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Queensland University of Technology
School of Physical and Chemical Sciences
Analysis of Dispersion and Propagation of Fine and Ultra Fine Particle Aerosols from a Busy
Road
Submitted by Galina GRAMOTNEV, School of Physical and Chemical Sciences, Queensland University of Technology in partial fulfilment of the requirements of the
degree of Doctor of Philosophy
January 2007
ii
Keywords Combustion aerosols, urban aerosols, outdoor aerosols, background aerosols, nano-
particles, ultra-fine particles, particle formation, aerosol evolution, busy road,
aerosol dispersion, air quality, transport emissions, emission factors, canonical
correlations analysis, multi-variate analysis, degradation processes, turbulent
diffusion, atmospheric monitoring, hydrodynamics, statistical mechanics,
probability, particle deposition.
iii
Statement of original
authorship
The work contained in this Thesis has not been previously submitted for a degree or
diploma at any other higher education institution. To the best of my knowledge and
belief, the Thesis contains no material previously published or written by another
persons except where due reference is made.
Galina Gramotnev
iv
Acknowledgements
I note my appreciation of financial support for this research from the Queensland
University of Technology (QUT), Faculty of Science, School of Physical and
Chemical Sciences, and QUT Office of Research.
I would like to express my sincere gratitude and appreciation to Dr. Richard J.
Brown for very helpful discussions, support, useful directions, and introduction to
the theory of turbulent atmospheric processes. I also thank Mr Pierre Madl and Ms
Maricella Yip for their substantial help and consultations with respect to monitoring
equipment, and all my friends from the International Laboratory for Air Quality and
Health for their support during my PhD studies.
Special thanks go to my husband, Dr Dmitri K. Gramotnev, for the
comprehensive support during my studies and suggested ideas.
v
Abstract
Nano-particle aerosols are one of the major types of air pollutants in the urban
indoor and outdoor environments. Therefore, determination of mechanisms of
formation, dispersion, evolution, and transformation of combustion aerosols near the
major source of this type of air pollution – busy roads and road networks – is one of
the most essential and urgent goals. This Thesis addresses this particular direction of
research by filling in gaps in the existing physical understanding of aerosol
behaviour and evolution.
The applicability of the Gaussian plume model to combustion aerosols near busy
roads is discussed and used for the numerical analysis of aerosol dispersion. New
methods of determination of emission factors from the average fleet on a road and
from different types of vehicles are developed. Strong and fast evolution processes
in combustion aerosols near busy roads are discovered experimentally, interpreted,
modelled, and statistically analysed.
A new major mechanism of aerosol evolution based on the intensive thermal
fragmentation of nano-particles is proposed, discussed and modelled. A
comprehensive interpretation of mutual transformations of particle modes, a strong
maximum of the total number concentration at an optimal distance from the road,
increase of the proportion of small nano-particles far from the road is suggested.
Modelling of the new mechanism is developed on the basis of the theory of turbulent
diffusion, kinetic equations, and theory of stochastic evaporation/degradation
vi
processes.
Several new powerful statistical methods of analysis are developed for
comprehensive data analysis in the presence of strong turbulent mixing and
stochastic fluctuations of environmental factors and parameters. These methods are
based upon the moving average approach, multi-variate and canonical correlation
analyses. As a result, an important new physical insight into the
relationships/interactions between particle modes, atmospheric parameters and
traffic conditions is presented. In particular, a new definition of particle modes as
groups of particles with similar diameters, characterised by strong mutual
correlations, is introduced. Likely sources of different particle modes near a busy
road are identified and investigated. Strong anti-correlations between some of the
particle modes are discovered and interpreted using the derived fragmentation
theorem.
The results obtained in this thesis will be important for accurate prediction of
aerosol pollution levels in the outdoor and indoor environments, for the reliable
determination of human exposure and impact of transport emissions on the
environment on local and possibly global scales. This work will also be important
for the development of reliable and scientifically-based national and international
standards for nano-particle emissions.
vii
LIST OF AUTHOR PUBLICATIONS
1. Refereed journal papers
[A1]. Gramotnev, G., Brown, R., Ristovski, Z, Hitchins, J., Morawska, L. 2003.
Determination of emission factors for vehicles on a busy road. Atmospheric
Environment, 37, pp. 465-474 (Number 13 out of 25 most downloaded papers in
2004).
[A2]. Gramotnev, G., Ristovski Z., Brown, R., Madl, P. 2004. New methods of
determination of emission factors for two groups of vehicles on a busy road,
Atmospheric Environment, vol.38, pp.2607-2610.
[A3]. Gramotnev, G., Ristovski, Z. 2004. Experimental investigation of ultra fine
particle size distribution near a busy road, Atmospheric Environment, vol.38,
pp.1767-1776.
[A4]. Gramotnev, D.K., Gramotnev, G. 2005. A new mechanism of aerosol
evolution near a busy road: fragmentation of nano-particles, Journal of Aerosol
Science, vol.36, pp.323-340. (Number 9 out of 25 most downloaded papers in 2005).
[A5]. Gramotnev, D.K., Gramotnev, G. 2005. Modelling of aerosol dispersion from
a busy road in the presence of nanoparticle fragmentation, Journal of Applied
Meteorology, vol.44, pp.888–899.
[A6]. Gramotnev, G., Gramotnev, D.K. Multi-channel statistical analysis of
combustion aerosols. Part I: Canonical correlations and sources of particle modes
Atmospheric Environment (accepted 9 January 2007).
[A7]. Gramotnev, D.K., Gramotnev, G. Multi-channel statistical analysis of
combustion aerosols. Part II: Anti-correlations of particle modes and fragmentation
theorem. Atmospheric Environment (accepted 9 January 2007).
[A8]. Gramotnev, D.K., Gramotnev, G. Kinetics of stochastic degradation /
viii
evaporation processes in polymer-like systems with multiple bonds, J. Appl. Phys.
(submitted).
[A9]. Gramotnev, D. K., Mason, D. R., Gramotnev, G., Rasmussen A. J. Thermal
tweezers for surface manipulation with nano-scale resolution. Appl. Phys. Lett.
(accepted 2 January 2007).
[A10]. Gramotnev, G., Madl, P., Gramotnev, D. K., Urban background aerosols:
Anti-correlations of particle modes and fragmentation mechanism. Geophysical
Research Letters (submitted).
2. Full-length refereed conference papers
[A11]. Gramotnev, G., Brown, R., Ristovski, Z., Hitchins, J., Morawska, L. 2002.
Estimation of fine particles emission factors for vehicles on a road using Caline4
program. Proceedings of 4th Queensland Environmental Conference, Brisbane,
Australia, 30 & 31 May 2002, pp. 43-48.
[A12]. Gramotnev, G., Ristovski, Z., Brown, R., Morawska, L, Jamriska, M.,
Agranovski, V. 2003. A new method for obtaining fine particles emission factors
with validation from measurements near a busy road in Brisbane. Proceedings of
National Environmental Conference, Brisbane, Australia, 18 & 20 June 2003, pp.
206-211.
3. Conference papers in refereed journals
[A13]. Gramotnev, G., Ristovski, Z., Brown, R., Morawska, L., Madl, P. 2003.
New method of determination of emission factors for different types of vehicles on a
busy road. Journal of Aerosol Science, EAC 2003, vol.34s, S259-S260.
[A14]. Gramotnev, G., Ristovski, Z. 2003. Nanoparticles near a busy road:
experimental observation of the effect of formation of a new mode of particles.
Journal of Aerosol Science, EAC 2003, vol.34s, S255-S256.
ix
[A15]. Gramotnev, G., Ristovski, Z. and Gramotnev, A. 2003. Dependence of
concentration of nanoparticles near a busy road on meteorological parameters:
canonical correlation analysis. Journal of Aerosol Science, EAC 2003, vol.34s,
S257-S258.
[A16]. Gramotnev, G., Ristovski, Z., Morawska, L., Thomas, S. 2003. Statistical
analysis of correlations between air pollution in the city area and temperature and
humidity. Journal of Aerosol Science, EAC 2003, vol.34s, S715-S716.
[A17]. Gramotnev, G. 2004. Determination of the average emission factors for
three different types of vehicles on a busy road. Journal of Aerosol Science, EAC
2004, vol.35, S1089-S1090.
[A18]. Gramotnev, D.K., Gramotnev, G. 2004. A new mechanism of aerosol
evolution near a busy road: fragmentation of nanoparticles. Journal of Aerosol
Science, EAC 2004, vol.35, S221-S222.
[A19]. Gramotnev, D.K., Gramotnev, G. 2004. Modelling of aerosol dispersion
from a busy road in the presence of nano-particle fragmentation. Journal of Aerosol
Science, EAC 2004, vol.35, S925-S926.
4. Other conference publications
[A20]. Gramotnev, G., Brown, R., Ristovski, Z, Hitchins, J., Morawska, L. 2002.
Dispersion of fine and ultra fine particles from busy road: the comparison of
experimental and theoretical results, in Chiu-Sen Wang (Ed) Proc. of Sixth
International Aerosol Conference, Taipei, Taiwan (September 9 – 13, 2002), pp.
839-840.
[A21]. Gramotnev, G., Thomas, S., Morawska, L., Ristovski, Z. 2002. Canonical
correlation analysis of fine particle and gaseous pollution in the city area, in Chiu-
Sen Wang (Ed) Proc. of Sixth International Aerosol Conference, Taipei, Taiwan
(September 9 – 13, 2002), pp. 873-874.
[A22]. Gramotnev, D.K., Gramotnev, G. 2004. Fragmentation of nanoparticles near
x
a busy road: Justification and modelling. Proceedings of 8th International
Conference on Carbonaceous Particles in the Atmosphere, Vienna, Austria, 14-16
September 2004, H3.
[A23]. Gramotnev, G., Gramotnev, D.K. 2004. New statistical method of
determination of particle modes in the presence of strong turbulent mixing.
Proceedings of 8th International Conference on Carbonaceous Particles in the
Atmosphere, Vienna, Austria, 14-16 September 2004, H4.
[A24]. Gramotnev, G., Gramotnev, D. K. 2005. Theoretical analysis of multiple
thermal fragmentation of aerosol nanoparticles from a line source: Evolution of
particle modes. Biannual AIP Congress, Canberra, Australia, February, 2005, p.210.
[A25]. Gramotnev, G., Gramotnev, D. K. 2005. Numerical and experimental
investigation of thermal fragmentation of aerosol nano-particles from vehicle
exhaust. Biannual AIP Congress, Canberra, Australia, February, 2005, p.210.
[A26]. Gramotnev, D. K., Gramotnev, G. 2005. Combustion nano-particle aerosols:
Mechanisms of evolution and modelling, Aerosol Workshop, 30 March – 1 April
2005, Sydney, Australia (invited talk).
[A27]. Gramotnev, D. K., Gramotnev, G. 2005. Time delays during multiple
thermal fragmentation of nanoparticles: evolution of particle modes. European
Aerosol Conference (EAC 2005), Ghent, Belgium, p. 690.
[A28]. Gramotnev, G., Madl, P. 2005. Multi-channel statistical analysis of
background fine particle aerosols, European Aerosol Conference (EAC 2005),
Ghent, Belgium, p. 697.
[A29]. Gramotnev, D. K., Bostrom, T. E., Devine, N., Gramotnev, G. 2005.
Experimental investigation of deposition of aerosol particles near a busy road.
European Aerosol Conference (EAC 2005), Ghent, Belgium, p. 696.
[A30]. Mason, D.R., Gramotnev, D.K., Rasmussen, A., Gramotnev, G. 2005.
Feasibility of thermal tweezers for effective manipulation of nano-particles on
surfaces. ACOLS’05, 6 December, Christchurch, New Zealand, ThC6.
xi
[A31]. Gramotnev, D. K., Bostrom, T. E., Gramotnev, G., Goodman, S. J.
“Deposition of Composite Aerosol Particles on Different Surfaces near a Busy
Road”, 7th International Aerosol Conference (IAC 2006), 10-15 September 2006, St.
Paul, Minnesota, USA, p.616-617.
[A32]. Gramotnev, D. K., Gramotnev, G. “Multiple thermal fragmentation of
nanoparticles: evolution of particle total number concentration”, 7th International
Aerosol Conference (IAC 2006), 10-15 September 2006, St. Paul, Minnesota, USA,
p.107-108.
[A33]. Gramotnev, G., Gramotnev, D. K. “Multi-channel statistical analysis of
combustion aerosols: Canonical correlations and sources of particle modes”, 7th
International Aerosol Conference (IAC 2006), 10-15 September 2006, St. Paul,
Minnesota, USA, p.177-178.
[A34]. Gramotnev, D. K., Gramotnev, G. “Anti-correlations of particle modes and
fragmentation theorem for combustion aerosols”, 7th International Aerosol
Conference (IAC 2006), 10-15 September 2006, St. Paul, Minnesota, USA, p.734-
735.
[A35]. Gramotnev, G., Madl, P., Gramotnev, D. K. “Anti-symmetric correlations of
particle modes in urban background aerosols”, 7th International Aerosol Conference
(IAC 2006), 10-15 September 2006, St. Paul, Minnesota, USA, p.1764-1765.
[A36]. Mason, D. R., Gramotnev, D. K., Gramotnev, G., Rasmussen, A. J.
“Thermal tweezers with dynamic evolution of the heat source”, 17th AIP Congress,
December 2006, Brisbane, Australia, abstract 461.
[A37]. Gramotnev, D. K., Bostrom, T. E., Mason, D. R., Gramotnev, G., Burchill,
M. J. “Deposition and Surface Evolution of Composite Aerosol Particles”, 17th AIP
Congress, December 2006, Brisbane, Australia, abstract 796.
[A38]. Gramotnev, D. K., Gramotnev, G. “Anti-Symmetric Correlation Pattern for
Particle Modes in Combustion and Background Aerosols: Fragmentation Theorem”,
17th AIP Congress, December 2006, Brisbane, Australia, abstract 795.
xii
[A39]. Gramotnev, G., Gramotnev, D. K. “Multi-Channel Statistical Analysis for
the Detailed Investigation of Combustion Aerosols”, 17th AIP Congress, December
2006, Brisbane, Australia, abstract 797.
[A40]. Gramotnev, D. K., Flegg, M. B., Gramotnev, G. “Stochastic
evaporation/degradation processes in complex structures with multiple bonds”, 17th
AIP Congress, December 2006, Brisbane, Australia, abstract 748.
xiii
LIST OF FIGURES
Fig. 3.1. Monitoring place 49
Fig. 3.2. Average wind parameters 50
Fig. 3.3. Theory and experiment (linear scale) 60
Fig. 3.4. Theory and experiment (logarithmic scale) 61
Fig. 3.5. Consultancy example 64
Fig. 5.1. Monitoring place 83
Fig. 5.2. Size distribution near the kerb 85
Fig. 5.3. Size distributions with experimental points (20 November 2002) 86
Fig. 5.4. Comparison of size distributions (20 November 2002) 87
Fig. 5.5. Size distributions with experimental points (23 December 2002) 89
Fig. 5.6. Comparison of size distributions (23 December 2002) 91
Fig. 5.7. Total number concentration 92
Fig. 5.8. Size distributions with experimental points (24 November 2002) 94
Fig. 5.9. Number concentrations (8 January 2003) 95
Fig. 6.1. Monitoring place 103
Fig. 6.2. Average wind parameters (25 November 2002) 105
Fig. 6.3. Size distributions with experimental points (25 November 2002) 106
Fig. 6.4. Moving average correlation coefficients 109
Fig. 6.5. Error curves 110
Fig. 6.6. Size distributions (20 November 2002) 116
Fig. 6.7. Size distributions (23 December 2002) 119
Fig. 6.8. Evolution pattern 123
Fig. 7.1. Geometry of the problem 132
Fig. 7.2. Fragmentation rate coefficient 135
Fig. 7.3. Total number concentrations (theoretical dependencies) 138
Fig. 7.4. Total number concentrations (comparison with experiment) 144
Fig. 8.1. Size distributions; moving average approach (25 November 2002) 159
Fig. 8.2. Moving average correlation coefficients 161
Fig. 8.3. Simple correlations with traffic 169
xiv
Fig. 8.4. Canonical correlation coefficients 173
Fig. 8.5. Canonical weights and loadings for heavy trucks 175
Fig. 8.6. Canonical weights and loadings for cars 176
Fig. 8.7. Canonical weights and loadings for temperature 187
Fig. 8.8. Canonical weights and loadings for solar radiation 188
Fig. 9.1. Moving average cross-correlation coefficients 195
Fig. 9.2. Anti-symmetric correlation pattern 197
Fig. 9.3. Anti-correlations with 13.6 nm mode 201
Fig. 9.4. Anti-correlations with 7 nm mode 202
Fig. 9.5. Anti-symmetric correlation pattern (later evolution stage) 203
Fig. 9.6. Fragmentation theorem 206
Fig. 10.1. Evolution of the 3-particle from the 1-2 state 215
Fig. 10.2. Random graph representation 216
Fig. 10.3. Particle concentrations (no dispersion) 226
Fig. 10.4. Particle concentrations (with dispersion) 227
Fig. 11.1. Monitoring place 230
Fig. 11.2. Background size distribution (before sunset) 231
Fig. 11.3. Comparison of size distributions before and after sunset 232
Fig. 11.4. Moving average correlation coefficients for background 233
Fig. 11.5. Anti-symmetric correlation pattern for background 236
xv
Contents
Abstract v
List of Author Publications vii
List of Figures xiii
Contents xv
1. Introduction 1
1.1. Aims 7
2. Background and Theory 10
2.1. Ambient aerosols and their origins 10
2.2. Turbulent dispersion of air pollutants 15
2.2a. Taylor theorem and asymptotic properties
of the diffusing cloud 18
2.2b. Turbulent diffusion from a point continuous sources 20
2.2c. Continuous ground level line source 23
2.3. Dispersion of fine particles from a busy road 26
2.4. Monitoring equipment 34
2.5. Statistical approaches: correlation techniques in data analysis 36
3. Determination of average emission factors for vehicles on a busy road 45
3.1. Introduction 45
3.2. CALINE4 model 46
3.3. Experimental measurements 48
3.4. Model adaptation 50
xvi
3.4.1. Model emission factors 52
3.4.2. Determination of the emission factor 54
3.5. Comparison of numerical and experimental results 58
3.6. An example of application of the model for road design 64
3.7. Conclusions 65
4. New methods of determination of average particle emission factors
for two groups of vehicles on a busy road 68
4.1. Introduction 68
4.2. Emission factors for two different groups of vehicles 69
4.3. Constrained optimization 73
4.4. Three types of vehicles on the road 74
4.5. Turbulent corrections to the w-factors 77
4.6. Conclusions 80
5. Experimental investigation of ultra fine particle size distribution
near a busy road 81
5.1. Introduction 81
5.2. Experimental procedure 82
5.3. Experimental results and discussion 84
5.4. Level of confidence and errors 96
5.5. Conclusions 99
6. A new mechanism of aerosol evolution near a busy road:
fragmentation of nano-particles 101
6.1. Introduction 101
6.2. Modes of particle size distribution 102
xvii
6.3. Maximum of the total number concentration 111
6.4. Failure of the conventional mechanisms of the aerosol evolution 113
6.5. Fragmentation model of aerosol evolution 120
6.6. Conclusions 126
6.7. Appendix for Chapter 6 127
7. Modelling of aerosol dispersion from a busy road
in the presence of nano-particle fragmentation 130
7.1. Introduction 130
7.2. Dispersion as a chemical reaction 131
7.3. Fragmentation of particles 133
7.4. Existence conditions for the maximum
of the total number concentration 140
7.5. Comparison with the experimental results 143
7.6. Applicability conditions 150
7.7. Conclusions 154
8. Multi-channel statistical analysis of aerosol particle modes
near a busy road 155
8.1. Introduction 155
8.2. Experimental data and particle modes 156
8.3. Moving average approach and the canonical correlation analysis 163
8.4. Sources of particle modes 169
8.5. Meteorological parameters 183
8.6. Conclusions 189
9. Correlations between particle modes: fragmentation theorem 191
xviii
9.1. Introduction 191
9.2. Moving average approach for particle modes 192
9.3. Numerical results and their discussion 194
9.4. Fragmentation Theorem 204
10. Probabilistic time delays during multiple stochastic
degradation/evaporation processes 213
10.1. Introduction 213
10.2. Time delays 214
10.3. Evolution time and kinetics of degradation 218
11. Multi-channel statistical analysis of background fine particle aerosols 229
12. Conclusions 239
List of main results 240
Bibliography 243
1
CHAPTER 1
INTRODUCTION
Rapid development of high-technology industry, transport, and ever increasing
consumption of energy have resulted in increasing changes to our environment, climate,
atmosphere, natural resources, etc. (Seinfeld and Pandis, 1998). All these changes
should prompt a rapid and decisive response, if we want to stop adverse effects of our
technological activities on the quality of life, environment, and health. Finding such a
response is one of the major aims of modern science, including all of its mainstream
branches such as environmental sciences, engineering, physics, chemistry, medicine,
and applied mathematics.
Transport emissions are one of the major sources of atmospheric and
environmental pollution with the global effect on climate, environment, and quality of
life (Whelan, J. 1998, Schauer, et al, 1996, Shi, et al, 1999, Shi, et al, 2001). Choking
atmospheres in major world cities and reducing air quality in residential areas of large
metropolitan centres require urgent measures on reduction, control, and effective
prediction of air pollution levels from busy roads and road networks. One of the major
types of pollutants from modern transport and road networks is combustion aerosols
comprising fine and ultra-fine particles with diameters from several nanometres to
several hundreds of nanometres (Schauer, et al, 1996, Shi, et al, 1999). It is long known
that such aerosols may have an effect on climate, mainly through cloud formation and
rainfall patterns (Seinfeld and Pandis, 1998, Jacobson, 1999). In addition, during the last
decade, researchers have established links between fine and ultra-fine particle aerosols
and noticeable health risks for humans in city areas (Pope, et al, 1995, Van Vliet, et al,
1997).
During the last several years, numerous studies have observed health effects of
particulate air pollutants. Compared to early studies that focused on severe air pollution
2
episodes (Beaver, H., 1953), recent research is more relevant to understanding health
effects of pollution at levels common to contemporary cities in the developed world.
