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Hygroscopic growth and evaporation in an aerosol with boundary heat and mass transfer
Aerosol Science 38 (2007) 1 – 16 www.elsevier.com/locate/jaerosci
Hygroscopic growth and evaporation in an aerosol with boundary heat and mass transfer Rawad Saleh, Alan Shihadeh∗ Aerosol Research Laboratory, Department of Mechanical Engineering, American University of Beirut, Beirut, Lebanon Received 9 May 2006; received in revised form 28 July 2006; accepted 31 July 2006
Abstract This study demonstrates an experimental method for using temperature measurements as a means for validating computations of particle size distribution in a growing or evaporating high-concentration aerosol flowing in a tube with wall heat and mass transfer. The method is based on the premise that aggregate growth or condensation from an ensemble of aerosol droplets can be inferred by comparing temperature evolution of an aerosol-laden and aerosol-free flow through a heated or cooled tube. The difference in bulk temperature is used as an indicator of latent heat effect which is directly related to condensational and evaporative particle size changes. Dimensional analysis is used to derive the conditions under which such an approach can be used. Two parameters, the “coupling number” and the dimensionless mass concentration are found to govern the sensitivity of continuous phase temperature to aggregate evaporation or condensation of the droplet ensemble, and the sensitivity of temperature to droplet diameter changes, respectively. Experimental data for an aqueous saline aerosol flowing through a heated, constant wall temperature tube are presented and compared to predictions derived using a Lagrangian plug-flow model with a fully moving particle bin structure. Measured and predicted bulk phase temperatures agree to within 3%. Using sensitivity analysis, it is shown that hygroscopic particle diameter changes will be at least as accurate. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Hygroscopic growth; Heat and mass transfer; Plug flow; Aerosol measurement; Nebulizer; Particle size distribution
1. Introduction Predicting changes in size distribution accompanying condensation or evaporation of hygroscopic aerosol droplets is fundamental in studies of inhalation toxicology, aerosol medicine, and atmospheric chemistry. The present work was motivated by an ongoing study of aerosol dynamics in tobacco smoke traversing the flow passages of the narghile waterpipe (aka hooka, shisha), where boundary heat and mass transfer can be important drivers of droplet growth and shrinkage. This problem, like that for inhaled pharmaceutical aerosols, involves a high number and mass concentration aerosol whose droplet phase can significantly impact the saturation ratio of the bulk phase, rendering the evaporation or condensation process two-way coupled. Following the approach of Finlay and Stapleton (1995), we implemented as a first step a computational model for the canonical problem of a polydisperse dilute aqueous saline aerosol in plug flow through a constant wall temperature tube. However, while analytical solutions (Seinfeld & Pandis, 1997; ∗ Corresponding author. Tel.: +961 1 350 000; fax: +961 1 744 462.
E-mail address: [email protected] (A. Shihadeh). 0021-8502/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jaerosci.2006.07.008
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Nomenclature Cm Cp Cr CMD D d h hm hfg k L Le LGR m M m ˙ m ˙ MMD mf n P Q ˙ Q ˙ Q Re Ru S SR t T u¯ x
Fuchs correction factor specific heat (kJ/kg K) Courant number count median diameter vapor diffusion coefficient in air (m2 /s) diameter (m) enthalpy (kJ/kg); heat transfer coefficient (W/m2 K) mass transfer coefficient (m/s) latent heat of vaporization (kJ/kg) thermal conductivity (W/m K) length (m) Lewis number liquid generation rate (g/hr) mass (kg); number of bins molar mass (kg/kmol) mass flow rate (kg/s) mass flow rate per unit volume (kg/m3 s) mass median diameter (m) mass fraction number concentration of droplets (m−3 ) pressure (Pa) volumetric flow rate (m3 /s) heat transfer rate (W) heat transfer rate per unit volume (W/m3 ) Reynolds number universal gas constant (kJ/kmol K) correction factor for solute effect saturation ratio time (s) temperature (K) flow average velocity (m/s) molar concentration (mol/m3 )
Greek letters
g k
thermal diffusivity (m2 /s) temperature depression due to droplet evaporation (K) density (kg/m3 ) geometric standard deviation kolmogorov time scale (s)
Subscripts a b d in i lm m p s t v w
air bulk droplet inlet size bin index log mean measured predicted salt tube vapor wall
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Park, Lee, Shimada, & Okuyama, 2001) can be found in the literature for one-way coupled cases, means for validating computations of size distributions for the two-way coupled problem—with or without boundary effects—are not available. The overarching purpose of this study, then, is to develop an experimental method to validate the model. Several challenges are faced in measuring particle size distribution (PSD) in high concentration volatile aerosols. One approach is to sample the aerosol and send it to a sizing instrument such as a differential mobility analyzer or cascade impactor. If the instrument temperature is different than the sampled aerosol, however, additional phase changes can occur during the measurement (Finlay & Stapleton, 1999). Some studies (e.g. Henning, Massling, Brechtel, & Wiedensohler, 2005) have dealt with this problem by immersing the sizing instrument and sampling lines in a constant temperature bath maintained at the aerosol inlet temperature. While this solves the problem of artificial growth or evaporation, there may be cases where the continuous and droplet phases are not in thermodynamic equilibrium at the sampling location, and will therefore continue to evolve during sample transport and measurement. Furthermore, in the case of a cascade impactor, inter-stage pressure drops induce a directly proportional reduction in local saturation ratio, causing droplet evaporation in the instrument (Hicks, Pritchard, Black, & Megaw, 1989). A second approach is to use non-intrusive optical methods for particle sizing in situ (Sommerfeld & Qiu, 1993). While unobtrusive, this approach suffers from problems of low scattering intensity for sub-micron particles, and coincidence errors at high number densities. Given the difficulties in measuring changes in PSD directly, we consider using an aggregate measure—the amount of aerosol evaporated from the ensemble of droplets—to validate the computations. One option is to measure vapor concentration changes as the aerosol flows down a heated or cooled tube. However, this requires filtering the particle phase upstream of the measurement location, and heating the resulting vapor phase to reduce the saturation ratio since humidity or dew point measurements near saturation carry large uncertainties. The potential cumulative errors involved in this procedure led us to consider the possibility of using continuous phase temperature, a relatively easy and robust measure, as the diagnostic against which the computational model would be validated. The question, then, was under what conditions can gas temperature be used to validate a model of 2-way coupled heat and mass transfer in a plug flow? Put another way: under what conditions is the effect of the aerosol droplets on the temperature of the flow significant enough that it can be discerned above the effect of boundary transport? To elucidate these conditions, we have non-dimensionalized the conservation and rate equations governing the droplet and bulk phases, and derived a non-dimensional parameter, the coupling number, to indicate the degree to which the particle phase impacts the bulk phase. Informed by this dimensional analysis, an experiment was designed to validate the two-way coupled plug flow model. The model formulation, dimensional analysis, experimental setup, and results are presented below. 2. Theory 2.1. Problem description and model Consider an aerosol composed of a volatile droplet and continuous phase initially in thermodynamic equilibrium. When this aerosol is introduced into a tube whose wall temperature and/or vapor concentration is different than that of the aerosol, changes in the droplet size distribution will ensue as a result of evaporation or condensation. The wall thus drives changes in the aerosol size distribution. When an initially cool aerosol enters a heated, dry tube, the wall transfers heat to the bulk phase, which responds in turn with a temperature rise, and therefore a transfer of heat to the droplet phase. The temperature increase of the droplets is accompanied by an increase in surface vapor concentration. The warmed droplets now find themselves in a bulk environment whose vapor concentration is lower than that at the droplet surfaces, causing them to evaporate towards equilibrium. Due to the latent heat required, the droplets experience a temperature drop during evaporation, causing further heat transfer from the relatively warm bulk phase. If the droplet concentration is great enough, the heat transferred to the droplets, and the vapor released from them, significantly reduce the temperature of the bulk phase and increase its vapor concentration, in turn reducing the impetus for further droplet evaporation. The presence of the droplets thus damps the temperature response of the bulk fluid to the boundary heating; without the aerosol, the bulk temperature rise would be higher. This state of affairs where the droplets are affected by the bulk and vice versa is referred to as two-way coupled.
