4b). For n-heptane with Rep = 107.3and T∞ = 272 K, differences between the constant and variablespecific heat models remain evident but are less pronouncedthan in Figure 4a. The constant property assumption of the ODEand RMM1 solutions results in a marginal overprediction ofthe evaporation rate. In contrast, the models that do account forvariable gas-phase specific heat result in a marginal underpre-diction of the evaporation rate. Furthermore, the reduced evap-oration rate has also largely eliminated differences among theRMM2, RMM3, and DLM solutions.

The effects of further decreasing the evaporation rate byconsidering a much lower volatility compound are shown inFigure 4c. Due to the high boiling point of n-decane, evapo-ration at T∞ = 272 K and Rep = 94.1 requires on the order of400 s for the droplet surface area to be reduced by a factor oftwo. Under these conditions, differences among the evaporationmodels considered are much less pronounced. Moreover, allmodels considered appear to predict the evaporation rateeffectively.

Evaporation results for a 570 µm droplet consisting of a50:50 or 50% heptane-decane mixture are shown in Figure 4din comparison to the results of Runge et al. (1998). Due to con-siderable overlap in the results, ODE and DLM solutions arenot shown. For the relatively low temperature environment (T∞= 272 K) and moderate Reynolds number (Rep = 107) con-sidered, all models match experimental results and indicate twodistinct phases of evaporation. First, a majority of the lighterheptane evaporates resulting in a rapid diameter reduction dur-

ing the initial 30 s. Thereafter, the droplet is primarily com-posed of heavier (or less volatile) decane, resulting in a moregradual evaporation rate. Beyond 250 s, the models moderatelyoverpredict the evaporation rate. However, this minor discrep-ancy is consistent with the results for pure decane (Figure 4c).Inclusion of mass concentration gradients within the dropletwill slow the initial evaporation of lighter components, therebyincreasing droplet evaporation at later times. This increase indroplet evaporation rate will further separate the numeric andexperimental results. Therefore, effects of internal mass gradi-ents are expected to be negligible for this scenario.

Considering droplet temperature, rapid cooling occurs un-til the droplet wet-bulb condition is reached, and then remainsrelatively constant due to the balance between evaporative heatloss and convective effects. In comparison to the experimen-tal results of Runge et al. (1998) for the n-heptane dropletconsidered in Figure 4a, the RMM2 solution matches the ob-served droplet temperatures very well (Figure 5a). Similar per-formance is also observed for all other solutions considered.For the 50% heptane-decane mixture, cooling associated withrapid n-heptane evaporation initially results in a large tempera-ture decline (Figure 5b). As the heptane is depleted, evaporativecooling effects are reduced, and the droplet temperature rises.Eventually, the droplet temperature remains constant at the wet-bulb state for a decane droplet evaporating under the specifiedconditions.

Due to relatively cold temperatures and resulting slowevaporation rates, little difference was observed among theevaporation models applied to the heptane-decane mixture con-sidered in Figure 4d. However, comparison of Figures 4a andb indicates the significance of temperature on factors affectingdroplet evaporation. The computed evaporation rate of a mul-ticomponent 50% heptane–decane droplet for near body tem-perature conditions (T∞ = 298 K) and Rep = 107 is shownin Figure 6. For this variable volatility configuration, signifi-cant differences are observed in the simulated evaporation ratedue to inclusion of concentration-dependent gas-phase specific

Figure 5. Variable property (RMM2) estimates of droplet temperature over time for: (a) n-heptane with T∞ = 272 K and Rep =107.3; and (b) multicomponent 50:50 heptane–decane mixture with T∞ = 272 K and Rep = 107.0.

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

1:54

03

May

201

3

134 P. W. LONGEST AND C. KLEINSTREUER

Figure 6. Semiempirical and resolved-volume simulations of droplet evaporation for a multicomponent 50:50 heptane–decanemixture with T∞ = 298 K and Rep = 107. Results for RMM3 and the DLM are practically indistinguishable from the RMM2solution.

heat (i.e., RMM1 versus RMM2). Results for the RMM3 andthe DLM solutions were practically indistinguishable from theRMM2 approximation. Therefore, it may be necessary to ac-count for variable gas-phase specific heat when computingmulticomponent respiratory droplet evaporations where rel-atively high volatility components are present at significantconcentrations.