Observed health effects include increased respiratory symptoms, decreased lung
function, increased hospitalizations and other health care visits for respiratory and
cardiovascular disease, increased respiratory morbidity as measured by absenteeism
from work and school, or other restrictions in activity, and increased cardiopulmonary
disease mortality. These health effects have been observed at levels common to many
U.S. cities including levels below current U.S. National Ambient Air Quality Standards
for particulate air pollution (Pope, et al, 1995).
It has also been found that those children who have been living within 100 m of
a freeway had significantly more coughs, wheezes, runny noses, and doctor-diagnosed
asthmas (Van Vliet, et al, 1997). In addition, the same study identified a significant
association between truck traffic density and black smoke concentration on the one hand
and chronic respiratory symptoms on the other.
Until recently, the main concern has been related to emission of relatively large
particles with diameters > 1 μm (Friedlander, 1977). Therefore the current emission
standards establish the limits on emission of overall particulate mass, rather than
concentration of particles. However, recent investigations have made it apparent that
fine and ultra-fine aerosol particles (within the ranges < 1 μm and < 0.1 μm,
respectively) emitted from combustion sources may present a significant health risk for
humans (Wichmann, and Peters, 2000, Zhiqiang, et al, 2000, Ziesenis, et al, 1998,
Borja-Aburto, et al, 1998), especially for people with specific health problems (e.g.,
heart, vascular, respiratory, etc. problems (Borja-Aburto et al, 1998). Moreover, it is
now clear that adverse health effects related to ultra-fine (< 100 nm) particles with large
number concentration but small overall mass appear to be significantly stronger than the
effects from larger (fine) particles with diameters between ~ 100 nm and ~ 1 μm (Stone,
3
2000, Brown, et al 2000). For example, proinflammatory response is greater for ultra-
fine particles, and is directly proportional to the surface area of the particles (Brown, et
al, 2001). Therefore, one of the possible explanations of increased health effects of
ultra-fine aerosol particles is related to the fact that decreasing particle diameters and
increasing their number concentrations results in a strong increase of particle surface
area per unit volume (Peters, 1997, Brown, et al, 2001, Nemmar, et al, 2002). This is
the surface area of the particles that probably drives inflammation in the short term,
resulting in significantly larger effect from ultra-fine particles having very large number
concentrations and surface area (Nemmar, et al, 2002). During a study of the
penetration of pollutant particles into the blood stream, it was found that ultra-fine
aerosol particles penetrate into the blood just in ~ 1 minute (Nemmar, et al, 2002). The
concentration in the blood reaches a maximum within ~ 10 – 20 minutes, and remains at
this maximal level for up to ~ 60 minutes (Nemmar, et al, 2002). One of the reasons for
these enhanced and fast effects is probably related to the fact that fine and ultra-fine
particles tend to penetrate much deeper into the respiratory tract (Siegmann, et al,
1999). However, the complete understanding of the observed health problems and risks
related to fine and ultra-fine particle aerosols still needs further studies including
research into physical mechanisms of particle transformation and evolution, in order to
understand which types of particles tend to play a predominant role in human exposure.
As mentioned above, the current particulate emission standards restrict the
overall particulate mass emissions. These standards are thus focusing only on PM10 and
PM2.5 (i.e., the overall particulate mass concentration for particle diameters < 10 μm and
< 2.5 μm, respectively). They are obviously of little use for the development of
regulations and policies when it comes to the strong adverse effects of fine and ultra-
fine particles, because the contribution of such particles to the overall aerosol mass is
negligible. Therefore, new standards for fine and ultra-fine particle aerosols are
4
required, based on number concentrations rather than overall particulate mass. This will
also require detailed and comprehensive understanding of the major mechanisms of
formation and evolution of combustion aerosols, transformation of particle modes,
determination of their possible sources, possible places of enhanced health risks,
mechanisms of removal and self-removal of particles from the atmosphere, etc. At the
same time, our current knowledge about fine and ultra-fine aerosol particles, their
possible sources and mechanisms of transformation is fairly limited and some times
inconsistent with experimental observations (for more detail see Chapter 2).
It is also clear that the development of adequate standards for fine and ultra-fine
particle aerosols may only help to determine and identify the existing and potential
problems with air pollution and transport and industry emissions. Solution of these
problems will be another very complex task that will require new approaches for
effective reduction and control of air pollution levels (including particulate pollutants)
and improvement of the air quality in major metropolitan centres. And this is again not
possible without the detailed understanding of processes of aerosol formation,
interaction, evolution, and eventual removal and/or self-removal from the atmosphere.
As a result, significant efforts of a number of aerosol scientists have recently
been focused on the advancement of our fundamental knowledge of behaviour of
combustion aerosols and their prediction in the urban environment. In particular,
detailed understanding of dispersion of nanoparticle aerosols is one of the most
important goals for achieving reliable and accurate forecast of aerosol pollution levels
and the resultant human exposure. One of the major physical mechanisms of dispersion
of air pollutants (including nanoparticle aerosols) in the atmosphere is turbulent
diffusion (Seinfeld & Pandis, 1998, Jacobson, 1999). If only this mechanism is taken
into account, dispersion of aerosols and gasses can be described by the Gaussian plume
model (Csanady, 1980, Pasquill and Smith, 1983, Zannetti, 1990). Several successful
5
software packages for different types of sources including point sources (industry)
(Bowers & Anderson, 1981), area sources (bushfires) (Hanna, et al, 1984), line sources
(busy roads) (Benson, 1992) have been developed for non-reactive pollutants. However,
modelling of dispersion of reactive gasses and rapidly evolving aerosols is a much more
complex problem (Bilger, 1978, Fraigneau, et al, 1995).
Previously, it was fairly commonly assumed that fine and ultra fine particle
aerosols do not undergo significant and rapid transformations (Shi et al, 1999). In this
case, particle size distributions should be more or less constant within a significant
period of time, and the Gaussian plume approximation should be applicable for the
approximate description of aerosol dispersion from different sources. In this case the
above-mentioned software packages should be applicable (after the appropriate re-
scaling) for the prediction of aerosol dispersion in the atmosphere. Therefore, the main
interest of aerosol scientists has been focused on the study of decay of the total number
concentration of particles with distance from a source, e.g., a busy road (Shi, et al,
1999, Hitchins, et al, 2000, Zhu, et al, 2002a,b). In particular, exponential decay laws
were used for the description of the total number concentration of fine particles as a
function of distance from the road (Zhu, et al, 2002a,b).
However, several recent experimental observations have suggested that the
Gaussian plume approximation is not always applicable, especially for smaller particles
within the range < 30 nm. Noticeable deviations of the size distributions of fine and
ultra fine particles near a busy road from those predicted by the Gaussian plume model
have been observed by Zhu, et al (2002a,b). This suggests that there are significant
processes of evolution of particles during their transport away from the road – see also
(Ketzel and Berkowicz, 2004). Such evolution processes may include particle formation
by means of homogeneous and heterogeneous nucleation (Alam, et al, 2003, Kulmala,
et al, 2000, Kerminen, et al, 2002, Lehtinen and Kulmala, 2003, Pirjola, 1999),
6
coagulation (Jacobson, 1999, Kostoglou & Konstandopoulos, 2001, Piskunov &
Golubev, 2002), deposition (Jacobson, 1999, Meszaros, 1999), condensation and
evaporation (Zhang et al, 2005, Uvarova, 2003).
Nevertheless, there are still noticeable discrepancies between the theoretical
predictions based on the mentioned mechanisms of aerosol evolution and the
experimental observations and monitoring data near busy roads. For example, Zhu, et al
(2002a,b) have observed a shift of one of the particle modes (maximums of the particle
size distribution) towards smaller particle diameters when the distance from the road is
increased. This observation is in obvious contradiction with the suggested coagulation
mechanism, of evolution of the particle size distribution (Zhu, et al, 2002a,b).
Contradictory suggestions regarding the nature of combustion nanoparticles have been
presented in the literature. Some of the researchers assume that particles with diameters
< 30 nm are mostly volatile (Sakurai, et al, 2003), whereas others suggest that they are
predominantly solid – graphite, carbon, or metallic ash (Pohjola, et al, 2003, Abdul-
Khalek, et al, 1998, Bagley, et al, 1996). Very few experiments on direct particle
observation and determination of their properties and structure under field conditions
have been undertaken so far, while laboratory analysis may give significantly different
results from the real-world situations with stochastically varying atmospheric conditions
and natural variability of the source (different types of vehicles, their maintenance, etc.).
Problems with such field experiments are well known. They are related to significant
fluctuations/dispersion of monitoring data associated with strong natural stochastic
processes, such as atmospheric turbulence, variability of temperature, humidity, solar
radiation, traffic conditions, etc. Therefore, deriving sensible conclusions about the
nature of different types of aerosol particles and their evolution in the presence of strong
turbulent mixing requires the development of new extensive and complex methods of
statistical analysis.
7
As a result, a number of important questions about the nature of particle modes
in combustion aerosols and their evolution/transformation and physical and chemical
structure in the real-world environment have so far been left unanswered. Some of these
questions can be listed as follows. (1) What is the predominant nature of the exhaust
nanoparticles? Are they mainly solid or volatile? (2) What are the dominant sources (if
any) of different particle modes? (3) How can we determine emission factors from
different types of vehicles on an actual road (these factors are essential for accurate
prediction of aerosol pollution levels)? (5) How do particle modes evolve with time and
distance from the source at different atmospheric, physical, and climate conditions? (6)
Are the known mechanisms sufficient for the complete description of aerosol evolution,
or we are missing something?
Detailed investigation of these and other questions is essential for accurate
forecast of aerosol pollution in the urban environment, establishment of working
emission standards and, ultimately, reduction or elimination of the impact of these
emissions on our environment, air quality and health.
Therefore, the general aim of this thesis is to gain better understanding of
behaviour of nanoparticle aerosols by means of detailed experimental, statistical and
theoretical investigation of evolution mechanisms, dispersion, and deposition of
combustion airborne nanoparticles in the real-world environment, and develop new
predictive models and statistical methods of data analysis in the presence of natural
variability of the source and environmental conditions.
The specific aims of the project can be listed as follows.
1. Adaptation of the currently available models for the analysis of dispersion of non-
reactive air pollutants from a busy road (CALINE4 model) for the reliable forecast
of aerosol pollution levels.
8
2. Development of new methods for the experimental determination of the average
emission factors per one vehicle on the road in the real-world environment, based on
the monitoring data for the total number concentration at just one point near a busy
road.
3. Development of new methods for the determination of average emission factors
from different types of vehicles on a road on the basis of monitoring data on
different days of observation.
4. Detailed experimental investigation of combustion aerosols near busy roads.
Investigation of particle modes and their evolution as the aerosol is transported
away from the road.
5. Development of new statistical methods of identification of particle modes and
analysis of mechanisms of their rapid evolution near a busy road, based on the
moving average approach in combination with the simple correlation and canonical
correlation analyses.
6. Development of a new major mechanism of aerosol evolution based on intensive
thermal fragmentation of nanoparticles. Comprehensive interpretation of a complex
pattern of aerosol evolution near a busy road.
7. Statistical determination of possible sources of nanoparticle modes in combustion
aerosols near a busy road. Determination and interpretation of mutual correlations
between different particle modes.
8. Statistical analysis of urban background aerosols, including mode analysis and their
correlations.
9. Development of a new model of aerosol dispersion near a busy road on the basis of
the theory of particle fragmentation. Determination of the applicability conditions
for the proposed model and derivation of conditions for a maximum of the total
number concentration at an optimal distance from the road.
9
10. Development of a theory of stochastic evaporation/degradation processes in
composite aggregate structures. Determination of substantial time delays during
fragmentation of composite nanoparticles. Investigation of formation of particle
modes during aerosol evolution.
10
CHAPTER 2
BACKGROUND AND THEORY
2.1 Ambient aerosols and their origins.
The term ‘aerosol’ refers to an assembly (suspension) of liquid or solid particles
in a gaseous medium. Such suspensions are usually the result of either natural processes
(e.g., marine aerosols, dust storms, etc.), or human activities mainly related to
combustion processes, use of fossil fuels, transport, airplanes, etc. (Seinfeld and Pandis,
1998, Hinds, 1982, Willeke & Baron, 1993, Fuch, 1989, Kaye, 1981).
Aerosols play a very important role in atmospheric behaviour, climate patterns,
global climate changes, air quality in large metropolitan centres, inside our homes and
at the workplace. For example, atmospheric aerosols may result in additional reflection
of sunlight from the Earth atmosphere, which may lead to a cooling effect on the global
climate. Atmospheric aerosols may lead to a substantial increase in the number of
precipitation centres, resulting in extensive cloud formation with smaller size of the
droplets. This may lead to decreased rainfall and increased reflectivity of sunlight
(increased brightness of the clouds (Seinfeld & Pandis, 1998, Jacobson, 1999). On the
other hand, aerosol particles that strongly absorb solar energy may result in an
additional heating effect, leading to intensified global warming (Seinfeld and Pandis,
1998, Jacobson, 1999). Which of these tendencies appear to be predominant in reality is
still to be determined.
On the more local scale, the most significant effect of aerosols is a decrease of
the air quality in our homes, at the workplace, and at other places of our everyday
activities and leisure. They are one of the most significant air polluting factors in
modern cities and large metropolitan centres, produced by a number of different
anthropogenic sources resulting from human activities. The most significant sources of
11
aerosol air pollution and reduction of the air quality in the urban environment are
transport and industrial emissions (Seinfeld and Pandis, 1998, Willeke & Baron, 1993).
These emissions have been shown to have a significant adverse effect on human health
and state of our environment.
Aerosols are normally classified in terms of size of the particles (Hinds, 1982,
Willeke & Baron, 1993), which usually ranges from ~ 1 nm to ~ 100 μm (Kaye, 1981).
This classification usually includes three major particle groups: coarse particles (with
diameters ≥ 1 μm), fine particles (with diameters between ~ 0.1 μm and ~ 1 μm), and
ultra-fine particles or nanoparticles (with diameters < 100 nm). Such a classification is
important because size of particles is one of the most important factors in the
determination of aerosol properties and behaviour (Mandelbrot, 1983). Moreover,
different physical laws and approaches should be used for their description and
characterisation (Cliff, et al, 1978).
Properties of the particles in the atmosphere have been of interest for physicists
and meteorologists since the late 1880s, when John Aitken measured for the first time
number concentrations of dust and fog particles. However, only during the last decade
has it become possible to measure concentrations of nano-scale particles in the
atmosphere. Aerosol number distributions have been widely measured in urban, rural
and remote environments to characterise properties of small particles starting from
diameters as small as ~ 3 nm (Seinfeld and Pandis, 1998). Unfortunately, the smallest
size range (with diameters < 3 nm) is still difficult to access, and there is no clear
understanding of concentration and composition of these particles.
Coarse particles can be detected and analysed relatively easily, and their
contribution to the overall particulate mass of an aerosol is predominant (Friedlander,
1977). They also produce significant visible effects (such as dust storms, smoke from
open fires, etc.). Aerosols of such particles may have significant health effects, cause
12
respiratory problems, result in poor visibility and equipment breakage, and lead to other
problems in the metropolitan and rural environments. Therefore, until recently, the main
efforts have been concentrated on the analysis, monitoring and forecasting of aerosols
consisting of coarse particles.
At the same time, due to relatively large size of coarse particles, they are
effectively stopped by the upper air ways of the human air tract, predominantly in the
nose, from where they can be removed very effectively (Siegmann, 1999). This
significantly reduces the adverse health effects of course particles on humans (unless
during relatively rare significant bushfire events, dust storms, etc.). On the contrary,
smaller particles may penetrate deeper into the respiratory tract (Siegmann, 1999). In
addition, the highest levels of concentration of trace elements and toxins from
anthropogenic sources are usually associated with very small particles mainly in the fine
and ultra-fine ranges, i.e., between ~ 1 nm and ~ 2.5 μm (Thomas, et al, 1997).
Commonly, urban aerosols are a mixture of emissions from industrial sources,
transport, power plants, natural sources, and particles from the gas-to-particle formation
processes. Therefore, the aerosol size distribution in urban environment is quite
variable. Extremely high concentrations of fine particles (up to ~106 cm-3) are found
close to sources, for example, near highways (Zhu et al, 2002ab), but the concentration
decreases rapidly with distance from the source (Zhu et al, 2002ab). Nevertheless,
typical particle concentrations in the urban aerosols are substantially (at least an order of
magnitude) higher than in the remote areas. The particle number distribution in the
urban environment is dominated by particles within the range less than ~ 100 nm. These
particles are of a special interest for aerosol scientists, because they are regarded to be
most harmful, their number concentrations are typically high, and a large proportion of
population in the developed countries is exposed to them on the daily basis (Hussein et
al, 2005).
13
Aerosols in rural areas are mainly of natural origin, but with moderate influence
of anthropogenic sources (Hobbs, et al, 1985) and secondary aerosol formation products
in remote continental areas (Bashurova, 1992). The number distribution is mainly
characterized by two modes ~ 0.02 μm and ~ 0.1 μm (Jaenicke, 1993), and the mass
distribution mode ~ 7 – 10 μm (Jaenicke, 1993). Similar modes are typical for remote
continental aerosols. Aerosol number concentrations are typically from ~ 2×103 cm-3 to
~ 104 cm-3 (Bashurova, 1992), with the PM10 concentrations (i.e., the mass
concentrations for particles with diameters less than 10 μm) being from ~ 10 μg/m3 to ~
20 μg/m3 (Koutsenogii & Jaenicke, 1994).
An important source of aerosols is particle nucleation and growth in relatively
clean air of rural and remote areas. The resultant particles are usually in the nanometre
range. This process is of a growing interest because of its possible effect on climate and
health. Although, despite more than 100 publications in the literature on this topic, the
nature of the nucleation process is still not entirely clear (Kulmala, et al, 2004).
Therefore, this topic of aerosol science is under intensive current investigation.
Particles over the remote oceans are largely of marine origin (Savoie, 1989). The
typical concentrations in marine atmospheric aerosols are normally within the range
between ~ 100 cm-3 and ~ 300 cm-3. The size of the particles is usually relatively large
an is typically characterized by three distinct modes: Dp < 0.1 μm, 0.1 < Dp <0.6 μm, Dp
> 0.6 μm (Fitzgerald, 1991). Particles in marine aerosols are usually of biological and
organic nature, iodine compounds, sea salt, or their combination (Fitzgerald, 1991). Due
to their usually low number concentration, these particles present little or no problems
for human health. Therefore, the main aim of their investigation is usually related to the
effect of these aerosols on climate, cloud formation, and possible spread of microscopic
species and their products in the atmosphere.
14
Background free tropospheric aerosols have received relatively little attention.
These aerosols are usually characterized by two different particle modes in the size
distribution. These are at ~ 0.01 μm and ~ 0.25 μm (Jaenicke, 1993). The middle
troposphere typically has more particles in the accumulation (~ 0.25 μm) mode,
compared to the lower troposphere. This was explained by precipitation, scavenging,
and deposition of smaller and larger particles (Leaitch and Isaac, 1991).
The described types of aerosols rarely appear separately from each other.
Typically, an observed aerosol is a mixture of these types, resulting in a complex
pattern of the corresponding particle size distribution within a huge range of particle
diameters of ~ 4 orders of magnitude. A number of different modes of particles can
appear mixing with each other and thus leading to results that may be difficult to
interpret and subdivide into separate groups originating from particular sources. On top
of this, we have the natural instability of the atmospheric and environmental parameters
(e.g., atmospheric turbulence and the corresponding strong stochastic fluctuations of the
measured parameters), which makes the aerosol analysis and interpretation especially
difficult.
Accurate prediction of concentrations and dispersion of aerosols from different
polluting sources in the urban environment is one of the major problems of modern
environmental science. Solution of this problem is complicated by changing size
distribution of the aerosol particles, caused by possible evolutionary and formation
processes. Therefore, clear understanding of these physical and chemical processes in
urban aerosols is an essential goal of modern aerosol science (Seinfeld and Pandis,
1998, Jacobson, 1999). This naturally leads us to the next section where we consider the
first (and probably the most important) mechanism of aerosol evolution – turbulent
dispersion in the atmosphere.
15
2.2 Turbulent dispersion of air pollutants
One of the major physical mechanisms of dispersion of air pollutants in the
atmosphere is turbulent diffusion (Csanady, 1980, Pasquill & Smith, 1983, Zannetti,
1990). Turbulence is stochastic (random) motion of the air caused by breakage of the
unstable laminar flow of the air/fluid due to its interaction with obstacles and/or uneven
heating (Landau & Lifshitz, 1987). Turbulent diffusion occurs when the diffusing
substance (aerosol particles) is transported by such random motion, which is similar to
random motion of separate molecules in the air, resulting in conventional molecular
diffusion (Csanady, 1980). Mathematical and physical consideration of turbulent
diffusion is the same as for molecular diffusion – the diffusing particles experience
random walk caused by their random transport by means of air parcels moving
randomly due to turbulence.
If only the mechanism of turbulent diffusion is taken into account, then
dispersion of the aerosol can be described in exactly the same way as for non-reactive
gasses – by the Gaussian plume model (Csanady, 1980, Pasquill & Smith, 1983,
Jacobson, 1999). This model is based upon the solution of the well-known diffusion
equation in a moving incompressible fluid
χ∇=χ∇•+∂χ∂ 2Dt
u , (2.1)
where χ is the volume density (or concentration) of the diffusing substance, u is the
velocity of the fluid, D is the coefficient of diffusion, and t is the time (see, for example,
Stull, 1989, Jacobson, 1999, Pasquill, 1983).
For the simplest case of one-dimensional diffusion in a uniform medium at rest
(u = 0, D = const), the solution to Eq. (2.1) can be written as:
DtxeDt
Q 4/2
2−
π=χ . (2.2)
16
This equation gives the time-dependent concentration of the diffusing
substance/particles at some moment of time t and the coordinate x. The constant Q is the
surface density (or surface concentration) of the diffusing substance at x = 0 at the
moment t = 0 (at this moment of time, all the diffusing substance is concentrated at the
plane x = 0).
