Simulation of aerosol formation due to rapid cooling of multispecies vapors
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Abstract
An extended classical nucleation approach is put forward with which aerosol formation from rapidly cooled, supersaturated multispecies vapor mixtures can be predicted. The basis for this extension lies in the treatment of the critical cluster that forms as part of the nucleation burst—a multispecies treatment of the thermodynamically consistent approach is proposed that can be solved efficiently with a Newton iteration. Quantitative agreement with Becker–Döring theory was established in case the equilibrium concentration of the critical clusters is properly normalized. The effects of nucleation, condensation, evaporation, and coalescence are consolidated in the numerical framework consisting of the Navier–Stokes equations with Euler–Euler oneway coupled vapor and liquid phases. We present a complete numerical framework concerning generation and transport of aerosols from oversaturated vapors and focus on numerical results for the aerosol formation. In particular, using adaptive timestepping to capture the wide range of time scales that lie between the nucleation burst and the slower condensation and coalescence, the aerosol formation of a system of up to five alcohols in a carrier gas is studied. The effects of the temperature levels, the cooling rate, and the composition of the vapor mixture under a constant temperature drop, on the formation and properties of the aerosol are investigated. A striking nonuniform dependence of the asymptotic number concentration of aerosol droplets on temperature levels was found. A decrease of the rate of cooling was shown to reduce the number concentration of aerosol droplets which asymptotically leads to significantly larger droplets. The simplification of the vapor mixture by removing the higher alcohols from the system was found to yield an increase in the asymptotic size of the droplets of about 15%, while the number density was reduced accordingly.
Keywords
Aerosol Alcohol mixtures Evaporation and condensation Multispecies Nucleation1 Introduction
The dynamics of an aerosol forming from a gaseous mixture of various chemical species is expressed by the detailed interplay between nucleation, evaporation, and condensation, as well as coalescence, interacting with vapor concentration, temperature, and velocity fields. We focus on an aerosol arising from nucleation in a supersaturated mixture of alcohol gases that is subjected to very rapid cooling. The prediction of the composition and size distribution of the droplets constituting the aerosol becomes particularly challenging when the number of species becomes large. The cornerstone approach in this field is the Becker–Döring (BD) theory [1], presenting a microscopic basis for the understanding of macroscopic properties of cluster formation in multispecies systems. However, in its full generality, this approach is unpractical for more than just a few species, requiring significant computational effort, even for steadystate cases, and the development of new solution methods [2] to reduce simulation costs. In this paper, we present and illustrate a modeling approach that is computationally much less demanding. It is based on an extension of classical nucleation theory, closely following [3, 4], that allows approximating aerosol properties in multispecies systems. The new formulation can handle systems containing tens to hundreds of species, as is shown by determining the multispecies nucleation rate. This extends significantly the capability of classical aerosol modeling. The method will be illustrated for spatially homogeneous systems, for which a tailored adaptive timestepping method is put forward to handle the very rapid nucleation burst, followed by the much slower evolution due to condensation and coalescence. We validate the new formulation by comparing full solutions of a ternary system of alcohol vapors based on the numerical solution to the BD equations [5]. The capability of the new approach is illustrated by studying aerosols from alcohol vapors containing up to five different species, for which the size distribution of the aerosol droplets as well as the chemical composition is computed under different initial temperatures and cooling rates at constant pressure drop.
Classical nucleation theory concentrates on the prediction of the socalled ‘critical cluster.’ This term designates a grouping of molecules from the gas phase that is large enough to stay coherent for long times with probability of one half. It signifies the ‘border’ of transient molecular aggregates; on average, smaller clusters are likely to disintegrate rather quickly into the gas phase, while larger clusters would likely grow on average. The critical cluster is identified as the key nucleation core from which droplets would grow due to condensation of molecules from the vapor. Virgin droplets that just nucleated are treated as if emerging with a certain startup diameter. Subsequently, the size can grow by several orders of magnitude, e.g., due to rapid cooling of the surrounding vapor inducing condensation. In this process the composition of the droplets also changes in accordance with the molar volume of the components and their saturation. This basic physical setting is presented in this paper, extending the ternary formulation presented in [3, 4] to an arbitrary number of components. The formulation requires a fast determination of the properties of the critical cluster in order to start up the aerosol evolution. This problem will be addressed in this paper.
A recent study based on Becker–Döring theory was presented in [5], considering the mixtures of alcohols in particular. A specialized multigrid solver was developed in order to efficiently solve the composition of droplets in ncomponent mixtures. The method was illustrated for up to five alcohols simultaneously, which constitutes the state of the art for a computational approach based on the Becker–Döring approach. Here, we do not follow the Becker–Döring methodology but rather consider a simpler theory, generalizing earlier work by Wilemski [4], in which multispecies nucleation is treated including the interaction between the species. This approach can be used for situations in which one of the components is dominant [6], but also for more complex nucleation problems with several components taking part in the nucleation process. The composition of the critical cluster can be determined on the basis of finding the root of the equation governing the molar fractions. Newton iteration was used to efficiently solve the molar fractions problem, solving the problem with little computational overhead for many components. We present the theory and illustrate the critical cluster properties for a number of characteristic situations.