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The problem modeled in this work is that of droplet growth/evaporation in a bounded plug flow. The flow is an air–water vapor mixture with droplets of a dilute aqueous salt solution forming a hygroscopic aerosol, and the boundary is a tube of constant diameter which can be heated or cooled with a constant wall temperature. Mass and energy flow between the bulk and the tube wall, and the droplets and the bulk; the directions and the rates of these flows are functions of the temperature and concentration gradients. The droplet diameters change by the amount of vapor condensed on them from the surroundings (growth) or liquid evaporated to the bulk (evaporation). The particles exhibit a polydisperse distribution spanning submicron and micron sizes and are sufficiently spaced such that local concentration and temperature gradients are unaffected by the presence of other droplets. They are small enough to be considered quiescent in the surrounding bulk, and mass transfer rates are small enough that transport phenomena may be described using Maxwell’s classical quasi-stationary single droplet diffusion equation (Maxwell, 1890), corrected for Knudsen number effects. Ferron (1977) has previously derived criteria for the applicability of the quasi-stationary single droplet equations. The net contribution to the continuous phase is given by summing the contributions of all the droplets in the aerosol. Finally, axial diffusion is neglected. With this formulation, the equations governing the evolution of the bulk and particle phases are given as follows: Bulk phase temperature: dTb ˙ d,i − m = [ni (−Q ˙ d,i Cpv (Td,i − Tb ))] + Q˙ w + m ˙ w Cpv (Tb − Tw ), dt m
(a Cpa + v,b Cpv )
(1)
i=1
˙ d,i = 2di ka (Tb − Td,i ) is heat transfer by conduction from one droplet; Q˙ w = 4 h(Tw − Tb ) is heat transfer where Q dt per unit volume by convection from the tube wall; m˙ w = d4t hm (cw − cb ) is mass transfer per unit volume by convection from the tube wall. Bulk phase vapor concentration: m dv,b (ni m ˙ d,i ) − m ˙ w . =− dt
(2)
i=1
The summations on the right-hand side of Eqs. (1) and (2) are over the droplet size bins. Droplet phase temperature: md C d
dTd,i ˙ d,i + m =Q ˙ d,i hfg . dt
(3)
Droplet mass: Mv dmd,i = −2Cm di D dt Ru
Pvb Pvd,i − Td,i Tb
,
(4)
where md = d 6d and Cm = (1 + Kn)/(1 + (4/3 + 0.377)Kn + 4/3Kn2 ) is the Fuchs correction factor for noncontinuum effects when particle diameter approaches the mean free path of the continuous phase molecules (Fuchs & Sutugin, 1970). The vapor pressure at the droplet surface, Pvd , is increased by curvature and reduced by the presence of salt ions. These are accounted for using the Kelvin correction factor and Raoult’s law, respectively. For the implementation of Raoult’s law for an aqueous NaCl solution, a van’t Hoff factor of 1.85 has been found to be more accurate than the ideal value of 2 (Finlay, 2001). Since coagulation is not allowed, the mass of salt in the droplets remains constant during growth/shrinkage. 3
2.2. Governing dimensionless groups Dimensional analysis was performed to elucidate the parameters governing this problem and to guide the experimental design. Eqs. (1)–(4) can be non-dimensionalized using the scheme shown in Table 1. The convective boundary transport time scale, dCpa / h, was chosen to non-dimensionalize time since the dynamics in this problem are driven by wall transport.
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Table 1 Variables in dimensionless form Variable
Dimensionless form
Time
t∗ =
Bulk temperature
Tb∗ =
ht dt a Cpa Tw − Tb Tw − Tin
Tw − Td Tw − Tin − v,b ∗v,b = v,w v,w − v,in Td∗ =
Droplet temperature Bulk vapor concentration Droplet surface vapor concentration
∗v,d =
Droplet diameter
d∗ =
v,w − v,d v,w − v,in
d din
Table 2 Dimensionless groups for a monodisperse aerosol C1 = 4
hm (v,w − v,in )Cpv h
C5 = 2
C2 = 2
nd in (a Cpa )dt h
C6 = 12
C3 = 2
nd in (v,w − v,in )Cpv Dd t h
C7 = 12
C4 = 4Len−1
C8 = 4
nd in Dd t hm (a Cpa )din [a Cpa dt / h]
w din3 Cpw hf g din D(v,w − v,in )[a Cpa dt / h]
v,w din3 Cpw (Tw − Tin )
din D(v,w − v,in )[a Cpa dt / h]
v,w din3
For the monodisperse case, neglecting the mass of vapor relative to that of air, the resulting equations for the dimensionless temperature and vapor concentration or droplet diameter of the continuous and droplet phases are: dTb∗ = −4Tb∗ + C1 ∗v,b Tb∗ + C2 d ∗ (Tb∗ − Td∗ ) − C3 d ∗ (∗v,b − ∗v,d )(Tb∗ − Td∗ ), dt ∗ d∗v,b dt ∗
= −C4 ∗v,b + C5 (∗v,d − ∗v,b )d ∗ ,
(5)
(6)
dTd∗ 1 1 = C6 ∗2 (Tb∗ − Td∗ ) + C7 ∗2 (∗v,b − ∗v,d ), dt ∗ d d
(7)
1 dd ∗ = C8 ∗ (∗v,d − ∗v,b ), dt ∗ d
(8)
where the dimensionless groups C1 –C8 are given in Table 2. The terms in Eq. (5) involving C1 and C3 are generally negligible since the energy required to bring the vapor leaving the wall or droplets to thermal equilibrium with the bulk is very small. Furthermore, if Lewis number is approximately unity, as for air–water vapor, C4 ≈ 1 and C5 ≈ C2 . Thus, the problem of hygroscopic growth or evaporation is usually governed by four dimensionless parameters only: C2 , and C6 .C8 . It can be seen from the bulk phase equations (Eqs. (6) and (7)) that C2 is the only parameter that links the droplet phase temperature and surface concentration to the state of the continuous phase. That is, if C2 were zero, the problem would be strictly one-way coupled;
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Fig. 1. Initial and final distributions of a growing initially lognormal aerosol for the numerical model and analytical solution by Seinfeld and Pandis (1997). The initial distribution was discretized using 20 bins.
conversely, the greater its magnitude, the stronger the coupling. C2 is therefore dubbed the “coupling number”. Physically, it represents the ratio of the rates of diffusive transport from the droplet phase to convective transport from the wall. Put another way, it represents the rate at which droplets can scavenge (or supply) vapor or heat relative to the rate at which the wall can supply (or absorb) it. When two-way coupling is important, the rates are of similar magnitude. 2.3. Numerical solution A Lagrangian approach is employed where a volume of aerosol with moving control surfaces is tracked in time. The volume is the local volumetric flowrate multiplied by the time step, and is geometrically represented as a differential cylinder that takes the diameter of the tube walls. At each time step, it moves by its height (l) (i.e. x = l) so that the upper control surface at ti is at the same position of the lower control surface at ti−1 . This assures that no two consecutive control volumes overlap, and thus a spatial distribution of properties can be obtained directly. It should also be noted that this scheme always results in a unity Courant number (Cr = t/x). The particle size distribution is discretized into m bins using a full-moving bin structure. Each bin has a constant number of droplets represented by a single diameter. During growth/shrinkage, the representative diameter of the bin increases/decreases in accordance with the conservation equations. The reader is referred to Jacobson (2000) for a presentation of bin structures and attendant numerical diffusion problems. The conservation equations are solved simultaneously for each bin and for the continuous phase, resulting in a system of 2m + 2 ordinary differential equations. The system of equations is discretized using a first order explicit finite difference scheme (higher order and implicit schemes were also implemented for comparison). The resulting computer code was checked for errors by comparing numerical results to analytical solutions for simplified cases. The first case was for an initially log-normal volatile aerosol released into a constant saturation ratio infinite environment (i.e., a one-way coupled problem). The analytical solution for the evolution of the particle size distribution is given by Seinfeld and Pandis (1997). As shown in Fig. 1, the numerical and analytical solutions are in agreement. The second case was for warm humid air flowing through a cooled tube with condensation at the wall (Brouwers, 1990). The numerical and analytical solutions were also in good agreement. The first case validated the particle size distribution discretization and particle heat/mass transfer in the code, while the second case validated continuous phase wall heat/mass transfer predictions.
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Fig. 2. Effect of droplets on continuous phase temperature.
3. Validation approach For the reasons given in the Introduction, the possibility of using an aggregate parameter, such as continuous phase temperature, to validate a model designed to predict size distribution changes due to growth or evaporation is attractive. The question is under what conditions is bulk temperature sufficiently sensitive to evaporative or condensational droplet size change that it can be used to verify changes in particle size distribution? More specifically, if the computational model over-predicted aggregate droplet evaporation by, say, 10% would the predicted and experimental continuous phase temperatures deviate enough so that the error could be detected? Consider a flow of air through constant wall temperature (Tw ) heated tube in which average temperature is measured at several axial locations downstream of the tube inlet, resulting in a temperature profile such as shown by curve A in Fig. 2. If the air were heavily laden with volatile aerosol droplets, and the flow characterized by a large coupling number, the temperature rise would be attenuated, resulting in curve B in Fig. 2. The temperature depression of the droplet-laden flow relative to the droplet-free flow is related to the latent heat required for evaporation, which in turn is related to the droplet mass evaporated. The greater the temperature depression of curve B relative to A, denoted by the symbol , the greater the amount of aerosol evaporated, and the greater the change in particle diameter. From an experimental perspective in which temperature measurements carry an inherent uncertainty, it is desirable that (1) grows rapidly, and (2) that is sensitive to particle diameter change resulting from condensation or evaporation. It can be shown that the variables invoked in conditions (1) and (2) can be calculated as ∗ dTb,A d∗ dTb,B = − = 2C2 d ∗ (Tb∗ − Td∗ ) ∗ ∗ dt dt dt ∗
(9)
and d∗ C2 ∗2 (Tb∗ − Td∗ ) = d . 2 C8 dd ∗ (Cd∗ − Cb∗ )
(10)
Thus, it can be concluded that the coupling number (C2 ) and its ratio to C8 (C2 /C8 ) are the key dimensionless groups for designing an experimental setup to validate the two-way coupled model where is the diagnostic. In particular, for to be measurable, the coupling number should be large, and for to be sensitive to diameter changes, C2 /C8 3 /( should be large. The ratio C2 /C8 can be expressed as C2 /C8 = nw din v,w − v,in )Le, which for Le = 1 is the ratio of the droplet mass to the mass which must be evaporated to bring the vapor phase to equilibrium with the wall. It can be regarded as the dimensionless droplet mass concentration. This ratio has been previously identified using a different approach by Finlay (1998) for analyzing diameter changes of inhaled pharmaceutical aerosols. A ratio larger than unity indicates that there is sufficient liquid in the aerosol to bring the system to equilibrium. Whereas the coupling number relates to the transport kinetics, the dimensionless mass concentration is related to thermodynamic equilibrium. The implications are further explored in the Results section below.
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Fig. 3. Schematic of the experimental setup.