It is possible to see that σ = 2 Dt is the typical distance within which the
substance (particles) have spread due to diffusion within the time interval t. This
distance can be regarded as a convenient scale of the width of the spatial distribution of
particles during their diffusion, and is termed as standard deviation for diffusing
particles from the position x = 0. Eq. (2.2) represents the Gaussian distribution that also
applies to turbulent diffusion in the atmosphere. However, in this case, the diffusion
coefficient D is much larger than that for molecular diffusion, simply because the scale
of random motion (given by the turbulence scale, i.e., typical size of the turbulent
eddies (Landau & Lifshitz, 1987, Csanady, 1980)) is much larger than for molecular
diffusion where it is given by the mean free path of the molecules. Therefore, in the
problems with turbulent diffusion in the atmosphere, molecular diffusion does not play
any noticeable role and thus can be neglected.
To determine σ in the case of turbulent diffusion of air pollutants in the real
environment, a statistical theory of diffusion has been developed, based on the history
of random-walk motion executed by the diffusing particles (Csanady, 1980). Turbulent
motion appears to be one of the most difficult problems of modern physics, and the
complete understanding of this process in the general case has not yet been achieved.
Historically, the first successful treatment of turbulent diffusion by Taylor (Taylor,
1922) was directly based on the statistical theory of Brownian motion that was
developed by Einstein in 1905 (Einstein, 1905).
17
In a process of random walk of particles (Brownian motion) or air parcels
(turbulent diffusion), the displacement of the diffusing particle is a function of time. At
the same time, the actual value of this displacement at any given time is random and can
only be specified in terms of probability distribution (Taylor, 1922). If this probability
distribution is time independent, then we have a steady-state stochastic process that has
some simple properties and can be analysed relatively easily. Turbulent diffusion is
steady-state if the temperature, mean velocity of the air, and turbulent intensity are
homogeneous over the whole turbulent field, i.e., over the whole space (Taylor, 1922,
Csanady, 1980). The concept of turbulent intensity was introduced to characterize the
“strength” of a turbulent state. Turbulent intensities are defined as follows. We
determine the average velocity of the air, i.e. the average wind velocity. Then, we
determine instantaneous values of wind components at some moment of time and the
considered point in space. The time average squares of the differences between the
instantaneous and average wind components 222 ,, wvu are called turbulent
intensities, where u, v and w are these differences between the instantaneous and
average wind components along the x-, y-, and z-axes, respectively.
If the magnitude of the mean velocity is U, we can introduce relative turbulent
intensities:
U
uix
2/12 ⎟⎠⎞⎜
⎝⎛
= ; U
viy
2/12 ⎟⎠⎞⎜
⎝⎛
= ; U
wiz
2/12 ⎟⎠⎞⎜
⎝⎛
= . (2.3)
The root-mean-square of the differences between the instantaneous and average wind
components 2/1
2 ⎟⎠⎞⎜
⎝⎛= uum ,
2/12 ⎟⎠⎞⎜
⎝⎛= vvm and
2/12 ⎟⎠⎞⎜
⎝⎛= wwm can be regarded as
characteristics of the eddies in the turbulent motion.
18
2.2.1. Taylor theorem and asymptotic properties of the diffusing cloud
Consider a point source of particles with the position vector x’. Let a particle be
released from the source at time t = 0 (the particle position vector at t = 0 is x’).
Suppose that during the diffusion time t the particle has moved to its new position x.
Then P(x – x’, t)dx is the probability that the displacement vector x – x’ ends within the
volume element dx around the point x at time t. If χ is the mean mass concentration of
the particles at the position x of the volume element dx, then
χ(x – x’, t) = QP(x – x’, t), (2.4)
where Q is the total mass of particles released at the point x’ at the moment of time t =
0.
If u is the component of the fluid velocity relative to a reference frame moving
with the average velocity of the fluid (i.e. U = 0), then in the steady-state case
0)( =tu ;
consttu =)(2 , (2.5)
with the same relationships holding for v and w.
An important characteristic of a steady-state stochastic process is its
autocorrelation function R(τ) that measures the “persistence” of a given value of a
random variable (e.g., velocity) within a time interval τ. In other words, the
autocorrelation function determines the likelihood that if a particle has a particular
velocity at some moment of time t, then this velocity will have similar magnitude and
direction at the moment of time t + τ (Csanady, 1980). It can be shown that the
autocorrelation function is given by the equation (Csanady, 1980):
2
)()()(u
tutuR τ+=τ , (2.6)
19
where the averages are taken in time. In a steady-state process, R(τ) is independent of
time t. At τ = 0 the autocorrelation function is equal to one, and R(t) → 0 when τ → +
∞.
Since the displacement of a particle is related to its velocity by
∫=t
dttutx0
')'()( ,
for the x-component (and similar for the y- and z-components), we have
∫=t
dttutudt
txd0
2')'()(2)]([ .
Taking an average over the whole ensemble of the diffusing particles of the both sides
of this equation, gives (Taylor, 1922, Csanady, 1980):
∫ ττ=t
dRudtxd
0
22
)(2 , (2.7)
where 2x is the square of displacement of a diffusing particle averaged over the whole
considered ensemble of the particles. Therefore, 2x is a function of time. On the
contrary, the other averages in Eq. (2.7) are taken over time (and thus are time-
independent).
Eq. (2.7) is the most important fundamental result of the random-walk theory of
diffusion (Brownian motion or turbulence). In the theory of turbulent diffusion it is
known as the Taylor theorem (Taylor, 1922).
For an ensemble of independently diffusing particles that are released at x = 0
and t = 0, the spread of the plume along the x-axis is given by the so-called radius of
inertia σx of the mean concentration distribution (Csanady, 1980):
∫∫∫ χ=σ dxdydztzyxxQx ),,,(1 22 . (2.8)
20
The length σx defined by Eq. (2.8) is a measure of the cloud size after some time of
turbulent diffusion (it has the same meaning as σ = 2 Dt introduced below Eq. (2.2)).
Assuming the χ(x,y,z,t) is the concentration formed by the independently
diffusing particles and substituting Eq. (2.4) into Eq. (2.8), gives:
222 ),,,( xdxdydztzyxPxx =∫∫∫=σ , (2.9)
where 2x is again the time-dependent ensemble average of the square of the particle
displacement. Therefore, using the Taylor theorem (Eq. (2.7)), Eq. (2.9) for the time-
dependent standard deviation (or the radius of inertia) of the cloud can be reduced as:
∫ τττ−=σt
x dRtut0
22 )()(2)( . (2.10)
Asymptotic properties of σx can be determined from the behaviour of R(τ) at small and
large values of τ. For example, when τ → 0 (and R → 1), Eq. (2.10) is reduced as
222 tux ≈σ (if t → 0) (2.11)
with the accuracy of ~ t4 – see (Csanady, 1980).
On the other hand, when τ → ∞, R(τ) → 0. In this case,
)(2 1022 tttux −≈σ (if t → ∞) (2.12)
where
∫ ττ=∞
00 )( dRt , ∫ τττ=
∞
001 )(1 dR
tt ).
Thus for small dispersion times (small t), the size of the plume σx increases linearly with
time, whereas for large t, it is proportional to the square root of time.
2.2.2. Turbulent diffusion from a point continuous source
Let the point source of particles be at the origin of the frame (x,y,z) and the wind
be parallel to the x-axis. We assume that the total mass of the particles released from the
source at the moment of time t = 0 is equal to Q. Then, assuming that the probability
21
distribution P(x,y,z,t) is Gaussian, the resultant average particle concentration as a
function of time and coordinates can be written as (Csanady, 1980):
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
σ−
σ−
σ−−
σσσπ=χ 2
2
2
2
2
2
2/3 222)(exp
)2(),,,(
zyxzyx
zyUtxQtzyx , (2.13)
where U is the wind speed.
If we have a continuous point source of particles, then such a source can be
represented as a number of releases of particle plumes at infinitesimally close moments
of time. In other words, we have a number of instantaneous point sources located at the
same point in space (frame origin) and releasing particles at different (infinitesimally
close) moments of time, thus producing a continuous particle release by a continuous
source.
One of the main assumptions of the mathematical analysis of turbulent diffusion
is that the motion of each of the air parcels (elements) is independent of the other
neighbouring parcels (Csanady, 1980). Therefore, in this approximation, turbulent
diffusion is a linear phenomenon, and the concentration field from multiple sources
(releasing particles at different moments of time) can be represented as a simple
superposition of the fields from each of the individual sources. Thus, the steady-state
concentration from a continuous point source is give as (Csanady, 1980):
zyxzyx
dtzyUtxqzyxσσσ⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
σ−
σ−
σ−−∫
π=χ
∞ '222
)'(exp)2(
),,( 2
2
2
2
2
2
02/3 (2.14)
where q is release rate, i.e., the total mass of the particles released per unit of time.
At short distances from the source, where the size of the cloud grows linearly
with time (see Eq. (2.11)):
σx ≈ umt, σy ≈ vmt, σz ≈ wmt,
and integration of Eq. (2.14) gives (Frenkiel, 1953):
22
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎟⎠
⎞⎜⎜⎝
⎛π+×⎟⎟⎠
⎞⎜⎜⎝
⎛−
π=χ
ruUx
ruxU
ruUx
uU
rwvquzyx
mmmmmm
m
21erfexp
21
2exp
)2(),,( 22
2
2
2
22/3
where
22
22
2
222 z
wuy
vuxr
m
m
m
m ++= (2.15)
However, for a point source that is not too high above the ground level (within ~
100 m), the application of this formula is non-trivial, since at these heights the wind
speed and the turbulence parameters (e.g., diffusivity) are height-dependent (Csanady,
1980). The detailed analysis of this situation is very complex and has not been done in
the general case. Usually, the wind speed is assumed to be approximately constant with
height (Benson, 1992a,b) or has a logarithmic profile (Stull, 1989), which in a number
of situations is associated with significant errors. Even if we assume that the turbulence
is homogeneous in the first approximation, we have to take into account the surface of
the ground that works as a rigid boundary. Usually, the ground is assumed to be a
perfect reflector, and it can be introduced by a mirror-image source placed below the
ground (Csanady, 1890, Jacobson, 1999, Seinfeld & Pandis, 1998). This approach has
been confirmed by the experimental evidence (see, for example, (Csanady, 1968,
Clauser, 1956)). Therefore, the mean concentration field due to a continuous elevated
point source of strength q can be written as (Csanady, 1980):
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
σ+−
σ−+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
σ−−
σ−
σσπ=χ 2
2
2
2
2
2
2
2
2)(
2exp
2)(
2exp
2),,(
zyzyzy
hzyhzyU
qzyx , (2.16)
where the frame origin is exactly under the point source that is located at the height h
above the ground level, and the wind speed U is assumed to be height-independent.
The practical importance of Eq. (2.16) is that it can be used for prediction of
ground level concentrations:
23
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
σ−
σ−
σσπ=χ 2
2
2
2
22exp)0,,(
zyzy
hyU
qyx . (2.17)
2.2.3 Continuous ground level line source
The considered case of dispersion from a point source is important for modelling
of dispersion of non-reactive pollutants (e.g., aerosols), for example, from an industrial
site. Another even more important situation is a line source of emission, such as a busy
road with traffic. Because traffic emissions from busy roads and road networks make a
predominant contribution to air pollution in large metropolitan areas, modelling of
aerosol dispersion from a line source is essential for our ability to forecast air pollution
levels and control air quality in the urban environment.
It is possible to think that the consideration of a line source can again be
conducted by subdividing this source into point sources and applying the equations
derived in the previous sections (Held, et al, 2001). However, application of Eqs. (2.12)
– (2.15) to the determination of mean particle concentrations from a continuous ground
line source is again substantially impeded by the fact that wind speed strongly depends
on height above the ground. Therefore Eqs. (2.12) – (2.15) are not applicable, since they
have been derived in the approximation of constant wind.
Therefore, the approximate analysis of dispersion from a ground level line
source (e.g., a busy road) is considered using the self-similarity theory and dimensional
approach (Csanady, 1980).
We choose the y-axis along the ground level line source and the z-axis
perpendicular to the ground. We also assume that the average wind velocity is parallel
to the x-axis, i.e., normal to the road. It has been shown that the vertical profile of the
wind (up to ~ 100 m above the surface of the earth) is given by the logarithmic law
(Stull, 1989):
24
0
lnzz
kuU ∗= , (2.18)
where ∗u is called friction velocity, and it depends on roughness of the surface z0 and
the Karman constant k (Stull, 1989). The roughness of the surface is approximately
estimated as 1/30 of the typical height of the obstacles on the surface (Stull, 1989).
The major parameter that determines turbulent diffusion of a plume is the length
scale Lc. The length Lc can be taken to equal the typical size of turbulent eddies, and
then the spread of the plume can be determined in terms of Lc, because the rate of
expansion of a plume (turbulent diffusivities) are determined by the typical size of
eddies. However, this choice may be ambiguous, because turbulence may have different
scales (Csanady, 1980, Landau & Lifshitz, 1987). Therefore, it is more convenient and
practical to define Lc directly as the typical spread of a plume emitted, for example,
from a point source within some particular time of dispersion. For example, we can
choose that Lc = σx, where σx is the radius of inertia of the plume – see above. It is
necessary, however, to keep in mind that the asymptotic behaviour of the radius of
inertia for a ground source is significantly different from that for an elevated point
source (Eqs. (2.8) – (2.12)). This is because of the mentioned dependence of the wind
on height above the ground and the corresponding non-homogeneity of the turbulence
(its scale, as well as the turbulent diffusivities, also changes with increasing height
above the ground) (Csanady, 1980, Stull, 1989, Zannetti, 1990, Pasquill & Smith,
1983).
Variation of Lc with distance x can then be derived as the universal “law of
growth” for the ground-level source (Csanady, 1980, ):
bkxezLL c
c =0
ln , (2.19)
where e = 2.71828, and b is the coefficient of proportionality.
Eq. (2.19) can be reduced to the form of the power law (Csanady, 1980):
25
γ×= xconstxLc )( , (2.20)
where Δ−=γ 1 , and )/ln(/1 0zLc=Δ . This form of Eq. (2.19) is more convenient for
practical use, because though the power γ depends on Lc (and thus on distance from the
road x), this dependence is usually relatively weak, because z0 << Lc. Therefore, within
reasonably large distance intervals near a road, Δ can be regarded as a constant that is
noticeably smaller than 1. As a result, the approximate dependence of plume spread is
indeed the power law of distance from the road with the power γ that is close to 1
(further confirmation of this statement will be provided in Chapter 3).
In this case, the concentration of particles in the plume will be the inverse power
law in distance from the road (Csanady, 1980):
μ−=χ Cx0 , (2.21)
where μ = 1 – Δ2, and C is the coefficient of proportionality (a constant). Because of the
same reasons as above, μ should be close to 1 and vary only weakly with distance from
the road.
This approximation and results (Eq. (2.21)) seem quite reasonable, because it is
well-known (for example, from the electrostatics) that a line source of some field results
in the decay of this field as r–1, where r is the distance from the source. In the case of
line source emitting particles, particle conservation requires approximately the same
dependencies. The small difference associated with μ being slightly smaller than 1 is
due to the presence of the boundary (ground surface). This boundary results in reflection
of particles from it, which is taken into account by introducing an additional imaginary
source below the ground level (Csanady, 1980). This is again very similar to
electrostatics considering field structure from charges in the presence of metallic
surfaces (Landau & Lifshitz, 1984).
26
As a result, dispersion from a busy road in the absence of particle
transformation/evolution and noticeable deposition is best described by the described
power law (Csanady, 1980, Zhang, et al, 2005), rather than exponential dependencies
that were some times used for this modelling (Hitchins, et al, 2000, Zhu, et al, 2002a,b).
At the same time, it is possible that deposition of nano-particles may result in a
significant loss of particles during their transport from the road, and the actual
dependence may be different (e.g., be closer to the exponential law).
2.3 Dispersion of fine particles from a busy road
Until recently, the major efforts in environmental science were focused on
gaseous air pollution or aerosols consisting of fairly large not evolving particles.
Dispersion of inert gasses (e.g., CO or NOx on the time scale of ~ tens of seconds) and
not evolving aerosols with the diameters ≥ 2.5 μm can be described by the conventional
theory of turbulent diffusion based on the Gaussian plume approximation, considered in
the previous section. The resultant most developed dispersion models for determination
concentrations of inert gasses emitted by a line source (highway) are CALINE4,
developed by California Department of Transport (Benson, 1992) and UCD2001 from
University of California (Held, et al, 2001).
CALINE4 has been designed for the analysis of carbon monoxide pollution near
a busy road, based on the knowledge of gaseous emission factors from stationary and
moving vehicles. The model considers the region directly above the road as a zone of
uniform emission that is called mixing zone (Benson, 1992). The aim of the
introduction of the mixing zone is to establish initial Gaussian dispersion parameters at
a reference distance near the edge of the roadway. CALINE4 model divides the
roadway into a series of elements from which incremental concentrations are computed
and summed (Benson, 1992). The UCD model (Held, et al, 2001) is based on an array
27
of point sources, rather than a sequence of sections of line sources (sections of the road)
used in the CALINE models. Similar to CALINE models, UCD 2001 assumes that all
pollutants are emitted from a mixing zone above the roadway (Held, et al, 2001).
However, even for the case of non-reactive pollutants that can be treated by
means of the CALINE4 or UCD models, the major problem is in the absence of
consistent experimental data on the average emission factors for fine and ultra-fine
particles from motor vehicles, that are required as an input for both the packages.
Indeed, experimental values of these emission factors presented by different researchers
vary by one or even two orders of magnitude, even for the same types of vehicles
(Graskow, et al, 1998, Gertler, et al, 2000, Gross, et al, 2000, Jamriska & Morawska,
2000). Typically, the emission factors lie within the intervals between ~ 1012 to ~ 1014
particles per vehicle per kilometre for gasoline (light-duty) vehicles, and between ~ 1014
to ~ 1015 for diesel (heavy-duty) vehicles. Thus, there is a strong need to develop
reliable methods for the determination of average emission factors for different types of
vehicles on a busy road.
Until recently (until the last couple of years), it was accepted that particle size
distributions do not change significantly with increasing distance from the road, i.e., the
particles were thought to be non-evolving (Shi, et al, 1999). Therefore, the main interest
of aerosol scientists has been focused on the study of decay of the total number
concentration of particles with distance from a busy road (Shi, et al, 1999, Hitchins, et
al, 2000, Zhu, et al, 2002a,b). In particular, exponential (Hitchins, et al, 2000, Zhu, et
al, 2002a,b) decay laws were used for the description of the total number concentration
of fine particles as a function of distance from the road (see also the end of the previous
section).
Shi, et al, (1999) measured the size distribution for particles in the range from ~
10 nm to ~ 352 nm at three distances downwind from a busy road. As a result, only an
28
insignificant shift (of only ~ 1 nm) of the size distribution was observed. However,
these measurements were conducted in an urban area with a number of other roads in
the vicinity of the measurement site, which made it difficult to see aerosol evolution
with increasing distance from the specific source (road). At the same time, in another
paper from the same group (Alam, et al, 2003), unusual strong bursts of concentration
of ultra-fine particles in the range ≤ 7 nm were registered at a significant distance from
the road. These maximums were associated with the traffic conditions and solar
radiation. An explanation of the obtained results by means of particle nucleation was
attempted. However, these measurements were not specifically focused on the
investigation of particle size distribution and its evolution with increasing distance from
a busy road (Alam, et al, 2003), which may be important for verification or disproval of
the nucleation model.
The first consistent attempt to investigate evolution of combustion aerosols near
a busy road has been undertaken in the two papers by Zhu, et al (2002a,b), where the
size distribution of particles in the range from 6 nm to 220 nm was investigated at six
distances from a busy road: 17 m, 20 m, 30 m, 90 m, 150 m, and 300 m. Concentrations
of particles were normalized to wind speed and direction (Zhu, et al, 2002a,b), and
averaged over different conditions during the observations. This research revealed
significant changes of the size distribution of fine and ultra-fine particles during their
transport from the road. A conclusion was made that both coagulation of particles and
turbulent dispersion contribute to the rapid decrease in particle number concentration
and mode transformation (Zhu, et al, 2002a,b). The process of coagulation was regarded
as the main reason for the variation of the size distribution with distance from a busy
road. Two main modes (strong maximums) of the size distribution (around ~ 10 nm and
~ 20 nm) that were observed at 17 m distance from the road, were claimed to shift
29
towards larger particle diameters as the aerosol was transported away from the road
((Zhu, et al, 2002a).
As mentioned above, concentrations were averaged over the atmospheric
conditions and normalized to wind speed and direction. This approach is important for
the determination of the general average tendencies of the aerosol evolution from a busy
road. However, it may mask effects that are sensitive to particular atmospheric
conditions and wind direction and speed. In addition, relatively large steps in distance
might have resulted in insufficient spatial and temporal resolution of the measurements.
The suggestion that particle coagulation may be used for the explanation of the
observed mode transformation (Zhu, et al, 2002a) seems questionable, since coagulation
at the considered particle concentrations is highly inefficient (Jacobson, 1999, Shi, et al,
1999, Zhang & Wexler, 2004, Zhang et al, 2004) and can hardly result in the observed
significant changes in the particle size distribution (Jacobson & Seinfeld, 2004).
It is clear that modelling of dispersion of evolving aerosol particles (or reactive
gasses) is a much more complex problem. Over the years, substantial efforts have been
made to understand chemical and physical processes inside plumes from point
(industrial) sources (see, for example, (Seigneur, et al, 1983, Kerminen and Wexler,
1995, Kumar & Russell, 1996, Bilger, 1978, Bilger, et al, 1991)). However, only a few
models have been published in the literature that could be applied for evaluation of
motor-vehicle aerosol transformation on the urban scale (Pilinis and Seinfield, 1987,
Jacobson, et al, 1996, Jacobson, 1997a,b). At the same time, application of any model to
dispersion of reacting and rapidly evolving particles will require detailed knowledge of
the actual processes occurring in the aerosol. Comprehensive understanding of these
processes has not been reached so far, and even the predominant nature of the exhaust
ultra-fine particles is still disputed by different authors (Alam et al, 2003, Sacurai et al,
30
2003, Fierz & Burtscher, 2003, Kittelson, 1998, Farrar-Khan, et al, 1992, Colbeck, et al,
1997, Wentzel, et al, 2003, Abdul-Khalek et al, 1998).
