Aerosol

For low values of the Reynolds number (<1), true for most aerosol motion, Stokes' law describes the force of resistance on a solid spherical particle in a fluid. However, Stokes' law is only valid when the velocity of the gas at the surface of the particle is zero. For small particles (< 1 μm) that characterize aerosols, however, this assumption fails. To account for this failure, one can introduce the Cunningham correction factor, always greater than 1. Including this factor, one finds the relation between the resisting force on a particle and its velocity:[19]

${\displaystyle F_{D}={\frac {3\pi \eta Vd}{C_{c}}}}$

where

${\displaystyle F_{D}}$ is the resisting force on a spherical particle

${\displaystyle \eta }$ is the dynamic viscosity of the gas

${\displaystyle V}$ is the particle velocity

${\displaystyle C_{c}}$ is the Cunningham correction factor.

This allows us to calculate the terminal velocity of a particle undergoing gravitational settling in still air. Neglecting buoyancy effects, we find:[20]

${\displaystyle V_{TS}={\frac {\rho _{p}d^{2}gC_{c}}{18\eta }}}$

where

${\displaystyle V_{TS}}$ is the terminal settling velocity of the particle.

The terminal velocity can also be derived for other kinds of forces. If Stokes' law holds, then the resistance to motion is directly proportional to speed. The constant of proportionality is the mechanical mobility (B) of a particle:[21]

${\displaystyle B={\frac {V}{F_{D}}}={\frac {C_{c}}{3\pi \eta d}}}$

A particle traveling at any reasonable initial velocity approaches its terminal velocity exponentially with an e-folding time equal to the relaxation time:[22]

${\displaystyle V(t)=V_{f}-(V_{f}-V_{0})e^{-{\frac {t}{\tau }}}}$

where:

${\displaystyle V(t)}$ is the particle speed at time t

${\displaystyle V_{f}}$ is the final particle speed

${\displaystyle V_{0}}$ is the initial particle speed

To account for the effect of the shape of non-spherical particles, a correction factor known as the dynamic shape factor is applied to Stokes' law. It is defined as the ratio of the resistive force of the irregular particle to that of a spherical particle with the same volume and velocity:[23]

${\displaystyle \chi ={\frac {F_{D}}{3\pi \eta Vd_{e}}}}$

where:

${\displaystyle \chi }$ is the dynamic shape factor

The aerodynamic diameter of an irregular particle is defined as the diameter of the spherical particle with a density of 1000 kg/m³ and the same settling velocity as the irregular particle.[24]

Neglecting the slip correction, the particle settles at the terminal velocity proportional to the square of the aerodynamic diameter, d_a:[24]

${\displaystyle V_{TS}={\frac {\rho _{0}d_{a}^{2}g}{18\eta }}}$

where

${\displaystyle \ \rho _{0}}$ = standard particle density (1000 kg/m³).

This equation gives the aerodynamic diameter:[25]

${\displaystyle d_{a}=d_{e}\left({\frac {\rho _{p}}{\rho _{0}\chi }}\right)^{\frac {1}{2}}}$

One can apply the aerodynamic diameter to particulate pollutants or to inhaled drugs to predict where in the respiratory tract such particles deposit. Pharmaceutical companies typically use aerodynamic diameter, not geometric diameter, to characterize particles in inhalable drugs.

The previous discussion focused 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 that move particles outside a volume of gas under study include diffusion, gravitational settling, and electric charges and other external forces that cause particle migration. A second set of processes internal to a given volume of gas include particle formation (nucleation), evaporation, chemical reaction, and coagulation.[26]

A differential equation called the Aerosol General Dynamic Equation (GDE) characterizes the evolution of the number density of particles in an aerosol due to these processes.[26]

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

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

Where:

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

${\displaystyle \mathbf {q} }$ is the particle velocity

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

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

As particles and droplets in an aerosol collide with one another, they may undergo coalescence or aggregation. This process leads to a change in the aerosol particle-size distribution, with the mode increasing in diameter as total number of particles decreases.[27] On occasion, particles may shatter apart into numerous smaller particles; however, this process usually occurs primarily in particles too large for consideration as aerosols.

The Knudsen number of the particle define three different dynamical regimes that govern the behaviour of an aerosol:

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

where ${\displaystyle \lambda }$ is the mean free path of the suspending gas and ${\displaystyle d}$ is the diameter of the particle.[28] For particles in the free molecular regime, K_n >> 1; particles small compared to the mean free path of the suspending gas.[29] 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:

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

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. The derivation of this equation assumes constant pressure and constant diffusion coefficient.

Particles are in the continuum regime when K_n << 1.[29] In this regime, the particles are big compared to the mean free path of the suspending gas, meaning that the suspending gas acts as a continuous fluid flowing round the particle.[29] The molecular flux in this regime is:

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

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 absolute units like kelvin), and P_A∞ and P_AS 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:

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

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

Condensation and evaporation

Aerosol partitioning theory governs condensation on and evaporation from an aerosol surface, respectively. Condensation of mass causes the mode of the particle-size distributions of the aerosol to increase; conversely, evaporation causes the mode to decrease. Nucleation is the process of forming aerosol mass from the condensation of a gaseous precursor, specifically a vapor. Net condensation of the vapor requires supersaturation, a partial pressure greater than its vapor pressure. This can happen for three reasons:

1. Lowering the temperature of the system lowers the vapor pressure.
2. Chemical reactions may increase the partial pressure of a gas or lower its vapor pressure.
3. The addition of additional vapor to the system may lower the equilibrium vapor pressure according to Raoult's law.

