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Aerosol Particle Formation:Kelvin effect

Dr. Alexandra Teleki

teleki@ptl.mavt.ethz.chphone: 044 632 39 52ML F 18

Particle Technology Laboratory,Department of Mechanical and Process EngineeringETH Zurich, www.ptl.ethz.ch

Materials Properties and Characterization

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Motion in Gases

Air molecules are in continuous motion

They collide among themselves because of their translational energy

The translational energy is proportional to the air temperature

Hinds, W. C., Aerosol technology: properties, behavior, and measurement of airborne particles, Wiley, New York (1999).

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Distribution of molecular velocities (Maxwell-Boltzmann distribution)

( )1 2 2

exp2 2

xx x x

mcmf c dc dckT kTπ

⎛ ⎞−⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

/ ak N=ℜBoltzmann constant:

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Distribution of Molecular Speed

c (m/s)

f(c)

( )3 2 2

24 exp2 2

m mcf c dc c dckT kT

ππ

⎛ ⎞−⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠

( )1 2

08 kTc c f c dc

mπ∞ ⎛ ⎞= =∫ ⎜ ⎟

⎝ ⎠

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1 23rms

TcMℜ⎛ ⎞= ⎜ ⎟

⎝ ⎠21 3

2 2aTKE N mc ℜ

= =

Kinetic energy of 1 mole of gas:

M: molecular weight of gasrms: root mean square

Molecular Velocity

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2

12z m

cn n d

λπ

= = nz: average number of collisions a particular molecule undergoes in one second

dm: molecule collision diametern: number density of molecules

1 2

2mkTπλ ν ⎛ ⎞= ⎜ ⎟

⎝ ⎠

Particle Mean Free Path

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2Kndλ

=

The ratio of the mean free path of the gas and the particle radius is the Knudsen number Kn:

Kn >> 1: Free Molecular Regime (Kinetic theory)Kn ~ 1: Transition RegimeKn << 1: Continuum Regime (Navier-Stokes)

In the Free Molecular Regime the exchange of momentum, heat & mass is determined by the kinetic theory of gases.

In the Continuum Regime the Navier-Stokes equations describe the momentum, heat and mass transfer between gas and aerosol.

Particle Knudsen Number

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Properties of Air

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Coagulation

Shrinkingby evaporationor dissolution

Fragmentation

Convection in Growth

by condensation or chemical reaction

Diffusion Settling

Convection out

Particle Dynamics

10S. K. Friedlander (1977) Smoke, Dust and Haze: Fundamentals of aerosol behavior, Wiley, New York.

Deposition of Particles in the Lung during Inhalation

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Theory: Population Balance Equation

coagulation

diffusion growth external force

( ) ( ) ( )∫ −−β+v

v~dv~vnv~nv~v,v~02

1 ( ) ( ) ( )∫∞

β−0

v~dv~nvnv~,v

convection

fragmentation

uDcβSγ

zyx u,u,u0unnuun ∇⋅+∇=⋅∇= gas velocity vector

= particle diffusivity

= velocity of particles of size v (e.g. settling)

= coagulation rate

= fragmentation rate

= fragment size distribution

continuity

( ) ( ) ( ) ( )∫∞

γ+−v

v~dv~nSv~,vvnvS

tn∂∂ un⋅∇+ nD∇⋅∇= ⎟

⎠

⎞⎜⎝

⎛∂∂

+tdvdn

vnc⋅∇−

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A phase transition is encountered in many industrial(e.g. crystallization, carbon black production) and environmental (e.g. smog formation) processes

Nucleation-Condensation

The goal is to determine:

1. the critical diameter for particle formation which isdictated by thermodynamics

2. the growth rate that is determined by thermodynamics and transport

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Key feature: The curved interface

The goal is to derive an expression relating the concentration (vapor pressure) of species A with a particle (droplet) of radius dP at equilibrium (Seinfeld, 1986)

If the interface was flat which is, for example, the tabulated equilibrium concentration or vapor pressure at a given temperature and pressure.

Consider the change in Gibbs free energy accompanying the formation of a single drop (embryo) of pure material A of diameter dP containing g molecules of A:

(1)ΔG G Gembryo system pure vapor= −

Critical Particle Size

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Now let’s say that the number of molecules in the starting condition of pure vapor is nT. After the embryo forms, the number of vapor molecules remaining is . Then the above equation is written as:

(2)

where GV and Gl are the free energies of a molecule in a liquid and vapor phases and σ is the surface energy

(3)

Noting that

Where vl is the volume occupied by a molecule in the liquid phase (equivalent sphere in liquid phase).

n n gT= −

vTPlv GndgGnGG −σπ++=Δ 2

( ) ( )ΔG g G G d dv

G G dl v PP

ll v P= − + = − +π σ

ππ σ2

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gv dl

P=π 3

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Before we go further let’s evaluate the difference in Gibbs free energy:

dG = VdP then dG = (vl - vv) dP

But vl << vv then dG = - vv dP

According to ideal gas law vv = kBT/P

Then

Where S is the saturation ratio.