For example, diesel exhaust particles consist mainly of highly agglomerated
solid carbonaceous material, ash, and volatile organic and sulphur compounds
(Kittelson, 1998). Solid carbon is the usual product of all combustion processes
including vehicle exhaust. According to (Kittelson, 1998), solid carbon agglomerates
constitute approximately 40% of particulate emission from diesel vehicles. Unburned
fuel particles (~ 7%), and evaporated lube oil (~ 25%) appear as volatile or soluble
organic compounds that form volatile particles or volatile cover for solid particles in the
exhaust (Kittelson, 1998). Soluble organic compounds contain polycyclic aromatic
compounds and sulphurs (~ 14%) (Farrar-Khan, et al, 1992). Metal compounds in the
fuel and lube oil result in a small amount of inorganic ash (~ 13% of the overall
particulate emission) (Kittelson, 1998). The fraction of the unburned fuel and lube oil
varies with engine design and operation condition. Most of the particle mass exists in
accumulation mode (0.1 – 0.3 μm) consisting mainly of carbonaceous agglomerates.
The coarse mode (> 1 μm) contains ~ 5 – 20% of the particle mass (Kittelson, 1998).
It has also been suggested that the nuclei mode (nanoparticles) that is typically ~
5 – 50 nm usually consists of volatile organic and sulphur compounds, and probably of
solid carbon and metal compounds (Kittelson, 1998). On the other hand, solid carbon
particles ~ 20 – 30 nm have readily been observed in the products of combustion of
different materials (Colbeck, et al, 1997), and in the vehicle exhaust (Meszaros, 1999,
Wentzel, et al, 2003). Several researchers have also reported the existence of a large
number of smaller solid particles around ~ 7 nm mobility diameter. For example,
Abdul-Khalek, et al (1998) reported the experimental observation of a strong mode of
particles from diesel exhaust within this particular range. The analysis of these particles
by means of a catalytic stripper has suggested that a large number of them are solid
31
(Abdul-Khalek, et al, 1998), and the authors assumed these particles to be metallic ash
formed from oil and fuel additives. On the other hand, Bagley, et al (1996) suggested
that this mode may consist primarily of carbonaceous particles. Recent investigation by
Sakurai, et al (2003), using a thermal desorption particle beam mass spectrometer, has
demonstrated the existence of solid nuclei particles in the diesel exhaust of ~ 3 – 7 nm.
The results have been explained by evaporation of a thick volatile layer from the surface
of the solid nuclei particles (Sakurai, et al, 2003). Yet another recent experiment on
separation of volatile and solid particles in the diesel exhaust by means of a hot dilution
system has also demonstrated the existence of a large number of solid (presumably,
carbon/graphite) particles of ~ 6 nm – 10 nm mobility diameter (Fierz & Burtscher,
2003).
The assumption of the above-mentioned predominantly volatile nanoparticles
inevitably raises questions about possible mechanisms of their formation and chemical
composition. In particular, it has been suggested that sulphur could play a significant
role in formation of the volatile compounds of the diesel aerosol (Kittelson, 1998).
However, more recent attempts to estimate the effects of nucleation of sulphur nano-
particles has suggested that this process may be significant only within just a few
seconds after the exhaust gasses leave the exhaust pipe (Pohjola, et al, 2003). This study
developed a model that was based on the specific composition of the vehicle exhaust of
~ 20% organic carbon and ~ 80% elemental carbon (Kittelson, et al, 1999). The traffic
flow was taken to be ~ 13,400 vehicle/day with ~ 5% of light duty diesel vehicles. The
model estimates the mutual importance of different aerosol processes during the first 25
s of evolution of particles emitted from the vehicle exhaust. The results showed that,
under the selected conditions, binary (H2SO4 – H2O) and ternary (H2SO4 – H2O – NH3)
nucleation can be neglected, as well as condensation of sulphate. Therefore, sulphur
compounds cannot play a noticeable role (at least at the considered traffic conditions) in
32
formation of the presumably volatile particles in the nucleation mode considered by
Kittelson, et al (1998).
Condensation of insoluble organic vapour is important if its concentration
exceeds ~ 1010 cm-3 (Pohjola, et al, 2003). Condensation or evaporation of water can
also be an important process. However, its influence is strongly dependent on the
hygroscopicity of particles. The effects of coagulation is negligible. A very important
finding of this work is that after 25 s of evolution the particulate population reaches a
quasi-steady state, i.e., most of the transformation processes have finished (Pohjola, et
al, 2003).
Another very recent work by K.M. Zhang and colleagues (Zhang & Wexler,
2004, Zhang et al, 2004) analysed the results of experiments conducted by Zhu, et al
(2002a,b). It is interesting to notice that the traffic conditions in this experiment were
strongly different from those in (Pohjola, et al, 2003): ~ 15,000 vehicle/hour with ~ 5%
of heavy-duty trucks for one road (Zhu, et al, 2002b), and ~ 12,500 vehicle/hour with ~
23% of heavy-duty trucks for another road (Zhu, et al, 2002a). Therefore, the traffic
density was ~ 10 times higher than that considered in (Pohjola, et al, 2003). At the same
time, whereas the width of the roads were very similar: 25 m in (Pohjola, et al, 2003)
and 26 – 30 m in (Zhu, et al, 2002a,b). The total number concentrations of particles in
the range 6 – 220 nm at the distances of 17 m and 30 m from the roads were very high:
~ 220×103 cm-3 and ~ (150 – 180)×103 cm-3, respectively (Zhu, et al, 2002a,b).
Two stages for dilution of the emitted exhaust were considered (Zhang &
Wexler, 2004, Zhang et al, 2004). The first stage dilution ~1000:1 is caused by the
traffic turbulence within ~ 1 – 3 s immediately after the emission from the exhaust pipe.
The second stage dilution 10:1 is mainly caused by the atmospheric turbulence, and this
process usually lasts ~ 3 – 10 min (Zhang & Wexler, 2004, Zhang et al, 2004). The pre-
dilution (“in-tailpipe”) aerosol composition measurements are not yet available.
33
Therefore, this composition was estimated from total mass of emissions, using ambient
measurements (Shauer, et al, 1999, Shauer, et al, 2002). Although, as it was shown by
Zhang & Wexler (2004) and Zhang et al (2004), particle growth is very sensitive to the
initial “in-tailpipe” processes, the results of Shauer, et al (1999) and Shauer, et al (2002)
have been used as best estimates for semi-volatile, gaseous and particulate components
of the exhaust emissions for medium-duty diesel and light-duty gasoline vehicles
(Zhang & Wexler, 2004, Zhang et al, 2004). The chemical species were assumed to be
sulphate and organic compounds since they are the likely candidates to trigger
nucleation or lead to significant particle growth in such a short time (Zhang & Wexler,
2004). The effective behaviour of organic compounds has been modelled by introducing
two different volatility classes: semi-volatile and low volatile (Zhang et al, 2004). Each
class was represented by a single carbon number. Their vapour pressures and molar
volumes were assumed to be the same as for alkanes of the same carbon number, whose
thermodynamic properties are well known (Zhang et al, 2004).
As a result, investigation and modelling of the aerosol dynamical processes such
as nucleation, condensation and coagulation have been carried out (Zhang & Wexler,
2004, Zhang et al, 2004). For the first dilution stage, it was shown that the nucleation
process can take place. However, extremely rapid dilution (within ~ 1 s) results in the
equally rapid decrease of vapour concentration for the gasses responsible for the
nucleation processes. Therefore, nucleation may only occur when the dilution ratio is
around 30 – 80 (Zhang & Wexler, 2004) – during the first second of dilution of the
exhaust gasses in the ambient air. Condensation of organic compounds also occurs at
the first dilution stage, which results in rapid growth of solid and liquid nanoparticles.
Coagulation has been shown to be negligible for both stages of dilution (Zhang &
Wexler, 2004). It has also been suggested that for the second dilution stage only
condensation/evaporation can affect the aerosol size distribution (Zhang et al, 2004).
34
The model demonstrated agreement with the measured particle size distribution at 3
distances from the road, though there were some discrepancies for smaller (< 10 nm)
particles and, in some cases, for particles larger than 100 nm (Zhang et al, 2004).
As a conclusion to this section, it can be noted that there is no current generally
accepted model and/or physical understanding of the complex processes that lead to the
formation and evolution of nanoparticle combustion aerosols at different stages of this
evolution. The suggested mechanisms do not seem to be entirely agreed upon and do
not present a clear and comprehensive picture of the aerosol behaviour. At the same
time, as has been mentioned, detailed understanding of these mechanisms of aerosol
evolution is essential for the development of reliable predicting models for aerosol
pollution levels, their impact on the environment and human health.
2.4 Monitoring equipment
One of the major types of instruments used in this work were Scanning Mobility
Particle Sizers (SMPS). The SMPS system measures the size distribution of
submicrometer and nano-particle aerosols, using an electrical mobility technique. The
SMPS spectrometer system is based on the principle of the mobility of a charged
particle in an electric field. Particles entering the system pass through a bipolar charger
(or neutralizer) with a radioactive source. Then they enter a Differential Mobility
Analyser (DMA) where the aerosol is classified according to electrical mobility, with
only particles of a narrow range of mobility diameter exiting through the output slit.
This monodisperse aerosol then goes to a Condensation Particle Counter (CPC) that
determines the particle concentration at that size.
The DMA consists of a cylinder, with a negatively charged rod at the centre, the
main flow through the DMA is the laminar flow of particle free 'sheath' air. The air with
aerosol particles is injected at the outside edge of the DMA, particles with a positive
35
charge move across the sheath flow towards the central rod, at a rate determined by their
electrical mobility. Particles of a given mobility exit through the sample slit at the top of
the DMA, while all other particles exit with the exhaust flow. The size of particle
exiting through the slit being determined by the particle size, shape, charge, central rod
voltage, and flow within the DMA. Exponentially scanning the voltage on the central
rod, a full particle size distribution is built up in the logarithmic scale of particle
diameters.
Two types of the SMPS spectrometer systems have been used within this
project. These are the Model 3071 Electrostatic Classifier (SMPS 3071) with the Model
3010 Condensation Particle Counter, and Model 3085 Electrostatic Classifier (SMPS
3936) with the Model 3025 Condensation Particle Counter. All the classifying
equipment was manufactured by TSI Incorporated. The first system (SMPS 3071 and
CPC 3010) have the measurement range of particle diameters from ~ 13 nm to ~ 763
nm. The second system (SMPS 3936 with CPC 3025) allowed concentration
measurements within the diameter range from ~ 4.6 nm to ~ 163 nm.
The SMPS software (available from TSI Incorporated) calculates particle
concentrations in each channel of the size distribution by using the raw counts in the
channel, calculations for single charge probability, corrections for multiple charges,
transfer function width, DMA flow rates, the CPC flow rate, the measurement time for
the size channel, slip correction for particles, the impactor cut-point, and user-defined
efficiencies for all the pieces of equipment involved. Therefore, no additional inversion
of the obtained data sets has been carried out.
Routine calibration of the SMPS system was performed by the manufacturer
(TSI Incorporated) on a regular basis. In addition, fine calibration of the system was
conducted before each of the monitoring campaigns by adjustment of the delay time
(which is the time that it takes for an aerosol to travel from the classifier column until it
36
is detected by CPC). This time, determining sizing accuracy of SMPS, was verified under
laboratory conditions by means of monodisperse polystyrene latex spheres of 100 nm diameter
An automatic weather station (Standard Weather Station from Monitor Sensors)
was used for monitoring of local atmospheric parameters such as temperature, humidity,
wind speed, wind direction and solar radiation at the point of aerosol monitoring. The
weather station was under manufacturer’s warranty, and its calibration was conducted
by the manufacturer. All the aerosol measurements were conducted simultaneously with
the monitoring of the local atmospheric parameters. The measurements of wind
direction were normally conducted every minute, while all other parameters were
measured every 20 s. The time scales for aerosol and atmospheric monitoring were
matched in order to enable detailed statistical analyses and correlations between particle
concentrations and atmospheric parameters.
2.5 Statistical approaches: correlation techniques in data analysis
Despite a substantial body of literature on computational modelling and
experimental investigation and measurements of various types of aerosols and gaseous
pollutants, as has been discussed above, there are still significant gaps in our
understanding of aerosol behaviour. One of the main such gaps is the lack of knowledge
and understanding of generation, evolution and propagation of fine and ultra-fine
particle aerosols, and their relationship with other types of pollutants such as gasses and
coarse particles under various atmospheric conditions (e.g., temperature, humidity, solar
radiation, etc.). The analysis of these relationships will involve simultaneous
consideration of a number of different variables, some of which vary unpredictably due
to the strong stochastic processes in the atmosphere and environment (such as, for
example, turbulent diffusion and the corresponding strong fluctuations of pollutant
37
concentrations). Therefore, one of the most effective and essential approaches to data
analysis and investigation of complex and unpredictable effects associated with air
pollution should be based on the extensive use of statistical methods.
Statistical methods include a number of specific techniques, including regression
analysis and various smoothing techniques, simple correlation analysis, time-series
analysis, principal component and canonical correlation analyses, etc. (Larsen & Marx,
1986, Box & Jenkins,1976, Diggle, 1990 & Rice,1992, Srivastava & Carter, 1983).
Statistical methods are expected to provide a significant aid in the analysis of multiple
data with substantial dispersion, determination of important relationships and tendencies
for different types of air pollutants, their mutual relationships and interactions under
different and ever changing atmospheric and environmental conditions.
For example, simple statistical methods of analysis could also be useful for the
determination of correlations between different components of aerosols, and between
these components and external physical and chemical factors, such as wind,
temperature, humidity, etc. (Meszaros, 1999, Morawska, et al, 1998, Campanelli, et al,
2003, Salvadora, et al, 2004, Paatero, et al, 2005).
Simple correlation analysis establishes correlations between just two variables
(Larsen & Marx, 1986). Though it can be used for the analysis of some relationships
between variables, e.g., particle concentration and solar radiation, etc., it may be
inaccurate and unreliable when it comes to the analysis of, for example, particle
concentration that depends not only on solar radiation, but also on temperature,
humidity, particle concentration in a neighbouring mode, etc. We simply do not know if
the dependence on solar radiation is due to solar radiation and not, for example, due to
humidity, etc. This may result in unreliable conclusions. Therefore, multi-variate
analysis should be used instead, where simultaneous correlations between a number of
variables can be determined.
38
Using different types of multiple correlations or multiple regression techniques,
it is possible to represent variables or their combinations in terms of linear combinations
of other variables (Dillon & Goldstain, 1984, Srivastava & Carter, 1983). Regression
analysis and correlation analysis are closely related but are conceptually very different.
Regression analysis attempts to estimate the mean value of the dependant variable on
the basis of the known values of one or more predictor variables (Dillon & Goldstain,
1984). Correlation analysis attempts to measure the strength of linear relationship
between two variables (Dillon & Goldstain, 1984, Srivastava & Carter, 1983).
This analysis is important for the accurate predictions of concentration of
various types of particles in the atmosphere with subsequent evaluation of their effects
on human health. In addition, the establishment of statistical correlations between
various components of aerosols and other pollutants (e.g., gaseous components) may
allow the determination of main sources of such pollutants and particles with
subsequent targeting technical solutions in terms of major improvement of the quality of
the indoor and outdoor air in our cities.
Only very few papers reported analysis and forecast of aerosol and/or gaseous
pollutants on the basis of multiple and canonical correlations (Cogliani, 2000, Hien, et
al, 2002, Paatero et al, 2005). However, these papers present only a starting point for
the extensive use of these powerful techniques in the area of air pollution and air
quality. So far, no one has attempted canonical correlation analysis of fine and ultra-fine
particle aerosols, determination of their origins and mutual interactions/transformations,
their relationships with the environmental and atmospheric parameters, etc. in the
presence of strong turbulent mixing. At the same time, such an analysis may be
expected to provide further important information about the physical nature of the
diverse processes of aerosol evolution and particle transformations in the real-world
urban environment.
39
The methods of time series analysis are another powerful tool of statistical data
analysis (Diggle, 1990, Box & Jenkins, 1976). These methods can be subdivided into
two classes: those that use the time domain procedure, i.e., simply present the
dependencies of some quantity versus time, and those that use the frequency domain to
investigate the periodic properties of the obtained dependencies (series of data), using
the Fourier analysis (Diggle, 1990). Plots of autocovariance or autocorrelation versus
time (Box & Jenkins, 1976, Everitt, 1994, Jones & Rice, 1992, Paatero, et al, 2005) are
useful for the determination of possible mechanisms underlying the series (e.g.,
mechanisms for the existence of dominant frequencies in the Fourier integrals of the
obtained time dependencies).
A problem in analysing data is that the number of variables in the real situation
may be excessively large. For example, if we have 7 variables (n = 7), there are 21
correlations that must be considered. With n = 15 the number of correlation coefficients
is 105, and it keeps increasing proportionally to n2, where n is the number of variables.
This obviously may make impossible such an analysis, and data reduction techniques
may be highly useful.
One of such techniques is called principal component analysis (Dillon &
Goldstain, 1984, Srivastava & Carter, 1983). This analysis transforms the original set of
variables into a smaller set of uncorrelated linear combinations (principal components)
that is responsible for most of the variance of the original set (Dillon & Goldstain, 1984,
Srivastava & Carter, 1983). The purpose of the principal component analysis is to
determine a set of the minimum possible number of principal components that would
account for the majority of the variation of the original set of variables (Dillon &
Goldstain, 1984, Srivastava & Carter, 1983, Everitt, 1994, Campanelli, et al, 2003,
Salvadora et al, 2004).
40
The most comprehensive statistical approach for multiple data with a number of
different dependent and independent variables is the canonical correlation analysis
(Dillon & Goldstain, 1984, Srivastava & Carter, 1983, Johnson, R.A., Wichern, 2002).
This approach allows the determination of complex and intricate relationships between
a number of different variables that may vary in an unpredictable and complex way –
just as happens in the real-world environment.
Canonical correlation analysis determines correlations between two groups of
variables in the case when the variables in each of the groups depend on each other. The
canonical correlation technique was initially developed by H. Hotelling to identify and
quantify the associations between two sets of variables (Hotelling, 1935, Hotelling,
1936).
The main idea of the canonical correlation analysis is to determine two linear
combinations (one for predictor set and one for criterion set of variables), such that the
correlation between them is maximal. These linear combinations (canonical variates)
are analogous to the factors in the principal component analysis (Dillon & Goldstain,
1984, Srivastava & Carter, 1983, Everitt, 1994). The difference is that in the canonical
correlation analysis, we have two groups of variables that we try to compare and
correlate with each other. A significant advantage of this method is that it can determine
the effect of each of the mutually dependent parameters (e.g., temperature, humidity,
etc.) on some variable or a group of variables (e.g., particle concentration in different
size intervals) (Dillon & Goldstain, 1984, Johnson & Wichern, 2002). Thus it can
uncover complex relationships that reflect the structure between the predictor and the
criterion variables.
When only qualitative information about main tendencies and relationships
(without determining levels of confidence of the obtained correlations) is required, there
are no restrictions on types of data distribution (e.g., normal, uniform, log-normal, etc.).
41
However, if we need to determine levels of confidence of the obtained correlations, the
data should meet the requirements of multivariate normality and homogeneity of
variance (Dillon & Goldstain, 1984, Johnson & Wichern, 2002).
The squared canonical correlation coefficient R2 determines the contribution of
the considered group of meteorological parameters (predictor variables) to the variation
of the particle concentration (the larger the contribution of the considered parameters to
the variation of the concentration, the larger the coefficient R). Increasing number of
predictor variables that the concentration depends on, should increase R.
The canonical correlation technique can be described briefly as follows. Let m
be the number of predictors and p be the number of criterion variables, with m ≥ p. We
denote by
(X(1))′ = (X1, X2, …, Xm) - the vector of predictor variables,
(Y(1))′ = (Y1, Y2, …, Yp) - the vector of criterion variables,
μx(1) and μy
(1) – the respective mean vectors associated with the set X and
Y.
We define the following variance-covariance matrices (Dillon & Goldstain,
1984, Johnson & Wichern, 2002):
})')({( )1()1()1()1(xxxx E μ−μ−=Σ XX ,
})')({( )1()1()1()1(yyyy E μ−μ−=Σ YY ,
})')({( )1()1()1()1(yxxy E μ−μ−=Σ YX . (2.22)
The objective of the canonical correlation analysis is to find a linear combination of the
m predictor variables (of X) that is maximally correlated with a linear combination of
the p criterion variables (of Y). Let
X(1) = (a(1))′x = a11x1 + a12x2 + … + a1m xm,
Y(2) = (b(1))′y = b21y1 + b22y2 + … +b1m ym (2.23)
42
be the respective linear combination. Then the correlation between X(1) and Y(2) is given
by (Dillon & Goldstain, 1984, Johnson & Wichern, 2002):
2/1)1()1()1()1(
)1()1()2()1(
)}))((){((
)(),(
∑′∑′
∑′=ρ
yyxx
xy
bbaa
baba (2.24)
Out of the infinite number of such linear combinations we find two linear combinations
which maximize the correlation ρ(a(1), b(1)). Since ρ(a(1), b(1)) is invariant under the
operation of multiplication of a(1) and b(1) by arbitrary constants (Larsen & Marx, 1986),
we can arbitrary normalize a(1) and b(1). We would like to require that
∑′ xx)1()1( )( aa = ∑′ yy
)1()1( )( bb = 1, (2.25a)
i.e., X(1) and Y(1) have unit variances, and
E(X(1)) = E(Y(1)) = 0. (2.25b)
This problem is equivalent to solving the following canonical equations:
(∑xx-1 ∑xy ∑yy
-1 ∑yx – λI)a = 0,
(∑yy-1 ∑yx ∑xx
-1 ∑xy – λI)b = 0. (2.26)
Here, I is the identity matrix and λ is the largest eigenvalue for the characteristic
equations
|∑xx-1 ∑xy ∑yy
-1 ∑yx – λI| = 0,
|∑yy-1 ∑yx ∑xx
-1 ∑xy – λI| = 0. (2.27)
In this case, this largest λmax = λ(1) will be equal to the squared canonical correlation
coefficient ρ(1) (Dillon & Goldstain, 1984, Johnson & Wichern, 2002). The eigenvectors
for ∑xx-1 ∑xy ∑yy
-1 ∑yx and ∑yy-1 ∑yx ∑xx
-1 ∑xy, associated with the eigenvalue λ(1), are the
vectors of coefficients a(1) and b(1). The elements of the vectors a(1) and b(1) are called
the canonical weights. The magnitudes of these weights indicate the importance of a
variable from one set with respect to the other set in obtaining a maximum correlation
between the sets. The canonical weights do not depend on original scale of
43
measurement and are expressed in standardized form (Dillon & Goldstain, 1984,
Johnson & Wichern, 2002). Under the normalization restrictions (2.25a,b), calculation
of the canonical variate is conducted using the obtained standardized weights together
with the standardized x and y variables.
