﻿ A toolbox for air ion and aerosol calculations

A toolbox for air ion and aerosol calculations

Version 20120205

Hannes.Tammet@ut.ee

The first version of the toolbox was published in web with aim to make easily available for colleagues the approximation algorithms, which were developed when compiling the numerical simulator of atmospheric aerosol nucleation bursts (Tammet and Kulmala, 2005). Afterwards, the toolbox was expanded with additional algorithms. Last (but not least) update is the function Z_approx_air, which motivation and explanation is described in the document Size-mobility-approx.pdf and handouts of a presentation. The present version of the toolbox contains 23 algorithms written in standard Pascal and meeting the requirements of Delphi, Free Pascal and Turbo Pascal. All algorithms are simultaneously presented as individual text files and as components of the source code of the test program. The test program is available as hypertext A_tools.htm , as a text file A_tools.txt, and as a compiled program file A_tools.exe. The last one can be obtained when unpacking the file A_tools.exe.zip. The source code can be made immediately acceptable for a compiler when the name of downloaded file A_tools.txt is changed to A_tools.dpr for Delphi or to A_tools.pas for Free Pascal or Turbo Pascal. Free Pascal and Turbo Pascal require deleting of some lines as explained in the text of the file.

The algorithms can be freely used, modified and translated into different programming languages. They are published without any warranty and some algorithms are not thoroughly checked. The author of the toolbox will be thankful for information about possible problems and solutions as well as for confirmation of successful applications.

The 23 algorithms are implemented as 22 functions and one procedure:

1. Function electrical_mobility_air

Arguments:

• air pressure (mb)
• air temperature (C)
• density of the particle matter ρ (g cm–3)
• particle number charge (dimensionless)
• mass diameter of the particle d = (6m/(πρ))1/3 (nm).

Result: Electrical mobility Z of the particle (cm2V–1s–1).

Comments: Algorithm is based on the model explained in the paper (Tammet, 1995). The published text consists of two errata that are indicated and corrected in the programs of the toolbox. Fortunately, these errata did not have any notable effect in typical applications. The output value is changed to the electrical mobility Z. Mechanical mobility and diffusion coefficient are

B (m s–1N–1) = 6.24×1014 Z (cm2V–1s–1)

D (m2 s–1) = 8.61×10–9 × T (K) × Z (cm2V–1s–1).

• gas parameters are included rigidly as air parameters (use function mechanical mobility in case of different gases),
• the original approximations of the air viscosity and free path are replaced by the approximations recommended in standard ISO15900
• the slip factor constants are replaced by the results of recent research:
A = 1.165, B = 0.480, C = 1.001.

2. Function mass_diameter_air

Arguments:

• air pressure (mb)
• air temperature (C)
• density of the particle matter (g cm–3)
• arithmetic number charge of the particle (dimensionless)
• electrical mobility of the particle (cm2V–1s–1).

Result: Mass diameter of the particle (nm).

Uses: function electrical_mobility_air.

Comment: The algorithm inverts the function electrical_mobility_air.

3. Function electrical_mobility_1_air

Arguments:

• air pressure (mb)
• air temperature (C)
• mass diameter of the particle d = (6m/(πρ))1/3 (nm).

Result: Electrical mobility Z of the particle (cm2V–1s–1).

Uses: function electrical_mobility_air.

Comment: . The result is the exactly same as electrical_mobility_air for a singly charged particle of density 2 g cm–3.The extra function is included for convenience of the user.

4. Function mass_diameter_1_air

Arguments:

• air pressure (mb)
• air temperature (C)
• electrical mobility of the particle (cm2V–1s–1).

Result: Mass diameter of the particle (nm).

Uses: function mass_diameter_air.

Comment: . The result is the exactly same as mass_diameter_air for a singly charged particle of density 2 g cm–3.The extra function is included for convenience of the user.

5. Function Z_approx_air

Arguments:

• air pressure (mb)
• air temperature (C)
• mass diameter of the particle (nm).

Result: Electrical mobility of the particle (cm2V–1s–1).

Restrictions: Particle should be singly charged and have density of about 2 g cm–3.

If the restrictions are not allowed then use electrical_mobility_air or mechanical_mobility instead.

Comment: This function is a mathematical approximation of electrical_mobility_air and its

advantages are simplicity of calculations (no iteration) and comprehensive interpretation in terms of Millikan equation.

6. Function d_approx_air

Arguments:

• air pressure (mb)
• air temperature (C)
• electrical mobility of the particle (cm2V–1s–1).

Result: Mass diameter of the particle (nm).

Uses: function Z_approx_air.

Restrictions: Particle should be singly charged and have density of about 2 g cm–3.