An important challenge for classical nucleation approaches of multispecies systems is the validation against wellcontrolled physical experiments. A prominent example is the laminar flow diffusion chamber (LFDC) in which an aerosol is formed due to rapid cooling under slow flow conditions. This requires the treatment of (a) the multispecies aerosol formation and (b) the capturing of spatial variations and details of the geometry of the experimental equipment. In this paper, we address the first element and focus on spatially homogeneous systems in which aerosol forms from a supersaturated vapor mixture, obtained by very rapid external cooling. The element of simultaneous laminar fluid flow will be integrated in later studies and is a subject of ongoing research. To allow at this stage a crossvalidation with fully resolved Becker–Döring theory [5], the method is illustrated for a system of alcohol vapors. Effects of (a) the temperature levels, (b) the cooling rate, and (c) the mixture composition will be investigated, and the consequences for the droplet composition and sizes are determined using the new model. Compared to full Becker–Döring theory, the approximate model is computationally inexpensive and applicable to large numbers of species.
The intention of this paper is twofold. First, we will provide a complete computational framework for multispecies aerosol including its formation, evolution, and transport taking into account all explicitly stated assumptions. Subsequently, the second part of this paper delivers detailed insight into the algorithm and accuracy of the presented nucleation model in a spatially homogeneous formulation. Further investigations that include spatially inhomogeneous transport for the coupled conservation equations are subject of ongoing research (see [7]). The organization of the paper is as follows. In Sect. 2, an extended model for nucleation of droplets from a supercritical vapor mixture and their subsequent evolution due to evaporation and condensation is presented, capable of treating systems with many chemical species. The formation of the critical clusters is at the root of the nucleation process and an algorithm for their chemical composition is discussed in Sect. 3, next to a method for efficient adaptive timestepping. The dynamics of the aerosol that emerges from a mixture of alcohol vapors is simulated under various cooling conditions and presented in Sect. 4. Concluding remarks are contained in Sect. 5.
2 Extended classical nucleation theory for many species mixtures
In this section, we present a general framework for multispecies aerosol formation and evolution, applicable to large numbers of species at low computational effort. First, we introduce the transport equations in Sect. 2.1, following the conventional ‘Euler–Euler’ setting. The source terms describing nucleation, condensation, and evaporation, as well as coalescence of the aerosol will be presented in Sects. 2.2, 2.3, and 2.4. In order to develop this model, the singlespecies aerosol formation was studied in detail in [6] standing as a base for further multispecies model extensions. Multispecies representation requires the redefinition of the aerosol formation (nucleation) model, in which a novel approach for the critical cluster composition and condensation rate was introduced. This applies also for the other submodels, e.g., the condensation/evaporation model must also take into account the multispecies character of the gas/liquid mixtures. The framework presented in this section serves as a complete source of information to build an Eulerian twomoment multispecies aerosol physics approach coupled with the computational fluid dynamics equations represented by the conservation laws. Furthermore, thermophysical properties of several basic acyclic alcohols, composing the aerosol formers for the nucleation model system studied in this paper, are discussed in Sect. 2.5. The dynamic system that governs the evolution of the aerosol in a spatially homogeneous system is summarized in Sect. 2.6.
2.1 Multispecies aerosol transport equations including phase transition
In this section, we formulate a general system of multiphase transport equations describing the evolution of a multispecies aerosol. We consider the total system to be composed of dispersed droplets, next to a mixture of carrier gas (e.g., air) and vapors that constitute the transporting medium and the source of chemical components, respectively. We adopt an Euler–Euler formulation in which the gaseous ‘carrier phase’, i.e., carrier gas and aerosol forming vapors, and the aerosol droplets are represented by continuous fields. We assume that the droplets are sufficiently small in order to precisely follow the flow, i.e., the relative velocity with respect to the carrier phase can be neglected. Likewise, the droplets are assumed to have immediate heat transfer with the carrier phase such that the temperature of all phases may be treated as being the same.
Mass fraction equations for \(\{ Y_i, Z_i\}\) in (1) describe the partitioning between gaseous and liquid aerosol phases containing mass transfer source terms that will be introduced momentarily. The particle number density equation for N gives further information concerning the characterization of the aerosol. Mass fraction equations must be consistent with the global mass conservation equation that specifies the overall density of the considered mixture. At the same time, the particle number density equation must be consistent with the liquid mass fraction equations to account for the aerosol transport in the system. We assume a fixed lognormal aerosol size distribution in the system, as is commonly done in moment equation models. Transport of liquid mass fractions together with the evolving particle number density provides information about the average aerosol droplet size. This may vary depending on the mass transfer (condensation/evaporation) and transport (convection and coalescence) processes in the system. Nucleation of aerosol affects both mass and particle number density. The assumed lognormal distribution excludes a closer connection to possible multimodal corrections as observed in some model studies [9]. In this paper, we focus on the accuracy of model predictions as governed by the temporal integration method, resolving very fast nucleation scales as well as much slower condensation and evaporation. The same lognormal approach implies that aerosol polydispersity will only influence the average diameter of the droplets and not the fluctuations. Finally, we assume a homogeneous temperature equilibration between the phases in which effects of phase changes are represented by the enthalpy of vaporization. More involved formulations that assign different temperatures to the gas phase and the liquid phase are not pursued here.