4. Experimental The experimental setup, shown in Fig. 3, consists of three main subsystems: an aerosol droplet generator, a quasi-plug flow heated wall reactor, and a sampling system. 4.1. Aerosol generation and sizing A 3-jet Collision nebulizer (BGI, Inc.) is used to generate an aqueous NaCl test aerosol. The nebulizer is kept in an ice bath to reduce effects of ambient temperature fluctuations during an experiment, and to minimize changes in salt concentration resulting from water evaporation. The pressurized air supply is provided via an air filter/diffusion dryer (TSI model 3074B) which provides particle-free, zero humidity air. The flow rate of the supplied air is monitored by a digital mass flow meter (Omega model FMA-1600A). To measure the size distribution of the generated droplets, the residual technique is employed as described by Roth and Gebhart (1996). The jet exiting the nebulizer is isokinetically sampled at a rate of 0.5 SLPM and diluted by a 1.5 SLPM filtered nitrogen stream. This is sufficient to remove the moisture from the droplets. The resulting dry salt particles are then sent to a QCM cascade impactor (California Measurements, Model 2H) for sizing, and the resulting data inverted to obtain the original droplet size distribution based on the known salt concentration. The mass concentration of the droplets is also obtained using a residual technique. The nebulized aerosol is sampled superisokinetically via an open face filter situated at the outlet of the nebulizer, and the mass of the generated salt is obtained gravimetrically after the filter is dried in an IR moisture analyzer (AND model MX-50). 4.2. Plug flow reactor The plug flow reactor is constructed from a copper tube (ID = 6.7 mm) concentric with a plexiglass heating jacket, through which hot water circulates to provide the constant tube wall temperature. Hot water circulation is provided by an automatically controlled heater/chiller (Julabo model F33), and the wall temperature is monitored by K-type thermocouples fixed to the external tube wall at three axial locations. Quasi-plug flow is achieved by turbulence-driven radial mixing. Fig. 8 in the Results section shows typical centerline temperature evolution with distance downstream of the inlet, where it can be seen that the temperature rises from
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Table 3 Experimental conditions (mean ± SEM)
(W/m2 K)
Tube heat transfer coeff. Nebulizer liquid generation rate (g/h) Nebulizer MMAD (m) Geometric standard deviation Droplets number concentration (/m3 ) Non-dimensional parameter Coupling number (C2 ) C6 C7 C8 Mass concentration (C2 /C8 )
17 SLPM
10 SLPM
39.3 ± 0.5 6.29 ± 0.36 1.10 ± 0.05 1.68 1.2 × 1013
19.1 ± 0.3 4.98 ± 0.05 1.19 ± 0.07 1.41 1.5 × 1013
64 1500 2662 132 0.484
175 3200 5552 276 0.635
the outset, indicating rapid mixing between the wall and core consistent with the plug-flow assumption. The droplets are expected to remain locally quiescent in the surrounding bulk despite the turbulent flow. This can be deduced by comparing the magnitudes of the particle relaxation time to the Kolmogorov time scale, k . It can be conservatively ¯ −0.5 (Pope, 2000) which for the baseline 17 slpm experimental flow rate corresponds to approximated as k ∼ dt /uRe −4 a value of the order [10 s], or two orders of magnitude greater than the relaxation time of a 1 m particle. It can be assumed therefore that the particles are capable of following the smallest turbulent eddies in the flow. By comparing the aerosol dry mass concentration at the inlet and outlet of the reactor, it was confirmed that deposition on the tube walls was negligible. 4.3. Measurement of temperature and heat transfer coefficient As shown in Fig. 3, centerline temperature was measured using a J-type thermocouple probe inserted from the outlet of the plug flow reactor to any axial position. The probe is supported by a series of guides attached to it that also serve to maintain the probe tip at the tube centerline. The thermocouple probe was extensively tested for conduction errors resulting from contact of the probe supports with the heated wall. It was also calibrated against an NIST-traceable high-accuracy thermistor. The average wall heat transfer coefficient at each flow rate for use in Eq. (1) was obtained by measuring temperature change of dry air flowing in the tube. The heat transfer coefficient was calculated as h=
m ˙ a cp (Tin − To ) /4dt2 LTlm
(11)
where Tin and To are measured inlet and outlet temperatures, and Tlm = Tin − To / ln(Tin /To ) the log mean temperature difference for which T = Tw − T . 4.4. Experimental conditions The setup was run at two flow rates, 10 and 17 SLPM, corresponding to flow Reynolds numbers of 2100 and 3600, respectively. The reactor wall temperature was maintained at 35 ◦ C and the solution in the nebulizer was 1 wt% NaCl for all the experiments. Humidification of the nebulizer supply air caused negligible changes in salt concentration during each experiment, as determined by electrical conductivity measurements. The temperature of the nebulizer varied from one experiment to another due to variations in the compressed air and ambient temperatures, but this variation is accounted for in the model predictions via the input inlet condition of the aerosol. For model predictions, the droplet and bulk phases at the tube inlet were assumed to be in phase equilibrium. Three experiments were performed at each flow rate, and each experiment resulted in nine axial temperature readings spaced 5 cm apart. The experimentally determined heat transfer coefficients as well as the nebulizer output and MMAD (both measured immediately downstream of the reduction fitting that couples the nebulizer to the heated tube) are given in Table 3. The heat transfer coefficients fall between the values predicted for laminar and fully turbulent cases which are 14 W/m2 K
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Fig. 4. Temperature depression versus droplet diameter for three monodisperse aerosols. A steeper slope indicates greater sensitivity of to changes in d ∗ . (a) C2 = 64, C2 /C8 = 0.5, (b) C2 = 64, C2 /C8 = 0.13, (c) C2 = 2.6, C2 /C8 = 0.5.
and 42.1 W/m2 K for the 10 SLPM case and 14 W/m2 K and 64.4 W/m2 K for the 17 SLPM case (Incropera & De Witt, 2002), as expected of flows with transition regime Reynolds numbers used in this study. 5. Results and discussion 5.1. Predicted coupling number and dimensionless mass concentration effects The effect of coupling number and dimensionless mass concentration on temperature depression and its relationship to dimensionless droplet diameter were explored with the model using three different monodisperse model aerosols: (a) the high coupling number/high concentration base-case (n = 1.2 × 1013 /m3 , d = 1.1 m) experimental aerosol produced at a flow of 17 SLPM (C2 = 64, C2 /C8 = 0.5), (b) an aerosol with two-times the number concentration and half the diameter, resulting in a high coupling number/low mass concentration aerosol (C2 = 64, C2 /C8 = 0.13), and (c) a low coupling number/high concentration (C2 = 2.6, C2 /C8 = 0.5) aerosol with a number concentration 1/125 times, and a diameter 5 times that of the base case. Fig. 4 shows the predicted relationship between temperature depression and dimensionless diameter for the first 10 cm of the heated tube for the three cases. It can be seen that for the base case a temperature depression of almost 3 ◦ C is predicted for a reduction in d ∗ to 0.9. For the high coupling number/low concentration aerosol, it can be seen that the diameter change is much greater for approximately the same temperature depression as the base case (i.e., has become a considerably less sensitive to changes in d ∗ ). This derives from the fact that the lower the aerosol droplet concentration, the greater change in diameter is required to provide the mass vaporized. Finally, for the low coupling number/high concentration case, it can be seen that the trajectory of the curve is the same as for the base case, but that the process has proceeded to a lesser extent due to the slower kinetics. Thus, it can be seen that the sensitivity of to changes in d ∗ is determined by the dimensionless aerosol concentration, C2 /C8 , as predicted by Eq. (10) above. Fig. 5 shows the predicted temperature depressions for the same three cases. It can be seen that the temperature depression is insensitive to dimensionless aerosol concentration, but is highly sensitive to coupling number; the coupling number determines the temperature depression regardless of the dimensionless aerosol mass concentration. As mentioned above, the coupling number represents the characteristic rates of diffusive transport at the droplet surfaces relative to the convective transport at the wall. Thus the lower the coupling number, the less the droplets will be able to absorb the heat provided by the wall, and the further the continuous and discrete phases will deviate from thermodynamic equilibrium as the aerosol travels down the heated tube. In this case, a low coupling number will mean a surplus temperature build-up in the bulk phase.
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Fig. 5. Temperature depression versus distance from tube inlet. (a) C2 = 64, C2 /C8 = 0.5, (b) C2 = 64, C2 /C8 = 0.13, (c) C2 = 2.6, C2 /C8 = 0.5.
Fig. 6. Bulk temperature versus axial distance from tube inlet. Tube wall is heated until x = 0.09 m, beyond which it is adiabatic. Further evolution in bulk temperature is a measure of deviation from thermodynamic equilibrium due to low droplet-phase transport rates. (a) C2 = 64, C2 /C8 = 0.5, (b) C2 = 64, C2 /C8 = 0.13, (c) C2 = 2.6, C2 /C8 = 0.5.
This can be seen directly in Fig. 6, where the wall heat transfer is turned off after some length in the tube. As shown, for the high coupling number cases, the bulk phase temperature stabilizes almost immediately after the wall heat transfer is turned off, indicating that the droplets are always “fast” enough to absorb heat supplied by the wall. In contrast, the bulk temperature in the low coupling number case continues to evolve for the entire length of the tube as the large “temperature surplus” provided by wall heat transfer in the first 10 cm, is gradually dissipated into the droplets. For the same reason, the droplet diameter in the high coupling number cases does not continue to evolve once the tube becomes adiabatic, as shown in Fig. 7. In summary, it can be seen that the coupling number and dimensionless concentration are the key dimensionless parameters that determine whether (1) a significant temperature depression is achievable, and (2) the temperature depression is sensitive to changes in droplet diameter. For the experimental setup used in this work, the coupling
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Fig. 7. Droplet diameter versus axial distance from tube inlet. Tube wall is heated until x = 0.09 m, beyond which it is adiabatic. (a) C2 = 64, C2 /C8 = 0.5, (b) C2 = 64, C2 /C8 = 0.13, (c) C2 = 2.6, C2 /C8 = 0.5.
