Modeling of Droplet Evaporation from a Nebulizer

Modeling of Droplet Evaporation from a Nebulizer in an Inductively Coupled Plasma Craig M. Benson, Sergey F. Gimelshein, Deborah A. Levin and Akbar Montaser The George Washington University Washington, DC 20052 l Abstract. The evaporation rate of sample droplets in an inductively coupled plasma is investigated through the development of two models using the direct simulation Monte Carlo technique. A standard continuum evaporation model is contrasted with a kinetic technique designed to obtain correct results over a large range of Knudsen numbers. The droplet evaporation rates predicted by the continuum desolvation model are found to be in agreement with those of previous studies. We present the first predicted spatial distribution of droplet concentrations and evaporation rates in an ICP flow. INTRODUCTION Atmospheric pressure argon inductively coupled plasmas (ICPs) are electrical discharges that have dramatically altered the practice of elemental and isotopic ratio analysis. [1,2] An analyte is most commonly introduced into an ICP for mass spectrometric, atomic emission, or atomic fluorescence analysis via a spray of dilute aqueous solution. Numerical modeling can assist in predicting the fate of sample droplets, which will lead to a better understanding of the amount of detectable analyte in the nebulizer spray. To this end, a two-phase flow computational model has been developed to predict the desolvation rate of droplets in the plasma. [3] Analytical measurements of the droplet desolvation rate within the ICP are very limited due to experimental difficulties. Recent work addressed the issue of droplet desolvation in an ICP [4]. The effect of a specific distribution of particle sizes and velocities was not considered; thus, the model could not predict the behavior of the spray as a whole. In addition, our approach provides a more detailed description of the change in droplet velocities due to interactions with argon in the plasma and the two-dimensional size distribution of particles in the ICP. The nebulizer chosen for this study is the direct injection high efficiency nebulizer (DIHEN)[5-7], a concentric micronebulizer for applications in which the sample is limited, expensive, or toxic. A fast flow of argon gas causes the liquid sample to disperse into droplets ranging from 0.5 to 25 /xm in size with an average velocity of 12.4 m/s. These droplets are then injected directly into the base of the plasma. The initial set of plasma conditions (temperatures, number densities, and flow velocities) was generated with the high frequency induction plasma (HiFI) model. [8,9] The energy loss by the plasma due to droplet evaporation is not taken into account by the HiFI code; however, the plasma temperatures will not be significantly affected assuming the limited volume of sample introduced. The plasma is approximately 5 cm in length, and ranges in temperature from 3,000 K within the central channel and outlying portions of the plasma to greater than 10,000 K, as shown in Fig. 1. The continuum solution is translated into discrete particles of argon with positions and velocities consistent with the temperature and pressure gradients as predicted with HiFI. The output of this program is then read into the DSMC simulation as a set of background gas flow parameters. ^ This research was supported by the Army Research Office (BMDO) ASSERT Grant No. DAAG55-98-1-0209 and the US Department of Energy under Grant No. DE-FG02-93ER14320. CP585, Rarefied Gas Dynamics: 22nd International Symposium, edited by T. J. Bartel and M. A. 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àÚãÞ¡åâPßß¸has áfè0ß been âPâêÝÅÞ¡ë¡âíæâPëàsè0ß®follows. õuÞ2íÌíæÞlÿ-ßö The LáÚâÅðñÈòeisäæß8maintained Ý/è{äæêÜè0äÌêÚâ=ëatè{Ü"aè(steady ß£Üâ=è0ë¡îstate, ß£Üè{Üâ2ùÍand è{êÉëÀäÌÜ8ä}ß ãreasonable â=The è0ßÞ0êÉè evaporation íÌâ-toÜÞ/assume è0ßß£çÉÝ;âÆthat ÜáÉè{ÜLthe ÜáÚâ8plasma àÚí}è0ßÝ/èmodeled àproperties ÚãÞ0à?