Condensation Discussion¶

Isothermal and non-isothermal condensation processes are fundamental in aerosol dynamics. Condensation involves the transfer of gas-phase species to the particle phase, which can be reversible when the species evaporates back into the gas phase. This process is pivotal in the formation of cloud droplets and the growth of atmospheric particles, influencing climate and air quality.

Condensation Isothermal¶

In the isothermal case, we consider condensation processes where the temperature remains constant, and the latent heat of vaporization is neglected. This approximation is valid when the heat released or absorbed during condensation or evaporation is insufficient to cause significant temperature changes.

This follows Chapter 2 (EQ 2.41) by Topping, D., & Bane, M. (2022). Introduction to Aerosol Modelling (D. Topping & M. Bane, Eds.). Wiley. https://doi.org/10.1002/9781119625728. Also Chapter 12 and 13 (EQ 13.3) of Seinfeld, J. H., & Pandis, S. N. (2016). Atmospheric Chemistry and Physics: From Air Pollution to Climate Change (3^rd ed.). Wiley.

The isothermal condensation or evaporation process is defined by the following equation:

Equation 1: Rate of Mass Change

dmi/dt = N × k_cond × (pᵢ, gas − pᵢ, particle surface) × (molar massᵢ / (R × T))

Where:

dmi/dt: Rate of change of mass of species i in the particle phase.
N: Number of particles.
k_cond: Per-particle first-order condensation coefficient.
pᵢ, gas: Partial pressure of species i in the gas phase.
pᵢ, particle surface: Partial pressure of species i at the particle surface, accounting for curvature and activity effects.
molar massᵢ: Molar mass of species i.
R: Ideal gas constant.
T: Temperature.

Description:

This equation quantifies the net mass flux of species i from the gas phase to the particle phase (or vice versa) due to condensation or evaporation. The driving force is the difference in partial pressures (pᵢ, gas − pᵢ, particle surface), and it's scaled by the molar mass and thermodynamic constants to yield a mass rate.

Condensation with Latent Heat¶

When condensation results in significant heat release or absorption, the latent heat of vaporization must be considered. This scenario is critical in cloud droplet formation, where the heat effects can influence the condensation rate and local temperature.

Derivation from Topping, D., & Bane, M. (2022) equation 2.36.

Equation 5: Rate of Mass Change with Latent Heat

dm/dt = [N × 4 × π × radius_wet × Dᵢ × (pᵢ, gas − pᵢ, particle surface)] / { [ (Dᵢ × Lᵢ × pᵢ) / (κ × T) ] × [ (Lᵢ / (R × T)) − 1 ] + Rᵢ × T }

Where:

dm/dt: Rate of change of mass of the droplet.
m: Mass of the droplet.
radius_wet: Wet radius of the droplet.
Dᵢ: Diffusion coefficient of species i.
pᵢ, gas: Partial pressure of species i in the gas phase.
pᵢ, particle surface: Partial pressure at the particle surface.
Lᵢ: Latent heat of vaporization for species i.
κ: Thermal conductivity of air.
T: Temperature.
Rᵢ: Specific gas constant for species i (R / molar massᵢ).

Description:

This equation modifies the isothermal rate to include thermal effects due to latent heat. The denominator accounts for the additional resistance to mass transfer caused by the temperature gradient established from heat release or absorption during phase change.

Additional Parameters¶

First-Order Condensation Coefficient¶

Equation 2: Condensation Coefficient

k_cond = 4 × π × radius_particle × Dᵢ × f(Kn, αᵢ)

Where:

radius_particle: Radius of the particle.
Dᵢ: Diffusion coefficient of species i in the gas phase.
f(Kn, αᵢ): Correction factor accounting for the transition between free-molecular and continuum regimes.
Kn: Knudsen number.
αᵢ: Mass accommodation coefficient.

Description:

The condensation coefficient k_cond represents the flux of molecules to the particle surface per unit concentration difference. It combines geometric factors with diffusion dynamics and corrections for different flow regimes.

Correction Factor f(Kn, αᵢ)¶

Equation 3: Correction Factor

f = [0.75 × αᵢ × (1 + Kn)] / [Kn² + Kn + 0.283 × αᵢ × Kn + 0.75 × αᵢ]

Where:

αᵢ: Mass accommodation coefficient for species i.
Kn: Knudsen number.

Knudsen Number:

Equation 4: Knudsen Number

Kn = λᵢ / radius_particle

Where:

λᵢ: Mean free path of gas molecules for species i.
radius_particle: Particle radius.

Description:

The correction factor f(Kn, αᵢ) adjusts the condensation coefficient to account for the finite mean free path of gas molecules relative to the particle size. It ensures accurate depiction of mass transfer in both the free-molecular (high Kn) and continuum (low Kn) regimes.

Partial Pressures¶

Understanding the partial pressures in the gas phase and at the particle surface is essential for calculating the condensation rate.

Gas Phase Partial Pressure:

Equation 6: Gas Phase Partial Pressure

pᵢ, gas = concᵢ, gas × (R × T) / molar massᵢ

Where:

concᵢ, gas: Concentration of species i in the gas phase.