Fig. 8. Predicted and measured bulk temperature profiles. Profiles for the no-aerosol cases are also shown.
number and dimensionless mass concentration (Table 3) are such that a significant change in diameter should occur across the tube, and that this diameter change will be accompanied by a large, measurable temperature depression. 5.2. Measured and predicted temperature Fig. 8 shows typical experimental and predicted temperature for the two flow conditions. The agreement is good considering that the magnitude of the variations is small compared to the magnitude of the temperature depression relative to the no-aerosol case. For reference, Table 4 tabulates the measured and predicted centerline temperatures at x = 0, 10, 20 and 40 cm for the six experiments performed. Temperatures at nine axial locations were measured for each of the six experiments, resulting in a data set of 54 measurements which spanned t ∗ over a range of 0–0.24. For each point, the predicted and measured were calculated,
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Table 4 Measured and predicted bulk temperatures (◦ C) at inlet and three downstream locations Flow rate
10 10 10 17 17 17
x=0
6.0 5.7 6.1 5.2 5.4 7.4
x = 10 cm
x = 20 cm
x = 40 cm
Meas
Pred
Meas
Pred
Meas
Pred
9 8.2 8.5 8.1 8.2 10.2
8.6 8.4 8.7 8.6 8.7 10.1
11.1 10.7 11.2 10.8 10.6 12.0
10.8 10.5 10.8 11.2 11.3 12.5
13.8 13.8 13.8 15.7 15.7 17.0
14.3 14.1 14.3 15.7 15.8 16.7
Fig. 9. Predicted versus measured temperature depression for the six experiments performed. Each experiment has temperature readings at nine axial locations resulting in a total of 54 data points.
and the pooled results regressed to determine the average prediction error. As shown in Fig. 9, the predicted and measured temperature depressions are highly correlated, and the model under predicts the temperature depression by less than 3% on average. While it is clear that the model is capable of accurately predicting temperature evolution in a highly coupled evaporating aerosol flow, the significance of these results to particle size distribution can only be assessed in light of the sensitivity of temperature depression to mass of the droplet phase evaporated. That is, if the droplet diameters predicted by the model were erroneous by a certain factor, how would this affect the measured temperature depression? This was evaluated by perturbing the droplet evaporation rate by applying a multiplier (varied in value from 0.7 to 1.3) to the vapor concentration at the droplet surface (Eq. (4)), and recording the resulting temperature depression and droplet diameter of mean mass. In this way the predicted diameters could be manipulated without violating mass or energy conservation. This sensitivity analysis was performed at the midpoint (0.2 m) of the heated tube for the 17 SLPM case. It can be seen in Fig. 10 that a droplet diameter perturbation of ±5% results in a ±15% perturbation in temperature depression. Therefore less than 3% average error in temperature depression reported in Fig. 9 indicates less than 1% error in droplet diameter at x = 20 cm, for a monodisperse aerosol. Similar results are obtained at other axial locations and using polydisperse aerosols. It can therefore be deduced that the model is at least as accurate in predicting diameter changes as it is bulk temperature in a two-way coupled flow.
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Fig. 10. Sensitivity of temperature depression to droplet diameter at x =0.2 m. Subscript “base” corresponds to the case where the vapor concentration at the surface of the droplets is not perturbed.
Fig. 11. Calculated particle size distribution at several axial distances from the tube inlet for the 17 SLPM case.
5.3. Calculated particle size distribution and energy balance Fig. 11 shows the calculated evolution in particle size distribution for the 17 SLPM case. It is evident that while the particulate mass concentration decreases, the MMD actually increases during evaporation due to the more rapid evaporation of small droplets relative to large ones. For distributions with smaller geometric standard deviations, MMD decreased with distance downstream of the inlet, since the differences in evaporation rates between large and small particles in a narrower spectrum of sizes were not large enough to cause this effect. Also apparent in Fig. 11 is that the aerosol becomes bimodal by the time it exits the tube. The sub-micron particles attain such a high salt concentration that the attendant reduction in vapor pressure essentially arrests further evaporation, while the larger particles continue evaporating. The differences in salt concentration at various axial distances can be
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15
Fig. 12. Salt mass fraction distribution at different axial locations for the 17 SLPM case.
Fig. 13. Distribution of energy delivered by the heated wall to the flowing aerosol for the 17 SLPM case.
seen in Fig. 12. This effect has been previously noted by Finlay (2001) and is important in the context of regional lung deposition of pharmaceutical aerosols. Fig. 13 shows an energy balance for the same 17 SLPM condition. Essentially all the heat delivered by the tube wall is converted to latent heat and sensible heat of the air. Sensible heat to the water, whether in liquid or vapor phase, is negligible. This illustrates the justification of neglecting the terms involving C1 and C3 in Eq. (5), and also illustrates that is essentially a measure of the latent heat required of the evaporation process. If this were not the case, would not be as sensitive a measure of evaporation, and therefore, droplet diameter changes. 6. Conclusions While modeling hygroscopic growth or evaporation in flowing high concentration aerosols with boundary heat and mass transfer is important for understanding deposition and other dynamics in a variety of applications, experimental
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data against which computational models can be validated is not readily available. In this study, we have demonstrated a simple means by which the aggregate evaporation or condensation from an ensemble of aerosol droplets can be inferred by comparing temperature evolution of an aerosol-laden and aerosol-free flow through a heated or cooled tube. Dimensional analysis of the governing equations revealed two parameters which dictate the conditions under which temperature measurements can be used for this purpose. The first parameter, dubbed the “coupling number” is a non-dimensional ratio of the rate of aggregate transport (of heat or mass) to the droplet surfaces to the rate of transport from the tube wall. The second dimensionless parameter is the aerosol mass concentration normalized by the mass required to bring about phase equilibrium with the tube wall. The coupling number dictates the degree to which aerosol temperature is affected by evaporation, while the dimensionless mass concentration indicates whether the evaporation is accompanied by significant diameter changes. When coupling number is much greater than unity, and the dimensionless mass concentration of the order 1 or larger, large temperature effects will be associated with hygroscopic changes in the particle size distribution. Experiments with a 1 wt% saline solution in a quasi-plug flow reactor at a constant wall temperature of 35 ◦ C and two coupling number and dimensionless mass concentrations indicated that the plug flow model previously developed by Finlay and Stapleton (1995) provides accurate predictions of hygroscopic changes in particle size distribution. Acknowledgments The authors acknowledge Joseph Nassif, George Jurdi, Joseph Zoulikian, and Ramzi Safi of the Faculty of Engineering and Architecture Machine Shop for their roles in building the experimental setup. The authors also acknowledge Prof Issam Lakkis for his valuable comments on an earlier draft of this manuscript. This work was funded by the University Research Board at the American University of Beirut. References Brouwers, H. J. H. (1990). Film models for transport phenomena with fog formation: The classical film model. International Journal of Heat and Mass Transfer, 35, 1. Ferron, G. A. (1977). The size of soluble aerosol particles as a function of the humidity of the air. Application to the human respiratory tract. Journal of Aerosol Science, 8, 251. Finlay, W. H. (1998). Estimating the type of hygroscopic behavior exhibited by aqueous droplets. Journal of Aerosol Medicine, 11, 221. Finlay, W. H. (2001). The mechanics of inhaled pharmaceutical aerosols. New York: Academic Press. Finlay, W. H., & Stapleton, K. W. (1995). The effect of regional lung deposition of coupled heat and mass transfer between hygroscopic droplets and their surrounding phase. Journal of Aerosol Science, 26, 655. Finlay, W. H., & Stapleton, K. W. (1999). Undersizing of droplets from a vented nebulizer caused by aerosol heating during transit through an anderson impactor. Journal of Aerosol Science, 30, 105. Fuchs, N. A., & Sutugin, A. G. (1970). Highly dispersed aerosols. Ann Arbor: Ann Arbor Science Publ. Henning, T., Massling, A., Brechtel, F. J., & Wiedensohler, A. (2005). A tandem DMA for highly temperature-stabilized hygroscopic particle growth measurements between 90% and 98% relative humidity. Journal of Aerosol Science, 36, 1210. Hicks, J. F., Pritchard, J. N., Black, A., & Megaw, W. J. (1989). Measurement of growth due to condensation for some common aerosols. Journal of Aerosol Science, 20, 289. Incropera, F. P., & De Witt, D. P. (2002). Fundamentals of heat and mass transfer. New York: Wiley. Jacobson, M. Z. (2000). Fundamentals of atmospheric modeling. New York: Cambridge University Press. Maxwell, J. C. (1890). The scientific papers of Clerk Maxwell. London: Cambridge University Press. Park, S. H., Lee, K. W., Shimada, M., & Okuyama, K. (2001). Alternative analytical solution to condensational growth of polydisperse aerosols in the continuum regime. Journal of Aerosol Science, 32, 187. Pope, S. (2000). Turbulent flows. Cambridge, UK: Cambridge University Press. Roth, C., & Gebhart, J. (1996). Aqueous droplet sizing by inertial classification. Particle Systems Characteristics, 13, 289. Seinfeld, J. H., & Pandis, S. N. (1997). Atmospheric chemistry and physics. New York: Wiley. Sommerfeld, M., & Qiu, H. H. (1993). Characterization of particle-laden, confined swirling flows by phase Doppler anemometry and numerical calculation. International Journal of Multiphase Flow, 19, 1093.
