Expressions for the rate of particle growth.
In this chapter, the differential equations governing the growth of particles by condensation in the continuum regime (CR) and free molecular regime (FMR) are presented. Then, the methodology used to couple the solutions of the condensation equations with the MarkovMC simulation of the coagulation equation is described. This approach exploits the advantage of the MarkovMC method in allowing a crosslinking of deterministic and probabilistic models. In the present case, coagulation, which is modeled from a probabilistic approach, is combined with condensation, which is modeled deterministically. Since the two processes occur simultaneously, they are coupled through time. The coupling is carried out by using the time step for coagulation as the delta time for condensation. Thus, during every coagulating time step, the extent of growth by condensation of each particle in the aerosol is evaluated, and the particle size distribution is updated according to the new sizes of the particles. The results presented here initially focus on situations where only condensation is important, and later on, situations in which both processes, coagulation and condensation, occur simultaneously.
1. Condensation
When a droplet is embedded in a sufficiently supersaturated environment, the droplet grows by condensation of vapor on its surface. Diffusion theory and the kinetic theory are respectively used to calculate the rate of condensation in the CR and FMR. The expressions are known as the
Equation (l) is well known as the Kelvin equation. It is a relationship that expresses the vapor pressure over a curved interface in terms of the saturation pressure P_{s}, which is the vapor pressure of the same substance over a flat surface. The vapor pressure of a liquid is determined by the energy necessary to detach a molecule from the surface by overcoming the attractive force exerted by its neighbors. When a curved interface exists, as in a small droplet, there are fewer molecules immediately adjacent to a molecule on the surface than when the surface is flat. Consequently, it requires less energy for molecules on the surface of a small drop to escape and the vapor pressure over a curved interface is greater than that over a plane surface. [64] The Kelvin effect is significant for particles smaller than 0.1 μm in diameter.
There is a critical particle size r* where the concentration gradient becomes zero and thus the growth laws predict no net rate of condensation. Under this condition, the system is in a metastable condition, where the rate of evaporation equals the rate of condensation. The growth laws show that all smaller particles (r < r*) evaporate while all the larger ones (r > r*) grow by condensation. The growth laws are obtained assuming that the condensation rate is sufficiently slow for the latent heat of condensation to be dissipated without changing the droplet temperature. Appendix A develops a heat transfer model to evaluate the accuracy of this assumption, and Chapter 6 presents the implications on the condensation rate when the increase in the droplet temperature is not negligible. Here the discussion is limited to situations where the assumption is reasonable. The growth laws can be expressed in terms of nondimensional variables η, S, and τ as shown in Table 1 by using η = r/r*, S = P/P_{s}, and τ = t/τ_{cond}, where r* is critical particle size for the onset of condensation; P is the vapor pressure of the condensable phase; Ps its saturation pressure; and τ_{cond} the characteristic time for condensation. Physically, 3τ_{cond} is the time required to double the volume of a particle of size r* by condensation. Expressions for τ_{cond} are also presented in Table 1.