âãÜäæâPßÈare è{ãâ®not êÚÞ{Ücaffected è QâPå¼ÜâPICP ëwith ÿcäÌÜáthe ÜáÚâ®introduction äÌêÜãÞ¡ë¡çÉå¼ÜäæÞ0ê·ofÞ{õOaèUlimited íæäÌÝÅäÌÜâPë·amount è{ÝÅÞ2çÚêitÜÈofisÞ0õ ßè0Ý;àÉíÌâ2ö -âPêÉå¼â2ùlthe ÜáÉâÈè0ãé2Þ0ê"plasma àÚí}è0ßÝ/è-isä}ßOmodeled ÝÅÞ¡ë¡âPíÌâ=ëè0ß¶aè background è0å é0ãÞ0çÉêÉë"argon è{ãé0Þ2ê"gas é2è0ßOáÉèìäæêÚéÆallè{íæíthe ÜáÚâÈÝ/è0åãÞ2àÚãÞ2àfâPã£ÜäÌâ=ßN(density, ïYë¡âêÉßäÌÜî0ù sample. ìvelocity, 0âíæÞ¡å¼äÌÜî0ùQHence, èand {êÉë·temperature), ÜâÝÅà?âargon ãèlÜçÉãâó³ùQofÞ{õbthe ÜáÚâUcalculated åPè{í}å¼çÚí}èlÜâPë(asplasma. àÚí}è0ßÝ/èÚö Using ¸ß£äæêÚéaè background è0å é0ãÞ0having çÉêÉë·gas é2è0ßÆimplies äÌÝÅàÚíæäæmacroproperties âPß®certain å¼âãÜè0äÌêWsimplifications. ßäæÝ;àÉíÌä fåè{ÜäæÞ0êÉßPö Û-ãé0Þ0ê·atoms èlÜÞ0Ý/ßLare è{ãâ¸ßè{ÝÅàÚíæâPëonce Þ0êÉåâ¸àÚãäæÞ0ãÈtoÜÞUthe ÜáÚâ8Ý/è{äæê(simulation ß£äæÝçÚí}èlÜäÌÞ2ê+using çÉßäæêÚéUequilibrium â=ýçÚäÌíæä ãäæçÚÝ ëÚäæß£Üãä ç¡ÜäæÞ0êÉßLwith ÿcäÌÜáthe ÜáÉâ®locationíæÞ¡åè{ÜäæÞ0ê¡ô Argon ëdependent ¡âà?âêÉëÚâêÜparameters àÉè{ãè{ÝÅsampled âÜâãß®obtained Þ Üè{äæêÚâPëprior õfrom uãÞ0Ý HiFI. Æä ¶ð³ö main ¸Using ß£äæêÚé(equilibrium âPýçÚäæíÌä ãäÌçÉÝdistributions ë¡ä}ßÜãä ç¡ÜäÌÞ2êÉß8isä}ß8distributions ãreasonable â=è0ßÞ0êÉè íÌâÅsince ßäÌêÉåâ;the ÜáÚâ/plasma àÚí}è0ßÝ/è+isä}ß äÌêthe ÜáÚâÅcontinuum å¼Þ0êÜäÌêçÚçÚÝ regime. ãâé2äÌÝÅâ0ö These LáÚâPßâUargon è0ãé2Þ0êatoms èlÜÞ0Ý/ßÆare è0ãâthen ÜáÚâPê fixed ¡âPëWäÌêÜáÚâ DSMC `ñß£äæÝçÉíæè{ÜäæÞ0êÉß®such ß£çfåá(that ÜáÉè{Ü¸they ÜáÚâîdoë¡Þnot êÚÞ{Ü in Ýmove ;Þlì2â+over Þlì2âãUtime ÜäæÝ;âand è0êÉëdoë¡Þnot êÚÞ0ÜÅcollide åÞ0íæíÌä}ë¡â·with ÿcäÜá!each â=è0åá!other. Þ{ÜáÉâã=ö Therefore, LáÚâãâ¼inõuÞ2ãthe â2ùbthe ÜáÉâ·properties àÉãÞ2àfsimulations âPã£ÜäÌâ=ßpredicted àÚãâPëÚäæå¼ÜâPë!with ÿcäÌÜá HiFI -ä OðUremain ãâPÝ/è{äæê å¼Þ0êfßÜè{êÜ¸throughout ÜáÚãÞ0çÚé2áÚÞ0ç¡Ü®the ÜáÉâÅsimulation. ßäæÝçÚí}èlÜäÌÞ2êjö;Argon-argon ÛÆãé2Þ0ê¡ôrè{ãé0Þ2êcollisions å¼Þ0íæíæäæßäÌÞ2êÉß8are è{ãâUêÉÞ{Ü"modeled, ÝÅÞëÚâíæâPë#ùÍè0êÉëthe ÜáÚâÅà?Þ2ßäÜäÌÞ2êÉß8ofÞ{õ^the ÜáÚâ/argon è{ãé0Þ2ê constant èatoms lÜÞ2Ý/ß"doë¡Þnot êÉÞ{Üchange, åáÉè0êÚé0â2ùQthus ÜáçÉß8preserving àÚãâPßâãìäÌêÚéthe ÜáÉâ/number êçÉÝ âãdensities ë¡âPêÉß£äÌÜäæâPß"and è0êÉënot Ütemperatures âÝÅà?âãèlÜçÚãâ=ß"and åcalculated è0íæåçÚíæè{Üpositions â=ëWwith ÿcäÌÜá HiFI. -ä ¶ð³öÍNo þ-Þ(new êÚâPÿ ml^n poqrq so stuo vnxw 9ozy stjozy0y0w |{zw}l DSMC APPROACH AND NUMERICAL CHALLENGES ~ ;>= 0 6! G G;>= G;>= E 0<;= ; <;>= G ) ' O O , ( ' <;= ;>= ;>= ;>= 7; K * ? < , ? / C K<;>= 247 1, 1, argon is introduced or removed from the simulation domain. Such an approach circumvents any potential computational difficulties connected with calculating argon collisions at atmospheric pressure. The number of collisions calculated in a standard fashion is directly proportional to the total collision crosssections of the particles in the gas and species number densities. The droplets generated by the DIHEN range from approximately 0.5 pm to 25 p,m in diameter; therefore, the collision cross section of each droplet is correspondingly larger than a gas particle cross section. Since there is a high argon number density, one may conclude that to model each argon-droplet collision would be prohibitively large. To effectively model the collision process for droplets, a special procedure is implemented to reduce the number of droplet-argon collisions by a factor of approximately IxlO10, but also increase the energy transfer during each collision by the same factor. This allows an equivalent amount of energy to be transferred during a single timestep, thus preserving both the calculation of droplet evaporation rate and the change in velocity of the droplets, while at the same time reducing the number of collisions calculated for each droplet. The flow in the DIHEN is assumed to have an axial symmetry, and an axisymmetric SMILE code has therefore been used. To examine droplet