Hygroscopic growth and evaporation in an aerosol with boundary heat and mass transfer Rawad Saleh, Alan Shihadeh∗ Aerosol Research Laboratory, Department of Mechanical Engineering, American University of Beirut, Beirut, Lebanon Received 9 May 2006; received in revised form 28 July 2006; accepted 31 July 2006
Abstract This study demonstrates an experimental method for using temperature measurements as a means for validating computations of particle size distribution in a growing or evaporating high-concentration aerosol flowing in a tube with wall heat and mass transfer. The method is based on the premise that aggregate growth or condensation from an ensemble of aerosol droplets can be inferred by comparing temperature evolution of an aerosol-laden and aerosol-free flow through a heated or cooled tube. The difference in bulk temperature is used as an indicator of latent heat effect which is directly related to condensational and evaporative particle size changes. Dimensional analysis is used to derive the conditions under which such an approach can be used. Two parameters, the “coupling number” and the dimensionless mass concentration are found to govern the sensitivity of continuous phase temperature to aggregate evaporation or condensation of the droplet ensemble, and the sensitivity of temperature to droplet diameter changes, respectively. Experimental data for an aqueous saline aerosol flowing through a heated, constant wall temperature tube are presented and compared to predictions derived using a Lagrangian plug-flow model with a fully moving particle bin structure. Measured and predicted bulk phase temperatures agree to within 3%. Using sensitivity analysis, it is shown that hygroscopic particle diameter changes will be at least as accurate. 䉷 2006 Elsevier Ltd. All rights reserved. Keywords: Hygroscopic growth; Heat and mass transfer; Plug flow; Aerosol measurement; Nebulizer; Particle size distribution
1. Introduction Predicting changes in size distribution accompanying condensation or evaporation of hygroscopic aerosol droplets is fundamental in studies of inhalation toxicology, aerosol medicine, and atmospheric chemistry. The present work was motivated by an ongoing study of aerosol dynamics in tobacco smoke traversing the flow passages of the narghile waterpipe (aka hooka, shisha), where boundary heat and mass transfer can be important drivers of droplet growth and shrinkage. This problem, like that for inhaled pharmaceutical aerosols, involves a high number and mass concentration aerosol whose droplet phase can significantly impact the saturation ratio of the bulk phase, rendering the evaporation or condensation process two-way coupled. Following the approach of Finlay and Stapleton (1995), we implemented as a first step a computational model for the canonical problem of a polydisperse dilute aqueous saline aerosol in plug flow through a constant wall temperature tube. However, while analytical solutions (Seinfeld & Pandis, 1997; ∗ Corresponding author. Tel.: +961 1 350 000; fax: +961 1 744 462.
E-mail address: [email protected] (A. Shihadeh). 0021-8502/$ - see front matter 䉷 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.jaerosci.2006.07.008
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Nomenclature Cm Cp Cr CMD D d h hm hfg k L Le LGR m M m ˙ m ˙ MMD mf n P Q ˙ Q ˙ Q Re Ru S SR t T u¯ x
Fuchs correction factor specific heat (kJ/kg K) Courant number count median diameter vapor diffusion coefficient in air (m2 /s) diameter (m) enthalpy (kJ/kg); heat transfer coefficient (W/m2 K) mass transfer coefficient (m/s) latent heat of vaporization (kJ/kg) thermal conductivity (W/m K) length (m) Lewis number liquid generation rate (g/hr) mass (kg); number of bins molar mass (kg/kmol) mass flow rate (kg/s) mass flow rate per unit volume (kg/m3 s) mass median diameter (m) mass fraction number concentration of droplets (m−3 ) pressure (Pa) volumetric flow rate (m3 /s) heat transfer rate (W) heat transfer rate per unit volume (W/m3 ) Reynolds number universal gas constant (kJ/kmol K) correction factor for solute effect saturation ratio time (s) temperature (K) flow average velocity (m/s) molar concentration (mol/m3 )
Greek letters
g k
thermal diffusivity (m2 /s) temperature depression due to droplet evaporation (K) density (kg/m3 ) geometric standard deviation kolmogorov time scale (s)
Subscripts a b d in i lm m p s t v w
air bulk droplet inlet size bin index log mean measured predicted salt tube vapor wall
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3
Park, Lee, Shimada, & Okuyama, 2001) can be found in the literature for one-way coupled cases, means for validating computations of size distributions for the two-way coupled problem—with or without boundary effects—are not available. The overarching purpose of this study, then, is to develop an experimental method to validate the model. Several challenges are faced in measuring particle size distribution (PSD) in high concentration volatile aerosols. One approach is to sample the aerosol and send it to a sizing instrument such as a differential mobility analyzer or cascade impactor. If the instrument temperature is different than the sampled aerosol, however, additional phase changes can occur during the measurement (Finlay & Stapleton, 1999). Some studies (e.g. Henning, Massling, Brechtel, & Wiedensohler, 2005) have dealt with this problem by immersing the sizing instrument and sampling lines in a constant temperature bath maintained at the aerosol inlet temperature. While this solves the problem of artificial growth or evaporation, there may be cases where the continuous and droplet phases are not in thermodynamic equilibrium at the sampling location, and will therefore continue to evolve during sample transport and measurement. Furthermore, in the case of a cascade impactor, inter-stage pressure drops induce a directly proportional reduction in local saturation ratio, causing droplet evaporation in the instrument (Hicks, Pritchard, Black, & Megaw, 1989). A second approach is to use non-intrusive optical methods for particle sizing in situ (Sommerfeld & Qiu, 1993). While unobtrusive, this approach suffers from problems of low scattering intensity for sub-micron particles, and coincidence errors at high number densities. Given the difficulties in measuring changes in PSD directly, we consider using an aggregate measure—the amount of aerosol evaporated from the ensemble of droplets—to validate the computations. One option is to measure vapor concentration changes as the aerosol flows down a heated or cooled tube. However, this requires filtering the particle phase upstream of the measurement location, and heating the resulting vapor phase to reduce the saturation ratio since humidity or dew point measurements near saturation carry large uncertainties. The potential cumulative errors involved in this procedure led us to consider the possibility of using continuous phase temperature, a relatively easy and robust measure, as the diagnostic against which the computational model would be validated. The question, then, was under what conditions can gas temperature be used to validate a model of 2-way coupled heat and mass transfer in a plug flow? Put another way: under what conditions is the effect of the aerosol droplets on the temperature of the flow significant enough that it can be discerned above the effect of boundary transport? To elucidate these conditions, we have non-dimensionalized the conservation and rate equations governing the droplet and bulk phases, and derived a non-dimensional parameter, the coupling number, to indicate the degree to which the particle phase impacts the bulk phase. Informed by this dimensional analysis, an experiment was designed to validate the two-way coupled plug flow model. The model formulation, dimensional analysis, experimental setup, and results are presented below. 2. Theory 2.1. Problem description and model Consider an aerosol composed of a volatile droplet and continuous phase initially in thermodynamic equilibrium. When this aerosol is introduced into a tube whose wall temperature and/or vapor concentration is different than that of the aerosol, changes in the droplet size distribution will ensue as a result of evaporation or condensation. The wall thus drives changes in the aerosol size distribution. When an initially cool aerosol enters a heated, dry tube, the wall transfers heat to the bulk phase, which responds in turn with a temperature rise, and therefore a transfer of heat to the droplet phase. The temperature increase of the droplets is accompanied by an increase in surface vapor concentration. The warmed droplets now find themselves in a bulk environment whose vapor concentration is lower than that at the droplet surfaces, causing them to evaporate towards equilibrium. Due to the latent heat required, the droplets experience a temperature drop during evaporation, causing further heat transfer from the relatively warm bulk phase. If the droplet concentration is great enough, the heat transferred to the droplets, and the vapor released from them, significantly reduce the temperature of the bulk phase and increase its vapor concentration, in turn reducing the impetus for further droplet evaporation. The presence of the droplets thus damps the temperature response of the bulk fluid to the boundary heating; without the aerosol, the bulk temperature rise would be higher. This state of affairs where the droplets are affected by the bulk and vice versa is referred to as two-way coupled.