Rate of growth  

Nondimensional form  

Characteristic time for condensation  

Solution for 


Solution for 


Assuming constant vapor concentration and temperature, the nondimensional versions of the growth laws were numerically integrated for several values of S. The results are presented in Figure 1. Figure 1a shows the evolution of particle size in the FMR and Figure lb for the CR. Three characteristic regions can be distinguished in each figure. For small sizes,
Analytical solutions can be obtained in each of the two extreme cases. Table l presents these solutions. For the case of
Then, after neglecting higherorder terms (HOT)
2. Coupling the condensation solution to the MarkovMC simulation of coagulation
The following discussion applies to twocomponent aerosols composed of a fully condensed material (solid or liquid) and a condensable material. Thus, condensation of only one of the components needs to be considered. Situations where the two components are condensable phases, or where there are more than two components, can be handled by a direct extension of this method but they will not be considered in this work.
It is assumed that a particle in the twocomponent aerosol can be described by its size (r) and mass fraction (y) of the condensable material present in the particle. Size and composition of a particle are given by:
where m_{k} and ρ_{k} are the mass and density of species k in the particle. We will let k = 1 represent the fully condensed material and k = 2 represent the condensable material.
The particle size range (R_{1}R_{m}) is divided into
where
The particles within a sizesection have different sizes and different compositions. Only particles within a sizecompositionsection have same size and composition.
Mass is conserved since size and composition within a section are allowed to assume their instantaneous mean values.
There is no restriction about the structure of the particles. In Chapter 6, as a special case, it will be assumed that the particles consist of a solid core with a spherical coating.
Since coagulation and condensation processes occur at the same time, the MarkovMC simulation of the coagulation equation and the deterministic solution of the condensation equation must be coupled through time. This suggests the use of the following type of algorithm:
Perform a MarkovMC coagulating step as described in Chapter 4
Compute the time step for coagulation δt_{coll} from Equation (4 from Chapter 4) and use this time step as the elapsed time for condensation δt_{cond}
For each particle in the aerosol
Compute the growth during δt_{cond}
Update the particle size distribution by moving the particle to the appropriate section according to its size and composition
Update vapor concentration to account for the condensed mass
Thus, this technique requires the evaluation of the particle growth equation for all particles in the aerosol with size greater than r* for each coagulating time step. Consequently, a direct numerical integration of the growth laws for the specific conditions of each particle would be computationally prohibitive. Instead, the nondimensional version of the particle growth laws can be solved for a range of values of S, as described in the former section. These solutions can then be stored in tables or can be represented by curvefits for the different solutions. Thus, during the actual run, the direct numerical evaluation of the integral expression is replaced by a tablelookup process or an evaluation of the curvefitting expression. Evaluation of particle growth for an arbitrary initial particle size η_{o}, nondimensional time step δτ, and vapor concentration S using the generalized solution, through either table searching or evaluation of curvefitted expressions, can be obtained by:
where η_{oτ} is the tabulated universal value for growth due to condensation during τ nondimensional times. The process is illustrated in Figure 1.
3. Results
A twocomponent aerosol will behave quite differently when the characteristic time for coagulation is greater than or less than the characteristic time for condensation. Thus, we define J as:
Expressions for