evaporation in more detail, only a portion of the ICP was modeled adjacent to the nebulizer nozzle. The radius enclosed by the simulation is 4.0 mm from the center of the plasma, and extends 3.0 cm within the plasma. Although HiFI predicts temperatures and number densities for a plasma approximately 5 cm long with a 1 cm radius, all droplet evaporation occurs within the smaller computational domain, and thus any extension of the simulation domain beyond that point is unnecessary. The grid is uniform and consists of 40 cells in the y direction and 120 cells in the x direction. The time step is IxlO"6 s. The number of background argon molecules is about one million. DISCUSSION OF EVAPORATION MODELS Two droplet desolvation models are used to simulation particle evaporation and the effect of gas/droplet collisions on droplet temperature and diameters. The first one, a standard model for droplet evaporation, is widely used, especially in combustion applications [13-15]. Since a continuum regime is assumed, the model may not work properly in the limit of high Knudsen number. The second model uses a kinetic approach which yields equally accurate results for all Knudsen numbers [19,20]. The key assumption is that the droplet evaporation rate can be calculated by assuming small temperature and pressure changes from droplet /gas equilibria. The continuum evaporation model is described as follows. The mass flux of water from the surface of a droplet is governed by the following equation [13]: mv = 2nD - In(l + BM) (1) where mv is the mass flux of water from the surface of the droplet, D is the diameter for the droplet, k is the thermal conductivity and cp the heat capacity of the gas surrounding the droplet, and BM is the mass transfer constant, based upon the mass fraction of water vapor at the droplet surface. The droplet surface temperature is initially 298 K, and then increases as the droplet remains in the plasma. The rate of increase is dependent upon the amount of energy absorbed by the droplet over the course of a single timestep. The amount of energy absorbed is dependent in turn on the energy gained by the droplet through droplet-argon collisions and the energy lost to the surroundings via evaporation processes. At some point, the evaporation rate of the droplet will become sufficiently high such that the net energy gain by the droplet during a single timestep approaches zero. At this point, no further energy is available to heat the droplet, and it will therefore reach a quasi-steady temperature. This representative temperature is known as the wet-bulb temperature of the droplet. The above analysis yields a droplet evaporation model consistent with the D2 law [13], which states that a plot of the square of an evaporating droplet's diameter versus time will be linear, with the slope equal to the evaporation constant for the droplet. For a droplet with a 10-/xm initial diameter in a uniform 3,000 K environment, the above analysis indicates a wet-bulb temperature around 365 K, a droplet heating time of approximately 0.03 ms, and a total droplet evaporation time of 0.21 ms. The limitations of the standard evaporation model are the assumption of a continuum regime and the requirement of a unitary Lewis number. The Lewis number is the ratio of thermal and mass diffusivities in the droplet; a value of 1 indicates that the energy and mass transport via diffusion are equally effective in the 248 boundary layer surrounding the droplet. This is not necessarily the case for every system. The Young-Carey model [18-20], a kinetic approach, does not share these two limitations. Young [19,20] has proposed a general model to describe droplet evaporation occurring under nonequilibrium conditions: The factors &pv/pv,oo and AT/T^ represent the deviations from equilibrium associated with variations in pressure and temperature. Young derives the