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The problem modeled in this work is that of droplet growth/evaporation in a bounded plug flow. The flow is an air–water vapor mixture with droplets of a dilute aqueous salt solution forming a hygroscopic aerosol, and the boundary is a tube of constant diameter which can be heated or cooled with a constant wall temperature. Mass and energy flow between the bulk and the tube wall, and the droplets and the bulk; the directions and the rates of these flows are functions of the temperature and concentration gradients. The droplet diameters change by the amount of vapor condensed on them from the surroundings (growth) or liquid evaporated to the bulk (evaporation). The particles exhibit a polydisperse distribution spanning submicron and micron sizes and are sufficiently spaced such that local concentration and temperature gradients are unaffected by the presence of other droplets. They are small enough to be considered quiescent in the surrounding bulk, and mass transfer rates are small enough that transport phenomena may be described using Maxwell’s classical quasi-stationary single droplet diffusion equation (Maxwell, 1890), corrected for Knudsen number effects. Ferron (1977) has previously derived criteria for the applicability of the quasi-stationary single droplet equations. The net contribution to the continuous phase is given by summing the contributions of all the droplets in the aerosol. Finally, axial diffusion is neglected. With this formulation, the equations governing the evolution of the bulk and particle phases are given as follows: Bulk phase temperature: dTb ˙ d,i − m = [ni (−Q ˙ d,i Cpv (Td,i − Tb ))] + Q˙ w + m ˙ w Cpv (Tb − Tw ), dt m
(a Cpa + v,b Cpv )
(1)
i=1
˙ d,i = 2di ka (Tb − Td,i ) is heat transfer by conduction from one droplet; Q˙ w = 4 h(Tw − Tb ) is heat transfer where Q dt per unit volume by convection from the tube wall; m˙ w = d4t hm (cw − cb ) is mass transfer per unit volume by convection from the tube wall. Bulk phase vapor concentration: m dv,b (ni m ˙ d,i ) − m ˙ w . =− dt
(2)
i=1
The summations on the right-hand side of Eqs. (1) and (2) are over the droplet size bins. Droplet phase temperature: md C d
dTd,i ˙ d,i + m =Q ˙ d,i hfg . dt
(3)
Droplet mass: Mv dmd,i = −2Cm di D dt Ru
Pvb Pvd,i − Td,i Tb
,
(4)
where md = d 6d and Cm = (1 + Kn)/(1 + (4/3 + 0.377)Kn + 4/3Kn2 ) is the Fuchs correction factor for noncontinuum effects when particle diameter approaches the mean free path of the continuous phase molecules (Fuchs & Sutugin, 1970). The vapor pressure at the droplet surface, Pvd , is increased by curvature and reduced by the presence of salt ions. These are accounted for using the Kelvin correction factor and Raoult’s law, respectively. For the implementation of Raoult’s law for an aqueous NaCl solution, a van’t Hoff factor of 1.85 has been found to be more accurate than the ideal value of 2 (Finlay, 2001). Since coagulation is not allowed, the mass of salt in the droplets remains constant during growth/shrinkage. 3
2.2. Governing dimensionless groups Dimensional analysis was performed to elucidate the parameters governing this problem and to guide the experimental design. Eqs. (1)–(4) can be non-dimensionalized using the scheme shown in Table 1. The convective boundary transport time scale, dCpa / h, was chosen to non-dimensionalize time since the dynamics in this problem are driven by wall transport.
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Table 1 Variables in dimensionless form Variable
Dimensionless form
Time
t∗ =
Bulk temperature
Tb∗ =
ht dt a Cpa Tw − Tb Tw − Tin
Tw − Td Tw − Tin − v,b ∗v,b = v,w v,w − v,in Td∗ =
Droplet temperature Bulk vapor concentration Droplet surface vapor concentration
∗v,d =
Droplet diameter
d∗ =
v,w − v,d v,w − v,in
d din
Table 2 Dimensionless groups for a monodisperse aerosol C1 = 4
hm (v,w − v,in )Cpv h
C5 = 2
C2 = 2
nd in (a Cpa )dt h
C6 = 12
C3 = 2
nd in (v,w − v,in )Cpv Dd t h
C7 = 12
C4 = 4Len−1
C8 = 4
nd in Dd t hm (a Cpa )din [a Cpa dt / h]
w din3 Cpw hf g din D(v,w − v,in )[a Cpa dt / h]
v,w din3 Cpw (Tw − Tin )
din D(v,w − v,in )[a Cpa dt / h]
v,w din3
For the monodisperse case, neglecting the mass of vapor relative to that of air, the resulting equations for the dimensionless temperature and vapor concentration or droplet diameter of the continuous and droplet phases are: dTb∗ = −4Tb∗ + C1 ∗v,b Tb∗ + C2 d ∗ (Tb∗ − Td∗ ) − C3 d ∗ (∗v,b − ∗v,d )(Tb∗ − Td∗ ), dt ∗ d∗v,b dt ∗
= −C4 ∗v,b + C5 (∗v,d − ∗v,b )d ∗ ,
(5)
(6)
dTd∗ 1 1 = C6 ∗2 (Tb∗ − Td∗ ) + C7 ∗2 (∗v,b − ∗v,d ), dt ∗ d d
(7)
1 dd ∗ = C8 ∗ (∗v,d − ∗v,b ), dt ∗ d
(8)
where the dimensionless groups C1 –C8 are given in Table 2. The terms in Eq. (5) involving C1 and C3 are generally negligible since the energy required to bring the vapor leaving the wall or droplets to thermal equilibrium with the bulk is very small. Furthermore, if Lewis number is approximately unity, as for air–water vapor, C4 ≈ 1 and C5 ≈ C2 . Thus, the problem of hygroscopic growth or evaporation is usually governed by four dimensionless parameters only: C2 , and C6 .C8 . It can be seen from the bulk phase equations (Eqs. (6) and (7)) that C2 is the only parameter that links the droplet phase temperature and surface concentration to the state of the continuous phase. That is, if C2 were zero, the problem would be strictly one-way coupled;
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Fig. 1. Initial and final distributions of a growing initially lognormal aerosol for the numerical model and analytical solution by Seinfeld and Pandis (1997). The initial distribution was discretized using 20 bins.
conversely, the greater its magnitude, the stronger the coupling. C2 is therefore dubbed the “coupling number”. Physically, it represents the ratio of the rates of diffusive transport from the droplet phase to convective transport from the wall. Put another way, it represents the rate at which droplets can scavenge (or supply) vapor or heat relative to the rate at which the wall can supply (or absorb) it. When two-way coupling is important, the rates are of similar magnitude. 2.3. Numerical solution A Lagrangian approach is employed where a volume of aerosol with moving control surfaces is tracked in time. The volume is the local volumetric flowrate multiplied by the time step, and is geometrically represented as a differential cylinder that takes the diameter of the tube walls. At each time step, it moves by its height (l) (i.e. x = l) so that the upper control surface at ti is at the same position of the lower control surface at ti−1 . This assures that no two consecutive control volumes overlap, and thus a spatial distribution of properties can be obtained directly. It should also be noted that this scheme always results in a unity Courant number (Cr = t/x). The particle size distribution is discretized into m bins using a full-moving bin structure. Each bin has a constant number of droplets represented by a single diameter. During growth/shrinkage, the representative diameter of the bin increases/decreases in accordance with the conservation equations. The reader is referred to Jacobson (2000) for a presentation of bin structures and attendant numerical diffusion problems. The conservation equations are solved simultaneously for each bin and for the continuous phase, resulting in a system of 2m + 2 ordinary differential equations. The system of equations is discretized using a first order explicit finite difference scheme (higher order and implicit schemes were also implemented for comparison). The resulting computer code was checked for errors by comparing numerical results to analytical solutions for simplified cases. The first case was for an initially log-normal volatile aerosol released into a constant saturation ratio infinite environment (i.e., a one-way coupled problem). The analytical solution for the evolution of the particle size distribution is given by Seinfeld and Pandis (1997). As shown in Fig. 1, the numerical and analytical solutions are in agreement. The second case was for warm humid air flowing through a cooled tube with condensation at the wall (Brouwers, 1990). The numerical and analytical solutions were also in good agreement. The first case validated the particle size distribution discretization and particle heat/mass transfer in the code, while the second case validated continuous phase wall heat/mass transfer predictions.
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Fig. 2. Effect of droplets on continuous phase temperature.
3. Validation approach For the reasons given in the Introduction, the possibility of using an aggregate parameter, such as continuous phase temperature, to validate a model designed to predict size distribution changes due to growth or evaporation is attractive. The question is under what conditions is bulk temperature sufficiently sensitive to evaporative or condensational droplet size change that it can be used to verify changes in particle size distribution? More specifically, if the computational model over-predicted aggregate droplet evaporation by, say, 10% would the predicted and experimental continuous phase temperatures deviate enough so that the error could be detected? Consider a flow of air through constant wall temperature (Tw ) heated tube in which average temperature is measured at several axial locations downstream of the tube inlet, resulting in a temperature profile such as shown by curve A in Fig. 2. If the air were heavily laden with volatile aerosol droplets, and the flow characterized by a large coupling number, the temperature rise would be attenuated, resulting in curve B in Fig. 2. The temperature depression of the droplet-laden flow relative to the droplet-free flow is related to the latent heat required for evaporation, which in turn is related to the droplet mass evaporated. The greater the temperature depression of curve B relative to A, denoted by the symbol , the greater the amount of aerosol evaporated, and the greater the change in particle diameter. From an experimental perspective in which temperature measurements carry an inherent uncertainty, it is desirable that (1) grows rapidly, and (2) that is sensitive to particle diameter change resulting from condensation or evaporation. It can be shown that the variables invoked in conditions (1) and (2) can be calculated as ∗ dTb,A d∗ dTb,B = − = 2C2 d ∗ (Tb∗ − Td∗ ) ∗ ∗ dt dt dt ∗
(9)
and d∗ C2 ∗2 (Tb∗ − Td∗ ) = d . 2 C8 dd ∗ (Cd∗ − Cb∗ )
(10)
Thus, it can be concluded that the coupling number (C2 ) and its ratio to C8 (C2 /C8 ) are the key dimensionless groups for designing an experimental setup to validate the two-way coupled model where is the diagnostic. In particular, for to be measurable, the coupling number should be large, and for to be sensitive to diameter changes, C2 /C8 3 /( should be large. The ratio C2 /C8 can be expressed as C2 /C8 = nw din v,w − v,in )Le, which for Le = 1 is the ratio of the droplet mass to the mass which must be evaporated to bring the vapor phase to equilibrium with the wall. It can be regarded as the dimensionless droplet mass concentration. This ratio has been previously identified using a different approach by Finlay (1998) for analyzing diameter changes of inhaled pharmaceutical aerosols. A ratio larger than unity indicates that there is sufficient liquid in the aerosol to bring the system to equilibrium. Whereas the coupling number relates to the transport kinetics, the dimensionless mass concentration is related to thermodynamic equilibrium. The implications are further explored in the Results section below.
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Fig. 3. Schematic of the experimental setup.