Therefore, the first pair of canonical variables (first canonical variate pair) is the
pair of linear combinations X(1), Y(1) having unit variances, which maximize the
correlation ρ(a(1),b(1)).
In the same fashion, the second pair of canonical variables (second canonical
variate pair) is the pair of linear combinations X(2), Y(2) with unit variances, which
maximize the correlation ρ(2) = ρ(a(2),b(2)), and are uncorrelated with the first pair of
canonical variables X(1), Y(1). In this case, (ρ(2))2 is the second largest eigenvalue for the
characteristic equations (2.27), etc. Altogether, p canonical variate pairs could be
extracted with ρ(1) ≥ ρ(2) ≥ …≥ ρ(p) (Dillon & Goldstain, 1984, Srivastava & Carter,
1983, Johnson & Wichern, 2002, Hotelling, 1935, Hotelling, 1936).
The canonical weights (the coefficients in linear combinations (2.23)) indicate
the contribution of each original variable to the variance of the respective canonical
variate. In other words, the larger the value of the canonical weight for a particular
variable, the larger the contribution of this variable to the corresponding correlation
(dependence). However, it is important to understand that some variables may obtain
small weights, because the variance in a variable has already been accounted for by
some other variable(s). This does not necessarily mean that the considered variable is
not important. It is possible that this variable’s contribution has been taken into account
through other variables. Therefore, if a variable has a small weight, then it is possible to
eliminate it from the consideration only if this does not change significantly the
canonical correlation, otherwise, the variable cannot be neglected.
44
Another important aspect is that the canonical variates are not directly
observable. Therefore, many researchers recommend the use of canonical loading for
identifying the structure of the canonical relationships (Dillon & Goldstain, 1984,
Johnson & Wichern, 2002). A canonical loading gives the simple correlation of the
original variable and its respective canonical variate (linear combination). It reflects the
degree to which a variable is represented by a canonical variate. The canonical loadings
can be computed by correlating the original variables with the canonical variate scores
(Dillon & Goldstain, 1984, Johnson & Wichern, 2002). For a proper interpretation of
the canonical solution, the inspection of both the canonical weights and the
corresponding canonical loadings is recommended (Dillon & Goldstain, 1984, Johnson
& Wichern, 2002). Large differences in magnitude and different sign between weights
and loadings can be an indication of nonlinear relationships between the variables, or
the fact that some additional variables/processes, that have not been taken into account,
may be as important for the analysis.
This complex but comprehensive and highly informative statistical approach is
expected to provide valuable information about possible mutual relationships between
particle modes, meteorological, environmental and traffic conditions. Therefore, it is
expected to also give an excellent insight into major physical and chemical processes of
aerosol transformation/evolution in the real-world environment.
45
CHAPTER 3
AVERAGE EMISSION FACTORS FOR VEHICLES ON A BUSY ROAD ([A1, A11, A12, A20])
3.1. Introduction
It is known that busy roads are one of the main contributing sources to the
overall air pollution in the urban environment (Zhiqiang, et al, 2000). As mentioned in
Chapter 2, one of the most well developed models for the analysis of turbulent diffusion
of busy road pollution is CALINE4 (Benson, 1992) that was designed by the California
Transport for the analysis of carbon monoxide pollution on the basis of knowledge of
gaseous emission factors from stationary and moving vehicles.
However, there are two main difficulties with the application of the CALINE4
model for the simulation of dispersion of fine particle aerosols from a busy road. Firstly,
gas parameters in the CALINE4 model (such as the emission factor and concentrations)
cannot be directly replaced by the corresponding parameters for particles, due to
different units for these parameters for gasses and aerosols. The second (more general
and significant) problem arises from the absence of consistent experimental data on
particulate emission factors from motor vehicles, that would be required as an input for
the CALINE4 package. Indeed, experimental values of these emission factors presented
by different researchers vary by one or even two orders of magnitude depending on type
of vehicles and conditions of measurements (Jamriska & Morawska, 2001, Cadle et al,
2001, Graskow et al, 1998, Gertler et al, 2000, Gross et al, 2000). From the same
references, typically, the emission factors lie within the interval between ~ 1012 to ~
1014 particles per vehicle per kilometre for gasoline (light-duty) vehicles, and between ~
1014 to ~ 1015 for diesel (heavy-duty) vehicles. Note also that very few attempts have
46
been made to determine average emission factors for vehicles on a real road with rather
inconclusive results: large spread of the obtained data (Gertler et al, 2000, Gross et al,
2000) and significant experimental errors ~ 70% (Jamriska & Morawska, 2001).
Therefore, the aim of this Chapter is to develop a new effective and accurate
method for the determination of the average emission factors for vehicles (average fleet)
on a road (based on the knowledge of experimental total number concentration at one
point near the road), and simultaneously re-scale the CALINE4 package to make it
suitable for the analysis of propagation of fine particle aerosols from a busy road. The
predictions obtained by means of the re-scaled CALINE4 package, and the determined
emission factors will be compared with two sets of experimental measurements.
3.2. CALINE4 model
Though the CALINE4 package based on the Gaussian plume model (Csanady,
1980) has been specifically developed for the analysis of carbon monoxide pollution
(Benson, 1992), it can be adapted for the approximate simulation of aerosol dispersion
and predictions of the total number concentrations near busy roads. This statement is
based on the following aspects.
First, unlike particles of larger size (> 1 μm), that have sedimentation velocities
between ~ 0.01 m/min and ~ 10 m/min, sedimentation velocities for fine particles are ~
1 mm per hour or less (Jacobson, 1999). Velocities of dry deposition due to turbulent
diffusion increase with decreasing size of particles. However, these velocities are only:
~ 0.3 cm/s for particles with the diameter 30 nm, and ~ 0.03 cm/s for 120 nm particles
(Jacobson, 1999). As a result, fine particle aerosols can propagate large distances from a
source without noticeable sedimentation and/or deposition.
Second, though as demonstrated in Chapters 5 – 10 below, combustion nano-
particle aerosols experience strong evolutionary processes resulting in rapid
47
transformation of particle modes, it may be expected that the total number concentration
near a busy road can be approximately described and predicted by means of the
Gaussian plume model (Csanady, 1980). This is the basis for the use of CALINE4 for
such a modelling. The accuracy of this model, its validity and usefulness should be
confirmed by the comparison of the predicted results with the observation data (see
below). Nevertheless, one should always keep in mind significant limitations of
Gaussian plume model, including its failure to predict strong mode evolution [A3,A4]
and experimentally observed maximum of the total number concentration at an optimal
distance from the road [A3-A5].
Therefore, in this Chapter, we adapt the CALINE4 model for the approximate
prediction of the total number concentration near a busy road in the Gaussian plume
approximation, while its extension to non-Gaussian dispersion will be considered below
in Chapter 7.
As mentioned above, the CALINA4 software package (Benson, 1992) enables
calculation of concentrations of carbon monoxide at different distances from a road and
presents these concentrations in parts per million (the number of CO-molecules per
million molecules of the air).
The inputs for the package are: roadway geometry, meteorological parameters
(wind speed, wind direction and its standard deviation, temperature and humidity),
background concentration (concentration of the pollutant in the absence of the traffic on
the considered road) in parts per million (ppm), traffic volume (in vehicles per hour),
and receptor positions. The program also requires CO-emission factors for vehicles on
the road in mg per vehicle per mile.
To adapt the software CALINE4 for fine particle aerosols, we need to find
scaling coefficients for emission factors and concentrations, since particle
concentrations are measured in particles per cubic centimeter, and emission factors in
48
particles per vehicle per mile. This adaptation will be done simultaneously with the
determination of the average emission factor for vehicles on the road, using the
experimental measurements of total number concentration at some distance from the
road. After this, the scaled package will be tested by comparing its predictions with the
experimental results of concentration as a function of distance from the road. Therefore,
in the next section we will discuss the experimental measurements of particle
concentrations, that will be needed for the developed methods and their verification, and
then proceed to the adaptation of CALINE4 in section 4.
3.3. Experimental measurements
The experimental measurements were taken near Gateway Motorway in the
Brisbane area, Australia. The total width of the Motorway was 27 m (with four traffic
lanes: two in each direction, separated by grass area with width ~ 7 m in the middle of
the road). The analysed road, its geometry (that should be used as an input in
CALINE4), and the surrounding area are presented in Fig. 3.1. The total number
concentration of fine and ultra-fine particles in the range from 0.015 μm to 0.7 μm was
measured at the height h = 2 m above the ground by means of a Scanning Mobility
Particle Sizer (SMPS-3071) and a condensation particle counter (CPC-3010). 11 sets of
measurements of the total number concentration (five independent measurements in
each set) were conducted at the distances of 15 m, 55 m, 15 m, 135 m, 15 m, 215 m,
15m, 295 m, 15 m, 375 m, and 15 m from the curb of Gateway Motorway during four
hours of monitoring [Hitchins et al, 2000]. All the concentration measurements were
conducted simultaneously with the measurements of the traffic flow on the road
(recorded and counted from the video tape with 75% of cars, 11% light trucks and 14%
heavy-duty vehicles on average). A weather station was used to measure temperature,
49
wind speed and wind direction at the time of concentration measurements and at the
same height above the ground, i.e. at h = 2 m.
Fig. 3.1. Gateway Motorway and the sample road. Dashed arrow indicates the direction of the
wind for calculations of emission factor. Dashed lines represent the imaginary vertical planes
parallel to the road (used in the calculations of the particle flux). The scale of the map and the
direction to the North are as indicated. The points on the road indicate the straight sections of
the road used in the CALINE4 for calculating dispersion.
The results of the measurements of wind speed and wind direction are presented
by points in Fig. 3.2a,b. The solid curves in Fig. 3.2a,b were obtained by means of the
“super smoother” method available in the S-Plus statistical package. These
dependencies (solid curves in Fig. 3.2a,b) approximately correspond to one hour
average values of wind speed and wind direction, which are used as the input
parameters in the CALINE4 model.
50
Each set of the concentration measurements took 12 minutes. The average of the
measured concentrations in each set was assumed to be the one hour average that is
substituted into CALINE4. The corresponding one hour average values for the wind
direction and speed were taken from the curves in Figs. 3.2a,b at the moments of time
corresponding to the middle of each of the 12 minute intervals.
Fig. 3.2. The dependencies of the one hour average wind speed (a) and wind direction (b) on
time during the whole period of measurements (four hours).
3.4. Model adaptation
The adaptation of the CALINE4 model to combustion aerosols can briefly be
outlined as follows. Firstly, we take experimentally measured concentrations of
particles at some distance (e.g., 15 m) from the curb of the road. Secondly, we substitute
all the known meteorological and environmental parameters, and some arbitrary
51
numbers for the emission factors into the model. Formally changing these emission
factors, we adjust them so that the output concentration at the considered distance (15
m) from the curb of the road is equal to the experimental value of concentration divided
by 1000. By doing this, we actually assume that the model gives concentration in ×103
particles/cm3. Note however that the adjusted emission factors are not measured in
particles per vehicle per mile. These are not real emission factors, but rather some input
numbers for the CALINE4 model. Therefore, the adjusted emission factors will be
called model emission factors and denoted as Em.
Thirdly, substituting the determined values of Em and the known meteorological
and environmental parameters back into the CALINE4 model, we calculate the
concentration of particles (in ×103 particles per cm3) at the considered distance from the
road (15 m) as a function of height above the ground (vertical concentration profile).
Fourthly, taking into account the vertical concentration profile, we determine the total
(integral) flux of particles through a plane that is normal to the ground, parallel to the
road, and located on the downwind side of the road (plane 1 denoted by the dashed line
in Fig. 3.1). The same flux can easily be determined from the average (real and
unknown) emission factor from a vehicle on the road, and the number of vehicles. Thus,
equating these two fluxes, we determine the unknown average emission factor, E (in
particles per vehicle per mile). Comparing this average emission factor, E, with the
previously obtained model emission factor, Em, we determine the scaling coefficient η =
Em/E. Thus, the average emission factor E (in particles per vehicle per mile) should be
multiplied by η, before substituting it into the CALINE4 model, in order to obtain
concentration in ×103 particles per cm3.
The following two subsections will present more detailed analysis of the
outlined procedure, including the calculations for the Gateway Motorway (Fig. 3.1).
52
3.4.1. Model emission factors
As mentioned above, we calculate model emission factors for vehicles on the
road using the concentration measurements at 15 m distance from the curb of the road.
The average values of the measured parameters that were used as the inputs of the
CALINE4 model in this calculation are presented in Table 3.1 for each of the six sets of
measurements at the distance 15 m from the road.
Set number 1 2 3 4 5 6
Wind speed, (m/s) 0.7 1.0 1.5 1.6 1.5 1.6
Wind direction, (degrees from the North)
17.5 358.2 351.2 1.7 21.9 35
Standard deviation of wind direction, (degrees)
55.5 53.7 48.7 37.5 41.0 38.0
Temperature, (°C) 31.0 34.8 33.4 35.5 36.8 36.4
Traffic flow, (vehicles/hour) 2928 3096 3108 3456 3216 3504
Cars, (vehicles/hour) 2064 2400 2520 2688 2256 2820
Trucks, (vehicles/hour) 444 456 312 480 624 468
Light trucks, (vehicles/hour) 420 240 276 288 336 216
Average experimental concentration, (×103 particle/cm3)
49.9 26.5 19.7 20.8 23.5 22.0
Model emission factor, Em 774 490 534 409 443 408
Table 3.1.
Wind speed and direction (one hour averages) were determined from the graphs
in Figs.2a,b, as described in Section 3. Standard deviations of the wind speed,
temperature, and experimental concentrations are not needed for calculations using the
CALINE4 model (Benson & Pinkerman, 1989). Therefore, they are not presented in
Table 3.1. Atmospheric stability was of class 1 (Benson & Pinkerman, 1989).
The background concentration was estimated from a different set of
concentration measurements on the same (left in Fig. 3.1) side of the road, but with the
53
opposite wind direction. These measurements gave the concentrations ~ 3200
particles/cm3. The reason for using this estimate is related to similar densities of the
residential and road areas on both sides of the road (see also the error analysis in end of
this section).
Standard deviation of the wind direction is an input parameter for the model.
However, the weather station gives only average values for the wind direction during a
6 minute interval (a continuous measurement) with a standard deviation sj for the same
interval. The standard deviation, s, of one hour average wind direction (used as an input
for the model) is related to sj as (Larsen & Marx, 1986):
s2 = k s jj
k−
=∑1 2
1. (3.1)
Here, k is the number of continuous measurements undertaken within one hour period,
and sj is the standard deviation for each of these measurements. The standard deviations
for one hour average wind directions, calculated by means of Eq.(3.1), are presented in
Table 3.1.
Using the CALINE4 model and the data in Table 3.1, we obtain the
corresponding model emission factors Em for each set of measurements at 15 m distance
from the curb of the road. The results of these calculations are presented in the last row
of Table 3.1. The units for these emission factors do not matter, because these factors do
not have any physical meaning, but are simply some input numbers for the CALINE4
model.
From Table 3.1, we can also see that the value of Em for the first set of data is ≈
1.5 times bigger than for the other sets. This may be interpreted by the fact that the first
set of measurements was taken during a very busy traffic hour – between 8 and 9 am,
when vehicle speed was relatively small and changed frequently. This can increase
values of Em, because it takes more time for a vehicle to travel the distance of one mile.
In addition, changing speed, and frequent acceleration of vehicles obviously results in
54
enhanced emission of particles. The lowest traffic flow for this set of measurements
(Table 3.1) is related to smaller speed of the vehicles in the heavy traffic, indicating
smaller number of vehicle passing the point of monitoring per hour.
The analysis of propagation of errors demonstrates that the values of Em are
relatively stable with respect to variations of the background concentration. Indeed,
20% variation of the average background concentration results in only ~ 4% variation of
the corresponding Em, and 50% variation of the average background results in only ~
10% variation of Em.
3.4.2. Determination of the emission factor
According to the general outline of the method, described in the beginning of
section 4, each of the six values of model emission factors Em are substituted back into
the CALINE4 model, together with the corresponding values of wind speed, wind
direction standard deviation, temperature and traffic flow (see Table 3.1). To simplify
further calculations of particle fluxes, we assume that the background concentration is
equal to zero, and the wind direction is normal to the road (i.e. 72o to the North – see the
dashed arrow in Fig. 3.1) for all six sets of parameters from Table 3.1. That is, instead
of all the values in the boxes of the third row in Table 3.1 we use 72o. This can be done
since in the calculations of the flux we do not use the experimental values of
concentrations (the seventh row in Table 3.1), but rather the determined values of Em
(the last row in Table 3.1), which are independent of the background concentration and
wind direction.
As a result, concentrations of particles (in ×103 particles/cm3) were calculated at
the distance 15 m from the curb of the road as a function of height above the ground
(vertical concentration profile) on the west of the road – Fig. 3.1. It can be seen that at
the same distance (15 m), but on the upwind side of the road (plane 2 in Fig. 3.1), the
concentrations are zero within an accuracy of the program (this accuracy is ~ 100
55
particles/cm3). This is the case for all wind speeds presented in Table 3.1 (if the wind
direction is normal to the road). This means that the flux of particles, caused by
turbulent diffusion, into the direction opposite to the wind direction is negligible.
Therefore, the overwhelming contribution to particle fluxes is due to transport by wind,
and practically all particles that are emitted by vehicles on the road are carried by wind
through the vertical plane on the downwind side of the road.
In this case, the flux F through the plane on the downwind side of the road
(plane 1 in Fig. 3.1) per segment of the road of length l is given by the equation (the
contribution of the turbulent diffusion to this flux is neglected):
F l U h c h dh≈ ∫+∞
( ) ( )0
, (3.2)
where c(h) is the concentration of particles as function of height h, calculated by means
of CALINE4 and the procedure discussed above, and U(h) is the average wind speed at
the height h.
For all six sets of parameters from Table 3.1, the vertical concentration
decreases to the background level at the height of ~ 15 m. The average wind speed can
be assumed to be constant within this height (the same is assumed in CALINE4
(Benson, 1992)). Indeed, wind starts changing with height if h >> 100h0, where h0 is
the dynamic roughness coefficient that is approximately equal to 1/30 of the average
height of obstacles on the considered surface (Csanady, 1980). In our case, the road is
located in a more or less flat region with isolated bushes and scattered buildings, which
corresponds to h0 ≈ 0.5 m (Stull,1989). Therefore, at the considered heights (up to 15 m
above the ground) variation of the wind speed with height was neglected: U(h) ≈ U0,
where U0 is the experimentally measured wind speed at the height of 2 m above the
ground level.
56
In the same approximation, the flux F was simply given by the strength of the
line source q (in particles per metre per second) multiplied by the length of the segment
of the road:
F ≈ ql = afl/v, (3.4)
where f is the number of particles emitted by one vehicle per second, a is the traffic flow
in vehicles per second, v is the average speed of vehicles on the road.
Comparing Eqs. (3.2) and (3.4), gives
af/v = U0 c h dh( ) .0
+∞∫ (3.5)
This equation determines f – the average number of particles emitted by one
vehicle per second. If we multiply f by the average time that it takes for a vehicle to
travel the distance of one kilometre (or one mile), we obtain the average (real) emission
factor, E, in particles per vehicle per kilometre (or particles per vehicle per mile).
The values of E calculated by means of the described procedure are presented in
Table 3.2 for the six sets of measurements from Table 3.1.
Set of measurements 1 2 3 4 5 6
Real emission factors, E, [1014particles/vehicle/mile]
6.51 4.64 4.84 3.78 4.07 3.41
Scaling coefficient, η = Em/E (×10-12 g/cm3)
1.19 1.06 1.10 1.08 1.09 1.20
Table 3.2. Emission factors and scaling coefficients.
From this table, the mean value of the six presented emission factors <E> = (4.5
± 0.4)×1014 particle/vehicle/mile, and the average scaling coefficient <η> = (1.12×10-12
± 0.02) g/cm3.
An additional error of <E> is associated with uncertainty of the background
concentration. However, this additional error cannot be large due to only weak
57
sensitivity of the resultant emission factor E to the uncertainty of the background
concentration (see the end of Section 3.4.1). Therefore, accurate knowledge of
background concentration is not essential for the developed method of determination of
average emission factors from busy roads. It is usually sufficient to have just a
reasonable estimate of background concentration (with an acceptable error of up to ~
100%).
Note also that the determination of the scaling coefficient η can be carried out
without using any experimental measurements. That is, the described procedure of
determining η can be used with arbitrary (hypothetical) emission factors and
meteorological parameters. Indeed, when calculating the scaling coefficient in the
beginning of this section, we assumed that the wind is normal to the road, and the
background concentration is zero. In the same way, all other parameters in Table 3.1
can be chosen arbitrarily, including the model emission factors. It can be seen that the
subsequent calculation and comparison of the corresponding particle fluxes at some
distance from the road (for different input parameters) give similar scaling coefficients
as those presented in Table 3.2, with the same mean value <η> ≈ 1.12×10-12. The only
error of this result is related to the uncertainty of calculations by means of the
CALINE4 model (i.e., by the sensitivity of the model).
Input parameters
Smallest increment
Variation of concentration,
×103 particles/cm3 wind speed, [m/s] 0.1 1.6
wind direction, [deg] 1 0.2
Std. Dev. of wind direction, [deg] 1 0.2 E [particle/mile/vehicle] 2.7×1012 0.1
traffic flow, [vehicle/hour] 5 0.1 receptor position, [m] 1 1
temperature, [°C] 1 0.1 stability class 1 1.1
Table 3.3. Sensitivity of the model.