phenomenological coefficients, Lmm and Lmq, by analyzing the relative pressure and temperature jumps found at the droplet-continuum interfaces. A full analysis can be found in Young's work [19,20]; however, it is worth noting here that the primary differences between the Young-Carey model and the continuum model lies in the dependencies of the phenomenological coefficients on Sc, the Schmidt number; Pr, the Prandtl number; and Kn, the Knudsen number. The gas-droplet surface Knudsen number is defined with the characteristic length taken to be the diameter of the evaporating droplet, as Kn = A/D, and A is the gas mean free path. The use of a Knudsen number dependent on D causes Lmm and Lmq to be proportional to D. Thus the evaporation rate is found to have a linear dependence of D2 with time, which brings the kinetic approach into agreement with the continuum D2 law. RESULTS Although the results of primary interest are the final points of evaporation for droplets generated by a DIHEN in an ICP, the simulation can also yield valuable information in combustion and atmospheric condensation problems. The DSMC /evaporation model is therefore tested under a variety of conditions, as described in Table 1. TABLE 1. Initial conditions for simulations2 Case 1 2 3 4 5 6 7 Temperature 2000 K ICP 2000 K ICP ICP 2000 K ICP 2 Droplet Distribution 10-pm 10-pm DIHEN DIHEN DIHEN DIHEN DIHEN Model Continuum evaporation Continuum evaporation Continuum evaporation Continuum evaporation Continuum (with recondensation) Kinetic evaporation (Young-Carey) Kinetic evaporation (Young-Carey) Figure 2 3 4 5 All simulations use the DIHEN droplet velocity distribution. The simplest of the cases considered consists of a uniform background gas at a temperature of 2,000 K with no net gas flow or macroscopic pressure gradients and an initial droplet diameter of 10 jum. This case will help to illustrate the trends in droplet evaporation due to the initial particle velocity distribution. Figure 2 shows the results of the simulation. Although a small number of droplets can reach distances of 8 mm from the nebulizer tip due to their higher velocities relative to the other droplets in the plasma, 90 percent of the droplets are completely desolvated within 4 mm. On average, a droplet will travel 2.5 mm, representing a residence time in the simulation domain of 0.20 ms. Now consider simulation 2 (Table 1) with a constant droplet distribution but with a background gas consistent with an ICP. The pressure is close to atmospheric conditions; however, the temperature is much higher than 2,000 K. In addition, there is a large background gas flow in the positive x direction due to the 300 m/s nozzle exit velocity of the nebulizing gas (this high velocity is achieved due to the small DIHEN gas exit annulus, despite the relatively low volume flow of gas). The background flow increases the axial velocity of the droplets, leading to the result that some droplets survive to approximately 1.1 cm from the nozzle, more than 25 percent 249 Droplet Droplet Number Number Density Density 11 0.8 0.8 6.9 0.6 0.6 4.1 6E 1.39 +1 1 E+1 1 1.25E+12 0.4 0.4 4E +1 1 0.2 0.2 0.8 0.8 1 6 0.6 °' •D CO tt 0.4 0-4 04 E+ I 5E +0 4 E+ 0 5 £ 8.9 98 2. E 2.09 Radial Position, mm Droplet Droplet Evaporation Rate 0.2 0.2 1 ¥¦§/¸¯§¥²¨æ P¢ª c2d?eGfKg:hvØ 2 3 È¢ =2«¤O§{¯2ª®Ã¡¢Q¬2§2©r¥¦¤Y±-¥¦§8ª OP§¬ÆPP¡ P¢£¼¤¥¦ =§®¢£¼¤b¥§¸ª ¶© I¨æ P¢O fª ¯2§¥²¨} ¢ª¬0¢ P«¦¤Í¬2¥¦©y¤r¢¥¦Ã¯0¤¥¦ =§ 8Ä ÃP°£Á0´¢ =¯2§¬U´=P©¾ J Axial Position, Position, mm Axial &Ù>Ú &Ù>Ú ?Ù/Z Z[VIÒ FIGURE 2. Droplet number density in m~3 and evaporation rate in m~3 s~l for a 10-pm uniform droplet distribution Û\6[6[Y[ in a uniform 2000 K background gas. õè{ãÜáÚâPãIÜáÉè0êÜáÚâ-ë¡ãÞ0àÚíæâ¼ÜßIäÌê;ÜáÉâcíÌÞlÿÈâã¶ÜâPÝÅàfâPãè{ÜçÚãâ0ù{çÉêÚäõuÞ2ãÝ é2è0ßPöOÛÆêîäæêÉå¼ãâPè2ß£âÈäÌêÅâPìè0àfÞ2ãè{ÜäæÞ0êUãè{Üâcë¡çÚâÈÜÞ"áÉäÌé2áÚâã ÜâÝÅà?