4. Experimental The experimental setup, shown in Fig. 3, consists of three main subsystems: an aerosol droplet generator, a quasi-plug flow heated wall reactor, and a sampling system. 4.1. Aerosol generation and sizing A 3-jet Collision nebulizer (BGI, Inc.) is used to generate an aqueous NaCl test aerosol. The nebulizer is kept in an ice bath to reduce effects of ambient temperature fluctuations during an experiment, and to minimize changes in salt concentration resulting from water evaporation. The pressurized air supply is provided via an air filter/diffusion dryer (TSI model 3074B) which provides particle-free, zero humidity air. The flow rate of the supplied air is monitored by a digital mass flow meter (Omega model FMA-1600A). To measure the size distribution of the generated droplets, the residual technique is employed as described by Roth and Gebhart (1996). The jet exiting the nebulizer is isokinetically sampled at a rate of 0.5 SLPM and diluted by a 1.5 SLPM filtered nitrogen stream. This is sufficient to remove the moisture from the droplets. The resulting dry salt particles are then sent to a QCM cascade impactor (California Measurements, Model 2H) for sizing, and the resulting data inverted to obtain the original droplet size distribution based on the known salt concentration. The mass concentration of the droplets is also obtained using a residual technique. The nebulized aerosol is sampled superisokinetically via an open face filter situated at the outlet of the nebulizer, and the mass of the generated salt is obtained gravimetrically after the filter is dried in an IR moisture analyzer (AND model MX-50). 4.2. Plug flow reactor The plug flow reactor is constructed from a copper tube (ID = 6.7 mm) concentric with a plexiglass heating jacket, through which hot water circulates to provide the constant tube wall temperature. Hot water circulation is provided by an automatically controlled heater/chiller (Julabo model F33), and the wall temperature is monitored by K-type thermocouples fixed to the external tube wall at three axial locations. Quasi-plug flow is achieved by turbulence-driven radial mixing. Fig. 8 in the Results section shows typical centerline temperature evolution with distance downstream of the inlet, where it can be seen that the temperature rises from
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Table 3 Experimental conditions (mean ± SEM)
(W/m2 K)
Tube heat transfer coeff. Nebulizer liquid generation rate (g/h) Nebulizer MMAD (m) Geometric standard deviation Droplets number concentration (/m3 ) Non-dimensional parameter Coupling number (C2 ) C6 C7 C8 Mass concentration (C2 /C8 )
17 SLPM
10 SLPM
39.3 ± 0.5 6.29 ± 0.36 1.10 ± 0.05 1.68 1.2 × 1013
19.1 ± 0.3 4.98 ± 0.05 1.19 ± 0.07 1.41 1.5 × 1013
64 1500 2662 132 0.484
175 3200 5552 276 0.635
the outset, indicating rapid mixing between the wall and core consistent with the plug-flow assumption. The droplets are expected to remain locally quiescent in the surrounding bulk despite the turbulent flow. This can be deduced by comparing the magnitudes of the particle relaxation time to the Kolmogorov time scale, k . It can be conservatively ¯ −0.5 (Pope, 2000) which for the baseline 17 slpm experimental flow rate corresponds to approximated as k ∼ dt /uRe −4 a value of the order [10 s], or two orders of magnitude greater than the relaxation time of a 1 m particle. It can be assumed therefore that the particles are capable of following the smallest turbulent eddies in the flow. By comparing the aerosol dry mass concentration at the inlet and outlet of the reactor, it was confirmed that deposition on the tube walls was negligible. 4.3. Measurement of temperature and heat transfer coefficient As shown in Fig. 3, centerline temperature was measured using a J-type thermocouple probe inserted from the outlet of the plug flow reactor to any axial position. The probe is supported by a series of guides attached to it that also serve to maintain the probe tip at the tube centerline. The thermocouple probe was extensively tested for conduction errors resulting from contact of the probe supports with the heated wall. It was also calibrated against an NIST-traceable high-accuracy thermistor. The average wall heat transfer coefficient at each flow rate for use in Eq. (1) was obtained by measuring temperature change of dry air flowing in the tube. The heat transfer coefficient was calculated as h=
m ˙ a cp (Tin − To ) /4dt2 LTlm
(11)
where Tin and To are measured inlet and outlet temperatures, and Tlm = Tin − To / ln(Tin /To ) the log mean temperature difference for which T = Tw − T . 4.4. Experimental conditions The setup was run at two flow rates, 10 and 17 SLPM, corresponding to flow Reynolds numbers of 2100 and 3600, respectively. The reactor wall temperature was maintained at 35 ◦ C and the solution in the nebulizer was 1 wt% NaCl for all the experiments. Humidification of the nebulizer supply air caused negligible changes in salt concentration during each experiment, as determined by electrical conductivity measurements. The temperature of the nebulizer varied from one experiment to another due to variations in the compressed air and ambient temperatures, but this variation is accounted for in the model predictions via the input inlet condition of the aerosol. For model predictions, the droplet and bulk phases at the tube inlet were assumed to be in phase equilibrium. Three experiments were performed at each flow rate, and each experiment resulted in nine axial temperature readings spaced 5 cm apart. The experimentally determined heat transfer coefficients as well as the nebulizer output and MMAD (both measured immediately downstream of the reduction fitting that couples the nebulizer to the heated tube) are given in Table 3. The heat transfer coefficients fall between the values predicted for laminar and fully turbulent cases which are 14 W/m2 K
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Fig. 4. Temperature depression versus droplet diameter for three monodisperse aerosols. A steeper slope indicates greater sensitivity of to changes in d ∗ . (a) C2 = 64, C2 /C8 = 0.5, (b) C2 = 64, C2 /C8 = 0.13, (c) C2 = 2.6, C2 /C8 = 0.5.
and 42.1 W/m2 K for the 10 SLPM case and 14 W/m2 K and 64.4 W/m2 K for the 17 SLPM case (Incropera & De Witt, 2002), as expected of flows with transition regime Reynolds numbers used in this study. 5. Results and discussion 5.1. Predicted coupling number and dimensionless mass concentration effects The effect of coupling number and dimensionless mass concentration on temperature depression and its relationship to dimensionless droplet diameter were explored with the model using three different monodisperse model aerosols: (a) the high coupling number/high concentration base-case (n = 1.2 × 1013 /m3 , d = 1.1 m) experimental aerosol produced at a flow of 17 SLPM (C2 = 64, C2 /C8 = 0.5), (b) an aerosol with two-times the number concentration and half the diameter, resulting in a high coupling number/low mass concentration aerosol (C2 = 64, C2 /C8 = 0.13), and (c) a low coupling number/high concentration (C2 = 2.6, C2 /C8 = 0.5) aerosol with a number concentration 1/125 times, and a diameter 5 times that of the base case. Fig. 4 shows the predicted relationship between temperature depression and dimensionless diameter for the first 10 cm of the heated tube for the three cases. It can be seen that for the base case a temperature depression of almost 3 ◦ C is predicted for a reduction in d ∗ to 0.9. For the high coupling number/low concentration aerosol, it can be seen that the diameter change is much greater for approximately the same temperature depression as the base case (i.e., has become a considerably less sensitive to changes in d ∗ ). This derives from the fact that the lower the aerosol droplet concentration, the greater change in diameter is required to provide the mass vaporized. Finally, for the low coupling number/high concentration case, it can be seen that the trajectory of the curve is the same as for the base case, but that the process has proceeded to a lesser extent due to the slower kinetics. Thus, it can be seen that the sensitivity of to changes in d ∗ is determined by the dimensionless aerosol concentration, C2 /C8 , as predicted by Eq. (10) above. Fig. 5 shows the predicted temperature depressions for the same three cases. It can be seen that the temperature depression is insensitive to dimensionless aerosol concentration, but is highly sensitive to coupling number; the coupling number determines the temperature depression regardless of the dimensionless aerosol mass concentration. As mentioned above, the coupling number represents the characteristic rates of diffusive transport at the droplet surfaces relative to the convective transport at the wall. Thus the lower the coupling number, the less the droplets will be able to absorb the heat provided by the wall, and the further the continuous and discrete phases will deviate from thermodynamic equilibrium as the aerosol travels down the heated tube. In this case, a low coupling number will mean a surplus temperature build-up in the bulk phase.
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Fig. 5. Temperature depression versus distance from tube inlet. (a) C2 = 64, C2 /C8 = 0.5, (b) C2 = 64, C2 /C8 = 0.13, (c) C2 = 2.6, C2 /C8 = 0.5.
Fig. 6. Bulk temperature versus axial distance from tube inlet. Tube wall is heated until x = 0.09 m, beyond which it is adiabatic. Further evolution in bulk temperature is a measure of deviation from thermodynamic equilibrium due to low droplet-phase transport rates. (a) C2 = 64, C2 /C8 = 0.5, (b) C2 = 64, C2 /C8 = 0.13, (c) C2 = 2.6, C2 /C8 = 0.5.
This can be seen directly in Fig. 6, where the wall heat transfer is turned off after some length in the tube. As shown, for the high coupling number cases, the bulk phase temperature stabilizes almost immediately after the wall heat transfer is turned off, indicating that the droplets are always “fast” enough to absorb heat supplied by the wall. In contrast, the bulk temperature in the low coupling number case continues to evolve for the entire length of the tube as the large “temperature surplus” provided by wall heat transfer in the first 10 cm, is gradually dissipated into the droplets. For the same reason, the droplet diameter in the high coupling number cases does not continue to evolve once the tube becomes adiabatic, as shown in Fig. 7. In summary, it can be seen that the coupling number and dimensionless concentration are the key dimensionless parameters that determine whether (1) a significant temperature depression is achievable, and (2) the temperature depression is sensitive to changes in droplet diameter. For the experimental setup used in this work, the coupling
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Fig. 7. Droplet diameter versus axial distance from tube inlet. Tube wall is heated until x = 0.09 m, beyond which it is adiabatic. (a) C2 = 64, C2 /C8 = 0.5, (b) C2 = 64, C2 /C8 = 0.13, (c) C2 = 2.6, C2 /C8 = 0.5.