Accounting for variable gas-phase specific heat resultedin significantly slower evaporation rates in some cases dueto variation in droplet temperature conditions. Consideringthe n-heptane droplet at T∞ = 298 K and Rep = 30.2(Figure 4a), droplet temperatures calculated with RMM1 andRMM2 were 284.2 and 275.4, respectively. The reduced droplettemperature of the variable specific heat case results in lowersurface mass fractions, which produce a reduced evaporationrate. Viewed another way, the inclusion of variable gas-phasespecific heat for this high volatility compound effectively insu-lates the droplet, thereby reducing heat transfer from the sur-rounding environment, which drives evaporation. This variableproperty effect results in a significantly reduced Nusselt num-ber compared to predictions from the empirical correlationsconsidered. For instance, the Nusselt numbers computed fromconstant (RMM1) and variable (RMM2) property models ap-plied to the n-heptane droplet are approximately 4.0 and 2.1,respectively. In contrast, little variation was observed in cal-culations of the Sherwood number. It would be highly advan-tageous if this variable property effect could be approximatedusing the semiempirical ODE-based approach. However, incor-poration of a mass-fraction-weighted gas-phase specific heat on

the droplet surface, which enters the ODE solution through thePrandtl number, affected the final result by less than 2%. More-over, inclusion of blowing velocity had a negligible effect on allsolutions considered.

Relevant to respiratory aerosol conditions, Runge et al.(1998) considered the evaporation of a JP-8 droplet in a 294 Kenvironment with a Rep of 120.3. Solutions of selected evap-oration models (i.e., RMM1 and RMM2) have been evaluatedfor a 12-component surrogate mixture suggested by Edwardsand Maurice (2001) (Table 2) and are compared to the exper-imental results of Runge et al. (1998) (Figure 7). The signifi-cant discrepancy between the computed and empirical resultsis likely due to an incorrect surrogate mixture for droplet evap-oration or inaccuracy of the rapid mixing models for dropletsconsisting of a significant number of compounds. Inclusion ofconcentration gradients in the droplet will slow the initial evap-oration rate by limiting transport of high volatility compoundsto the surface, in the absence of Hill vortex formation. However,evaporation at later times will be increased due to continuedavailability of the lighter components. It appears that the slopesof the predicted evaporation curves remain greater than or equalto the experimental data. This indicates that internal gradientsmay not be responsible for the discrepancy between predictedand experimental values, which is consistent with the computedresults for a binary heptane–decane droplet. Based on calcula-tions with the RMM2 approximation, a surrogate mixture thataccurately predicts droplet evaporation is defined in Table 2,and evaporation rates are shown in Figure 7. For the suggestedsurrogate model, differences among the RMM1 and RMM2

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

1:54

03

May

201

3

MULTICOMPONENT AEROSOL EVAPORATION 135

Table 2Twelve component JP-8 surrogate

Wt % Edwards and Wt % ComputedComponent Maurice (2001) with RMM2

iso-octane 5.0 0.25Methylcyclohexane

5.0 0.25

m-xylene 5.0 0.25Cyclooctane 5.0 0.25Decane 15.0 7.0Butyl benzene 5.0 3.01,2,4,5 tetramethyl 5.0 14.0

benzeneTetralin 5.0 15.0Dodecane 20.0 23.01-methylnaphthalene

5.0 8.0

Tetradecane 15.0 17.0Hexadecane 10.0 12.0

solutions are largely negligible, which indicates that either areduced presence of volatile hydrocarbons or a reduced evapo-ration rate associated with multiple components minimizes theneed to account for variable gas-phase specific heat in this case.As such, the ODE-based semiempirical solution used with thesuggested surrogate mixture provides a computationally effec-tive method for evaluating the evaporation of potentially toxicJP-8 aerosols in the respiratory tract.