58
Table 3.3 shows the variations of concentration resulting from the smallest
(allowed by the model) increments of each of the input parameters separately. As can be
seen from the Table, normally, CALINE4 responds with the accuracy of ~ 100 – 200
particles/cm3 with respect to the smallest increments of the input parameters. The
largest sensitivity occurs for wind speed variations and stability class – see Table 3.3.
As soon as the scaling coefficient is known, it is easy to use the CALINE4
model for the determination of average emission factors from vehicles on different
roads. To do this, we only need to measure the average total number concentration at
just one point near the road, find the model emission factor (using the CALINE4 model)
that produces this concentration for given meteorological conditions, and divide the
model emission factor by the scaling coefficient.
3.5. Comparison of numerical and experimental results
Note again that the procedure for the determination of the emission factors and
scaling of the CALINE4 model for the analysis of fine particle aerosols from a busy
road has been developed on the basis of the Gaussian plume model and experimental
measurements of particle concentrations at a particular distance from the road (in our
case it was 15 m from the curb). The obtained emission factors are valid only for the
particular road under consideration (since they depend on the average speed and type of
the vehicles). At the same time, the scaling coefficient is correct for any road and is
characteristic for the software package (CALINE4).
The described procedures were based on the assumption that the CALINE4
model can be used for the analysis of dispersion of fine particle aerosols from a busy
road. Therefore, in this section, we verify this original assumption by means of
calculating a theoretical dependence of total number concentration on distance from the
road, and comparing it with the experimental measurements. The theoretical
dependence is obtained by substituting the average model emission factor and the
59
corresponding meteorological parameters, averaged over the four hour interval of
measurements, into the CALINE4 model and calculating average particle concentration
as a function of distance from the road. The average value of Em = 510 ± 60 is obtained
by averaging the six values in Table 3.1. The average wind speed for the period of four
hours is equal to 1.3 m/sec, with wind speed standard deviation being 0.5 m/sec. The
average wind direction for the same period is 17o to the North, and the standard
deviation ≈ 46o. The average traffic flow for the period of measurement is 3218 vehicles
per hour (with the error of the mean ≈ 90 vehicles per hour), and the average
temperature is 34.9°C (summer period). The background concentration was estimated to
be ~ 3200 particles/cm3 – see above.
In the input of CALINE4, coordinates of receptor points can only be integer
numbers (in metres) that have to be entered separately for each of the points. Therefore,
it is inconvenient to use this model for plotting an actual theoretical dependence of
concentration on distance from the road. Instead, we calculate concentrations only at the
distances, corresponding to the experimental measurements (see Section 3.3). After this,
we perform similar curve fitting procedure (see below) for both experimental and
theoretical points, and compare the resultant curves.
The experimental values of concentration at the different distances from the road
are presented in Fig. 3.3. The theoretical points are not presented in the figure, as they
all lie almost exactly on the corresponding theoretical curve (curve 2 in Fig. 3.3a).
The curve-fitting procedure for the experimental and theoretical points was
based on the self-similarity theory of concentration distribution (Csanady, 1980). This
theory approximates the concentration c (in ×103 particles/cm3) as a power law in
distance from the road:
c = Kd-μ + c0 , (3.6)
60
where d is the distance from the road in metres, K and μ are constants to be determined,
and c0 is the background concentration. The constants K = 289 and μ = 0.73 (for the
experimental curve), and K = 500 and μ = 0.88 (for the theoretical curve) were
calculated by means of the non-linear regression model in the S-Plus statistical package
(Venables & Ripley, 2000).
Fig. 3.3. The experimental (solid curves 1 and 4) and theoretical (dashed curves 2 and 3)
dependencies of the total number concentration c on distance from the centre of the road. The
two sets of the experimentally measured average total number concentrations are represented by
the small dots (for the summer measurements in 1999 – experimental curve 1) and big dots (for
the winter measurements in 2002 – experimental curve 4).
61
Fig. 3.4. The experimental (solid curves) and theoretical (dashed curves) dependencies of the
average total number concentrations without the background, c – c0, on distance from the
middle of the road in the logarithmic scale. The dotted curves give the standard errors for the
experimental (solid) lines. (a) The summer set of measurements in 1999; (b) the winter set of
measurements in 2002 at the same place near Gateway Motorway, Brisbane area, Australia.
The resultant experimental and theoretical dependencies of concentration on
distance from the middle of the road are presented in Fig. 3.3 by curves 1 and 2,
respectively. The significant scatter of experimental points around curve 1 can be
explained by changing average speed and direction of the wind during the period of
measurements. Indeed, as can be seen from Figs.2a,b, variations of wind direction
62
within the four hour interval are ~ 60o, and variations of wind speed are ~ 1 m/s for the
same period.
Curves 1 and 2 in Fig. 3.3 demonstrate a good agreement between the theory,
based on the approaches developed in this Chapter, and the experimental results for
dispersion of fine and ultra-fine particle aerosols from a busy road. However, using
curves 1 and 2 in Fig. 3.3, it is difficult to judge if the theoretical curve lies within the
error range for the experimental curve or not. To answer this question, we subtract the
background concentration c0 from all the values of the number concentration and draw
the resultant dependencies c – c0 on distance from the road in the logarithmic scale –
Fig. 3.4a. The dotted curves in Fig. 3.4a give the standard errors (Hamilton, 1991) for
the experimental (solid) line. It is clear that the theoretical (dashed) line indeed lies well
within the standard error of the experimental curve, and CALINE4 with the calculated
scaling coefficient can be used for the analysis of aerosol propagation from a busy road.
At the same time, this agreement has been demonstrated so far only for one set
of measurements. Therefore, in order to have a better confirmation for the theoretical
model, another set of measurements (taken on 30 July 2002) in the same size range (for
particles from 0.015 μm to 0.7 μm) at the same place near the road was considered. The
average wind speed for the period of four hours was 1.8 m/sec, with wind speed
standard deviation of 0.8 m/sec. The average wind direction for the same period was
131o to the North with the standard deviation ≈ 50o. The average temperature was
20.5°C (winter period). The atmospheric stability was of class one. This time, the
background was simultaneously measured to be ≈ 2400 particles/cm3 with the standard
deviation ≈ 240 particles/cm3 (note that this measured value also confirms the estimate
of the background for the previous set of measurements). The measurements of the total
number concentration were taken at the following distances from the curb of the road:
15 m, 40 m, 65 m, 90 m, 115 m, 190 m, and 265 m; (the width of the road is 27 m). The
63
corresponding experimental points are presented in Fig. 3.3 by small dots (the distance
on the horizontal axis is taken from the middle of the road).
Again, using the non-linear regression model in the S-Plus statistical package
(Venables, 2000), the corresponding theoretical and experimental curves were plotted
(curves 3 and 4 in Fig. 3.3). Both these curves are noticeably lower than curves 1 and 2
(for the previous set of measurements). This is expected, since the wind speed for the
second set of measurements was noticeably larger (1.8 m/s compared to 1.3 m/s). If the
wind is taken the same for both sets of measurements, all the curves appear to be very
close to each other.
Similarly to curves 1 and 2 (for the first set of measurements), curves 3 and 4 in
Fig. 3.3 clearly demonstrate good agreement between the theoretical model and
experimental results. The dependencies of (c – c0) on distance from the centre of the
road in logarithmic scale (Fig. 3.4b) again demonstrate that the theoretical line lies
within the error for the experimental dependence. The typical difference between the
theoretical and average measured number concentrations is estimated to be ~ 10%.
The average emission factor for vehicles on the road for the second set of
measurements was again calculated from the measured average number concentration at
the distance 15 m from the curb of the road and the counted traffic flow 4212 vehicles
per hour. The calculated value of the emission factor in this case was E ≈ 4.6×1014
particle/vehicle/mile (with the uncertainty ~ 16% mainly due to uncertainty of
concentrations). This is very close to the value of the emission factor, calculated from
the summer experiment (E ≈ 4.5×1014 particle/vehicle/mile - see above).
Both the obtained values of the average emission factor are in reasonable
agreement with the previous results obtained by means of a box model (Jamriska &
Morawska, 2001), where the average emission factor was estimated as ≈ 2.8×1014
particles/vehicle/mile. The discrepancies could be explained by the larger number of
64
heavy duty vehicles in our experiments (14% compared to just 4% in (Jamriska &
Morawska, 2001), and by the significant uncertainty of the results from the box model
(~ 70% (Jamriska & Morawska, 2001)).
3.6. An example of application of the model for road design
Fig. 3.5 shows an example of the existing (solid line) and the proposed (dashed
line) road. The proposed road was designed to be parallel to the existing road in order to
take half of the traffic load. We assume that originally the traffic on the existing road is
2000 vehicles per hour and the average emission factor is 4.5×1014
particles/vehicle/mile.
-100 100 200 300 400 500
500
1000
N
E
S
W
A B C
distance, m
dist
ance
, m
exist
ing
road
prop
osed
road
Fig. 3.5. An example of existing (solid line) and proposed (dashed line) roads with the receptor
points A, B, C. Arrows (solid for the case of one road and dashed for the case of two roads)
indicate the direction of the wind that corresponds to maximum concentrations for a given point.
The length of the arrow is proportional to the concentration.
The analysis of the worst-case angle (the direction of the wind that corresponds
to maximum concentration of particles at a given point of observation) was conducted
65
by means of the CALINE4 program. As shown in the table below, the new
configuration (two parallel roads instead of one) decreased the worst-case particle
concentrations at points A and B by 34% and 50%, respectively. At the same time, the
worst case angle concentration at point C increased by 90%. This type of analysis will
aid in scientifically-based decision-making and achieving an optimal design of the
proposed roads and road networks in modern metropolitan areas.
A B C
Worst case angle, o to the North
49 229 225 One road
Concentration, 103particles/cm3
3.2 3.2 1.1
Worst case angle, o to the North
49 2 229 Two roads
instead of one Concentration, 103particles/cm3
2.1 1.6 2.1
Table 3.4. Comparison of concentration of particles at points A, B and C for two different
configurations of the roads.
3.7. Conclusions
In this Chapter, the CALINE4 software package, that was originally designed
for calculation of concentrations of carbon monoxide near a busy road, was adapted for
the analysis of aerosols of fine and ultra-fine particles, generated by vehicles on a busy
road. As a result, the scaling procedure for the available CALINE4 model was
developed and justified. A scaling coefficient relating the model and real emission
factors was determined: η ≈ 1.12×10-12 g/cm3. A new method of determination of the
average emission factor for fine particle emission from a vehicle on a road was also
developed. This method was based on experimental measurements of particle
concentration at just one point at some distance from the road. The average emission
factor of ~ 4.5×1014 particle/vehicle/mile (or 2.8×1014 particle/vehicle/km) (with the
standard deviation of ~ 10% – 15%, depending mainly on uncertainty of concentration
66
measurements) was determined for the road under consideration (Gateway Motorway,
Brisbane, Australia). This value must obviously be different for different roads, since it
depends on average speed and type of the vehicles on the road. At the same time, the
determined scaling coefficient and the whole procedure of the analysis are correct for an
arbitrary road, and arbitrary meteorological and environmental conditions, as long as we
use the CALINE4 software package.
It is important that the determined scaling coefficient gives us an easy way to
calculate an average emission factor for vehicles on a road, using only measurements of
average concentration at one point in the vicinity of the road. When applied to different
roads, this method may lead to an estimate of emission factors for different types of
vehicles.
Good agreement between the experimental results for the two sets of summer
and winter measurements and the predicted theoretical dependencies of concentration
on distance from the road has confirmed the applicability of the CALINE4 package for
the approximate prediction of total number concentrations in fine particle aerosols near
busy roads. In particular, statistical analysis of the experimental and theoretical results
has also demonstrated that the concentration of fine and ultra-fine particles reduces as a
power law in distance from the road.
The main applicability conditions for the developed model are the same as for
the CALINE4 software package (Benson & Pinkerman, 1989, Benson, 1992) using the
line source approximation. For example, it is not applicable for roads with traffic lights,
roads in canyons and tunnels, roads with very low traffic flow (e.g., one vehicle per
several minutes), etc. Another significant limitation for the applicability and reliability
for application of the Gaussian plume model, and thus CALINE4, is related to the
existence of fast processes of evolution of nano-particles, especially within the range <
100 nm – see Sections 5 – 9 and [A3-A5]. These include such processes as particle
67
condensation, evaporation, deposition, and thermal fragmentation (Sections 5 – 9).
Therefore, more accurate and reliable model of aerosol dispersion near a busy road will
be developed in Section 7 of this thesis. Nevertheless, as has been demonstrated, use of
CALINE4 and the Gaussian plume approximation gives reasonable results in terms of
prediction of the total number concentrations at different distances from the road.
Therefore, it can be used in environmental and urban planning practice for simple
evaluation of particle concentrations near major highways and suburban roads, and
determine expected impact of aerosol pollution on population exposure and health. In
addition, the considered approach and adapted CALINE4 model will be essential for the
development of the new much more accurate model based on the theory of particle
fragmentation (Section 7).
68
CHAPTER 4
NEW METHODS OF DETERMINATION OF AVERAGE PARTICLE EMISSION FACTORS FOR TWO GROUPS OF
VEHICLES ON A BUSY ROAD ([A2, A12, A13, A17])
4.1. Introduction
One of the significant problems with the determination of the impact of busy
roads and the resultant aerosol air pollution on human health and exposure in the urban
environment has been the lack of consistent knowledge of emission factors from
different types of vehicles in the real-world environment. The values of the emission
factors obtained under laboratory conditions for different types of vehicles differ by up
to ~ 3 orders of magnitude (Graskow et al, 1998, Watson et al, 1998, Ristovski et al,
1998, Cadle et al, 2001), and lie within the intervals between ~1012 to ~1014
particles/vehicle/kilometre for gasoline (light-duty) vehicles, and ~1014 to ~1015
particles/vehicle/kilometre for diesel trucks. Gross et al (2000) also estimated during
on-road measurements that the ratio of the average emission factors for trucks and cars
is ~ 48. However, the actual values for the emission factors have not been determined.
The CALINE4 model, designed for calculation of concentrations of carbon
monoxide near a busy road (Benson, 1992), has been adapted for the analysis of
aerosols of fine and ultra-fine particles (see Chapter 3 and [A1, A11, A12, A20]). The
scaling procedure for this model has been developed and justified (Chapter 3), together
with the new method for the determination of emission factor for the average fleet on
the road, based on the experimental values of the total number concentration at some
distance from the road. However, this approach is not applicable for the determination
the emission factors of different types (groups) of vehicles on the road, for example,
heavy truck and light cars. At the same time, as mentioned above, this information is
69
important for the effective forecast of aerosol pollution levels and human exposure from
busy roads.
Therefore, in this Chapter, two new methods are developed for the determination
of the average emission factors of fine and ultra-fine particles for different groups of
vehicles on a busy road. These methods are based on the experimental measurements of
the total number concentration near the road. The values of these emission factors for
heavy-duty trucks and light-duty cars are calculated, discussed, and compared with the
previous results obtained mainly in laboratory conditions. The method is also extended
to three different types of vehicles on the road (cars, light trucks and heavy-duty
diesels).
4.2. Emission factors for two different groups of vehicles
The measurements were taken near the Gateway Motorway (Brisbane, Australia)
at different traffic conditions: 18.1% of heavy-duty trucks on 30 July 2002 (weekday)
and 2.7% of heavy-duty trucks on 24 November 2002 (weekend). The total number
concentration of fine and ultra-fine particles in the range from 14 nm to 710 nm was
measured at the distance of 15 m from the kerb at 2 m height above the ground by a
scanning mobility particle sizer (SMPS-3071) and a condensation particle counter
(CPC-3010). The concentrations were measured in 110 equal intervals (channels) of
Δlog(Dp), where Dp is the particle diameter in nanometres. Five and ten scans were
taken on the weekday and weekend, respectively, and the average total number
concentration was determined. Even with such low number of scans, the developed
approaches give quite reasonable errors (see below), which is a demonstration of their
effectiveness. The time intervals within which SMPS took the concentration
measurements in one channel were τ1 ≈ 2.73 s for the weekday, and τ2 ≈ 1.36 s for the
weekend (this results in the same overall sampling time for weekday and weekend
70
measurements). The meteorological parameters (wind speed, wind direction,
temperature and humidity) were measured every 20 seconds by a weather station.
The average traffic parameters and measured concentrations with their relative
standard deviations of the mean are presented in Table 4.1. The one hour average wind
speed and direction and their standard deviations (required as inputs in CALINE 4) are
also shown in Table 4.1 (specifically indicated as “Std. Dev”). Traffic flow for each
type of vehicles (light duty vehicles, light trucks and heavy duty vehicles) has been
counted (from the video tape) within 5 min intervals eight times during the period of
measurements of 40 min. Then the average traffic flows were calculated – see Table
4.1. These small standard deviations clearly indicate the high stability of the traffic. It is
also worth mentioning that the average speed on the motorway was approximately the
same for the period of measurements on both the days (100 km/h, which is the speed
limit on the road with no traffic congestion).
30 July 24 November Concentration at 15 m, cm-3 20.3×103 (±16%) 2.2×103 (±13%) Background concentration, cm-3 2.3×103 (±4%) 0.74×103 (±9%) Traffic flow, vehicle/hr 4295 (±2%) 3694 (±2.2%) Heavy-duty trucks, vehicle/hr 776 (±2.3%) 100 (±15%) Cars, vehicle/hr 3097 (±2.3%) 3337 (±2.3%) Light trucks, vehicle/hr 422 (±9%) 257 (±9%) Wind direction, o to the North 142 (Std. Dev. = 48) 28.54
(Std. Dev. = 39.43) Wind speed, m/s 2.3 (Std. Dev. = 8) 2.2 (Std. Dev. = 0.7) Temperature, °C 22 27 Humidity, % 33 35 Emission factor, particle/vehicle/km
2.8×1014 (±23%) 0.23×1014 (±24%)
Table 4.1
Using the concentrations, meteorological and traffic parameters from Table 4.1
and the roadway geometry (Fig. 3.1), the values of the emission factors Ef for the
71
average fleet on the road were calculated by means of the CALINE4 model (Chapter 3)
– see the last row of Table 4.1.
Let us now assume that there are two groups of vehicles on the road – heavy-
duty trucks and cars. The light trucks (Table 4.1) are included in the car group. In this
case the emission factors for the average fleet (in particle/vehicle/km) for the weekday
(index 1) and weekend (index 2) can be written as
Ef1 = wt1nt1et + wc1(1 – nt1)ec, (4.1)
Ef2 = wt2nt2et + wc2(1 – nt2)ec. (4.2)
Here, nt1 and nt2 (no units) are the fractions of heavy-duty trucks in the traffic flow, ec
and et are the emission factors for cars and heavy-duty trucks (in particle/vehicle/km)
(to be determined), wt1,2 and wc1,2 (no units) are the correction factors that are
introduced to compensate for the discreteness of the traffic flow (breach of the line
source approximation). The reasons for using these factors can be understood from the
following.
The concentration measurements were taken in 110 size channels in sequence
within the time intervals τ1 and τ2 per one channel. Let Nt1,2 and Nc1,2 be the numbers of
trucks (index t) and cars (index c) passing by within the time interval for a measurement
within one channel on the weekday (index 1) and weekend (index 2). If, for example,
Nt2 < 1, then the particle concentration will be affected by the passing trucks only in a
fraction of channels that equals N. Table 4.1 gives Nt2 = 0.04. Therefore, only one of ~
25 channels in one scan “feels” the presence of a heavy truck. This effectively reduces
the contribution of et to Ef2 by a factor 0.04. Therefore the values of et,c in Eqs. (4.1) and
(4.2) are multiplied by the additional correction factors wt1,2 = min{1,Nt1,2,}, and wc1,2 =
min{1,Nc1,2}. It follows from the traffic data (Table 4.1) that in our experiments, wc1,2 =
1, wt1 ≈ 0.6, and wt2 ≈ 0.04. Note also that this determination of the correction factors
72
has been carried out in the slender plume approximation, i.e., when turbulent dispersion
of the plume is neglected (further corrections to these factors, associated with turbulent
diffusion are determined in Section 4.5).
Eqs. (4.1), (4.2) can be solved with respect to et and ec. The theory of variance
(Larsen and Marx, 1986) gives that if ΔEf1 and ΔEf2 are the standard deviations of Ef1
and Ef2, then the standard deviations of et and ec are
Δet = [wc22(1 – nt2)2ΔEf2
2 + wc12(1 – nt1)2ΔEf1
2]1/2/D0, (4.3)
Δec = [wt12nt1
2ΔEf22 + wt2nt2
2ΔEf12]1/2/D0, (4.4)
where
D0 = |nt1wt1wc2(1 – nt2) – nt2wt2wc1(1 – nt1)|.
For example, for the values of Ef1,2 presented in Table 4.1, we obtain:
et = (25 ± 6)×1014 particle/vehicle/km,
ec = (0.21 ± 0.06)×1014 particle/vehicle/km. (4.5)
Note that another possible source of errors is related to the possibility of
different contributions of the light trucks to the overall flow of cars on the weekday and
weekend. To evaluate the upper limit of this error, we take the difference between the
flow of the light trucks on weekday and weekend (i.e. 165 vehicles/hour) and include it
into the number of heavy trucks. Thus we assume that the average emission factor of
165 light trucks on the weekday is equal to that of heavy-duty trucks (which is an
obvious exaggeration). The resultant emission factor for heavy trucks appeared to be
different from that given by Eq. (4.4) by ≈ 30%, while for the cars, this difference was ≈
4%. This gives the upper (exaggerated) limit for the possible error due to differences in
the flow of the light trucks on weekday and weekend.
73
4.3. Constrained optimization
The same results can be obtained from Eqs. (4.1), (4.2) by means of the method
of constrained optimization (Kreyszig, 1999, Wolfram, 1999). In this method, et and ec,
are regarded as variables, and we wish to find intervals of their variations that are
determined by linear constraints. The four constraints are given by the two linear
equations (4.1), (4.2), and the two intervals for the emission factors for the average fleet
Ef1,2, determined by their standard deviations (Table 4.1). Numerical solution of this
problem (Wolfram, 1999) with the considered constrains gives the intervals (in
particle/vehicle/km):
18.3×1014 ≤ et ≤ 31.4×1014, 0.14×1014 ≤ ec ≤ 0.27×1014 (4.6)
that are hardly different from Eq.(4.5).