âãèlÜçÚãâ=ßcíæâPè0ëÉßcÜÞÂè+ß£áÚÞ2ã£ÜâãcãâPßäæëÚâêÉåâ"ÜäæÝÅâÞ{õIÜáÉâë¡ãÞ0àÚíæâ¼ÜÆäæê(ÜáÉâàÚí}è0ßÝÅè ÉáÚÞlÿÈâì2âã=ùÜáÚä}ß-ä}ß-ÝÅÞ2ãâ8ÜáÉè{êÞ Qßâ¼Ü î·ÜáÚâ;è2ååâíæâãèlÜäæÞ0êàÉãÞlìä}ë¡âPë î·ÜáÉâUêÚâ çÚíæä Pâã8é2è2ßö LáÚâÅèì0âãè{é2âë¡ãÞ0àÚíæâ¼Ü®ÿcäÌíæíIàfâPêÚâ¼Üãè{ÜâÜáÚâ;àÚíæè2ß£Ý/èÂÜÞ`ø0ö /ÝÅÝ ïuäæêÉë¡ä}åèlÜäÌêÉéÂè{êèì0âPãè0é0âÆãâPßä}ë¡âêÉåâ8ÜäæÝÅâ"Þ{õ Úöæø=úUÝ/ßù ;àfâPãåâêÜÆß£áÉÞ0ãÜâãLÜáÉè{ê(äæê(ÜáÚâ"íæÞlÿhÜâÝÅà?âãèlÜçÚãâ"ßäÌÝUçÚí}èlÜäæÞ0ê?ó³ù ÿcáÚäÌíæâ Uà?âãå¼âêÜ-ÿcäæíÌíÍáÉèì0â®âìlè{à?Þ0ãèlÜâPëÿcäÜáÚäæê Éö ;ÝÅÝ(ö ¶äÌé2çÚãâ ÅßáÚÞlÿ-ßÈÜáÚâ"ãâPßçÚíÌÜßLÞ0õÍÜáÉâß£äæÝçÚí}èlÜäÌÞ2êjö LáÚâÜáÚäæãëWå¼Þ2êÉë¡äÌÜäæÞ0êWçÚêfë¡âã®ÿcáÚä}åáÜáÚâUÝÅÞëÚâí¶ÿÈè2ß-ãçÚêWä}ß-ÜáÉèlÜ®Þ{õbÜáÉâçÚêÚäÌõuÞ0ãÝ(ùQíÌÞlÿÈâãcÜâÝÅà?âãèlÜçÉãâ è0å é0ãÞ0çÉêÉë é2è0ßÆÿcäÜáÜáÉâUäæêÉå¼íæçÉßäÌÞ2êWÞ{õbÜáÚâ ¸ð ^þ ëÚäæß£Üãä ç¡ÜäæÞ0êWÞ0õNë¡ãÞ0àÚíæâ¼Ü8ß£ä â=ßï ¶äæéÉö ó¼ö LáÚâPãâäæß®èÂí}è{ãé0âßàÚãâPè0ë(Þ0õNë¡ãÞ0àÚíæâ¼Ü ë¡äæè0ÝÅâ¼ÜâPãßPù0ãè{êÉé0äæêÚé®õuãÞ2Ý íæâPßßbÜáÉè{ê Éö #Ý ÜÞé2ãâ=èlÜâãIÜáfè{ê+ú #Ý(ùãè0êÉë¡Þ2Ý;íæîUè2ßßäæé0êÚâ=ëÜÞâPè2åá/ë¡ãÞ2àÚíæâ¼ÜEäæê2ÜãÞ¡ë¡çfå¼âPë äÌêÜÞWÜáÚâ+ßäæÝçÚí}èlÜäÌÞ2êjö ¡àfâPãäæÝÅâêÜè0í^ëÚè{Üè(äæêÉë¡ä}åè{ÜâÂíæäÜÜíæâåÞ0ããâPíæè{ÜäæÞ0ê âÜÿ^âPâêë¡ãÞ0àÉíÌâÜß£ä â+è0êÉëºë¡ãÞ0àÚíæâ¼ÜUì0âíæÞ¡å¼äÌÜî0ù è{êÉë+ÜáçÉßLÜáÚâ8ÜÿÈÞ;àfè{ãè{ÝÅâ¼ÜâãßÈè0ãâ8ë¡â=å¼Þ0çÉàÚíÌâ=ë#ö LáÚâíÌÞ¡åè{ÜäæÞ0êèlÜ¸ÿcáÚäæåá(ÜáÚâÝ/è Þ2ãäÌÜî+Þ0õEë¡ãÞ2àÚíæâ¼ÜßÆë¡âPßÞ0íæìlèlÜâ8õuÞ2ã-ÜáÚäæß¸ß£âÜÆÞ{õbÞ2àfâPãè{ÜäæêÚéÂå¼Þ2êÉë¡äÌÜäæÞ0êÉßcä}ß-íæâPßßÆëÚâ ÉêÚâ=ë#ö Û ßä =è íæâWõuãè0å¼ÜäæÞ0ê Þ{õÜáÚâ`ë¡ãÞ2àÚíæâ¼ÜßáÉèì2âèßç Âå¼äæâêÜíæîÖíÌÞlÿ ßä PâÀßçÉåá ÜáÉèlÜ·âìlè{à?Þ0ãèlÜäÌÞ2ê ÞåPå¼çÚãßÂÿcäÌÜáÉäÌêeß£âPì0âãè{í Ý;ä}å¼ãÞ2ßâPåÞ0êÉëÉß^Þ0õ¶âPêÜâãäæêÚé;ÜáÚâàÉíæè2ß£Ý/èÚö LáÉâPßâë¡ãÞ2àÚíæâ¼ÜßcëÚÞ/êÚÞ{Ü-áÉèì2â"ß£ç ÂåäÌâPê2ÜcÜäÌÝÅâ"ÜÞÂß£äæé0êÉä fåPè{êÜíæîÂà?âêÚâÜãèlÜâ8ÜáÚâ àÚíæè2ß£Ý/è â¼õuÞ0ãâë¡â=ß£Þ2íÌìlèlÜäÌÞ2êÞ¡åPå¼çÚãßö ÆÞlÿ^âPì0âã=ùÜáÚâ"Ý/è Þ0ãäÜîÂÞ0õOÜáÚâë¡ãÞ0àÚíæâ¼Üßcè{ãâ8íæè0ãé2âãLäÌêßä Pâ0ùÉè0êÉë+ÜáÚâßäæÝçÚí}èlÜäÌÞ2ê ß£áÚÞlÿ-ß^ÜáÉèlÜ Uà?âãå¼âPêÜLÞ0õOÜáÚâë¡ãÞ0àÚíæâ¼ÜßLâìlè{à?Þ0ãèlÜâ¸ÿcäÌÜáÉäÌêúÚö UÝÅÝ Þ{õ¶ÜáÉâ"êÚÞ íæâ®â ¡äÜ=ö ¶äæêÉè{íæíÌî2ùÜáÚâ8ß£äæÝçÉíæè{ÜäæÞ0êÂÿLè0ßNãçÚêçÉßäÌêÚéÜáÚâ è2å é2ãÞ2çÚêÉë;é2è2ß^åÞ0ããâ=ß£à?Þ0êfë¡äÌêÉé8ÜÞUÜáÚâÆÜâÝÅà?âãèlÜçÚãâ=ßùàÚãâPßßçÚãâ=ßùè{êÉë üÉÞlÿ-ßàÚãâPßâêÜäæêºÜáÉâ+ðñÈòÿcäÌÜáºÜáÚâ ¸ð ^þ ë¡ä}ß£Üãä çÚÜäæÞ0êÞ0õcëÚãÞ2àÚíÌâÜßïåè2ß£â Éùbß£âPâ ¶äÌéfö 2ó³öÛ¸ßÿcäÌÜáºÜáÚâø #Ý ë¡ãÞ2àÚíæâ¼ÜÆåè2ß£â=ßùÜáÚâà?Þ0äæêÜÆèlÜ-ÿcáÚä}åá ;à?âãå¼âêÜ-Þ0õOÜáÚâë¡ãÞ0àÚíæâ¼ÜßLáÉèì0â"âPìlè{à?Þ0ãèlÜâ=ë+ä}ßLõè0ã£ÜáÚâãLõuãÞ0Ý ÜáÚââ ¡äÜ-êÚÞ PíÌâ"Þ0õ ÜáÚâ ¸ð ^þeÜáÉè0ê+õuÞ2ãLÜáÚâåPè0ßâ¸Þ0õ¶çÚêÉäõuÞ2ãÝ(ù ç¡Ü-íæÞlÿÈâãÈÜâPÝ;à?âãèlÜçÚãâ0ù è2å é2ãÞ2çÚêÉë/éè0ßPöIðrêÜáÚä}ßcåè0ßâ0ùÜáÚâ Uà?âãå¼âêÜ ë¡ãÞ2àÚíæâ¼ÜÆâPìè0àfÞ2ãè{ÜäæÞ0êà?Þ0äæêÜ¸ä}ßÆíæÞ¡åèlÜâPë ÉöÌøÝ;Ý äæêÜÞÂÜáÚâUßäæÝçÚí}èlÜäÌÞ2êWë¡Þ0Ý/è{äæêjö -ÞlÿÈâì0âPãPùfíæè0ãé2âãÆè0êÉë·õè0ß£ÜâPã®ëÚãÞ2àÚíÌâÜß åè{ê·ãâÝ/è{äæêäæê·ÜáÚâ"àÉíæè2ß£Ý/è;çÚà·ÜÞ(ø0ö Åå¼Ý õuãÞ2Ý ÜáÚâ"êÉâ çÚíæä âãLÜäÌàÍö âPíÌâ=å³ÜâPë åè0ßâPßÜáfèlÜ/çÉßâPëºÜáÚâßÜè{êÉëÉè{ãëºâìlè{à?Þ0ãèlÜäÌÞ2êÝÅÞ¡ë¡âícÿÈâãâ+ãâà?âPè{ÜâPëçÉßäÌêÚéÀÜáÉâ EÞ2çÚêÚé0ôñÈè0ãâPî ï äæêÚâÜä}åPó Ý;Þ¡ë¡âPíEÞ{õLë¡ãÞ0àÚíæâ¼ÜâPìè0àfÞ2ãè{ÜäæÞ0êjö LáÚâ äæêÚâ¼ÜäæåÅâPìè0àfÞ2ãè{ÜäæÞ0êÀÝ;Þ¡ë¡âPíEÿLè0ß8çÉßâPëÜÞWåPè{í}å¼çÚí}èlÜâUÜáÚâ/âìlè0àfÞ2ãè{ÜäæÞ0êWÜäÌÝÅâ/Þ0õ ë¡ãÞ2àÚíæâ¼ÜßLÿcäÌÜá·ÜáÚâ Æð Nþßä Pâ"ë¡äæß£Üãä ç¡ÜäæÞ0ê(äæê·è/úÚù çÚêÚäÌõuÞ0ãÝ âPêìäæãÞ0êÚÝÅâêÜcè0ßLÿÈâíæíjè0ßcè ¸ð NþëÚãÞ2àÚíÌâÜ-ß£ä â ë¡äæß£Üãä ç¡ÜäæÞ0êäÌêÜáÉâ¸ðñÈòIö LáÚâ®àfÞ2äÌêÜcèlÜLÿcáÚä}åá Uà?âãå¼âêÜLÞ{õjÜáÚâ8ë¡ãÞ2àÚíæâ¼ÜßLë¡â=ß£Þ2íÌìlè{ÜâÆä}ßÈíÌÞ¡åPèlÜâ=ë+è{ÜcúÚö úÝ;Ý õuãÞ2Ý ÜáÚâ êÚâ çÉíÌä âPãÆÜäÌàõuÞ2ãÆÜáÚâðñÈò è0å é0ãÞ0çÉêÉë·é2è0ß-ÜâPÝ;à?âãèlÜçÚãâPßÆè0êÉëWè{ÜUø0ö²úÅÝÅÝ õuãÞ2Ý ÜáÚâUêÚâ çÉíÌä âPãÆÜäÌàõuÞ0ã-ÜáÚâ;ú çÚêÚäÌõuÞ0ãÝ è2å é2ãÞ2çÚêÉë/éè0ßcåÞ0êÉë¡äÌÜäæÞ0êÍö LáÚâ`âìlè0àfÞ2ãè{ÜäæÞ0êÖÜäæÝÅâ`è2ß+Þ Üè0äÌêÉâPë ÿcäÌÜáTÜáÚâ EÞ0çÚêÉé{ô£ñÈè{ãâî Ý;Þ¡ë¡âPí®ëÚä Qâãß(ß£äæé0êÚä fåPè{êÜíæîõuãÞ0Ý ÜáÉèlÜ·Þ Üè{äæêÚâPë çÉß£äæêÚé+ÜáÚâÅå¼Þ2ê2ÜäÌêçÚçÉÝ ÝÅÞ¡ë¡âPíYö LáÚâ EÞ2çÚêÚé0ôñÈè0ãâPîÂÜãâPè{ÜÝÅâêÜ®ÿÈè2ß¸ë¡âPãäæì0â=ë·õuÞ0ã8è+åPè0ßâäÌêÀÿcáÚä}åáÜáÚâÅë¡âPìä}èlÜäÌÞ2ê(õuãÞ2Ý è{ê`âPýçÚäæíÌä ãäÌçÚÝ ß£äÌÜçfèlÜäæÞ0êä}ß"ãâí}èlÜäæì0âPíÌîWßÝ/è{íæí-÷ø lûyö LáÚâÂßäÌÝUçÚíæè{ÜäæÞ0êÉß"ßáÚÞlÿ ÜáÉè{Ü îíÌÞlÿÈâãäÌêÉé+ÜáÚâ è0å é2ãÞ2çÚêÉëWéè0ß ÜâÝÅà?