DISCUSSIONIn this study, semiempirical and resolved-volume evapora-

tion models were assessed based on comparisons to empiri-cal data for conditions consistent with inhalable multicompo-nent aerosols in the upper respiratory tract. The effects of im-plementing empirical correlations, variable gas-phase specificheats, blowing velocity, and internal droplet temperature gra-dients in computing the evaporation of multicomponent respi-ratory aerosols were evaluated. Of the parameters considered,variable gas-phase specific heat had the largest effect on evap-oration, which was realized for high volatility compounds atnear body temperature conditions. Accounting for this phe-nomenon, an appropriate representative or surrogate mixture ofa complex multicomponent hydrocarbon fuel (JP-8) was iden-tified for future toxicological analysis. In addition, other fac-tors that may influence the evaporation of multicomponent in-haled respiratory aerosols include hygroscopic and rarefied floweffects, as well as reduced internal droplet convective transportfor aerosols smaller than those considered in this study.

Other studies have shown the importance of including vari-able gas-phase properties for evaporating volatile droplets inmoderate-to-high temperature combustion applications (T ≥400 K; Chen et al. 1997; Chiang and Kleinstreuer 1993). How-ever, due to a relatively low temperature environment, vari-able gas-phase properties are typically neglected for respira-tory aerosols (Finlay 2001; Zhang et al. 2004). Results of thisstudy indicate that concentration dependent gas-phase specificheat should be taken into account for respiratory aerosols thatcontain species of similar volatility to heptane at significantconcentrations. For instance, variable gas-phase specific heatis expected to influence the evaporation rate of droplets with

Figure 7. Computational estimates of normalized droplet surface area (d2/do2) over time compared to the experimental results

of Runge et al. (1998) for a twelve-component JP-8 surrogate mixture.

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

1:54

03

May

201

3

136 P. W. LONGEST AND C. KLEINSTREUER

significant amounts of toxic single-ring aromatic compounds,e.g., benzene, and medical aerosol propellants, e.g., HFA 134a.However, evaporation of multicomponent JP-8 aerosols was notlargely affected by variable gas-phase properties for the condi-tions studied. This is largely due to the wide range of volatili-ties present in which heavier compounds limit the availability ofthe lighter fractions on the surface, via Equation (13), therebyreducing the evaporation rate. As such, a semiempirical ODEsolution is appropriate for the evaporation of potentially toxicJP-8 aerosols in the respiratory tract. To compute effectively theevaporation of other aerosols for which variable gas-phase spe-cific heat does play a significant role, semiempirical methodolo-gies are in need of further development. Furthermore, effects ofa solid particle core surrounded by a liquid layer, as may oc-cur with aerosols generated by metered dose inhalers, warrantsadditional investigation.

Due to wide use and potentially toxic effects of JP-8 fuel,other studies have modeled the low temperature evaporationof these aerosols. Catoire et al. (1999) and Benaissa et al.(2002) compared results of the homogeneous mixture modelof Gauthier (1996) for JP-8 to the low-temperature suspended-droplet evaporation results of Runge et al. (1998). Results oftheir homogeneous mixture model were in moderate agree-ment (10–30%) over an indicated time period of 80 s. However,the predicted evaporation curve diverges from the experimentalresults such that significant discrepancies will occur over thelonger time period measured by Runge et al. (1998). To correctthis divergence, Zhang et al. (2004) implemented a modifiedcomposite gaseous diffusion coefficient. Still, the homogeneousmixture model is unable to differentiate among the evaporationof lighter potentially toxic components and heavier inert carri-ers. In contrast, the evaporation models assessed in this study,in conjunction with the surrogate mixtures, can be used to ef-fectively compute multicomponent droplet evaporation, whichaffects deposition, and identify the vaporization of individualpotentially toxic markers. As such, the local deposition and ab-sorption of individual chemical species can be modeled, whichis a critical component for accurate dosimetry estimates andtoxicokinetic analysis.