 (i)Mass transfer between the liquid and the vapor of species i, denoted by \(S_{i}^{l \rightarrow v}\). This contains two contributions: the nucleation mass flow rate \(S_{i}^\mathrm{{nuc}}\), characterizing vapor changing into liquid, and the mass flow rate due to evaporation of vapor from already formed droplets minus that due to condensation onto these droplets, \(S_{i}^{\mathrm{{e}}\mathrm{{c}}}\), such that$$\begin{aligned} S_{i}^{l \rightarrow v}=S_{i}^\mathrm{{nuc}}+S_{i}^{\mathrm{{e}}\mathrm{{c}}}. \end{aligned}$$(5)
 (ii)The heat flow rate due to phase change, denoted by \(S_\mathrm{{h}}\). This can be computed by summing the products of \(S_{i}^{l \rightarrow v}\) of species i with the heat of evaporation of that species \(\Delta h_{i}^\mathrm{{vap}}\):$$\begin{aligned} S_\mathrm{{h}}= \sum _{i=1}^{n} \Delta h_{i}^\mathrm{{vap}} S_{i}^{l \rightarrow v}. \end{aligned}$$(6)
 (iii)The rate of change of the number concentration of dropletsin which we distinguish the nucleation rate \(J_\mathrm{{N}}\), the coalescence rate \(J_\mathrm{{c}}\), and the rate of complete droplet evaporation \(J_\mathrm{{ev}}\). Complete evaporation in an Eulerian aerosol model is a subject in its own right. In this paper, we will not address this issue and concentrate on situations in which droplets, once formed, undergo size changes through condensation and evaporation, but are assumed to never fully disappear because of evaporation. This assumption is well satisfied, e.g., in situations in which only cooling of a freshly nucleated aerosol would take place, a process dominated by condensation. We consider \(J_\mathrm{{ev}}=0\) in this paper. Coalescence is often negligible during fast nucleation bursts. It does play a role in the much slower dynamics of aerosol evolution on longer time scales. In this paper, we include a simple model for \(J_\mathrm{{c}}\) as specified in Sect. 2.4.$$\begin{aligned} S_\mathrm{{N}}=J_\mathrm{{N}}J_\mathrm{{c}}J_\mathrm{{ev}}, \end{aligned}$$(7)
2.2 Nucleation of aerosol
In this section, we present a generalization of classical multicomponent theory for homogeneous nucleation, starting from work on ternary nucleation by Arstila et al. [3]. This theory is applicable both to situations in which one of the components is dominant in super saturation as well as to situations in which a number of species engage in the nucleation simultaneously. In situations where only one component dominates in saturation, one might also adopt homogeneous nucleation theory for that species [6] to a good approximation. The more complete situation in which several species contribute to the nucleation process requires that one accounts for the coupling between the nucleating species as well, e.g., expressed by ‘competition’ for energy. For example, energy that is released by the nucleation of a particular species, will affect the local temperature and thereby the nucleation rates of other species as well. It is not easy to determine whether one component is dominant in saturation or not. In fact, two components with very different partial vapor pressures and very different saturation vapor pressures can have similar saturations (also called ’activities’ in some literature [5] in case a mixture of species is concerned). Therefore, a general classical multispecies nucleation theory is formulated here.
 1.The first element in the nucleation model concerns the formation of the virgin core from which aerosol droplets will form later. The composition of the socalled ‘critical cluster’ is described next, i.e., the cluster whose size is such that it has an equal probability to subsequently grow or shrink due to evaporation and condensation to and from the surrounding vapors. For that purpose, we introduce [4] the actual vapor pressure of species i, denoted by \(p_{i}\), and the corresponding saturation vapor pressure of species i in the current mixture, \(p_{i, \mathrm{{sat}}}^\mathrm{{mix}}\). In addition, we denote the partial molar volume by \(v_{i}\). Formally extending the thermodynamically consistent classical nucleation theory [3] to an arbitrary number of species suggests that the composition of the critical cluster is such thatThe commonly used saturation pressure of the pure component i, which arises above a surface of the pure liquid of that component held at temperature T, denoted by \(p_{i}^\mathrm{{sat}}\), can now be used to define the saturation of species i, i.e., \(S_{i} \ge 0\), and the mole fraction of species i in the critical cluster, i.e., \(0 \le w_{i} \le 1\) as$$\begin{aligned} f_{1}=f_{2}=f_{3} = \cdots = f_{n},~~~~~\text{ where }~~~ f_{i}=\frac{1}{v_{i}}\ln \left( \frac{p_{i}}{p_{i, \mathrm{{sat}}}^\mathrm{{mix}}}\right) . \end{aligned}$$(8)This relation applies if Raoult’s law holds [11]. With these definitions we may express the conditions from which the composition of the critical cluster in the ncomponent mixture can be obtained as$$\begin{aligned} p_{i}=S_{i} p_{i}^\mathrm{{sat}}; \quad