âãèlÜçÚãâ·ÜÞ`íæâPßßUÜáÉè0êø0ù ÅùIÜáÚâãâPßçÚíÌÜß;Þ{õ¸ÜáÚâ äæêÚâ¼Üäæå·âPìè0àfÞ2ãè{ÜäæÞ0ê!Ý;Þ¡ë¡âPí-å¼Þ2êì2âãé0âÿcäÜá!ÜáÚÞ2ßâ(Þ{õ¸ÜáÚâ farther than the droplets in the lower temperature, uniform gas. Any increase in evaporation rate due to higher Æ $ temperatures leads to a shorter residence time of the droplet in the plasma; however, this is more than# offset < by the acceleration provided by the nebulizer gas. The average droplet will penetrate the plasma to 1.5 mm )! P! (indicating an average residence time of 0.12ms, 40 percent shorter than in the low temperature simulation), :.$! ² ! 2, while 90 percent will have evaporated within 6.0 mm. Figure 3 shows the results of the simulation. ? The third condition underGwhich the model was run is that of the uniform, lower temperature background +, < gas with the inclusion of the DIHEN distribution of droplet sizes (Fig. 4). There is a large spread of droplet (! <% !% diameters, ranging from less than 0.5 /xm to greater than 20 /xm, randomly assigned to each droplet introduced C) into the simulation. Experimental data indicate little correlation between droplet size and droplet velocity, and thus the two parameters are* decoupled. The desolvate for this set of operating conditions is less defined. location at which the majority of droplets > A sizable fraction of the droplets have a sufficiently low size such within several 2 > that evaporation occurs microseconds of entering the plasma. These droplets do not have sufficient time to significantly penetrate the ) plasma before the droplets$are larger in size, and the simulation .4! desolvation occurs. However, the majority of 6 shows that 90 percent of the droplets evaporate? within 2.3 mm of the nozzle exit. , Finally, the simulation was run using the background gas correspondingto thetemperatures, pressures, and , Y!C % flows present in the ICP with the distribution of droplets (case 4, see Fig. 5). As with the 410 7.$DIHEN ! /xm droplet cases, the point at which 90 percent of the droplets have evaporated is farther from the exit : ? :.$! nozzle of the DIHEN than for the case of uniform, but lower temperature, background gas. In this case, the 90 percent 7 < droplet evaporation point is located 3.1 mm into the simulation domain. However, larger and faster droplets can; remain in the plasma up to 1.3 cm from the nebulizer tip. ³ + Selected cases that used the&standard evaporation model were repeated using the Young-Carey (kinetic) model of droplet :evaporation. The kinetic evaporation model was used to calculate the&evaporation time of !4!$!K5 droplets with the DIHEN size distribution in a 2,000 K uniform environment as well as a DIHEN droplet size > .$! distribution in the ICP. The point at which 90 percent of the droplets desolvate is located at 2.2 mm from the5 ? $!$!4! nebulizer tip? for the ICP background gas temperatures and at 1.2 mm from the nebulizer tip for the 2000 K uniform background gas condition. }³ The evaporation time as obtained with the Young-Carey model differs significantly from that obtained :³ using the continuum model. The Young-Carey was derived for a case in which the deviation from Y. treatment ( ? an equilibrium situation is relatively small [19]. TheCsimulations show that by lowering the background gas !$!$!