Variations in the computed JP-8 surrogate blend and the for-mulation suggested by Edwards and Maurice (2001) may be at-tributed to several factors. The surrogate of Edwards and Mau-rice (2001) was established for combustion applications and isbased on the oxidation characteristics of JP-8, which does notensure an equivalent distillation curve or comparable dropletevaporation rates. In addition, the composition of a JP-8 fueldroplet is continually changing due to evaporation. In the ex-periments of Runge et al. (1998), droplet evaporation data maynot have been recorded before some of the lighter componentswere removed by ambient diffusion from either the droplet orthe original batch fuel. This observation emphasizes the needto model both aerosol and vapor transport for accurate doseassessments.

Compared to the time required for droplet evaporation, meanresidence times in the upper airways are very brief (e.g., 30 ms).However, Zhang et al. (2004) showed that droplet evaporationhad a significant effect on the deposition of 5 and 10 µm JP-8fuel aerosols in the upper respiratory tract. This effect is largelydue to the increased residence time of near-wall particles incomparison to mean flow. The presence of the wall creates anadditional drag force on the particle (Dahneke 1974; Loth 2000;Longest et al. 2004), which effectively increases the particleReynolds number and results in enhanced evaporation rates.Flow characteristics within the upper respiratory tract may alsoalter droplet evaporation rates. Enhanced conduction and dif-fusion due to near-wall proximity and shear stress have beenshown to significantly increase heat and mass transfer in largedroplets (Longest and Kleinstreuer 2004). Furthermore, turbu-lent eddies that may arise in the upper respiratory tract willlikely increase evaporation rates, potentially leading to appre-ciable internal heat and mass gradients.

For the droplets considered in this study, thermal Biot num-bers are on the order of unity, which indicates that the rate ofconduction within the liquid droplet is comparable to externalconvection. In contrast, comparisons of the numerical modelsto empirical results indicate that thermal concentration gradi-ents within the droplets considered are largely negligible. Thevalidity of the rapid mixing assumption is therefore due to con-vective mixing effects within the droplets driven by inertial orviscous surface forces. However, Hill vortex formation is notexpected for the droplet sizes and Reynolds numbers consid-ered due to some degree of surface contamination (Clift et al.1978; Sadhal et al. 1997). Therefore, an alternative internal con-vention mechanism may exist within the droplet that resultsin well-mixed temperature and concentration profiles. Furtherstudies are required to quantify internal convective mechanismsand evaluate their persistence for droplets smaller than thoseconsidered here. Should the prediction of internal convectiveeffects become necessary, an effective conduction model couldbe formulated to enhance numerical efficiency.

Both hygroscopic and rarefied flow effects, which frequentlyarise for respiratory aerosols, can be accounted for with themodels evaluated in this study. Considering a droplet in air,noncontinuum effects begin to become significant for diame-ters less than 5 µm (Clift et al. 1978). Similar to the Cunning-ham correction factor for drag in rarefied flows, theoretical andempirical expression are available for heat and mass transfer.Clift et al. (1978) suggested an empirical correlation that re-duces the Nusselt number value of a 5 µm droplet in air byapproximately 5%. Alternatively, corrections for the heat andmass flux terms are proved by Qu et al. (2001a, b). The pres-ence of water vapor will also impact droplet evaporation ratesin the respiratory tract. Gas-phase conditions, including specificheat, can be approximated from knowledge of the spatially andtemporally variable local respiratory humidity. Diffusion coef-ficients for each hydrocarbon in an air-water mixture can be

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

1:54

03

May

201

3

MULTICOMPONENT AEROSOL EVAPORATION 137

approximated from the Stefan-Maxwell relations (Bird et al.1960). Furthermore, water vapor condensation onto an evap-orating droplet may be approximated as an additional speciesin the multicomponent models described.