p_{i, \mathrm{{sat}}}^\mathrm{{mix}}=w_{i}p_{i}^\mathrm{{sat}}. \end{aligned}$$(9)where \(\alpha \) is a priori unknown and specified by the auxiliary condition that$$\begin{aligned} f_{i}=\frac{1}{v_{i}}\ln \left( \frac{S_{i}}{w_{i}} \right) = \alpha ; \quad i=1, 2, \ldots , n, \end{aligned}$$(10)Combined, the system of equations (10) and (11) constitute \(n+1\) equations for the \(n+1\) unknowns \(\{ \alpha , w_{1}, \ldots , w_{n} \}\). The mole fractions in the critical cluster can be obtained once the saturations of all species \(S_{i}\) and their partial molar volumes \(v_{i}\) are specified. We may extract the mole fraction of species i from (10) as$$\begin{aligned} \sum _{i=1}^{n} w_{i}=1; \quad w_{i} \ge 0. \end{aligned}$$(11)Since we require the solution to be also a partition of unity, we obtain a consistency relation for \(\alpha \):$$\begin{aligned} w_{i}=S_{i}\exp (\alpha v_{i}). \end{aligned}$$(12)This is also referred to as the mole fraction equation. We may solve this problem for \(\alpha \) iteratively as shown and analyzed in Sect. 3 in which we also consider the dependence of the composition on the saturation and the partial molar volume.$$\begin{aligned} f(\alpha )=\left( \sum _{i=1}^{n} S_{i}\exp (\alpha v_{i}) \right) 1= 0. \end{aligned}$$(13)
 2.The second step in the determination of the nucleation rate is the calculation of the equilibrium concentration of critical clusters, denoted by \(c_\mathrm{{eq}}\). We adopt an ideal mixture approximation in which the surface tension \(\sigma \) of the critical cluster may be written asin which \(\sigma _{i}\) denotes the surface tension of component i. These quantities depend on temperature, for which accurate approximations are available in literature for a wide range of components [13]. The specification of the thermophysical properties of a range of the smaller acyclic alcohols is postponed until Sect. 2.5. The radius of the critical cluster r can be calculated from [3]$$\begin{aligned} \sigma =\sum _{i=1}^{n} w_{i}\sigma _{i}, \end{aligned}$$(14)in which we introduced the average molecular volume v through an ideal mixture law \(v \equiv \sum _{i=1}^{n}w_{i} v_{i}\).$$\begin{aligned} r=\frac{2\sigma v}{kT\sum _{i=1}^{n} w_{i}\ln (S_{i}/w_{i})} = \frac{2 \sigma }{kT\alpha }, \end{aligned}$$(15)After these preparations, the Gibbs free energy barrier \(\Delta G\) of the critical cluster, measuring the energy needed for the formation of a critical cluster with radius r and a surface tension \(\sigma \), can be expressed asThis allows calculating the equilibrium concentration \(c_\mathrm{{eq}}\) of critical clusters. In fact, from statistical mechanics, \(c_\mathrm{{eq}} \sim \exp (\Delta G/kT)\) with normalization still to be specified. The correct normalization is subject of much discussion in literature [14]. We consider two options for completing the expression for \(c_\mathrm{{eq}}\). A crude approximation, based on the partial vapor pressures of the species involved in the nucleation can be written as$$\begin{aligned} \Delta G=\frac{4}{3}\pi r^{2} \sigma . \end{aligned}$$(16)where H denotes Heaviside’s function \(H(z)=1\) if \(z > 0\) and 0 otherwise. In this expression only the species actually contained in the critical cluster, i.e., with \(w_{i} > 0\), contribute a factor \(p_{i}^{v}/kT\) and the total represents a sum over monomer concentrations [3]. This normalization is know to be physically inconsistent in some limiting cases [5]. A refinement for the determination of the equilibrium concentration \(c_\mathrm{{eq}}\) is obtained by adopting a socalled ‘selfconsistent’ normalization [14], which is mathematically consistent in the limits of singlespecies conditions [2]. In fact, introducing the single species, or ‘monomer,’ surface area$$\begin{aligned} c_\mathrm{{eq}}=\exp \left(  \frac{\Delta G}{kT} \right) \sum _{i=1}^{n} \frac{H(w_{i})p_{i}^{v}}{kT}, \end{aligned}$$(17)the equilibrium concentration is expressed as$$\begin{aligned} s^\mathrm{{mon}}_{i} = \left( 36 \pi \right) ^{1/3} v_{i}^{2/3}, \end{aligned}$$(18)in terms of the species saturation pressure \(p_{i}^\mathrm{{sat}}\).$$\begin{aligned} c_\mathrm{{eq}}=\exp \left(  \frac{\Delta G}{kT} \right) \prod _{i=1}^{n} \left( \frac{p_{i}^\mathrm{{sat}}(T)}{kT} \exp \left( \frac{s^\mathrm{{mon}}_{i} \sigma _{i}}{kT} \right) \right) ^{w_{i}}, \end{aligned}$$(19)
 3.The third step leading toward the completion of the nucleation rate is the specification of the Zeldovich factor, which characterizes the contribution of Brownian motion to the formation of the critical cluster [15]. As proposed in [3] this factor can be approximated asin terms of the number of components that is actually involved in the nucleation, \(n_\mathrm{{nuc}}\). The latter can conveniently be expressed as$$\begin{aligned} Z=\left( \frac{\sigma v^{2}}{kT 4 \pi ^2 r^{4}}\right) ^{1n_\mathrm{{nuc}}/2}, \end{aligned}$$(20)$$\begin{aligned} n_\mathrm{{nuc}}=\sum _{i=1}^{n} H(w_{i}). \end{aligned}$$(21)