G G k T dPP

k T PP

k T Sv l BP

P

BA

AB

A

A− = − = − = −∫

0 0ln ln

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Now equation 3 becomes:

Now plot ΔG as a function of dP

S <1 monotonic increase in ΔGS > 1 positive and negative contributions at small dP the surface tension dominates and the behavior of ΔG as a function of dP is close to that for S <1. For larger dP the first term becomes important.

ΔG

dP

S < 1

dP∗

S > 1

droplet at equilibriumwith surrounding vapor

ΔG dv

k T S dP

lB

volume free energy of an embryo

P

surfacefree energy

= − +π

π σ3

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ln

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At

This is the minimum possible particle size.

This equation relates the equilibrium radius of a droplet of a pure substance to the physical properties of the substance and the saturation ratio of its environment. It is called also the Kelvin equation and the critical diameter is called the Kelvin diameter.

∂∂ΔGdP

= 0 ⇒ d vk T SP

l

B

∗ =4σ

ln

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The Kelvin equation states that the vapor pressure over a curvedinterface always exceeds that of the same substance over a flat surface:

See the anchoring of the surface molecules on a flat and a curved surface. Surface molecules are anchored on two molecules on the layer below flat surfaces while on curved interfaces some are anchored on just one!These can easily escape (evaporate) from the condensed (liquid or solid) phase.

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The mechanism for particle growth refers to droplet or particle growth from gas (condensation), to crystal growth from solution etc.. In all cases mass should be transported to the particle surface.

In principle, two steps are required, a diffusional step followed by a surface reaction or rearrangement step.In condensation the former is dominant while in crystallization is the latter. In many processes both can be dominant.

Particle Growth

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Consider a single droplet growing by condensation without convection at rather dilute conditions. The goal is to determine the flux of mass to its surface. For this the vapor concentration profile around the droplet is needed at steady state:

(1)

D = vapor diffusivityC = vapor concentration

(moles/cm3)

dP

droplet

vapor molecules

0rCr

rrD

tC 2

2 =⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

=∂∂

Mass transfer to a particle surface (continuum)

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With boundary conditions:at r = dP/2 C = Cd the equilibrium concentration at the droplet surface

at r = ∞ C = C∞ bulk vapor concentration

Solving the above equation for C as a function of r gives:

(2)

Then the rate of condensation F towards the surface is(negative r direction):

(3)

r2d1

CCCC P

d

d −=−−

∞

( )( )

2P2

P

Pd d

2d2d0CCD π

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−= ∞

( ) Pd dCCD2 π−= ∞

Pdr =2

CF= -Dr

∂⎛ ⎞ Α⎜ ⎟∂⎝ ⎠

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And the rate of particle volume growth is:

where MW and ρP are the molecular weight and density of the condensing material

So the diameter growth rate is (molecules/cm2):

(4)( )

PP

dPd

MWCCD4dtdd

ρ−

= ∞

( ) ( )P

Pd

P

3P dMWCCD2MWF

dt6dd

dtdv

ρπ−

=ρ

=π

= ∞

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The collision rate per unit area is:

(5)

where c and m1 are the molecular velocity and mass and NAV the Avogadro number

so z becomes (6)( ) 21

1

BdAVm

Tk84

CCNz ⎟⎟⎠

⎞⎜⎜⎝

⎛π

−= ∞

4cCNz AV=

Mass transfer to a particle surface (free molecule)

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Then the rate of condensation F to particle surface is:

(7)

And the rate of particle volume growth is:

(8)

So the diameter growth rate is:

(9)

( )d2P

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1

BAV CCd

m2TkN/areazF −π⎟⎟⎠

⎞⎜⎜⎝

⎛π

=⋅= ∞

( )P

d2P

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1

B

P

MWCCdm2TkMWF

dtdv

ρ−π⎟⎟

⎠

⎞⎜⎜⎝

⎛π

=ρ

= ∞

( )d

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1

B

P

P CCm2TkMW2

dtdd

−⎟⎟⎠

⎞⎜⎜⎝

⎛πρ

= ∞

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For particle growth from the free molecule to continuum regime, the expression for the continuum regime is extended by an interpolation factor:

(10)

where the Knudsen number is Kn= 2λ/dP

This is called the Fuchs effect.

( )⎟⎟⎠

⎞⎜⎜⎝

⎛

++

+ρ−

= ∞2

PP

dP

Kn33.1Kn71.11Kn1

dMWCCD4

dtdd

Mass transfer to a particle surface (entire spectrum)

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Summary of the lecture

Motion in Gases

Particle DynamicsNucleation: Kelvin effectParticle Growth: Condensation

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Next lecture

Selected Fundamentals of Particle Formation and Motion

Diffusion

Heat Transfer of Matter