In conclusion, effective models for predicting multicompo-nent aerosol evaporation in the upper respiratory airways thatare capable of identifying individual compounds for a rangeof volatilities were assessed. Resolved-volume simulations in-dicated that a semiempirical ODE solution including a rapidmixing model assumption is appropriate for large respiratoryaerosols except in cases where high volatility compounds af-fect the local gas-phase specific heat. In conjunction with ap-propriately selected surrogate mixtures, the models consideredprovide an effective method for computing droplet evaporationof complex multicomponent hydrocarbon mixtures in a man-ner that can track individual chemical species for computationaldosimetry analysis. The significant effect of variable gas-phasespecific heat on respiratory aerosol evaporation identified in thisstudy obviates the need for improved semiempirical models thatcan account for this phenomenon. Further experimental and nu-merical studies are also necessary to better assess the evapora-tion behavior of multicomponent respiratory aerosols smallerthan those considered in this study, especially for low particleReynolds number conditions in which internal convection maybe reduced.

REFERENCESAmerican Conference of Industrial Hygienists (ACGIH). (1997). Threshold

Limit Values and Biological Exposure Indices. ACGIH, Cincinnati, OH.Agency for Toxic Substances and Disease Registry (ATSDR). (1998). Toxi-

cological Profile for Jet Fuels JP-5 & JP-8. U.S. Department of Health andHuman Services, Public Health Service, Atlanta, GA.

Aggarwal, S. K., Tong, A. Y., and Sirignano, W. A. (1984). A Comparison ofVaporization Models in Spray Calculations, AIAA J. 22:(10)1448–1457.

Bakshi, K. S., and Henderson, R. F. (1998). Permissible Exposure Levels forSelected Military Fuel Vapors, Inhal. Toxicol. 10:955–961.

Bardon, M. F., and Rao, V. K. (1991). Estimating Instantaneous Properties ofVaporizing Light Petroleum Fractions, Combust. Flame 84:427–431.

Benaissa, A., Gauthier, J. E. D., Bardon, M. F., and Laviolette, M. (2002). Mod-eling Evaporation of Multicomponent Fuel Droplets under Ambient Temper-ature conditions, J. Inst. Energy 75:19–26.

Bird, R. B., Stewart, W. E., and Lightfoot, E. N. (1960). Transport Phenomena,John Wiley & Sons, New York.

Broday, D. M., and Georgopoulos, P. G. (2001). Growth and Deposition ofHygroscopic Particulate Matter in the Human Lungs, Aerosol Sci. Technol.34:144–159.

Catoire, F., Gauthier, J. E. D., Bardon, M. F., and Benaissa, A. (1999). Quasi-Steady State Evaporation Model for Real Multi-Component Fuel Droplets,J. Inst. Energy 72:134–142.

Chen, G., Aggarwal, S. K., Jackson, T. A., and Switzer, G. L. (1997). Exper-imental Study of Pure and Multicomponent Fuel Droplet Evaporation in aHeated Air Flow, Atomization Sprays 7:317–337.

Chiang, H., and Kleinstreuer, (1992a). Computational Analysis of InteractingVaporizing Fuel Droplets on a One-Dimensional Trajectory, Combust. Sci.Technol. 86:289–309.

Chiang, H., and Kleinstreuer (1992b). Transient Heat and Mass Transfer ofInteracting Vaporizing Droplets in a Linear Array, Int. J. Heat Mass Transfer35:(10)2675–2682.

Chiang, H., and Kleinstreuer, (1993). Numerical Analysis of Variable-Fluid-Property Effects on the Convective Heat and Mass Transfer of Fuel Droplets,Combust. Flame 92:459–464.

Clift, R., Grace, J. R., and Weber, M. E. (1978). Bubbles, Drops, and Particles.Academic Press, New York.

Cohen Hubal, E. A., Kimbell, J. S., and Fedkiw, P. S. (1996). Incorporationof Nasal-Lining Mass-Transfer Resistance into a CFD Model for Predictionof Ozone Dosimetry in the Upper Respiratory Tract, Inhal. Toxicol. 8:831–857.

Dahneke, B. (1974). Diffusional Deposition of Particles, J. Colloid InterfaceSci. 48:(3)520–522.