}5 temperature to less than 1,000 K, the results of the kinetic evaporation model converge with those of the 250 Droplet Number Density Droplet Evaporation Rate 1.2 E 1 0.8 o 0- 0.6 i "O 0.4 CO DC 0.2 2 3 Axial Position, mm FIGURE 3. Droplet number density in m~3 and evaporation rate in m~3 s~l for a 10-pm uniform droplet distribution in a background gas with temperature gradients identical to that of an ICP. continuum evaporation model. This highlights the need for a kinetic-based evaporation model designed to give adequate solutions for droplet evaporation in situations that are far from equilibrium. CONCLUSIONS The study presented in this report demonstrated the ability to formulate and construct a general gas-particle evaporation model and simulation technique general to a wide range of Knudsen numbers and non-equilibrium flow conditions. Several numerical algorithms were developed to efficiently model high pressure gas-surface interactions and have been incorporated into the DSMC method. Two specific droplet evaporation models were considered and compared. The numerical scheme is sufficiently general to provide a framework for future extensions of the two phase flow modeling that will incorporate droplet condensation. The results of the simulations show that the depth to which a droplet penetrates the plasma is, in general, less than 1 cm. The total time of evaporation agrees with that predicted in reference 6. However, the model in reference 6 did not include droplet transport and thus a full comparison is not possible. The plasma, however, is approximately 5 cm in length, and therefore desolvation processes can occupy 20 percent of the analyte's lifetime in the plasma. Considering that the analyte must then undergo the processes of evaporation to the gas phase, atomization, excitation, and ionization before it can be detected, a shorter time to desolvation would be preferred. It can be seen that the high plasma temperatures lead to efficient desolvation; however, high gas velocities cause the analyte to penetrate the plasma to a higher degree. Although a lower gas velocity would lead to slower droplets, the nebulizer would be less efficient, producing larger droplets. Further studies of the design trade-offs between these two effects are necessary. Finally, it can be seen that a more inclusive droplet evaporation model is required. Although the continuum model is likely to give accurate results, a particle-based method would be more comprehensive. Since the situation under study is dominated by continuum gas conditions, the continuum evaporation model can be used to give an acceptable desolvation rate. However, the influence of smaller droplets (higher Knudsen number) in terms of ICP/DIHEN performance and related plasma and combustion processes is still under investigation. 251 Droplet Number Density 1 0.8 0.6 0.4 0.2 Droplet Evaporation Rate E S I 1 0.8 0.6 0.4 0.2 Axial Position, mm FIGURE 4. Droplet number density in m~3 and evaporation rate in m~3 s~l for a droplet size distribution representative of the DIHEN in a uniform 2000 K background gas. REFERENCES 1. A. Montaser, Ed., Inductively Coupled Plasma Mass Spectometry, VCH, New York, NY, 1998, 964 pages. 2. A. Montaser and D. W. Golightly, Eds., Inductively Coupled Plasmas in Analytical Atomic Spectrometry, 2nd Ed., VCH, New York, NY, 1992, 1017 pages. 3. C. M. Benson, S. F. Gimelshein, D. A. Levin, and A. Montaser, Simulation of Droplet-Gas Interactions in an Inductively Coupled Plasma Using the Direct Simulation Monte Carlo Method, AIAA Paper 2000-2431, June 2000. 4. J. A. Horner and G. M. Hieftje, Computerized Simulation of Mixed-Solute-Particle Vaporization in an Inductively Coupled Plasma, Spectrochimica Acta 53B, 1235-1259 (1998). 5. J. A. McLean, Hao Zhang, and A. Montaser, A Direct Injection High-Efficiency Nebulizer for Inductively Coupled Plasma Mass Spectrometry, Anal. Chem. 70, 1012-1020 (1998). 6. J.S. Decker, H. J. Dietze, J. A. McLean, and A. Montaser, Ultratrace and Isotope Analysis of Long-Lived Radionuclides by Inductively Coupled Plasma Quadrupole Mass Spectrometry Using a Direct Injection High Efficiency Nebulizer, Anal. Chem. 71, 3077-3084 (1999). 7. B. W. Aeon, J. A. McLean, and A. Montaser, A Large Bore Direct Injection High Efficiency Nebulizer for Inductively Coupled Plasma Spectrometry, Anal. Chem. 72, 1885-1893 (2000). 