 4.The fourth and final preparation step toward an expression for the nucleation rate concerns the determination of the average growth rate. The total number of molecules in the critical cluster can be expressed as \(N_\mathrm{{tot}}=(4/3)\pi r^{3}/v\), which allows to compute the number of molecules of component i in such a cluster as \(N_{i}=N_\mathrm{{tot}}w_{i}\), and the total mass of the cluster as \(m=\sum _{i=1}^{n} N_{i}m_{i}\). Under the assumption that cluster–cluster collisions can be neglected, i.e., in the sufficiently dilute state, the condensation rate \(K_{ii}\) of component i can be found from [3]Extending the ternary expression for the average growth rate \(R_\mathrm{{av}}\) as proposed in [3] to a general system of n components, we formally arrive at$$\begin{aligned} K_{ii}=\left( \frac{p_{i}^{v}}{kT}\right) \left( \frac{3}{4\pi }\right) ^{1/6}\big ( 6kT\big )^{1/2} \left( \frac{1}{m_{i}}+\frac{1}{m} \right) ^{1/2}\left( \left( \frac{m_{i}}{\rho _{i}^\mathrm{{l}}}\right) ^{1/3} + \left( \frac{4 \pi }{3} \right) ^{1/3} r \right) ^{2}. \end{aligned}$$(22)$$\begin{aligned} R_\mathrm{{av}}=\frac{\sum _{i=1}^{n} N_{i}^{2}}{\sum _{i=1}^{n} N_{i}^{2}/K_{ii}}=\frac{\sum _{i=1}^{n} w_{i}^{2}}{\sum _{i=1}^{n} w_{i}^{2}/K_{ii}}. \end{aligned}$$(23)
The expression for the nucleation rate \(J_\mathrm{{N}}\) is typical for currently adopted classical nucleation theory in the sense that it rests largely on phenomenological physics and scaling arguments clarifying the main dependencies and mechanisms and capturing the expected order of magnitudes. In all this, the normalization of the equilibrium concentration \(c_\mathrm{{eq}}\) is very important to the final level of quantitative agreement with physical reality that is achieved. A key point of reference for gaging the phenomenological expression for \(J_\mathrm{{N}}\) is the seminal Becker–Döring theory [1, 16], which constitutes a fundamental microscopic treatment of the nucleation process. It is computationally rather expensive for systems with many species but in some cases the ncomponent Becker–Döring equations can be solved in full detail [2], thereby allowing to crossvalidate the developed model for the nucleation rate \(J_\mathrm{{N}}\) with the full numerical solution. We return to this crossvalidation and motivation of the appropriate normalization of \(c_\mathrm{{eq}}\) in Sect. 3.1.
2.3 Evolution of aerosol by evaporation and condensation
Evaporation and condensation are two sides of one mechanism—that of gas–liquid mass transfer. While evaporation relates to net mass transfer from the liquid droplets to the gas phase, condensation is net mass transfer from the gas phase to the droplet phase. Evaporation (or condensation) will make the droplets shrink (or grow), but it will not change the number of droplets. For multispecies evaporation and condensation in the dilute regime, as considered here, one may treat all components independently as proposed by Friedlander [17], Wilck and Stratmann [18]. A more elaborate treatment including full coupling between the various species as proposed in [19] is not used here.
In order to derive an expression for the evaporation and condensation mass flow rate \(S_{i}^{\mathrm{{e}}\mathrm{{c}}}\) of component i, we need to collect expressions of a number of constituting factors. The desired mass flow rate is proportional to the number density of droplets N and should be sensitive to whether the saturation is larger or smaller than the equilibrium saturation. In addition, specific transport properties such as diffusion should be incorporated to quantify \(S_{i}^{\mathrm{{e}}\mathrm{{c}}}\). We specify the various elements next.
2.4 Coalescence effects in aerosol dynamics
2.5 Thermophysical properties of acyclic alcohols
Parameters defining thermophysical properties of the several acyclic alcohols \(\mathrm{{C}}_{i}\mathrm{{H}}_{2i+1}\mathrm{{OH}}\) considered in this paper: ethanol (\(i=2\)), propanol (\(i=3\)), butanol (\(i=4\)), pentanol (\(i=5\)), and hexanol (\(i=6\))
i  \(\frac{p_{\mathrm{{c}},i}}{10^{6}}\)  \(\alpha _{i}\)  \(\beta _{i}\)  \(\gamma _{i}\)  \(\delta _{i}\)  \(T_{\mathrm{{c}},i} \)  \(A_{i}\)  \(B_{i}\)  \(10^{3} a_{i} \)  \(10^{3} b_{i}\) 

2  6.13  \(\)8.69  1.18  \(\)4.88  \(\)1.59  513.92  1060.6  0.96  24.05  0.083 
3  5.17  \(\)8.54  1.96  \(\)7.69  \(\)2.95  536.78  1050.1  0.85  25.26  0.078 
4  4.42  \(\)8.41  2.23  \(\)8.25  \(\)0.71  563.05  1050.3  0.88  27.18  0.090 
5  3.91  \(\)8.98  3.92  \(\)9.91  \(\)2.19  588.15  1049.8  0.79  27.54  0.087 
6  3.47  \(\)9.49  5.13  \(\)10.58  \(\)5.15  610.70  1044.1  0.77  26.44  0.087 
2.6 Dynamic model for multispecies aerosol evolution
The composition of the critical cluster is a key element in the description of a nucleation burst. In the next section, we turn to this problem and illustrate the effects of changing the saturation of the various components on the composition of the cluster.