Note however, that though in the considered example the method with
constraints is equivalent to the direct solution of Eqs. (4.1) and (4.2), it may be very
important for the determination of the emission factors when the traffic conditions
during the two sets of measurements are similar: nt1 ≈ nt2. In this case, the slopes of the
two lines, given by Eqs. (4.1), (4.2) in the (et, ec) space, may be too close, and the point
of intersection of these lines (the solution to Eqs. (4.1), (4.2)) is highly sensitive to
experimental errors. As a result, the obtained values of et and ec suffer from substantial
errors (increasing when nt1 → nt2) – see Eqs. (4.3), (4.4).
In this case, using the method of constrained optimization (Kreyszig, 1999,
Wolfram, 1999), we can determine the average emission factors for two groups of
vehicles from only one set of measurements and the typical ratios for the emission
factors determined in the laboratory conditions: et/ec ≈ 36 (Watson et al, 1998), and et/ec
≈ 377 (Ristovski et al, 1998). The on-road value of et/ec ≈ 48 was estimated by Gross et
74
al (2000). The inconsistency of these results is obvious. However, they determine the
constraint 36 < et/ec < 377.
Suppose that we have only one set of measurements on the weekday (30 July
2002). Thus the second constraint is given by Eq. (4.1), while the third is again obtained
from the standard deviation of Ef1 (Table 4.1): 2.2×1014 ≤ Ef1 ≤ 3.4×1014. The solution
of this problem with constraints (Wolfram, 1999) gives in particle/vehicle/km: 17×1014
≤ et ≤ 32×1014, and 0.06×1014 ≤ ec ≤ 0.75×1014.
If we take the other set of measurements, then Eq. (4.2) is the second constraint,
whereas the third is 0.17×1014 ≤ Ef2 ≤ 0.29×1014 (Table 4.1). In this case, 6.2×1014 ≤ et ≤
78.9×1014, and 0.13×1014 ≤ ec ≤ 0.28×1014 (in particle/vehicle/km). Note however, that
in this case the accuracy of et and ec is lower than in Eq. (4.5). This is due to the fairly
loose constraint on the laboratory results for et/ec.
Note that the second method automatically gives the comparison of the
determined emission factors with the previously obtained results from the
measurements mainly in laboratory conditions (Watson et al, 1998, Ristovski et al,
1998, Gross et al, 2000).
4.4. Three types of vehicles on the road.
In this section, we extend the above approach to the case of three different types
of vehicles on the road: cars, light trucks and heavy-duty diesels. The average emission
factors for particles for these types of vehicles within the range from 14 nm to 710 nm
will be determined under field conditions. Associated errors of the results will be
evaluated.
The analysis is based on the same set of data shown in Table 4.1. On 30 July
2002 (weekday), the fractional composition of the traffic was 0.181 of heavy-duty
trucks, 0.098 of light trucks, and 0.721 of cars, while on 24 November 2002 (weekend)
it was 0.027, 0.069, and 0.903, respectively. As mentioned above, the average speed on
75
the motorway was approximately the same during the measurements on both the days
(100 km/h).
Using the average total number concentrations at the distance of 15 m from the
kerb, meteorological and traffic parameters, the average emission factors for the average
fleet were determined (see Chapter 3 and [A1]):
Ef1 = 2.8×1014 particle/vehicle/km (±23%) for weekday,
Ef2 = 0.23×1014 particle/vehicle/km (±23%) for weekend. (4.7)
If we assume that there are three types of vehicles on the road (heavy-duty
diesels, light trucks and cars), then the emission factors for the average fleet can be
written as:
Efi = wdindied + wtintiet + wcinciec, (4.8)
where ndi, nti, nci are the fractions of heavy-duty trucks, light trucks and cars in the
traffic flow, respectively, ed, et, ec are their emission factors, and wdi, wti, wci are the
correction factors that have been introduced to compensate for the breach of the line
source approximation if less than one vehicle of a particular type passes by within the
period of time corresponding to a measurement in one channel (see section 4.2), i = 1,2.
In principle, if we had experimental measurements on three different days with
different traffic conditions (i.e., i = 1,2,3), then Eqs. (4.8) could have been directly
solved for unknown emission factors ed, et, ec. Unfortunately, this often does not give
suitable results, since the traffic conditions are usually not that different on all three
days of measurements. It can be seen that in this case the resultant emission factors are
highly sensitive to experimental and statistical errors, and we may get zero or even
negative results, which is obviously a nonsense.
Therefore, the method of constrained optimization (Kreyszig, 1999) is used.
This method will allow determination of these three emission factors ed, et, ec only from
two sets of measurements, i.e. two Eqs. (4.8) (i = 1,2), and the constraints on the ratios
76
ed/ec and et/ec. The constraint for ed/ec is obtained from the literature data by taking the
maximum and minimum values of this ratio ed/ec ≈ 377 (Ristovski et al, 1998) and ed/ec
≈ 36 (Watson et al, 1998):
36 < ed/ec < 377 (4.9)
The constraint on the ratio et/ec has been determined from (Maricq, et al, 1999):
1.2 < et/ec < 3.7. (4.10)
Eqs. (4.7) determine the error intervals of the average emission factors, and they
are considered as two other constraints on Ef1 and Ef2. Eqs. (4.8) are also regarded as
two (functional) constraints on ed, et, ec. This gives us 6 constraints on three functions
ed, et, ec. In this case, the method of constrained optimization (Wolfram, 1999) gives
minimum and maximum values of the functions ed, et, ec, The resultant intervals for
possible values of ed, et, ec represent the error intervals for the emission factors.
Using the described procedure, the three average emission factors and their
uncertainties were determined as follows:
ed = (24.5 ± 6.5)×1014 particles/vehicle/km,
et = (0.6 ± 0.4)× 1014 particles/vehicle/km,
ec = (0.22 ± 0.07)× 1014 particles/vehicle/km. (4.11)
Note that the described method may even be useful if only one set of
measurements is available. In this case, we have only four constraints. As a result, the
determined emission factors for two of three types of vehicles usually have large errors.
However, the emission factor for the third type of vehicles usually has reasonable
uncertainty similar to those in Eqs. (4.11). Thus, in this case, only one of three emission
factors can be determined with reasonable accuracy.
If we assume that there are only two types of vehicles on the road (light trucks
and cars are put in one group), then ed = (25 ± 6)×1014 particles/vehicle/km, ec+t = (0.21
77
± 0.06)×1014 particles/vehicle/km (see above). Comparison of these results with Eqs.
(4.11) suggests that from the view-point of forecasting particulate pollution levels near a
busy road, subdivision of the traffic into heavy-duty diesels, light trucks, and cars (and
further) is rather excessive. Subdivision into just two groups of vehicles seems quite
sufficient within the limits of the experimental errors.
4.5. Turbulent corrections to the w-factors.
As has been mentioned in Section 4.2, the correction factors wd1,2, wt1,2 and wc1,2
were calculated under the assumption that turbulent dispersion of the plume is
neglected. That is, it was assumed that each truck/car passing by the monitoring point
affects the concentration only in one channel of the size distribution. This is a fairly
rough approximation, because it assumes that the plume emitted by any single truck/car
does not experience increase in size (dilution) due to turbulent diffusion as it is
transported from the point of emission to the point of monitoring. In reality, this is not
the case, and the emitted plume will experience two significant stages of dilution. First,
the emitted plume will experience rapid dilution in the mixing zone on the road. That is,
its width is assumed to “instantaneously” become equal to the width of the road (a
typical assumption for the CALINE4 model (Benson, 1992)). Second, the resultant
plume of the width that is equal to the width of the road will experience further increase
due to turbulent diffusion as it is transported from the road to the monitoring point
(Csanady, 1980).
As a result, every passing truck/car is likely to affect not only one channel in the
particle size distribution, but M channels, the total time of concentration measurements
in which will be equal to the time that it takes for the expanded (due to turbulent
diffusion processes) plume to pass through the monitoring point. This should result in
increasing the correction factors determined in Section 4.2.
78
It can be understood that that a correction w-factor determined in Section 4.2 is
equal to the probability for a particular vehicle (e.g., a truck) to pass by the point of
monitoring during the time that it takes for particle concentration measurement in one
channel. In the case of turbulent diffusion and increasing size of the emitted plume, this
will become the probability for a particular vehicle to pass by the point of monitoring
during the time that it takes for concentration measurements in M channels. As before,
this is correct only if the obtained w-factor is less than one. If it appears to be larger than
one, then it should be taken to equal one (similar to Section 4.2).
Dispersion of the plume in the direction of the wind can be estimated as σx ≈ umt,
where 2/1
2 ⎟⎠⎞⎜
⎝⎛= uum is the root-mean-square of turbulent wind fluctuations (Section
2.2.1). Therefore, for the typical relative turbulent intensity in the direction parallel to
the wind (Csanady, 1980)
i = um/U ≈ 0.1 (4.12)
(U is the average wind speed), the dispersion of the plume travelling the distance x form
the road to the point of observation in the direction of the wind can be estimated as
(Csanady, 1980):
σx ≈ ix ≈ 0.1x (4.13)
As a result, the total time for the expanded plume to pass through the monitoring
point is determined as T = (Lm + σx)/U, where Lm is the width of the mixing zone (in our
particular case, we will take Lm ≈ 10 m, which is the width of one of the two separated
sections of the road).
The correction factors wd1,2, wt1,2 or wc1,2 for each type of vehicle are equal to the
ratio of time T for the dispersed plume to pass through the monitoring point (which is
simultaneously the time for concentration measurements in M channels: τ1,2M) to the
average time interval between two vehicles of the considered type to pass by this point.
79
Using the wind and traffic parameters from Table 4.1, the values of the correction
factors were calculated for both weekday and weekend:
wd1,t1,c1 = {1; 0.72; 1} (weekday), (4.14)
wd2,t2,c2 = {0.13; 0.33; 1} (weekend). (4.15)
Applying this method for two types of vehicles (heavy trucks and cars) and
assuming that Lm = σx = 0, we immediately obtain the same values of the correction
factors as in Section 4.2 (as expected).
Using the average emission factors for the average fleet for weekday and
weekend, meteorological and traffic parameters, and applying the method of constrained
optimization (Sections 4.3 and 4.4), the three average emission factors for different
types of vehicles and their uncertainties were determined as follows:
ed = (15 ± 4)×1014 particles/vehicle/km,
et = (0.54 ± 0.40)× 1014 particles/vehicle/km,
ec = (0.19 ± 0.08)× 1014 particles/vehicle/km. (4.16)
Assuming that there are only two types of vehicles on the road (light trucks and
cars are included in one group), then ed = (15 ± 4)×1014 particles/vehicle/km, ec+t =
(0.18 ± 0.07)×1014 particles/vehicle/km. Comparison of these results with Eqs. (4.16)
again suggests that subdivision of the traffic into heavy-duty diesels, light trucks, and
cars (and especially any further) is rather excessive. Subdivision into just two groups of
vehicles is sufficient within the limits of the experimental errors. At the same time,
neglecting the dispersion of the mixing zone results in overestimation of the emission
factors for heavy trucks, whereas determined in sections 4.3 and 4.4 values for emission
factors for light trucks and cars are not affected.
80
4.6. Conclusions
In this Chapter, two new methods have been developed for the determination of
the average emission factors of fine and ultra-fine particles for two and three groups of
vehicles (heavy-duty trucks and cars) on a busy road. The first method requires
experimental measurements of particle concentrations at different traffic conditions
(e.g., on a weekday and on a weekend), whereas the second method is applicable when
the traffic conditions are not changing. However, the second method requires some
knowledge (typical range of variation) of the ratio of the average emission factors for
heavy trucks and cars (e.g., from the literature). The values of the emission factors have
been determined during the on-road measurements. Both the methods have been shown
to yield very similar results, which clearly demonstrates the advantage of the proposed
methods compared to the laboratory approaches giving strongly dispersed results.
The correction factors compensating for the discreteness of the traffic flow (i.e.,
for the breach of the line source approximation) have also been introduced and
discussed.
81
CHAPTER 5
EXPERIMENTAL INVESTIGATION OF ULTRA FINE PARTICLE SIZE DISTRIBUTION NEAR A BUSY ROAD ([A3, A14])
5.1. Introduction. As was shown in Chapter 3, total number concentrations in combustion aerosols
near busy roads may be approximately estimated by means of the Gaussian plume
model. However, there is still a questions whether this model is applicable for the
analysis of separate particle modes, and how accurate and reliable it is even in terms of
predictions of the total number concentration. This question results from the insufficient
knowledge about possible interactions and transformations of aerosol nano-particles
during their transport from the road. If rapid and significant transformations take place,
then Gaussian plume model cannot be used, because it is only applicable for non-
reactive pollutants (Benson & Pinkerman, 1989, Benson, 1992).
The answer to this question lies in the detailed experimental investigation of the
particle size distribution and its evolution near a busy road. As indicated in Chapter 2,
experimental evidence has recently been obtained, demonstrating significant evolution
of particle modes in combustion aerosols near busy roads (Zhu et al, 2002a,b). This
confirms inapplicability of the Gaussian plume model for the description of dispersion
of separate particle modes in combustion aerosols. However, these results were only
first indications of possible rapid evolutionary processes in such aerosols. Moreover, as
indicated in Chapter 2, the undertaken experimental approach may not be suitable for
the detailed investigation of specific processes and the effects of separate external
parameters on particle modes in combustion aerosols. Further extensive experimental
investigation of evolution of combustion aerosols near busy roads is essential for proper
understanding and description of the mechanisms involved.
82
Therefore, the aim of this Chapter is in detailed experimental investigation of the
evolution of the particle size distribution in ultrafine aerosols near a busy road. Several
observed modes of the size distribution will be analysed. In particular, we will show
that some of the modes (e.g., the 30 nm mode and 20 nm mode) tend to shift towards
smaller particle diameters. For the first time, clear evidence of an increase of the total
particle number concentration at an optimal distance from the road will be presented.
Strong and unusual correlation between different modes of the size distribution will also
be demonstrated.
5.2. Experimental procedure.
The experimental measurements were taken near the Gateway Motorway in the
Brisbane area, Australia. The analyzed four-lane road (of the total width 27 m) and the
surrounding area are presented in Fig. 5.1. The height of the Motorway above the
surrounding ground level is ≈ 2 m. There are no any buildings around the measurement
area that is practically flat grass field with isolated scattered bushes and trees. On the
other (upwind) side of the Motorway, there is a small residential area with a parkland
(Fig. 5.1).
The total number concentration and size distribution of fine and ultra-fine
particles in the range from 4 nm to 710 nm was measured at the height h = 2 m above
the ground level (i.e. approximately at the level of the Motorway). Number
concentration and size distribution of particles were measured by means of two
scanning mobility particle sizers (SMPS) in two size ranges: from 4 nm to 163 nm in
100 equal intervals of Δlog(Dp) (where Dp is particle diameter in nanometres), and
from 14 nm to 710 nm in 110 equal intervals of Δlog(Dp). Concentrations of particles
within the larger range were measured by means of SMPS 3934 (the differential
mobility analyzer DMA 3071A and the condensation particle counter CPC 3010).
Concentrations of particles in the smaller range were measured using SMPS 3936
83
(model 3080 classifier with nano DMA 3085 and CPC 3025). The time for one full up-
scan was 2.5 minutes. An automatic weather station was used to measure temperature,
humidity, wind speed, wind direction and solar radiation at the same height h = 2 m and
~ 45 m from the middle of the road every 20 sec during the whole period of the
concentration measurements.
Fig. 5.1. Gateway Motorway with a few examples of receptor points. The wind direction is
indicated by the solid arrows (for 20 November 2002 and 23 December 2002) and dashed arrow
(for 8 January 2003). The scale of the map and the direction to the North are as indicated. The
insert presents a section of the map of the area of measurements. The arrow on the insert shows
the approximate place of measurements.
The measurements were conducted on four different days: 20 November 2002,
23 December 2002, 8 January 2003 (weekdays), and 24 November 2002 (weekend). On
20 November 2002, both SMPSes were used simultaneously, while other measurements
were taken only by means of SMPS with the smaller size range from 4 nm to 163 nm.
The concentration and size distribution measurements were taken at various distances
84
from the road with increments from 6 m to 25 m within the range between 25 m and
307 m from the centre of the road. At each distance from the road, five consecutive
measurement of the size distribution were conducted, from which the average size
distribution was determined (see the figures in the next section).
The concentration and size distribution measurements were conducted
simultaneously with counting of the traffic flow. All the measurements were taken at
approximately the same time intervals (from 11 am to 3 pm), when the traffic conditions
on the Motorway were stable and variations of meteorological parameters were small
(see below).
The main idea of the designed experiment was to determine the main features of
the aerosol evolution at given (and sufficiently constant) meteorological and traffic
parameters, such as humidity, temperature, wind speed, etc. This was the reason for
taking the measurements at different distances from the road within a relatively short
period of time (within several hours) when these parameters are approximately constant.
The validity of such a method is clearly confirmed by the high level of confidence of all
predicted features of the particle mode evolution (see Section 5.4).
5.3. Experimental results and discussion.
The average values of the meteorological and traffic parameters and their
standard deviations are presented in Table 5.1.
On 8 January, the wind was almost parallel to the road (the dashed arrow in Fig.
5.1), while on other days it had a noticeable component normal to the road (see Table 1
and the solid arrows in Fig. 5.1). In addition, the wind speed during all the
measurements was relatively small (Table 5.1).
Typical average size distributions for various distances within the range from 25
m to 307 m from the centre of the road are plotted in Figs. 5.2 – 5.4 for 20 November
85
2002. Standard deviation of the mean values of the concentrations within different
channels are ~ 20 – 25%. The Friedman super smoother method (Venables & Ripley,
2000) was used for plotting the corresponding curves.
Meteorological parameters
November 20
November 24
December 23
January 8
Wind direction (degrees to the North)
24 (±40)
25 (±45)
37 (±33)
7 (±27)
Wind speed (ms-1)
2.05 (±0.9)
2.2 (±0.8)
2.3 (±0.8)
2.6 (±1.0)
Normal wind component (ms-1)
1.15 (±0.5)
1.33 (±0.5)
2.05 (±0.7)
0.96 (±0.4)
Temperature (°C)
26.5 (±0.9)
27.1 (±0.4)
29.4 (±0.5)
28 (±1)
Humidity (%)
35 (±3)
36.7 (±0.9)
36 (±3)
42 (±4)
Solar radiation (W/m2)
800 (±300)
1000 (±200)
900 (±200)
800 (±400)
Traffic (vehicles per hour)
3900 3700 5000 4300
Number of trucks (%)
20 3 23 (morning) 10 (afternoon)
16
Table 5.1. Average meteorological and traffic conditions
Fig. 5.2. The typical size distribution in the immediate proximity to the road (12 m from the
kerb). The measurements were taken on 20 November 2002. The solid curve was plotted by
means of the Friedman super smoother (Venables and Ripley, 2000). The dashed curve was
plotted by means of the moving average approach with 5 channels in the moving interval. The
shaded band represents the standard errors of the moving average curve.
86
Fig. 5.3. The particle size distributions with the experimental values of the average (over five
scans) concentrations in each of the 100 channels on 20 November 2002 (midweek) at the
following distances from the road: (a) 45 m, (b) 57 m, (c) 70 m, (d) 82 m, (e) 107 m, (f) 132 m.
All the dependencies in these figures were plotted using Freedman super smoother The
meteorological and traffic parameters, and their standard deviations, are presented in Table 5.1.
For example, Fig. 5.2 presents the experimental results for the average (over
three scans) particle concentrations in different channels at the distance 12 m from the
kerb, as a function of particle diameter. Although the standard deviation for the points
in Fig. 5.2 is ~ 20%, the super smoother curve reveals all the major, statistically
significant features of the size distribution, i.e. maxima and minima. Indeed, for
comparison, Fig. 5.2 also presents the particle size distribution plotted by means of the
moving average approach with 5 channels in the moving interval (dashed curve). The
87
shaded band presents the standard errors associated with the moving average technique.
It can be seen that all the statistically significant maxima and minima on the moving
average dependence are also correctly displayed by the Freedman super smoother (solid
curve). However, the height of the maxima on the super smoother curve (especially in
the range from ~ 10 nm to ~ 30 nm) is noticeably smaller than what is suggested by the
experimental points and the moving average curve. Similar situation occurs for all other
dependencies below.
Fig. 5.4. The same size distributions as in Fig. 5.3 (re-drawn for comparison) without the
experimental points.
It can be seen that six different maxima (particle modes) can be seen in the size
distribution at about 7 nm, 12 nm, 20 nm, 30 nm, 50 nm, and 100 nm (Fig. 5.2). All
88
these modes, except probably for the 7 nm mode, were previously observed by different
researchers. For example, the 12 nm, 20 nm, 30 nm, and 50 nm modes were reported
previously by Zhu et al (2002a). Further evolution of all these modes as the aerosol is
transported by the wind away from the road is presented in Figs. 5.3 and 5.4.
For example, Fig. 5.3 presents the experimental results for the particle size
distributions at six distances from the road. Several distinct particle modes can be
observed in this figure. It can be shown that all of these modes are statistically
significant (for more detail see Section 5.4).
The dependencies in Figs. 5.3 were compared with each other in order to
identify the evolutionary effects on combustion aerosols. The main grounds for such
comparison are related to stability of the environmental parameters and discussed in
Section 5.4. For the convenience of the comparison and clear demonstration of the
major features of mode evolution, the obtained smoothed dependencies of particle size
distributions are re-plotted in the two figures 5.4a and 5.4b without the experimental
points. As a result, the four distinct modes at the particle diameters ~7 nm, ~12 nm, ~30
nm, and ~50 nm are displayed by the curves in Figs. 5.4a,b. As can be seen from Figs.
5.4a,b, the 12 nm mode is fairly stable in position, and shifts only insignificantly with
changing distance from the road (generally, this is in agreement with the results of Zhu
et al (2002a)). However, what is more important, is that not too far from the road this
maximum clearly increases in height with distance (curves 1 – 3 in Fig. 5.4a), despite
the fact that dispersion must result in the opposite tendency. As far as we know, this
result has not been described in the literature, and it is contrary to what was indicated
previously in (Zhu et al, 2002a).