8. J. Mostaghimi, P. Proulx, and M. I. Boulos, An Analysis of the Computer Modeling of the flow and temperature fields in an Inductively Coupled Plasma, Numerical Heat Transfer 8, 187-201 (1985). 9. M. Cai, D. A. Haydar, A. Montaser, and J. Mostaghimi, Computer Simulation of Argon-Nitrogen and ArgonOxygen Inductively Coupled Plasmas, Spectrochimica Acta 52B, 369-386 (1997). 10. G. A. Bird, "Molecular Gas Dynamics and the Direct Simulation of Gas Flows," Clarendon Press, Oxford, 1994. 11. M. S. Ivanov, G. N. Markelov, and S. F. Gimelshein, Statistical Simulation of Reactive Rarified Flows: Numerical Approach and Applications, AIAA Paper 98-2669, June 1998. 12. Ivanov, M.S., and Rogasinsky, S.V., "Analysis of the numerical techniques of the direct simulation Monte Carlo method in the rarefied gas dynamics," Soviet J. Numer. Anal. Math. Modelling, Vol. 3, No. 6, 1988, pp. 453-465. 13. A. H. Lefebvre, "Atomization and Sprays," Hemisphere Publishing, New York, 1989. 14. W. C. Hinds, "Aerosol Technology: Properties, Behavior, and Measurement of Airborne Particles," John Wiley &; Sons, New York, 1982. 15. N. C. Clampitt and G. M. Hieftje, Investigation into the Mechanism of Desolvation of Sample Droplets in Flame Spectrometry, Anal. Chem. 44(7), 1211-1219 (1972). 16. E. M. Sparrow and J. L. Gregg, The Variable Fluid-Property Problem in Free Convection, Tram. ASME 80, 879-886 (1958). 252 Droplet DropletNumber NumberDensity Density 1 0.8 E+1 0.4 3. 82 6.36 1.14E+12 0.6 E+ 1 1.2 1 7E +1 1 1 0.2 Radial Position, mm DropletEvaporation Evaporation Rate Rate Droplet 1 E 0.8 E+ 0. 05 _ 0.4 0.4 (0 QC 5. 55 1.67E+05 89 3. 0.6 E+ 0 4 0.2 * 0.2 Droplet È¢ =2«¤bnumber §{¯ª®Ã¡¢¶density ¬0§©r¥²¤Y±"¥¦in§Uªm~ and §¬"Axial PP¡Position, P¢£¼¤¥¦ =§mm ¢£¤È¥¦in§ªm~ s~l © N¨} ¢EaÆdroplet ¬0¢ =2«¦¤I©r¥ cdistribution ¬0¥¦©y¤r¢¥Ã2¯2¤¥¦ =§"represen¢0¢©r§ FIGURE 5. evaporation ¤tative £¼¤¥¦cof P¨Qthe ¤2 DIHEN LÎ in¥¦§Åa®background ÃP°£Á{´P¢ =¯2§¬gas ´P©EµEwith ¥²¤;¤temperature ª"¡¢£¼¤¯2¢L´gradients ¢£=rate ¬0¥§{¤©E¥identical ¬2§{¤¥¦°³«Éfor ¤to ¸¤that ¤N of¨jPan§Åsize ÎyICP. »IÏQ¾ 11 c2d?eGfKg:hvø ^J ¾ 0¾ 0¾ 0¾ J ÙÂÚ 2 Axial Position, mm 3 3 Ù>Ú 3 Ù Z ßÞ $V k^QRT ¸¾ j¾ ^¯Ã2Ã¢£¬ ¸¾ b¾ È§§l± {§¬¿Æ¾ ¶¾¹;¥«¦«¦© ^¢ =«¦¤ ÍPP¡ ¢£¤¥¦ =§ Ú°¤©b ¨jf¢£§©r¥¦§l¤©EP§¬ O¢¥PÃ2«¦cÏj¢ P¡¢r¤¥¦© £|2~ -orwP~ wx£x8msrwP|2x ³os ¾ Æ¾2Ï#¾»I¢± ¡¾¹¾ ^±0¯ª"¥ 0P§¬ Ú¾2¿Nª"³¬ 0ÏQ =©y¤ §{¯°«¦³¼¤¥ P§ ^¢ ³µb¤Å P¨?Ê¤¢^¹;¥°¢ {¬0¢ =2«¦¤©È¥¦§ {¯¡¢©¼¤¯2¢£¼¤³¬ L© ¹;¥ {¤¯2¢© ¶¿ ¹; =«¦°¯2«¢ 0¥¦ª®¯2«Ì¼¤¥¦ =§ l¤¯¬0± £|2~ -orwP~ +wPxxmsrwP|2x ³os ¾ 0¾ ^¾ ¶ P¯§´ NI+»¶ P§¬2§©¤¥¦ P§P§¬ ÍPP¡ ¢£¤¥¦ =§ P¨ É¥½{¯2¥¬ È¢ P«¦¤©¥§ÏÍ¯0¢ ¶¡ =¯2¢U¼¤Å¿E¢Ã¥²¤r¢£¢r±Äc§{¯¬0©r§ ^¯ª¸Ã¡¢ £|~ ÆorwP~ wPxx"msywP|x ³os ¾ 0¾ ^¾ O =¯2§´ lI^»¶ =§¬2§2©¤¥¦ =§P§¬ ÍPP¡ ¢£¤¥¦ =§ ¨ É¥½{¯2¥¬ È¢ =2«¤©I¤¶¿N¢Ã2¥²¤r¢£¢r±8Äc§{¯¬2©r§ ^¯ª¸Ã¡¢j¥¦§8¤NÏj¢©r§° ¨ÍP§UÎA§¢r¤ LP© £|2~ -orwP~ wx£x8msrwP|2x ³os ¾ Z Ö 2S 4ï 4k àì 4Q 4J *à 4_ ?àÂJ )Q Q 0ì ?à 17. á G.ú L.$úHubbard, V, E. Denny, and A. F. Mills, Droplet Evaporation: Effects of Transients and Variable Properties, ÿ /ê îÆàÆZ[6[YaVÑZ?[Y[6XFZ?W Ö U6L Int. J. Heat*à>Mass Transferöà 18,N 1003-1008 (1975). Z?X ì N ÂB à çV (S N &S 18. V.#÷ P. Carey, S. M. Oyumi, and S. Ahmed, Post-nucleation Growth of Water Microdroplets in Supersaturated Gas ^N (N *àPá ú$úÃÿ /ê Ý>üÃFZ[ Là>\6a6WYaV\b [ Ô FZW6W Ö L Mixtures: A$à Molecular Simulation Study, Int. J. HeatMass Transfer 40(10), 2393-2406 (1997). Z?W )ô )H Õ sQ ï }J ì 19. T J. B. Young, The Condensation and Evaporation of Liquid Droplets in a Pure Vapour at Arbitrary Knudsen àPá ú$úÃÿ Rê ~Ü$Ý`F Ö LàÃZ Ô bYWVZ Ô6Ô Z0FZW6W$ZL Int. (1991). \[ )Number, ô H Õ 4à J. Heat Mass Transfer Q 34(7), 1649-1661 ï J T 20. J. B. Young, The Condensation and Evaporation of Liquid Droplets Arbitrary Knudsen Number in the Presence ES ?àÆá ú$úÿ /ê ~ÜÃFZ6ZLà>\6W6b$ZV\6WYU Ô FZW6WYaYat L of an Inert Gas, Int. J. Heat Mass Transfer 36(11), 2941-2956 (1993). 253

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