3 Numerical methods for efficient timestepping and critical cluster computations
In this section, we present and illustrate the method used to compute the critical cluster composition (Sect. 3.1) and discuss the adaptive timestepping method used for efficient simulation of the multiscale problem that includes a rapid nucleation burst next to evaporation and condensation, as well as much slower coalescence effects (Sect. 3.2).
3.1 Critical cluster composition
The Newton iteration can also be adopted to solve the critical cluster equation in cases in which many species make up the gas phase. There are many possible situations that one can use to illustrate the current formulation. We restrict ourselves to a situation in which \(\phi _{i} =1\), denoting systems for which the molar volume \(v_{i}\) does not depend on the index. We consider two situations in which species ‘1’ is dominant and supersaturated, e.g., \(S_{1}=2\) and consider subdominant species with (a) \(S_{i}=\beta 2^{i}\) and (b) \(S_{i}=\beta /(2i)\) for \(i \ge 2\), with \(\beta =5\). In Fig. 2, we show the mole fractions \(\{w_{i} \}\) for these two situations as function of the number of species included in the formulation. In both situations, the Newton iteration was found to converge very rapidly. We notice that in case (a), the subdominant contributions for \(n \gtrsim 10\) can effectively be neglected and a good representation of the critical cluster appears not to require more species in the model. In case (b), the subdominant species do not reduce in saturation as rapidly and the composition of the critical cluster is much richer and many more species need to be included before an accurate prediction of the composition can be achieved.
The key step of computing the composition of the critical cluster on the basis of Newton iteration was found to be possible even for large numbers of species. This makes it possible to study the extended classical nucleation theory for many species in a dynamic setting, including also evolution of the nucleating aerosol due to evaporation and condensation.
3.2 Adaptive timestepping for rapid nucleation bursts and slow coalescence
In order to efficiently simulate the evolution of the aerosol subject to rapid nucleation during the initial stages as well as to the long term much slower coalescence, it is natural to adopt a timestepping method that adapts to the actual instantaneous time scale. We maintain control over the timeaccuracy by keeping the size of the timestep appropriately small during the rapid nucleation stages, while increasing the size of the timestep in case the dynamics so allows.

Using the basic propagation algorithm for performing an explicit timestep we may compute the numerical solution in the next two instants of time, i.e., \(u_{\delta t} (t_{n+1})\) and \(u_{\delta t}(t_{n+2})\) with \(t_{n+1}=t_{n}+\delta t\) and \(t_{n+2}=t_{n}+2\delta t\). Here, we added the subscript \(\delta t\) to indicate that the numerical solution was obtained using timestep \(\delta t\).

The solution at \(t_{n+2}\) may also be approximated using one timestep of size \(2 \delta t\), denoted by \(u_{2\delta t}(t_{n+2})\).

Likewise, we may approximate the solution at \(t_{n+2}\) by taking four timesteps of size \(\delta t /2\), denoted by \(u_{\delta t/2}(t_{n+2})\).

We determine the relative differences \(\epsilon _{2 \delta t} = \Vert u_{2 \delta t}(t_{n+2})u_{\delta t}(t_{n+2})\Vert /\Vert u(0)\Vert \) and \(\epsilon _{\delta t/2} = \Vert u_{\delta t/2}(t_{n+2})u_{\delta t}(t_{n+2})\Vert /\Vert u(0) \Vert \). Specifically, we concentrate on the ethanol component in our model system to monitor the accuracy of the time integration, i.e., we take \(u=Y_{1}\) in this paper.
We adopted Euler forward timestepping as basic propagation algorithm in the simulations. Obviously, this is not a strict requirement and the same general approach can be combined with other, higherorder explicit timestepping methods. In the simulations we use a timestep stretching factor of \(a=1.2\) when increasing or decreasing the timestep size. We limit the timestep to \(\delta t_\mathrm{{max}}\) such that the solution remains stable and timeaccurate even for the slowest time scales. The initial size of the timestep is taken sufficiently small to capture the brief nucleation burst that occurs during the sharp cooling ramp. Finally, the value of the tolerance needs to be specified to control the adaptation of the timestep. How precisely to specify these numerical parameters is a matter of some experimentation in actual examples. We turn to the evolution of the aerosol emanating from a mixture of alcohol vapors in the next section and illustrate the specification of the numerical control parameters.