Description:

This equation relates the concentration of a gas-phase species to its partial pressure using the ideal gas law, adjusted for the molar mass of the species.

Particle Surface Partial Pressure:

Equation 7: Particle Surface Partial Pressure

pᵢ, particle surface = pᵢ^pure × γᵢ × xᵢ × kᵢ, Kelvin

Where:

pᵢ^pure: Saturation vapor pressure of pure species i (also denoted as pᵢ^sat, pᵢ^vap, or pᵢ^0).
γᵢ: Activity coefficient of species i in the particle phase.
xᵢ: Mole fraction of species i in the particle phase.
kᵢ, Kelvin: Kelvin effect correction factor.

Description:

This equation adjusts the pure saturation vapor pressure to account for solution non-idealities (via γᵢ and xᵢ) and curvature effects (via kᵢ, Kelvin).

Kelvin Effect Correction Factor¶

Equation 8: Kelvin Effect

kᵢ, Kelvin = exp( kᵢ, Kelvin radius / radius_particle )

Where:

kᵢ, Kelvin radius: Kelvin radius factor.

Equation 9: Kelvin Radius Factor

kᵢ, Kelvin radius = [2 × σ_surface × molar massᵢ] / [ R × T × density ]

Where:

σ_surface: Surface tension of the particle.
density: Density of the particle.

Description:

The Kelvin effect expresses how vapor pressure over a curved surface differs from that over a flat surface. Small particles exhibit increased vapor pressure due to curvature, influencing condensation and evaporation rates.

Variable Descriptions¶

Understanding the Parameters:

Mass Accommodation Coefficient (αᵢ):
Represents the probability that a molecule colliding with the particle surface will stick and be incorporated into the particle.
Values range from 0 (no sticking) to 1 (all molecules stick upon collision).
Influenced by surface properties, temperature, and species-specific interactions.
Diffusion Coefficient (Dᵢ):
Indicates how quickly species i diffuses through the gas phase.
Dependent on temperature, pressure, and molecular characteristics.
Higher Dᵢ leads to faster mass transfer to the particle surface.
Mean Free Path (λᵢ):
Average distance a gas molecule travels before colliding with another molecule.
Inversely proportional to pressure; decreases as pressure increases.
Important for calculating the Knudsen number and determining the appropriate flow regime.
Knudsen Number (Kn):
Dimensionless number that characterizes the flow regime.
- Kn << 1: Continuum regime; diffusion dominates.
- Kn >> 1: Free-molecular regime; ballistic motion dominates.
Essential for selecting the correct correction factor f(Kn, αᵢ).
Latent Heat of Vaporization (Lᵢ):
Energy required to convert species i from liquid to vapor without temperature change.
Affects the heat balance during condensation and influences the condensation rate when significant.
Thermal Conductivity (κ):
Measures the ability of air to conduct heat.
Determines how quickly heat generated or absorbed at the particle surface is dissipated.
Activity Coefficient (γᵢ):
Accounts for non-ideal interactions between molecules in the particle phase.
Deviations from ideality can significantly impact the equilibrium vapor pressure.
Surface Tension (σ_surface):
Affects the Kelvin effect.
Dependent on particle composition and temperature.
Influential for small particles where curvature effects are pronounced.

Applications and Implications:

Aerosol Growth: These equations are vital for predicting how aerosols grow through condensation, impacting visibility, climate forcing, and human health.
Cloud Formation: Understanding condensation with latent heat is essential for cloud microphysics, influencing cloud droplet activation and lifetime.
Air Quality Modeling: Accurately modeling gas-particle partitioning helps in predicting pollutant behavior and secondary aerosol formation.

Assumptions and Limitations:

Isothermal Assumption: In the isothermal equation, neglecting latent heat is valid only when temperature changes are negligible. For processes involving significant heat exchange, the non-isothermal equation should be used.
Spherical Particles: The equations assume particles are spherical, which may not hold true for all aerosols (e.g., fractal soot particles).
Uniform Composition: Assumes homogeneous particle composition. In reality, phase separation or gradients may exist within particles.

Further Considerations:

Multicomponent Systems: In mixtures, interactions between different species can complicate calculations. Mutual diffusion coefficients and interactive effects need to be considered.
Dynamic Conditions: Environmental factors like fluctuating temperature and pressure can affect condensation rates. Real-world applications may require time-dependent modeling.
Parameter Estimation: Accurate values for parameters like Dᵢ, αᵢ, and γᵢ are necessary for precise predictions but can be challenging to obtain, especially for complex organic species.

Conclusion¶

By reviewing the equations and expanding on the descriptions, we enhance the understanding of condensation processes in aerosol dynamics. The interplay between mass transfer, thermodynamics, and kinetics is critical for accurately modeling aerosol behavior. Recognizing the importance of each parameter and the assumptions inherent in these equations allows for more informed application and interpretation in research and environmental modeling.

References¶

Topping, D., & Bane, M. (2022). Introduction to Aerosol Modelling. Wiley. DOI: 10.1002/9781119625728
Seinfeld, J. H., & Pandis, S. N. (2016). Atmospheric Chemistry and Physics: From Air Pollution to Climate Change (3^rd ed.). Wiley.