Fig. 8. Predicted and measured bulk temperature profiles. Profiles for the no-aerosol cases are also shown.
number and dimensionless mass concentration (Table 3) are such that a significant change in diameter should occur across the tube, and that this diameter change will be accompanied by a large, measurable temperature depression. 5.2. Measured and predicted temperature Fig. 8 shows typical experimental and predicted temperature for the two flow conditions. The agreement is good considering that the magnitude of the variations is small compared to the magnitude of the temperature depression relative to the no-aerosol case. For reference, Table 4 tabulates the measured and predicted centerline temperatures at x = 0, 10, 20 and 40 cm for the six experiments performed. Temperatures at nine axial locations were measured for each of the six experiments, resulting in a data set of 54 measurements which spanned t ∗ over a range of 0–0.24. For each point, the predicted and measured were calculated,
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Table 4 Measured and predicted bulk temperatures (◦ C) at inlet and three downstream locations Flow rate
10 10 10 17 17 17
x=0
6.0 5.7 6.1 5.2 5.4 7.4
x = 10 cm
x = 20 cm
x = 40 cm
Meas
Pred
Meas
Pred
Meas
Pred
9 8.2 8.5 8.1 8.2 10.2
8.6 8.4 8.7 8.6 8.7 10.1
11.1 10.7 11.2 10.8 10.6 12.0
10.8 10.5 10.8 11.2 11.3 12.5
13.8 13.8 13.8 15.7 15.7 17.0
14.3 14.1 14.3 15.7 15.8 16.7
Fig. 9. Predicted versus measured temperature depression for the six experiments performed. Each experiment has temperature readings at nine axial locations resulting in a total of 54 data points.
and the pooled results regressed to determine the average prediction error. As shown in Fig. 9, the predicted and measured temperature depressions are highly correlated, and the model under predicts the temperature depression by less than 3% on average. While it is clear that the model is capable of accurately predicting temperature evolution in a highly coupled evaporating aerosol flow, the significance of these results to particle size distribution can only be assessed in light of the sensitivity of temperature depression to mass of the droplet phase evaporated. That is, if the droplet diameters predicted by the model were erroneous by a certain factor, how would this affect the measured temperature depression? This was evaluated by perturbing the droplet evaporation rate by applying a multiplier (varied in value from 0.7 to 1.3) to the vapor concentration at the droplet surface (Eq. (4)), and recording the resulting temperature depression and droplet diameter of mean mass. In this way the predicted diameters could be manipulated without violating mass or energy conservation. This sensitivity analysis was performed at the midpoint (0.2 m) of the heated tube for the 17 SLPM case. It can be seen in Fig. 10 that a droplet diameter perturbation of ±5% results in a ±15% perturbation in temperature depression. Therefore less than 3% average error in temperature depression reported in Fig. 9 indicates less than 1% error in droplet diameter at x = 20 cm, for a monodisperse aerosol. Similar results are obtained at other axial locations and using polydisperse aerosols. It can therefore be deduced that the model is at least as accurate in predicting diameter changes as it is bulk temperature in a two-way coupled flow.
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Fig. 10. Sensitivity of temperature depression to droplet diameter at x =0.2 m. Subscript “base” corresponds to the case where the vapor concentration at the surface of the droplets is not perturbed.
Fig. 11. Calculated particle size distribution at several axial distances from the tube inlet for the 17 SLPM case.
5.3. Calculated particle size distribution and energy balance Fig. 11 shows the calculated evolution in particle size distribution for the 17 SLPM case. It is evident that while the particulate mass concentration decreases, the MMD actually increases during evaporation due to the more rapid evaporation of small droplets relative to large ones. For distributions with smaller geometric standard deviations, MMD decreased with distance downstream of the inlet, since the differences in evaporation rates between large and small particles in a narrower spectrum of sizes were not large enough to cause this effect. Also apparent in Fig. 11 is that the aerosol becomes bimodal by the time it exits the tube. The sub-micron particles attain such a high salt concentration that the attendant reduction in vapor pressure essentially arrests further evaporation, while the larger particles continue evaporating. The differences in salt concentration at various axial distances can be
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Fig. 12. Salt mass fraction distribution at different axial locations for the 17 SLPM case.
Fig. 13. Distribution of energy delivered by the heated wall to the flowing aerosol for the 17 SLPM case.
seen in Fig. 12. This effect has been previously noted by Finlay (2001) and is important in the context of regional lung deposition of pharmaceutical aerosols. Fig. 13 shows an energy balance for the same 17 SLPM condition. Essentially all the heat delivered by the tube wall is converted to latent heat and sensible heat of the air. Sensible heat to the water, whether in liquid or vapor phase, is negligible. This illustrates the justification of neglecting the terms involving C1 and C3 in Eq. (5), and also illustrates that is essentially a measure of the latent heat required of the evaporation process. If this were not the case, would not be as sensitive a measure of evaporation, and therefore, droplet diameter changes. 6. Conclusions While modeling hygroscopic growth or evaporation in flowing high concentration aerosols with boundary heat and mass transfer is important for understanding deposition and other dynamics in a variety of applications, experimental
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data against which computational models can be validated is not readily available. In this study, we have demonstrated a simple means by which the aggregate evaporation or condensation from an ensemble of aerosol droplets can be inferred by comparing temperature evolution of an aerosol-laden and aerosol-free flow through a heated or cooled tube. Dimensional analysis of the governing equations revealed two parameters which dictate the conditions under which temperature measurements can be used for this purpose. The first parameter, dubbed the “coupling number” is a non-dimensional ratio of the rate of aggregate transport (of heat or mass) to the droplet surfaces to the rate of transport from the tube wall. The second dimensionless parameter is the aerosol mass concentration normalized by the mass required to bring about phase equilibrium with the tube wall. The coupling number dictates the degree to which aerosol temperature is affected by evaporation, while the dimensionless mass concentration indicates whether the evaporation is accompanied by significant diameter changes. When coupling number is much greater than unity, and the dimensionless mass concentration of the order 1 or larger, large temperature effects will be associated with hygroscopic changes in the particle size distribution. Experiments with a 1 wt% saline solution in a quasi-plug flow reactor at a constant wall temperature of 35 ◦ C and two coupling number and dimensionless mass concentrations indicated that the plug flow model previously developed by Finlay and Stapleton (1995) provides accurate predictions of hygroscopic changes in particle size distribution. Acknowledgments The authors acknowledge Joseph Nassif, George Jurdi, Joseph Zoulikian, and Ramzi Safi of the Faculty of Engineering and Architecture Machine Shop for their roles in building the experimental setup. The authors also acknowledge Prof Issam Lakkis for his valuable comments on an earlier draft of this manuscript. This work was funded by the University Research Board at the American University of Beirut. References Brouwers, H. J. H. (1990). Film models for transport phenomena with fog formation: The classical film model. International Journal of Heat and Mass Transfer, 35, 1. Ferron, G. A. (1977). The size of soluble aerosol particles as a function of the humidity of the air. Application to the human respiratory tract. Journal of Aerosol Science, 8, 251. Finlay, W. H. (1998). Estimating the type of hygroscopic behavior exhibited by aqueous droplets. Journal of Aerosol Medicine, 11, 221. Finlay, W. H. (2001). The mechanics of inhaled pharmaceutical aerosols. New York: Academic Press. Finlay, W. H., & Stapleton, K. W. (1995). The effect of regional lung deposition of coupled heat and mass transfer between hygroscopic droplets and their surrounding phase. Journal of Aerosol Science, 26, 655. Finlay, W. H., & Stapleton, K. W. (1999). Undersizing of droplets from a vented nebulizer caused by aerosol heating during transit through an anderson impactor. Journal of Aerosol Science, 30, 105. Fuchs, N. A., & Sutugin, A. G. (1970). Highly dispersed aerosols. Ann Arbor: Ann Arbor Science Publ. Henning, T., Massling, A., Brechtel, F. J., & Wiedensohler, A. (2005). A tandem DMA for highly temperature-stabilized hygroscopic particle growth measurements between 90% and 98% relative humidity. Journal of Aerosol Science, 36, 1210. Hicks, J. F., Pritchard, J. N., Black, A., & Megaw, W. J. (1989). Measurement of growth due to condensation for some common aerosols. Journal of Aerosol Science, 20, 289. Incropera, F. P., & De Witt, D. P. (2002). Fundamentals of heat and mass transfer. New York: Wiley. Jacobson, M. Z. (2000). Fundamentals of atmospheric modeling. New York: Cambridge University Press. Maxwell, J. C. (1890). The scientific papers of Clerk Maxwell. London: Cambridge University Press. Park, S. H., Lee, K. W., Shimada, M., & Okuyama, K. (2001). Alternative analytical solution to condensational growth of polydisperse aerosols in the continuum regime. Journal of Aerosol Science, 32, 187. Pope, S. (2000). Turbulent flows. Cambridge, UK: Cambridge University Press. Roth, C., & Gebhart, J. (1996). Aqueous droplet sizing by inertial classification. Particle Systems Characteristics, 13, 289. Seinfeld, J. H., & Pandis, S. N. (1997). Atmospheric chemistry and physics. New York: Wiley. Sommerfeld, M., & Qiu, H. H. (1993). Characterization of particle-laden, confined swirling flows by phase Doppler anemometry and numerical calculation. International Journal of Multiphase Flow, 19, 1093.