4 Dynamics of aerosol formation from rapidly cooled alcohol vapors
In this section, we first specify the computational model with which the aerosol evolution under rapid cooling of a mixture of alcohol vapors can be simulated; we will address the connection with the Becker–Döring theory by exploiting the detailed numerical solution as presented by Van Putten et al. [2], and quantify the relevance of the Wilemski normalization for this situation (Sect. 4.1). The resulting computational model is subsequently applied to analyze the aerosol dynamics of the multispecies alcohol vapors under different cooling rates and temperature levels, focusing on droplet sizes and their number density (Sect. 4.2).
4.1 Numerical model for aerosol formation from alcohol vapors
A key element in the aerosol formation is the nucleation rate \(J_\mathrm{{N}}\). As discussed earlier, the normalization of the equilibrium concentration of critical clusters \(c_\mathrm{{eq}}\) is subject to discussion in literature. In order to find out the accuracy of the proposed models, we compare (17) and (19) with the reference Becker–Döring result for a ternary system of ethanol, propanol, and hexanol in air, as obtained in [2]. In Fig. 3, we show the predicted \(J_\mathrm{{N}}\) at three different temperatures. To obtain this result, we closely follow [5] and consider a system at pressure \(p=66.76\) kPa. We vary the saturation of the species while keeping their ratios unchanged, i.e., \(S_\mathrm{{propanol}}/S_\mathrm{{ethanol}}\) and \(S_\mathrm{{hexanol}}/S_\mathrm{{ethanol}}\) are kept constant at 1.2 and 4, respectively. The mass fractions follow from (53), using Dalton’s law. We plot \(J_\mathrm{{N}}\) as a function of \(\mathbf{{S}}=\sqrt{S_\mathrm{{ethanol}}^{2}+S_\mathrm{{propanol}}^{2}+S_\mathrm{{hexanol}}^{2}}\).
All models included display quite similar trends but there is a significant underestimation of \(J_\mathrm{{N}}\) in case the normalization as presented in Eq. (17) is adopted. As the classical nucleation approach is not a ‘firstprinciple’ theory but largely a phenomenological model, various corrections/normalizations are applied to it that better suit the validation purposes for certain chemical compounds or that are mathematically appealing in an asymptotic limit. For example, the underestimation of two to three orders of magnitude of nucleation rate is almost completely corrected when Eq. (19) is adopted. The corresponding predictions correspond quite closely with the numerical solution to the full Becker–Döring theory as obtained in [2]. The current extended classical nucleation theory with normalization as in Eq. (19) yields an accurate agreement with the full Becker–Döring approach, at very low computational cost in contrast to the full ncomponent Becker–Döring (NBD) equations [5], for which comparative results were obtained at a substantial computational cost using multigrid methods. We notice that the steadystate classical nucleation theory based on Reiss [24] almost coincides with our prediction for \(J_\mathrm{{N}}\), which is based on earlier work of Stauffer et al. [25], establishing that both the Reiss and Stauffer approaches yield quite similar results for \(J_\mathrm{{N}}\).
We complete the investigation of the timestepping approach by considering the effect of the timestep stretching factor a. For that purpose we set \(\epsilon _\mathrm{{tol}}=10^{5}\) and use \(\delta t_{0}=10^{10}\) s. In Fig. 6a, we show the direct effect of the stretching a on the evolution of N. We observe that an increase in the stretching factor, i.e., adaptation leading to very rapid growth and reduction of the timestep, yields an overestimation of the asymptotic value of N by 35% in case \(a=4\). This is reminiscent of the effect of a relatively high value of \(\epsilon _\mathrm{{tol}}\) as shown in Fig. 5. The timestep variations in the course of time are shown in Fig. 6b. We observe that a large value of a not always implies highest \(\delta t\). Rather, the variations are highest as is the increase in \(\delta t\) initially. The algorithm can also yield rapidly oscillating behavior in \(\delta t\), as observed in case \(a=2\). Although this is not diminishing the overall accuracy in case the Euler forward scheme is used, it could lead to unwanted numerical errors when higherorder methods would be adopted. Adhering to an overall accuracy level in the asymptotic value of N of better than about 5% we select as stretching factor \(a=1.2\) in the sequel.
After these exploratory investigations of the numerical treatment of the nucleation burst, we proceed by investigating the physical implications of changing the temperature level and the cooling rate on the vapor and liquid phases as well as on the aerosol droplet properties such as number density and size.
4.2 Aerosol dynamics in multispecies alcohol vapor mixtures
In order to illustrate the capability of the proposed model to deal with nucleation, condensation, and coalescence of multiple species under different process conditions, we collect several characteristic illustrations in this section. First the effect of the overall temperature level is investigated, keeping the cooling rate fixed. Subsequently, the effect of cooling rate on the aerosol size distribution and chemical composition of the droplets is considered. Finally, we present an example in which we compare aerosol properties arising in alcohol mixtures containing different numbers of species.
Next to the system pressure, the temperature and the rate of cooling are decisive of the response of the system. Throughout, the system pressure is kept constant at 66.75 kPa. We are interested in the effect of rapid cooling of a vapor mixture, leading to supersaturation and the formation of an aerosol following a rapid nucleation burst. For this purpose, we consider a reference case in which we decrease the temperature by \(\Delta T=75\) K within 0.01 s. The properties of an aerosol that forms strongly depend on the thermodynamic state and properties of the system. For example, results obtained at various cooling rates show a significantly different temporal behavior in formation of the liquid phase (droplets) due to nonlinearity of the nucleation process. Figures are presented in a way that we feel gives the best illustration of the nonlinear behavior and shows the sensitivity to the chosen conditions.