Comment: The algorithm inverts the function Z_approx_air. The iterative procedure may not converge in some unrealistic extreme situations. In this case the result will be 0.

7. Function plain_Millikan_mobility

Arguments:

• air pressure (mb)
• air temperature (C)
• mass diameter of the particle (nm).

Result: Electrical mobility of the particle (cm2V–1s–1).

Restrictions: Particle should be singly charged..

Comment: This function follows ISO15900 and uses corresponding approximations for air viscosity and free path. Mobility is calculated according to Millikan equation without diameter complement that is not recommended when particle diameter is less than 10 nm. For smaller particles a diameter complement is required. Fernandez de la Mora et al. (2003) recommended a complement of 0.53 nm. Ku et al. (2009) studied particles of diameter down to 1 nm and used the complement of 0.3 nm. A flexible size-dependent complement is implemented in function Z_approx_air.

8. Function plain_Millikan_diameter

Arguments:

• air pressure (mb)
• air temperature (C)
• electrical mobility of the particle (cm2V–1s–1).

Result: Mass diameter of the particle (nm).

Restrictions: Particle should be singly charged.

Comment: This function is an inverse of plain_Millikan_mobility and user should consider the comments to this function. The iterative procedure of inversion may not converge in some unrealistic extreme situations. In this case the result will be 0.

9. Function attachment_coefficient (ion-to-particle)

Arguments:

• number of elementary charges on the particle, attracting with sign –, repelling with +
• mass diameter of the ion (nm)
• density of the ionic matter (g cm–3)
• mass diameter of the particle (nm)
• air temperature (K)
• air pressure (mb).

Result: Attachment coefficient (synonym: combination coefficient) (cm3 s–1).

Uses: function electrical_mobility_air.

Restriction: Ion is expected to be singly charged.

Comments: The function is a minor modification of the algorithm explained in the paper (Tammet and Kulmala, 2005, Equation 7). It is designed as a mathematical approximation of numerical results by Hoppel and Frick (1986) and experimental results by Reischl et al. (1996). The paper (Tammet and Kulmala, 2005) contains an erratum in Equation 4, where a coefficient is written 8.616×10–8, but should be 8.616×10–9. Fortunately, this erratum had no effect in the numerical algorithms.

10. Function coagulation_coefficient (particle-to-particle, according to Sahni approximation)

Arguments:

• mass diameter of the small neutral particle (nm)
• mass diameter of the large charged or neutral particle (nm)
• density of the small particle matter (g cm–3)
• number charge of the large particle (dimensionless)
• polarizability of the small particle (Å3)
• air temperature (K)
• air pressure (mb).

Result: Coagulation coefficient (cm3 s–1).

Uses: function electrical_mobility.

The coagulation coefficient K is calculated for mutual coagulation of two different particles dN1/dt = dN2/dt = –K N1 N2. The algorithm is proposed and explained in the paper (Tammet and Kulmala, 2005, see Equations 26 and 31). It is based on a proposal by Fuchs and Sutugin (1971) to use the Sahni method for interpolation between the exact models of continuum regime and free molecule regime.

As an addition, the effect of van der Waals interaction is included in the first approximation by adding an extra distance h to the collision distance. The value of the extra distance is estimated 0.115 nm (Tammet, 1995).

The second addition is including the effect of electric polarization of the small particle in case of charged large particle according to the (∞–4) theoretical model (see McDaniel and Mason, 1973) and using an approximation by Tammet (1995).

Polarizability of a small particle is often roughly estimated as equal to r3 and polarizability of a cluster is estimated as n Å3, where n is the number of atoms in the cluster.

11. Function Boltzmann_probability (of charge qe on a particle of diameter d)

Arguments:

• algebraic number of elementary charges (dimensionless)
• diameter of particle (nm)
• air temperature (K).

Result: Probability to carry given algebraic number of charges in the steady state.

Comment: Boltzmann distribution is valid only in case of relatively large particles, see textbooks e.g. (Fuchs and Sutugin, 1971). If d < 100 nm then function charge_probability is recommended as replacement.

12. Function Wiedensohler_probability (of charge qe on a particle of diameter d)

Arguments:

• algebraic number of elementary charges q (dimensionless)
• diameter of particle (nm)

Result: Probability to carry given algebraic number of charges in the steady state.

Uses: included 6×5 table of numeric coefficients.

Restriction: q must be in range of –2 to + 2.

Presumptions: see Wiedensohler (1988).

Comment: A simple and quick procedure that well approximates theoretical results by Hoppel and Frick (1986).

