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Aerosol Dynamics

The previous discussion focussed on single aerosol particles. In contrast, aerosol dynamics explains the evolution of complete aerosol populations. The concentrations of particles will change over time as a result of many processes. External processes which move particles across the wall of a volume of gas include diffusion, gravitational settling and migration caused by external forces such as electric charges. A second set of processes are internal to a given volume of gas are particle formation (nucleation), evaporation or chemical reaction and coagulation.

The evolution of the aerosol due to these process can be characterized by a differential equation called the Aerosol General Dynamic Equation (GDE).

${\frac {\partial {n_{i}}}{\partial {t}}}=-\nabla \cdot n_{i}\mathbf {q} +\nabla \cdot D_{p}\nabla _{i}+\left({\frac {\partial {n_{i}}}{\partial {t}}}\right)_{growth}+\left({\frac {\partial {n_{i}}}{\partial {t}}}\right)_{coag}-\nabla \cdot \mathbf {q} _{F}n_{i}$

Change in time = Convective transport + brownian diffusion + gas-particle interactions + coagulation + migration by external forces

Where:

$n_{i}$ is number density of particles of size category $i$

$\mathbf {q}$ is the particle velocity

$D_{p}$ is the particle Stokes-Einstein diffusivity

$\mathbf {q} _{F}$ is the particle velocity associated with an external force

Coagulation

When particles are present in an aerosol they collide with each other. During that they may undergo coalescence or aggregation. This process leads to a change in the aerosol number/size distribution function, with the mode growing in diameter and decreasing in number .

Aerosol dynamics regimes

There are three different dynamical regimes which govern the behaviour of an aerosol, which can be defined by the Knudsen number of the particle

$K_{n}={\frac {2\lambda }{d}}$

where $\lambda$ is the mean free path of the suspending gas and $d$ is the diameter of the particleTemplate:(Baro n and Willeke 200 I. Particles are in the free molecular regime when K_n >> 1, that is particles are small compared to the mean free path of the suspending gas. In this regime, particles interact with the suspending gas through a series of 'ballistic' collisions with gas molecules. As such, they behave similarly to gas molecules, tending to follow streamlines and diffusing rapidly through Brownian motion. The mass flux equation in the free molecular regime is:

$I={\frac {\pi a^{2}}{k_{b}}}\left({\frac {P_{\infty }}{T_{\infty }}}-{\frac {P_{A}}{T_{A}}}\right)\cdot C_{A}\alpha$

where a is the particle radius, P_∞ and P_A are the pressures far from the droplet and at the surface of the droplet respectively, k_b is the Boltzmann constant, T is the temperature, C_A is mean thermal velocity and α is mass accommodation coefficient. It is assumed in the derivation of this equation that the pressure and the diffusion coefficient are constant.

Particles are in the continuum regime when K_n << 1 . In this regime, the particles are big compared to the mean free path of the suspending gas, meaning that the suspending gas can be though of as a continuous fluid flowing round the particle Template:DeCarlo. The molecular flux in this regime is:

$I_{cont}\sim {\frac {4\pi aM_{A}D_{AB}}{RT}}\left(P_{A\infty }-P_{AS}\right)$

where a is the radius of the particle A, M_A is the molecular mass of the particle A, D_AB is the diffusion coefficient between particles A and B, R is the ideal gas constant, T is the temperature in kelvins and P are the pressures at infinite and at the surface respectively.

The transition regime contains all the particles in between the free molecular and continuum regimes or K_n ≈ 1. The forces experienced by a particle are a complex combination of interactions with individual gas molecules, and macroscopic interactions. The semi-empirical equation describing mass flux is:

$I=I_{cont}\cdot {\frac {1+K_{n}}{1+1.71K_{n}+1.33K_{n}^{2}}}$

where I_cont is the mass flux in the continuum regime. This formula is called the Fuchs-Sutugin interpolation formula. These equations don’t take into account the heat release effect.

Aerosol partitioning

Aerosol partitioning theory governs the condensation and evaporation of substances to and from and aerosol surface, respectively. Condensation of mass causes the mode of a aerosol number/size distribution to grow to larger diameters. Conversely, evaporation moves the mode to smaller diameters. Nucleation is the process of forming aerosol mass from the condensation of a gaseous precursor, specifically a vapour. In order for the vapour to condense, it must be supersaturated i.e. its partial pressure must be greater than its vapour pressure. This can happen for three reasons:

1. If the vapour pressure is lowered by lowering the temperature of the vapour
2. If chemical reactions increase the partial pressure of a gas, or lower its vapour pressure
3. If the addition of another vapour lowers the equilibrium vapour pressure due to the Raoult Effect

There are two types of nucleation processes. Gases will preferentially condense onto pre-existing surfaces (e.g. aerosol particles), which is known as heterogeneous nucleation. This causes the number/size distribution mode diameter to increase with the total number concentration remaining constant. If the supersaturation is high enough, and no suitable surfaces are present, particles may condense in the absence of a pre-existing surface, which is known as homogeneous nucleation. The emergence of a number/size distribution mode growing from a diameter of zero.

Activation

Aerosols are said to be activated when they become coated by water, usually in the context of forming a cloud droplet. Following the Kelvin effect (based on the curvature of liquid droplets) smaller particles need a higher ambient relative humidity to maintain equilibrium than bigger ones would. Relative humidity (%) for equilibrium can be determined from the following formula:

$RH={\frac {p_{s}}{p_{0}}}\times 100\%=S\times 100\%$

where $p_{s}$ is the saturation vapor pressure above a particle at equilibrium (around a curved liquid droplet), p₀ is the saturation vapor pressure (flat surface of the same liquid) and S is the saturation ratio.

Kelvin equation for saturation vapor pressure above a curved surface is:

$\ln {p_{s} \over p_{0}}={\frac {2\sigma M}{RT\rho \cdot r_{p}}}$

where r_p droplet radius, σ surface tension of droplet, ρ density of liquid, M molar mass, T temperature, and R molar gas constant.