The approach developed in this paper can also be adopted to investigate the effect of the cooling rate on the developing multispecies aerosol. For this purpose, we set the initial temperature to \(T_{1}=255 K\) and traverse the temperature drop within a time interval of 0.02, 0.04, 0.08, and 0.16 seconds. We observe in Fig. 8a that, with decreasing cooling rate the number concentration of aerosol droplets also strongly decreases. Simultaneously, we notice that the onset of nucleation is also delayed somewhat with decreasing cooling rate. This behavior is similarly observed in the evolution of the vapor concentrations, e.g., shown by the decay of ethanol in Fig. 8b, in which a slightly delayed nucleation can be observed as well at lower cooling rates. Moreover, we notice that decreasing the cooling rate leads to a slowing down of the consumption of the alcohol vapors in the aerosol formation. The total process appears to slow down proportionally to the size of the time interval during which the temperature drop takes place.
The consequences of the mixture composition on the number concentration and size of the developing aerosol are illustrated in Fig. 11. The asymptotic number concentration in Fig. 11a is seen to decrease by up to about 1 / 3 by simplification of the mixture. This has also consequences for the size of the developing aerosol droplets. As may be observed in Fig. 11b, simplification of the mixture leads to slightly delayed nucleation and smaller droplets initially, while asymptotically the droplet size is seen to be higher for the simplified mixtures.
5 Concluding remarks
The problem of multicomponent nucleation was formulated in terms of an Euler–Euler model with continuous velocity, temperature, and components fields, both for the vapor and the liquid phases. In the classical multicomponent approach as provided by the Becker–Döring theory [1], it is challenging to treat systems with large numbers of components. The current state of the art [5] is limited to modest numbers of components. Recent model studies discussed systems with up to five components [3, 5]. Experimentally, binary and ternary species nucleation is a challenging topic of ongoing investigations. We addressed the fully coupled classical nucleation theory as put forward in [3, 4] and developed a computational model that is capable of treating systems of considerably higher complexity being aware of the limitations concerning predictions coming from the classical nucleation theory. The current method can be used to treat systems with many components, as illustrated by the evaluation of the multispecies nucleation rate \(J_\mathrm{{N}}\) and the characteristics of the critical cluster in case up to 25 species were incorporated.
No principal nor computational reason appears not to include more complex vapor mixtures—application to environmental aerosols and to aerosols stemming from the processing of biomass and food products come within reach as soon as the thermophysical basic information of the individual species is available. Nucleation of multispecies gas mixtures from oversaturated vapors is a very challenging topic. Experimental results display a wide range of prediction. Moreover, it is well known that the introduction of certain additives/traces may significantly alter the nucleation process [26]. Such aspects are not taken into account in classical nucleation theory. Hence, application of classical nucleation theory must be considered with care, particularly for systems with many species. Mathematically, our approach does not have limitations in the number of species as for example presented in [27] and certainly more sophisticated methods can be used for the timestepping algorithms as it can be found in [28].
The level of complexity required to capture a certain fraction of the total mass of components in the droplets can be illustrated by considering the convergence of the critical cluster as function of the number of components retained in the model. Where one might be tempted to take a rather small number of components in the model in view of practical limitations, the extended classical nucleation approach allows to compute many species and identify the number \(\widetilde{n}\) of components needed to describe the aerosol formation with sufficient accuracy. This is basic to developing coarsened descriptions of aerosolforming vapor mixtures that include a very large number of species, e.g., in atmospheric conditions over urban areas or in consumption of smoking articles and tobaccorelated research. The Newton iteration that was used to solve the mole fraction equation was found to converge to machine accuracy within 5–8 iterations. For a wide range of characteristic nucleation conditions, the convergence was found to be quite independent of the number of species that was included in the problem.
In order to capture the wide range of time scales in the system, a simple adaptive timestepping method was developed and used in conjunction with Euler forward time integration. A range of 5–6 orders of magnitude between the size of the initial timesteps and timesteps that can be used after the nucleation burst, in case condensation and coalescence dominate, was found to be achievable, leading to a significant saving of computation time. Further improvements may be possible by combination of higherorder timestepping and the inclusion of specific physical simplifications that provide accurate approximations during certain stages of the evolution, e.g., during the late stages of dominant condensation/coalescence, but also during the initial phase where nucleation characterizes the full response to the employed cooling.
The illustration of the effects of (a) temperature level, (b) cooling rate, and (c) mixture composition on the properties of the developing aerosol show that the extended classical nucleation approach adopted in this paper can be used effectively for spatially homogeneous systems. This capability will be integrated with laminar flow of the aerosolforming species and hence constitute a comprehensive computational model with which physical experiments conducted in LFDC equipment can be simulated. This is subject of ongoing research and will provide a full comparison with experimental data, allowing a validation of the method.
Notes
Acknowledgements
The research presented in this work was funded by Philip Morris Products S.A. (part of Philip Morris International group of companies).
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