Dwyer, H. A., Stapf, P., and Maly, R. (2000). Unsteady Vaporization and Igni-tion of a Three-Dimensional Droplet Array, Combust. Flame 121:181–194.

Edwards, D. A., and Dunbar, C. (2002). Bioengineering or TherapeuticAerosols, Annu. Rev. Biomed. Eng. 4:93–107.

Edwards, T., and Maurice, L. Q. (2001). Surrogate Mixtures to RepresentComplex Aviation and Rocket Fuels, J. Propul. and Power 17:(2)461–466.

Finlay, W. H. (2001). The Mechanics of Inhaled Pharmaceutical Aerosols.Academic Press, San Diego, CA.

Finlay, W. H., and Stapleton, K. W. (1995). The Effect on Regional Lung De-position of Coupled Heat and Mass Transfer between Hygroscopic Dropletsand Their Surrounding Phase, J. Aerosol Sci. 26:(4)655–670.

Gauthier, J. E. D., Bardon, M. F., and Rao, V. K. (1996). General Flame-Propagation Model for Fuel Droplet, Particle and Vapor Mixtures in Air,J. Inst. Energy 69:59–67.

Hawkins, I. R., and Wilkes, N. S. (1991). Moving Grids in Harwell-FLOW3D,AEA-InTec-0608.

Hiller, F. C. (1991). Health Implications of Hygroscopic Particle Growth in theHuman Respiratory Tract, J. Aerosol Med. 4:1–11.

Hinds, W. C. (1999). Aerosol Technology: Properties, Behavior, and Measure-ment of Airborne Particles, 2nd ed. John Wiley & Sons, New York.

International Commission on Radiological Protection (ICRP). (1994). HumanRespiratory Tract Model for Radiological Protection. ICRP Publication 66,Elsevier Science Ltd., New York.

Kimbell, J. S., Overton, J. H., Subramaniam, R. P., Scholsser, P. M., Morgan,K. T., Conolly, R. B., and Miller, F. J. (2001). Dosimetry Modeling ofInhaled Formaldehyde: Binning Nasal Flux Predictions for Quantitative RiskAssessment, Toxicol. Sci. 64:111–121.

Kleinstreuer, C., Comer, J. K., and Chiang, H. (1993). Fluid Dynamicsand Heat Transfer with Phase Change of Multiple Spherical Droplets ina Laminar Axisymmetric Gas Stream, Int. J. Heat Fluid Flow 14:(3)292–300.

Landis, R. B., and Mills, A. F. (1974). Effects of Internal Resistance on the Va-porization of Binary Droplets. Fifth International Heat Transfer Conference,Tokyo, Japan Paper B7-9.

Law, C. K. (1982). Recent Advances in Droplet Vaporization and Combustion,Prog. Energy Combust. Sci. 8:171–201.

Longest, P. W., and Kleinstreuer, C. (2004). Interacting Effects of UniformFlow, Plane Shear, and Near-Wall Proximity on the Heat and Mass Transferof Respiratory Aerosols, Int. J. Heat Mass Transfer 47:4745–4759.

Longest, P. W., Kleinstreuer, C., and Buchanan, J. R. (2004). Efficient Compu-tation of Micron-Particle Dynamics Including Wall Effects, Comput. Fluids33:577–601.

Loth, E. (2000). Numerical Approaches for Motion of Dispersed Particles,Droplets and Bubbles, Prog. Energy Combust. Sci. 26:161–223.

Martonen, T. B., Bell, K. A., Phalen, R. F., Wilson, A. F., and Ho, A. (1982).Growth Rate Measurements and Deposition Modeling of HygroscopicAerosols in Human Tracheo-Bronchial Models, Ann. Occup. Hyg. 26:93–108.

Martonen, T. B., and Schroeter, J. D. (2003). Risk Assessment DosimetryModel for Inhaled Particulate Matter: II. Laboratory Surrogates, Toxicol.Lett. 138:133–142.