The condition
3.1. Results for J → ∞
The performance of the MarkovMC method for simulating an aerosol subject to condensation but not coagulation was evaluated. A lognormal distributed aerosol with parameters r_{m} = 3.0 nm, N = l0 [18] particles/m^{3}, and σ_{g} = 1.45 was taken as the initial particle size distribution.
Condensation was assumed to occur in the FMR and a vapor with S = 1.2 and r_{o}* = 3.9 nm was considered.
Figure 2 shows the results obtained with a) S = constant and no Kelvin effect, b) vapor phase mass depletion and no Kelvin effect, and c) vapor phase mass depletion and Kelvin effect. Since r* is the critical size above which condensation occurs, there is a discontinuity in the particle size distribution at r = r_{o}*. When the Kelvin effect is neglected and the concentration of the vapor remains constant, the tail of the particle distribution above r* is observed to steadily increase, appearing as equally spaced parallel curves (Figure 2a). This result is expected because particle growth is independent of size in the FMR when the Kelvin effect is negligible. Figure 2a also shows that the MarkovMC method is able to handle discontinuities in size distribution and does not suffer from numerical dispersion or diffusion as is the case with the sectional method. [60]
Figure 2b shows the corresponding evolution when the mass of the condensable vapor is depleted as condensation occurs. Two additional effects are observed. First, as condensation takes place, the rate of growth slows down and then, the distributions are no longer equispaced in time. Second, as condensation proceeds, S decreases, r* becomes larger (Figure 3), and then the growth of the particles with sizes r_{o}* < r < r* diminishes and the particle size is frozen.
For the same conditions but including the Kelvin effect, Figure 2c shows the slow growth of particles with sizes close to r*. The evolution of the aerosol can be described as the superposition of the lognormal distribution tail over the exponential variation with size of the Kelvin effect, which leads to bimodal distributions. Due to the Kelvin effect, small particles (r
3.1.1. Evolution of a single particle
To have a better understanding of the effect of gasphase depletion on particle size evolution, tagged particles have been observed during the condensation process. Figure 4 shows their behavior. For a particle with initial size close to r_{o}*, say η = r/r_{o} *= 1.01, the growth is strongly affected by the Kelvin effect and the particle grows very slowly. As the particle becomes larger, the Kelvin effect is less important, and the particle grows faster. As the condensable material is depleted, particle growth slows down and the particle starts to approach its asymptotic size which is reached when the condition S = l reached. However, before that equilibrium condition is reached r* can be larger than the particle size, and consequently its growth is frozen. In this model reevaporation processes has not been considered and thus the particle maintains its maximum size. The larger particles, e.g., η_{o} = 1.18 and η_{o} = 1.35, exhibit similar behavior initially. However, the cumulative influence of the Kelvin effect separates particle sizes.
When condensation occurs in the FMR over all the particles in the aerosol, i.e., when r_{m} >> r*, the particle size distribution is not affected by condensation provided that S = const. However, when vaporphase mass depletion is considered, the condensation process widens the particle size distribution during the final stages of the process. On the other hand, condensation in the CR is proportional to (1/r) and then condensation causes the particle size distribution to narrow.
The literature review of Section 3.3.2 concluded that most of the problems of the numerical methods for modeling aerosols arise when coagulation and condensation occur simultaneously. The next section describes the results obtained when the MarkovMC method is used to simulate simultaneous coagulation and condensation in a singlecomponent aerosol. Results for twocomponent aerosols will be presented in Chapter 6.
3.2. Results for J = 0.18
The results presented are for an atmospheric aerosol in the FMR. Initially, the particles have a lognormal distribution with r_{m} = 1.85 nm and σ_{g} = 1.56. The vapor concentration is assumed to remain constant with parameters S = 1.2 and r* = 5.9 nm. The results were obtained using a linear grid with 1000 sections to cover the range 1–20 nm. The evolution of the aerosol is described in terms of the parameters r, τ = t/τ_{cond}, n/N, and
For
3.3. Results for J = 1.18
Keeping the same set of conditions, but decreasing N_{0} up to 10 [20] particles/m^{3} so that
During the evolution of the aerosol, the model keeps track of the composition of the particles. For each particle, it distinguishes the mass coming from condensation from the mass coming from coagulation. For singlecomponent aerosols, as in this case, particle composition is defined as the ratio of the mass of the particle due to condensation over the total mass of the particle. Figure 7 shows the average sizecomposition distribution of the aerosol at τ = 3320. It shows that particles with r < r* are formed through pure coagulating processes. Particles with sizes slightly larger than the critical size are affected by the Kelvin effect and thus the content of the condensed phase is small but increases with particle size. A maximum is reached and then for larger particles, the mass ratio decreases due to further coagulation with smaller particles, i.e., with particles that have not received mass through condensation.
4. Concluding remarks
It has been shown that the MarkovMC method is an effective approach to modeling aerosol dynamics under conditions of pure coagulation, pure condensation, and when both coagulation and condensation occur simultaneously. Because the MarkovMC method does not have restrictions in the number of sections used, the particle size distribution of aerosols can be represented with high resolution. It also can be applied to broad particle size ranges and can evaluate the evolution of aerosols over long time periods.
The method is accurate in solving the condensation equation in multicomponent aerosols because it evaluates the growth of each particle in the aerosol at the time, and in doing such evaluation it uses generic exact solutions of the condensation equation. The errors introduced come from the methodology used to store the exact solutions of the particle growth equation. However, this process can be optimized up to any degree of accuracy as desired. Furthermore, when the Kelvin effect can be neglected, analytical expressions are available and no errors are introduced during this step. This approach can be extended to include other aerosol processes such as source/removal, and chemical reactions. The assumptions made for coagulation and condensation are summarized as follows:
Particles are spherical and stick together upon collision
Rate of condensation and coagulation independent of particle composition
Particles within an (i, j) section have the same, but not constant, size, and composition
In the following chapter, the MarkovMC method will be applied to a twocomponent aerosol to study the particle encapsulation process.