There are two types of nucleation processes. Gases preferentially condense onto surfaces of pre-existing aerosol particles, known as heterogeneous nucleation. This process causes the diameter at the mode of particle-size distribution to increase with constant number concentration.[30] With sufficiently high supersaturation and no suitable surfaces, particles may condense in the absence of a pre-existing surface, known as homogeneous nucleation. This results in the addition of very small, rapidly growing particles to the particle-size distribution.[30]

Water coats particles in an aerosols, making them activated, usually in the context of forming a cloud droplet. Following the Kelvin equation (based on the curvature of liquid droplets), smaller particles need a higher ambient relative humidity to maintain equilibrium than larger particles do. The following formula gives relative humidity at equilibrium:

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

where ${\displaystyle p_{s}}$ 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:

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

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

There are no general solutions to the general dynamic equation (GDE);[31] common methods used to solve the general dynamic equation include:[32]

• Moment method[33]
• Modal/sectional method,[34] and
• Quadrature method of moments[35][36]/Taylor-series expansion method of moments,[37][38] and
• Monte Carlo method.[39]

Some available in situ measurement techniques include:

• Aerosol mass spectrometer (AMS)
• Differential mobility analyzer (DMA)
• Electrical aerosol spectrometer (EAS)
• Aerodynamic particle sizer (APS)
• Aerodynamic aerosol classifier (AAC)
• Wide range particle spectrometer (WPS)
• Micro-Orifice Uniform Deposit Impactor(MOUDI)
• Condensation particle counter (CPC)
• Epiphaniometer
• Electrical low pressure impactor (ELPI)
• Aerosol particle mass-analyser (APM)
• Centrifugal Particle Mass Analyser (CPMA)

Remote sensing approaches include:

• Sun photometer
• Lidar
• Imaging spectroscopy

Particles can deposit in the nose, mouth, pharynx and larynx (the head airways region), deeper within the respiratory tract (from the trachea to the terminal bronchioles), or in the alveolar region.[47] The location of deposition of aerosol particles within the respiratory system strongly determines the health effects of exposure to such aerosols.[48] This phenomenon led people to invent aerosol samplers that select a subset of the aerosol particles that reach certain parts of the respiratory system.[49] Examples of these subsets of the particle-size distribution of an aerosol, important in occupational health, include the inhalable, thoracic, and respirable fractions. The fraction that can enter each part of the respiratory system depends on the deposition of particles in the upper parts of the airway.[50] The inhalable fraction of particles, defined as the proportion of particles originally in the air that can enter the nose or mouth, depends on external wind speed and direction and on the particle-size distribution by aerodynamic diameter.[51] The thoracic fraction is the proportion of the particles in ambient aerosol that can reach the thorax or chest region.[52] The respirable fraction is the proportion of particles in the air that can reach the alveolar region.[53] To measure the respirable fraction of particles in air, a pre-collector is used with a sampling filter. The pre-collector excludes particles as the airways remove particles from inhaled air. The sampling filter collects the particles for measurement. It is common to use cyclonic separation for the pre-collector, but other techniques include impactors, horizontal elutriators, and large pore membrane filters.[54]

Two alternative size-selective criteria, often used in atmospheric monitoring, are PM₁₀ and PM_2.5. PM₁₀ is defined by ISO as particles which pass through a size-selective inlet with a 50% efficiency cut-off at 10 μm aerodynamic diameter and PM_2.5 as particles which pass through a size-selective inlet with a 50% efficiency cut-off at 2.5 μm aerodynamic diameter. PM₁₀ corresponds to the “thoracic convention” as defined in ISO 7708:1995, Clause 6; PM_2.5 corresponds to the “high-risk respirable convention” as defined in ISO 7708:1995, 7.1.[55] The United States Environmental Protection Agency replaced the older standards for particulate matter based on Total Suspended Particulate with another standard based on PM₁₀ in 1987[56] and then introduced standards for PM_2.5 (also known as fine particulate matter) in 1997.[57]

• Volcanic eruptions release large amounts of sulphuric acid, hydrogen sulfide and hydrochloric acid into the atmosphere. These gases represent aerosols and eventually return to earth as acid rain, having a number of adverse effects on the environment and human life.[59]
• Aerosols interact with the Earth's energy budget in two ways, directly and indirectly.

E.g., a direct effect is that aerosols scatter and absorb incoming solar radiation.[60] This will mainly lead to a cooling of the surface (solar radiation is scattered back to space) but may also contribute to a warming of the surface (caused by the absorption of incoming solar energy).[61] This will be an additional element to the greenhouse effect and therefore contributing to the global climate change.[62]

The indirect effects refer to the aerosols interfering with formations that interact directly with radiation. For example, they are able to modify the size of the cloud particles in the lower atmosphere, thereby changing the way clouds reflect and absorb light and therefore modifying the Earth's energy budget.[59]

There is evidence to suggest that anthropogenic aerosols actually offset the effects of greenhouse gases, which is why the Northern Hemisphere shows slower surface warming than the Southern Hemisphere, although that just means that the Northern Hemisphere will absorb the heat later through ocean currents bringing warmer waters from the South.[63]

• When aerosols absorb pollutants, it facilitates the deposition of pollutants to the surface of the earth as well as to bodies of water.[62] This has the potential to be damaging to both the environment and human health.
• Aerosols in the 20  μm range show a particularly long persistence time in air conditioned rooms due to their "jet rider" behaviour (move with air jets, gravitational fall out in slowly moving air);[64] as this aerosol size is most effectively adsorbed in the human nose,[65] the primordial infection site in COVID-19, such aerosols may contribute to the pandemic.
• Aerosol particles with an effective diameter smaller than 10 μm can enter the bronchi, while the ones with an effective diameter smaller than 2.5 μm can enter as far as the gas exchange region in the lungs,[66] which can be hazardous to human health.