13. Function charge_probability (of charge qe on a particle of diameter d)

Arguments:

• algebraic number of elementary charges (dimensionless)
• diameter of particle (nm)
• diameter of ions (nm)
• air temperature (K).

Result: Probability to carry given algebraic number of charges in the steady state.

Uses: function attachment_coefficient.

Presumptions: ion density 2 g cm–3 and air pressure 1013 mb.

Comment: Recommended when the particle diameter is less than 100 nm. In case of large particles the Boltzmann distribution can be applied, see textbooks e.g. (Fuchs and Sutugin, 1971).

14. Procedure ion_equilibrium

Arguments:

• mobility of positive small ions (cm2V–1s–1)
• mobility of negative small ions (cm2V–1s–1)
• ionization rate (cm–3s–1)
• air temperature (K)
• number concentration of background aerosol particles (cm–3)
• mean diameter of background aerosol particles (nm).

Results:

• Number concentration of positive small ions (cm–3)
• Number concentration of negative small ions (cm–3).

Restriction: Ions are expected to be singly charged.

Procedure solves the system of balance equations:

dn+/dt = I – αn+n – sb+n+,

dn/dt = I – αn+n – sbn,

where α is the recombination coefficient (estimated as 1.5×10–6 cm3s–1) and s is sink of small ions on background aerosol particles calculated according to (Tammet and Kulmala, 2007, Equation 6). Space charge of the background aerosol can be calculated afterwards as it equals to (nn+) e.

Application: calculating of air ion balance considering the deposition of ions on particles background aerosol.

15. Function Knudsen_growthrate (free molecule regime)

Arguments:

• mass diameter of the small neutral particle considered as the growth unit (nm)
• density of the growth unit (g cm–3)
• number concentration of growth units (cm–3)
• air temperature (K).

Result: Plain Knudsen growth rate (nm h–1).

Comments: Nucleating particles are expected to grow by condensation of a substance, which particles are called the growth units. Growth units are molecules or clusters. Plain Knudsen growth rate assumes the free molecule regime when growth rate does not depend on the size of the growing particle. The model is implemented as discussed in (Tammet and Kulmala, 2005, Equation 37).

Application: modeling of aerosol nucleation.

16. Function growthfactor0 (non-evaporating substance, method of effective polarizability)

Arguments:

• mass diameter of the small neutral particle considered as the growth unit (nm)
• mass diameter of the large charged or neutral particle considered as the growing particle (nm)
• density of the growth unit (g cm–3)
• number charge of the large particle (dimensionless)
• polarizability of the growth unit (Å3)
• air temperature (K)
• air pressure (mb).

Result: Growth factor = (Growth rate) / (Plain Knudsen growth rate).

Uses: function mechanical_mobility.

Comments: The function is recommended for growth units of d < 0.7 nm, e.g. molecules of sulfuric acid. The algorithm is proposed and explained in the paper (Tammet and Kulmala, 2005) discussing the process of aerosol nucleation. Nucleating particles are expected to grow by condensation of a substance, which particles are called the growth units. Growth units are molecules or clusters. Plain Knudsen growth rate assumes the free molecule regime. The model discussed in (Tammet and Kulmala, 2005) considers the growth as coagulation of growing particles with the growth units and the present function is an application of the coagulation algorithm.

The distinctive features of the function growthfactor0 are: 1) the condensing substance, e.g. sulfuric acid, is expected not to evaporate back into air, 2) the electrical forces are considered according to the method of effective polarizability, which is used in the function coagulation_coefficient. Polarizability of a cluster in Å3 is often close to the number of atoms in the cluster. However, there are considerable exceptions, e.g. polarizability of a molecule of sulfuric acid is about 149 Å3.

Application: modeling of aerosol nucleation.

17. Function growthfactor1 (non-evaporating substance, method of Nadykto and Yu)

Arguments:

• mass diameter of the small neutral particle considered as the growth unit (nm)
• mass diameter of the large charged or neutral particle considered as the growing particle (nm)
• density of the growth unit (g cm–3)
• number charge of the large particle (dimensionless)
• air temperature (K)
• air pressure (mb)
• first dipole enhancement coefficient by Nadykto and Yu (dimensionless)
• second dipole enhancement coefficient by Nadykto and Yu (dimensionless).

Result: Growth factor = (Growth rate) / (Plain Knudsen growth rate).

Uses: function mechanical_mobility.

Comments: The function is recommended for growth units of d < 0.7 nm, e.g. molecules of sulfuric acid. The algorithm is proposed and explained in the paper (Tammet and Kulmala, 2005) discussing the process of aerosol nucleation. Nucleating particles are expected to grow by condensation of a substance, which particles are called the growth units. Growth units are molecules or clusters. Plain Knudsen growth rate assumes the free molecule regime. The model discussed in (Tammet and Kulmala, 2005) considers the growth as coagulation of growing particles with the growth units and the present function is an application of the coagulation algorithm.