Miller, F. J., Overton, J. H., Jaskot, R. H., and Menzel, D. B. (1985). A Modelof the Regional Uptake of Gaseous Pollutants in the Lung: I. The Sensitivity

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

1:54

03

May

201

3

138 P. W. LONGEST AND C. KLEINSTREUER

of the Uptake of Ozone in the Human Lung to Lower Respiratory Tract Se-cretions and Exercise, Toxicol. Appl. Pharmacol. 79:11–27.

Morrow, P. E. (1986). Factors Determining Hygroscopic Aerosol Depositionin Airways, Physiol. Rev. 66:330–343.

Niazmand, H., Shaw, B. D., Dwyer, H. A., and Aharon, I. (1994). Combust.Sci. Tech. 103:219–233.

Overton, J. H., Kimbell, J. S., and Miller, F. J. (2001). Dosimetry Modeling ofInhaled Formaldehyde: The Human Respiratory Tract, Toxicol. Sci. 64:122–134.

Press, W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, W. T. (1992).Numerical Recipes, 2nd ed. Cambridge University Press, Cambridge, UK.

Qu, X., Davis, E. J., and Swanson, B. D. (2001a). Non-Isothermal DropletEvaporation and Condensation in the Near-Continuum Regime, J. AerosolSci. 32:1315–1339.

Qu, X., and Davis, E. J. (2001b). Droplet Evaporation and Condensation in theNear-Continuum Regime, J. Aerosol Sci. 32:861–875.

Ranz, W. E., and Marshall, W. R. (1952). Evaporation from Drops, Chem. Eng.Prog. 487:141–146, 173–180.

Renksizbulut, M., Nafziger, R., and Li, X. (1991). A Mass Transfer Correla-tion for Droplet Evaporation in High-Temperature Flows, Chem. Eng. Sci.46:(9)2351–2358.

Runge, T., Meske, M., and Polymeropoulou, C. E. (1998). Low-TemperatureVaporization of JP-4 and JP-8 Fuel Droplets, Atomization Sprays 8:25–44.

Sadhal, S. S., Ayyaswamy, P. S., and Chung, J. N. (1997). Transport Phenom-ena with Drops and Bubbles. Springer, New York.

Sirignano, W. A. (1999). Fluid Dynamics and Transport of Droplets andSprays. Cambridge University Press, Cambridge, UK.

Spalding, D. B. (1953). The Combustion of Liquid Fuels, Fourth Symposiumon Combustion. Williams & Wilkins, Baltimore, MD, pp. 847–864.

Stahlhofen, W., Rudolf, G., and James, A. C. (1989). Intercomparison ofExperimental Regional Aerosol Deposition Data, J. Aerosol Med. 2:285–290.

Stapleton, K. W., Finlay, W. H., and Zuberbuhler, P. (1994). An in vitro Methodfor Determining Regional Dosages Delivered by Jet Nebulizers, J. AerosolMed. 7:325–344.

Tannehill, J. C., Anderson, D. A., and Pletcher, R. H. (1997). ComputationalFluid Mechanics and Heat Transfer, 2nd ed. Taylor and Francis, Washington,DC.

Violi, A., Yan, S., Eddings, E. G., Sarofim, A. F., Granata, S., Faravelli, T.,and Ranzi, E. (2002). Experimental Formulation and Kinetic Model for JP-8Surrogate Mixtures, Combust. Sci. Technol 174:(11–12)399–417.

Xin, J., and Megaridis, C. M. (1996). Effects of Rotating Gaseous Flows onTransient Droplet Dynamics and Heating, Int. J. Heat Fluid Flow 17:52–62.

Zhang, Z., Kleinstreuer, C., Kim, C. S., and Cheng, Y. S. (2004). VaporizingMicro-Droplet Inhalation, Transport and Deposition in a Human Upper Air-way Model, Aerosol Sci. Technol. 38:36–49.

Dow

nloa

ded

by [

Bro

wn

Uni

vers

ity]

at 0

1:54

03

May

201

3