The distinctive features of the function growthfactor1 are: 1) the condensing substance, e.g. sulfuric acid, is expected not to evaporate back into air, 2) the electrical forces are considered according to the method by Nadykto and Yu (2003) as explained in the paper (Tammet and Kulmala, 2005). The recommended values of the first and second enhancement coefficients for sulfuric acid are 14 and 1.43.

Application: modeling of aerosol nucleation.

18. Function growthfactor2 (includes evaporation of growth unit substance)

Arguments:

• mass diameter of the small neutral particle considered as the growth unit (nm)
• mass diameter of the large charged or neutral particle considered as the growing particle (nm)
• density of the growth unit (g cm–3)
• arithmetic number charge of the large particle (dimensionless)
• polarizability of the growth unit (Å3)
• air temperature (K)
• air pressure (mb).

Result: Growth factor = (Growth rate) / (Plain Knudsen growth rate).

Uses: function mechanical_mobility.

Comments: The model considers the growth as coagulation of growing particles with the growth units and the coagulation coefficient is calculated in like manner as in the function coagulation_coefficient. The function is recommended for growth units of d > 0.7 nm, e.g. molecules and clusters of organic substances, which are able to evaporate back into the air. The algorithm is proposed and explained in the paper (Tammet and Kulmala, 2005) discussing the process of aerosol nucleation.

The distinctive features of the function growthfactor2 are: 1) the growing particle are expected to have passed the threshold size of the nano-Köhler model (Kulmala et al., 2004) and the condensing substance can partially evaporate back into air, which is typical for some organic compounds, 2) the electrical forces are considered according to the method of effective polarizability in the same way as in the function coagulation_coefficient.

Application: modeling of aerosol nucleation.

19. Function ion_needlesink (–(dn/dt)/n)

Arguments:

• air ion mobility (cm2V–1s–1)
• diameter of needles (mm)
• total length of needles in a volume unit (m–2)
• wind speed (m s–1)
• air temperature (K)
• air pressure (mb).

Result: a component of sink in the equation of the air ion balance (s–1).

Restriction: ion is expected to be singly charged.

Comments: Algorithm is proposed and explained in the paper (Tammet and Kulmala 2007). The deposition of ions is calculated according to the Churchill-Bernstein heat transfer equation (see Incropera and Dewitt, 2002) translated into the terms of particle diffusion.

Application: calculating of air ion balance in a conifer forest.

20. Function particle_needlesink (–(dn/dt)/n)

Arguments:

• diameter of particles (nm)
• diameter of needles (mm)
• total length of needles in a volume unit (m–2)
• wind speed (m s–1)
• air temperature (K)
• air pressure (mb).

Result: a component of sink in the equation of the air ion balance (s–1).

Uses: function mechanical_mobility.

Restriction: particles are expected to be in the nanometer size range so that the deposition is controlled only by diffusion.

Comments: Algorithm is proposed and explained in the paper (Tammet and Kulmala, 2007). The deposition of particles is calculated according to the Churchill-Bernstein heat transfer equation (see Incropera and Dewitt, 2002) translated into the terms of particle diffusion.

Application: calculating of nanometer particle balance in a conifer forest.

21. Function ion_aerosolsink (–(dn/dt)/n)

Arguments:

• air ion mobility (cm2V–1s–1)
• polarity of ions (+1 or –1)
• number concentration of aerosol particles (cm–3)
• mean diameter of aerosol particles (nm)
• algebraic mean charge number of aerosol particles (dimensionless)
• air temperature (K).

Result: sink in the equation of the air ion balance (s–1).

Restriction: Ion is expected to be singly charged.

Comment: Algorithm presents an approximation proposed and explained in the papers (Tammet et al., 2006, Equation 10; Tammet and Kulmala, 2007, Equation 6).

Application: Calculating of air ion balance considering the deposition of ions on particles background aerosol.

22. Function mechanical_mobility

Arguments:

• molar mass of the carrier gas (amu)
• polarizability of gas molecules (nm3)
• constant a in the equation of the collision size (nm), see comments below
• constant b in the equation of the collision size (K)
• constant c in the equation of the collision size (dimensionless)
• gas pressure (mb)
• gas temperature (K)
• density of the particle matter (g cm–3)
• particle number charge (dimensionless)
• mass diameter of the particle (nm).

Result: Mechanical mobility B of the particle (m s–1 fN–1).

Algorithm is proposed and explained in the paper (Tammet, 1995), where it consists of two errata that are indicated and corrected in the presented files. Fortunately, these errata did not have any notable effect in typical applications. Collision size of a gas molecule is approximated as d = a (1 + (b/T)c), see (Tammet, 1995, Equation 3).

Typical values of the first six arguments are shown in the Pascal code.

Mass diameter is defined as d = (6m/(πρ))1/3 , where ρ is density of the particle matter.

If a numerical result is B then the numerical value of mechanical mobility in SI is 10–15B and in CGS 10–12B. The electrical mobility of a particle is 1.602×B cm2V–1s–1.

23. Function mechanical_diameter

Arguments:

• molar mass of the carrier gas (amu)
• polarizability of gas molecules (nm3)
• constant a in the equation of the collision size (nm), see comments below
• constant b in the equation of the collision size (K)
• constant c in the equation of the collision size (dimensionless)
• gas pressure (mb)
• gas temperature (K)
• density of the particle matter (g cm–3)
• particle number charge (dimensionless)
• mechanical mobility of the particle (m s–1 fN–1).

Uses: function mechanical_mobility.

Result: Mass diameter of the particle (nm).

Comment: The algorithm inverts the function mechanical_mobility. The iterative procedure may not converge in some extreme situations. In this case the result will be 0.

References

Fernandez de la Mora, J., de Juan, L., Liedtke, K. Schmidt-Ott, A. (2003) Mass and size determination of nanometer particles by means of mobility analysis and focused impaction, J. Aerosol Sci. 34, 79–98.

Fuchs N.A. & Sutugin A.G. (1971) High-dispersed aerosols. In: Hidy G.M & Brock J.R. (eds.), Topics in current aerosol research, Pergamon Press, pp. 1–60.

Hoppel, W.A. & Frick, G.M. (1986) Ion-aerosol attachment coefficients and steady-state charge distribution on aerosols in a bipolar ion environment. Aerosol Sci. Technol. 5, 1–21.

Kim, J.H., Mulholland, G.W, Kukuck, S.R., Pui, D.Y.H. (2005) Slip Correction Measurements of Certified PSL Nanoparticles Using a Nanometer Differential Mobility Analyzer (Nano-DMA) for Knudsen Number From 0.5 to 83. J. Res. Natl. Inst. Stand. Technol. 110, 31-54.

Ku B.K. and Fernandez de la Mora J. (2009). Relation between Electrical Mobility, Mass, and Size for Nanodrops 1-6.5 nm in Diameter in Air. Aerosol Science and Technology 43, 241-249.

Kulmala, M., Kerminen, V-M., Anttila, T., Laaksonen, A. & O'Dowd, C.D. (2004) Organic aerosol formation via sulphate cluster activation. J. Geophys. Res., 109, D04205, http://dx.doi.org/doi:10.1029/2003JD003961.

Incropera F.P. & Dewitt, D.P. (2002) Fundamentals of Heat and Mass Transfer, Wiley, New York.

McDaniel, E.W. & Mason, E.A. (1973) The mobility and diffusion of ions in gases. New York: John Wiley.

Nadykto, A.B. & Yu, F. (2003) Uptake of neutral polar vapor molecules by charged clusters/particles: Enhancement due to the dipole-charge interaction. J. Geophys. Res., 108(D23), 4717, http://dx.doi.org/doi:10.1029/2003JD003664.

Reischl, G.P., Mäkelä, J.M., Karch, R., & Necid, J. (1996) Bipolar charging of ultrafine particles in the size range below 10 nm. J. Aerosol Sci., 27, 931–949.

Tammet, H (1995) Size and mobility of nanometer particles, clusters and ions. J. Aerosol Sci., 26, 459–475.

Tammet H. & Kulmala M. 2005. Simulation tool for atmospheric aerosol nucleation bursts. J. Aerosol Sci. 36: 173–196. http://dx.doi.org/doi:10.1016/j.jaerosci.2004.08.004.

Tammet H., Hõrrak U., Laakso L., & Kulmala M. (2006) Factors of air ion balance in a  coniferous forest according to measurements in Hyytiälä, Finland. Atmos. Chem. Phys., 6, 3377–3390, http://www.atmos-chem-phys.net/6/3377/2006/

Tammet, H. & Kulmala, M. (2007) Simulating aerosol nucleation bursts in a coniferous forest, Boreal Env. Res., 12, 421–430, http://www.borenv.net/BER/pdfs/ber12/ber12-421.pdf.

Wiedensohler, A. (1988) An approximation of the bipolar charge distribution for particles in the submicron size range, J. Aerosol Sci., 19, 387–389.