Even more unusual behaviour is displayed by the 30 nm mode – see curve 1 in
Fig. 5.4a. As has been mentioned in the Introduction, this maximum has also been
observed by Zhu et al (2002a). However, their assumption that this maximum shifts
89
noticeably to the right towards larger particle size (due to coagulation) with increasing
distance from the road has not been confirmed by our results. For example, as
demonstrated by curves 1 – 3 in Figs. 5.4a,b, the 30 nm mode slightly decreases in
height with increasing distance from 45 m to 57 m (curves 1, 2), and then, at 70 m,
noticeably shifts to the left rather than to the right (curve 3). This is indicated by a
noticeable ”shoulder” (at ~ 20 nm) on the right of the 12 nm maximum (curve 3). A
significant dip is left in place of the 30 nm mode. Further increasing distance to 82 m
(curve 4) causes further increase of the 20 nm mode and its merger with the 12 nm
mode, which results in a single broad maximum of approximately the same height as the
12 nm mode on curve 3. After this, increasing distance results in a monotonic decrease
of both the modes (curves 5 – 7).
Fig. 5.5. The particle size distributions with the experimental values of the average
concentrations on 23 December 2002 (midweek) at the following distances from the road: (a) 45
m, (b) 57 m, (c) 64 m, (d) 77 m, (e) 89 m, (f) 95 m. The meteorological and traffic parameters
are presented in Table 5.1. Curves (d) – (f) were measured in the afternoon when the number of
heavy truck was significantly lower (see Table 5.1).
90
At the same time, on the left of the 12 nm mode, there appears another
maximum at ~ 7 nm (curve 5 in Fig. 5.4b). This is the same maximum as the main
maximum in the Fig. 5.2. The comparison of Fig. 5.2 with curve 1 in Fig. 5.4a (the next
measured size distribution from the road) suggests that within the considered interval of
distances, processes of particle formation (coagulation) have occurred, resulting in a
significant decrease of the 7 nm mode in Fig. 5.2. Indeed, according to the Gaussian
plume dispersion (Chapter 3 and [A1]), the decrease of concentration within the
distance interval from 25 m to 45 m from the road (corresponding to the curve in Fig.
5.2 and curve 1 in Fig. 5.4a) must only be ≈ 1.8 times. However, the decrease of
concentration of the 7 nm particles within the same distance interval appeared to be ≈ 3
times. This suggests that there was a significant additional outflow of particles from this
mode, possibly due to particle coagulation.
The last obvious mode in Figs. 5.3 and 5.4 is represented by the maximum at ~
50 nm that can be seen at distances ≥ 60 m (curves 3 – 6 in Fig. 5.4a,b). This is the
mode that was interpreted by Zhu et al, (2002a) as the original 30 nm mode (curve 3)
shifted substantially to the right due to particle coagulation. However, as has been
demonstrated, this is not the case, since the 30 nm mode shifts in the opposite direction
(curves 1 – 4 in Figs. 5.4a,b).
In order to confirm these unexpected results, two other sets of measurements
were undertaken. Very similar behaviour of the size distribution modes was observed on
23 December 2002 (Fig. 5.5 – 5.6 ). The normal component of the wind velocity in this
case was noticeably larger than that on 20 November 2002 (Table 5.1). As a result, the
30 nm mode has appeared at significantly larger distances (~ 89 m) from the road –
curve 5 in Fig. 5.6b. Otherwise, the behaviour of this mode on 23 December is similar
to that on 20 November (compare curves 1 – 3 in Fig. 5.4a with curves 5 – 6 in Fig.
91
5.6b). Very similarly, the mode at ~ 7 nm can also be noticed in Fig. 5.6b (compare
curve 7 from Fig. 5.6b with curve 2 from Fig. 5.4a).
The overall noticeably larger concentrations in Fig. 5.6a than those in Fig. 5.4a
are due to the larger traffic flow, especially at the beginning of the experiment when
curves 1 and 2 in Fig. 5.6a were obtained.
Fig. 5.6. The same size distributions as in Fig. 5.5 (re-drawn for comparison) without the
experimental points.
The observed increase of the 12 nm and 20 nm modes within the range between
45 m and 82 m (Fig. 5.4a) is expected to affect the behaviour of the total number
concentration as a function of distance from the road. As a result, a substantial increase
of the total number concentration at these distances from the road was observed on 20
92
November 2002 (squares in Fig. 5.7a). The maximum of the total number concentration
is achieved at ≈ 82 m from the middle of the road (Fig. 5.7a). The error bars correspond
to the standard deviation of the mean values of the total number concentrations, and
confirm that the maximum is statistically significant.
Fig. 5.7. The dependencies of the total number concentration of particles on distance from the
road for the measurements on (a) 20 November 2002, and (b) 23 December 2002. The ranges of
particles are as indicated. The error bars show the standard deviation of the average (over five
measurements) total number concentrations. The experimental measurements were taken in the
sequence from the smallest distance to the largest. The double points at the distances 57 m and
132 m (in Fig. 5.7a) indicate the control sets of measurements in the end of the experiment for
the confirmation of the reliability and repeatability of the obtained results.
The dispersion of larger particles (> 163 nm) takes place in the usual fashion,
i.e., results in a monotonic decay with increasing distance (triangles in Fig. 5.7a). Note
93
that the overall concentration of these larger particles is negligible compared to the
concentration within the range < 163 nm (Fig. 5.7a). Therefore, these larger particles do
not have any effect of the observed maximum of the total number concentration.
Moreover, as can be seen (dots in Fig. 5.7a), the total number concentration within the
range > 30 nm also decays monotonically with distance, which strongly suggests that
the observed maximum of the total number concentration is caused by some
physical/chemical mechanisms resulting in increasing number of particles within the
range ≤ 30 nm (for a detailed description of such mechanisms see Sections 6 and 7). It is
important to note that the thermal rise does not have a noticeable effect on aerosol
dispersion near a busy road (confirmed experimentally by Zhu & Hinds (2005)).
The two additional points at the distances 50 m and 125 m (Fig. 5.7a) represent
two control measurements of the total number concentrations at the end of the
experiment. This was done to confirm the reproducibility of the obtained results. The
point in Fig. 5.7a at ≈ 232 m from the road does not seem to be an outlier (though this
was indicated in [A3]). On the basis of distribution of the experimental points (Fig.
5.7a) around the theoretical dependencies derived below in Chapter 7 (and in [A5]),
discarding the point at ≈ 232 m from the road (Fig. 5.7a) does not seem reasonable.
Note that Fig. 5.7b also demonstrates a tendency towards a maximum of the
total number concentration at the distances between ~ 90 m and ~ 150 m from the road.
However, with only a couple of measurements at these distances, it is difficult to state
with certainty that this maximum really exists. Nevertheless, because the normal wind
component for Fig. 5.7b is ~ 2 times larger than that for Fig. 5.7a, it is possible to
expect that the maximum of the total number concentration may occur in Fig. 5.7b at
approximately 150 m from the road.
The obtained results and dependencies are strongly affected by the type of
vehicles on the road. For example, measurements on weekend (24 November 2002)
94
resulted in noticeably different dependencies – Fig. 5.8. In particular, the 12 nm
maximum is relatively small (Fig. 5.8), which is related to the significantly smaller
number of heavy duty diesel trucks in the traffic flow. This is in agreement with the
earlier conclusion that this mode is mainly due to diesel vehicles (Zhu et al, 2002a).
Otherwise, the dependencies in Figs. 5.4a,b, 5.6b and 5.8 are fairly similar.
Fig. 5.8. The particle size distributions on 24 November 2002 (Sunday), with significantly
lower contribution of trucks (see Table 5.1). The dependencies were plotted using the Freedman
super smoother technique.
The measurements on 8 January 2003 resulted in a substantial scatter of
concentrations within channels, not allowing repeatable size distributions. This is due to
the wind being almost parallel to the road (Table 5.1 and Fig. 5.1). In this case, large
tangential component of the wind results in increased turbulence and, therefore,
fluctuations of concentration. Thus, the evolution effects are masked by strong turbulent
fluctuations. In this circumstances, significantly more than five measurements of size
distribution are required to obtain a repeatable average size distribution.
The dependence of the total number concentration on distance from the road for
8 January 2003 is shown in Fig. 5.9 by the squares for the range of particles from 4 nm
to 163 nm. In this case, methods of curve fitting based on the exponential function c =
95
k1e-αd (Hitchins et al, 2001, and Zhu et al, 2002a) or the power law c = k2d-m (Chapter 3)
with k1 = 7.0×104, α = 0.015, and k2 = 2×105, m = 0.5 can be used. The t-values (Larsen
& Marx, 1986) in these cases are as high as 17, 10, 5, and 8 for k1, α, k2, and m,
respectively. This indicates that the level of confidence for fitting both exponential and
power curves exceeds 99%.
Fig. 5.9. The total number concentrations on 8 January 2003 (almost parallel wind) within the
particle ranges: 1) from 4.6 nm to 163 nm; 2) from 7 nm to 20 nm; 3) from 21 nm to 42 nm
(includes the 30 nm mode). The curves are plotted by the Loess regression (Venables and
Ripley, 2000).
On the other hand, in order to reveal small features of the dependencies of the
total number concentration on 8 January 2003, the Loess regression (Venables &
Ripley, 2000) was used. This method is more efficient (than the Friedman super
smoother) in drawing regression curves with small features represented by a few
experimental points (Venables & Ripley, 2000). The obtained regression curves are
shown in Fig. 5.9 for the total number concentrations within the three size ranges as
functions of distance from the road. In particular, it can be seen that curve 3
corresponding to the 30 nm mode displays a tendency to levelling at the distances
96
between 25 m to 70 m. This may indicate an inflow of particles into this mode. This is
probably the reason for the small bump on curve 1 at ~ 50 m from the road. As a result,
mode evolution observed in Figs. 5.3 – 5.8 has only a very limited effect in the case of
almost parallel wind.
5.4. Level of confidence and errors
There are several sources of possible errors in the obtained experimental results.
In this section, we will consider their contribution to the level of confidence of the
features of mode evolution and will demonstrate the validity of the results.
One of the questions that may arise is that the measurements undertaken in this
chapter were not simultaneous at different distances. From this point of view, sufficient
stability of the average environmental and traffic conditions is important (this is
discussed below). However, observation of the consistent features of particle size
distributions, total number concentration, and their evolution during at least 6
independent sets of measurements on different days under different conditions (plus
another independent confirmation by the group at Lancaster University, UK) seems to
be a sufficient demonstration of the general validity of the obtained results.
Specifically, the first source of errors can be associated with changing
atmospheric parameters during the course of measurements. However, as shown in
Table 5.1, standard deviations of such parameters as humidity and temperature are small
during each of the measurement periods. These small variations of humidity and/or
temperature can hardly affect the presented curves for size distributions. This, however,
does not mean that temperature and humidity do not affect the described mode
evolution. On the contrary, they are expected to be important for understanding the
behaviour of the modes – see Chapters 6, 8, 10 and [A4, A6, A8, A9].
97
The standard deviations of speed, direction, and normal component of the wind
are large (as expected) due to strong turbulence (Table 5.1). However, the associated
random errors of particle concentrations in this case are effectively reduced by taking
the average of five scans at every distance from the road (see Section 5.2). This
statement is confirmed by the limited dispersion of experimental (average) points
around the size distribution curves (Figs. 5.3 and 5.5).
Solar radiation changed fairly noticeably (Table 5.1). However, no direct and
obvious effect of solar radiation on the observed mode evolution has been detected.
Once again, this does not mean that there is no such an effect (which should be the
matter for further investigation), but rather that the appearance and transformation of the
modes in our experiment did not correlate with the solar radiation.
Standard deviations of the overall traffic (for the traffic flow measured in 5
minute intervals) on all days of measurements were less than 10%, and the standard
deviations for trucks ~ 20%. This is a clear indication of a highly stable traffic
conditions on the considered road. Insignificant contribution of traffic variations to the
errors of the measured particle size distributions and the total number concentrations is
also demonstrated by the control measurements conducted in the end of the
measurement period on 20 November 2003 (see the additional two points at the
distances 57 m and 132 m in Fig. 5.7a). The two size distributions at the distance 57 m
for the original (curve 2 in Fig. 5.4a) and control sets of five scans were hardly
different. This is also another confirmation of the high stability of the obtained results
with respect to the fluctuations of the meteorological parameters, including wind speed
and direction.
At the same time, it is important that on 23 December 2003, traffic conditions
changed substantially for the morning and afternoon measurements (due to the next day
holiday). The number of trucks in the afternoon dropped very noticeably (see Table
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5.1). This was one of the major reasons why the 30 nm mode and 20 nm modes in Fig.
5.6b were substantially smaller than those in Fig. 5.4a. This is also likely to be a reason
for the maximum of the total number concentration in Fig. 5.7b to be significantly
smaller than in Fig. 5.7a. Nevertheless, despite such a drastic variation in the traffic
flow, the major features of the mode evolution remained unchanged (compare Fig. 5.4a
with Fig. 5.6b). This clearly demonstrates that the discovered unusual features of the
aerosol evolution are highly resilient to variations of traffic flow, and that they are not
an artefact of changing atmospheric and traffic conditions.
In addition, levels of confidence for all of the maxima of the size distributions
(modes) in Figs. 5.4, and 5.6 were determined by means of known statistical methods
(Bowman and Azzalini, 1997). Using S-Plus, the experimental point near each of the
modes are approximated by a polynomial of the fifth order (to include the neighbouring
modes). The level of confidence of the considered mode is then determined using the
two neighbouring minima and the error of the approximation (Bowman and Azzalini,
1997). The resultant values of level of confidence for the modes in Figs. 5.2 – 5.6 are
presented in Tables 5.2 and 5.3, clearly demonstrating the validity of the obtained
results and conclusions.
25 m 45 m 57 m 70 m 82 m 107 m 132 m 307 m 7 nm >99 - - - 85 91 - >99 12 nm 83 >99 >99 >99 >99 >99 >99 - 20 nm 83 - - - - - - - 27 nm >99 >99 >99 - - - - - 50 nm >99 - - 99 81 91 >99 79
Table 5.2. Levels of confidence in % for different modes on the curves in Figs. 5.2 –
5.4 (20 November 2003). The modes are shown in the first column, while the distance
from the road (corresponding to different curves in Figs. 5.2 – 5.4) are shown in the first
row.
99
45 m 57 m 64 m 77 m 89 m 95 m12 nm >99 >99 >99 >99 >99 >99 20 nm - - - - - 89 30 nm - - - - >99 - 40 – 50 nm - >99 99 >99 - -
Table 5.3. Levels of confidence in % for different modes on the curves in Figs. 5.5 –
5.6 (23 December 2003).
Finally, it can be shown that the effect of Wynnum road (Fig. 5.1) is negligible.
Indeed, the Gaussian plume approximation gives that particle concentrations should
decay as a power function of distance from the road (Chapter 3 and [A1]). Taking into
account the typical traffic flow on Wynnum road (2127 vehicle/hour with 1.8% of
heavy duty trucks), the distance to the place of measurements (600 – 800 m), and the
average emission factors for light cars and heavy-duty trucks (Chapter 4 and [A2]), we
obtain that the typical concentrations due to Wynnum road are at least ~ 10 times less
than those from the Gateway Motorway at all sampling points.
5.5. Conclusions
In this Chapter, for the first time, a detailed experimental investigation of the
evolution of particle size distribution modes in the fine and ultra-fine ranges has been
carried out near a busy road. A number of unexpected and unusual effects have been
observed. These are related to mutual transformation of different modes, resulting in
appearance, growth and disappearance of these modes at particular distances from the
road. For example, the first clear and consistent experimental evidence of existing of a
strong 7 nm mode of particles has been obtained. A strong mode at ~ 30 nm particle
diameter appears at a particular distance from the road. This mode has been shown to
consistently shift to the left, i.e. towards smaller particle diameters. As a result, the 30
nm mode is transformed into the 20 nm mode, and then into the 12 nm mode. At the
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same time, it has been shown that during the whole process of mode conversion and
transformation, the particle concentration in the 7 nm mode changes only insignificantly
(increasing at some stages) with increasing distance from the road. Therefore, since
dispersion must obviously result in a significant reduction of particle concentrations,
there is a substantial influx of particles into the 7 nm mode during the aerosol evolution
and its transport away from the road.
For the first time, clear experimental evidence of a maximum of the total
number concentration at a particular distance from the road has been obtained. This
maximum has been shown to occur due to a strong increase of particle concentration
within the range < 30 nm. The observed maximum has been demonstrated to decrease
and move away from the road with increasing normal (to the road) component of the
wind. The effect of normal wind component on the particle size distribution has also
been investigated. For example, increasing normal component of the wind results in
approximately proportional increase of the distance at which the 30 nm mode appears.
If the average wind is approximately parallel to the road, then the observed modes are
shown to significantly fluctuate due to the turbulence and stochastic nature of
fluctuations of wind direction.
Statistical analysis of the errors associated with the experiments have been
conducted, and the levels of confidence for the observed modes of particle size
distribution have been determined, clearly confirming the obtained results and
conclusions.
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CHAPTER 6
A NEW MECHANISM OF AEROSOL EVOLUTION NEAR A BUSY ROAD: FRAGMENTATION OF NANO-PARTICLES ([A4, A18, A22,
A23, A25, A26])
6.1. Introduction
As has been demonstrated in the previous Chapter 5, combustion nano-particle
aerosols may experience rapid evolutionary processes near their major source – busy
roads. Therefore, investigation and modelling of the specific mechanisms responsible
for these processes are essential for understanding and reliable prediction of aerosol
dispersion in the urban environment. However, serious doubts about the possibility of
using only the conventional mechanisms of evolution of combustion aerosols, e.g., such
as particle formation and coagulation, arise if we attempt a comprehensive explanation
of the experimental features of the aerosol evolution and particle mode transformation
(Chapter 5 and [A3]). For example, because at the considered particle concentrations
coagulation is highly inefficient (Jacobson, 1999, Jacobson and Seinfeld, 2004, Pohjola
et al, 2003, Zhang & Wexler, 2004, Zhang et al, 2004), it can hardly result in the
observed strong changes in the particle size distribution. It is also difficult to expect that
only particle formation and coagulation can be used for the interpretation of the
observed evolution of the 30 nm mode into the 20 nm mode and then into the 12 nm
mode (Chapter 5).
In this Chapter, we present further experimental evidence that the conventional
mechanisms of aerosol evolution are insufficient for a comprehensive interpretation of
the experimental observations of evolution of combustion aerosols near busy roads. A
new statistical method of analysis will be developed for the investigation of particle
modes. An alternative definition of particle modes will be presented. As a result, a
number of distinct modes strongly interacting with each other will be shown to exist in
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particle size distributions in the presence of strong turbulent mixing. A new mechanism
of mode evolution, based on thermal fragmentation of nano-particles, will be introduced
and justified. A complex pattern of the aerosol evolution will be suggested, including
several stages of particle formation, coagulation, and fragmentation.
6.2. Modes of particle size distribution
The new method of statistical analysis of particle modes near a busy road is
demonstrated and developed on an example of 50 independent scans of particle size
distribution, obtained on 25 November 2002 within the time interval of 3 hours at the
distance of 40 m from the centre of the road (Gateway Motorway, Brisbane area,
Australia). The description of the road and the environment at the point of monitoring is
presented in Chapter 5 (see also Fig. 6.1 with the indicated wind directions during this
monitoring campaign).
The size distributions of fine and ultra-fine particles in the range from 4.6 nm to
163 nm were measured at the height h = 2 m above the ground level (i.e. approximately
at the level of the Motorway). The measurements were taken by means of a scanning
mobility particle sizer SMPS 3936 and the condensation particle counter CPC 3025 in N
= 100 equal intervals of Δlog(Dp) (where Dp is particle diameter in nanometres). The
time for one full scan was 2.5 minutes. An automatic weather station was used to
measure temperature, humidity, wind speed, wind direction and solar radiation at the
same height h = 2 m and ~ 45 m from the middle of the road every 20 sec during the
whole period of the concentration measurements.
103
Fig. 6.1. The area of measurements near Gateway Motorway, Brisbane, Australia. The indicated
receptor point is at the distance 40 m from the centre of the road. The scale of the map and the
direction to the North are as indicated (the distances on the axes are given in meters). The
crosses indicate the receptor point.
The time dependencies of the wind direction and speed during the
measurements, plotted by means of the Friedman super smoother (Venables & Ripley,
2000), are presented in Figs. 6.2a,b. It can be seen that these dependencies give
approximately constant average wind directions and speeds within the time intervals
between 14:05 and 14:42 (indicated by the dashed vertical line in Fig. 6.2a,b) and
between 15:06 and 16:16 (indicated by the two dotted vertical lines in Figs. 6.2a,b).
These time intervals correspond to following two sets of scans: (i) scans from 1 to 11
(between 14:05 and 14:42), and (ii) scans from 19 to 38 (between 15:06 and 16:16). The
104
average normal and parallel (to the road) components of the wind for these sets are then
determined as: (i) v⊥ = 1.65 ± 0.16 ms-1, v|| = 1.6 ± 0.2 ms-1; (ii) v⊥ = 0.8 ± 0.1 ms-1, v|| =
2.4 ± 0.3 ms-1. We will consider these sets of scans separately because they correspond
to significantly different (on average) stages of the aerosol evolution. Indeed, since the
normal wind component for set (i) is approximately two times larger than that for set
(ii), the time that it takes for the aerosol to reach the point of observation during set (i) is
approximately twice as small. The difference between the average evolution times for
the two sets of measurements is thus ~ 25 s. The other meteorological and traffic
parameters are presented in Table 6.1 for both the sets of scans.
It is important to understand that here we consider average wind components
rather than their instantaneous values. Consideration of the instantaneous values cannot
be justified due to stochastic nature of the atmospheric turbulence (Csanady, 1980).
These strong stochastic turbulent fluctuations are typically taken into account in the
analysis through turbulent diffusion in the atmosphere, which represents a type of
averaging of these fluctuations and the resultant diffusive transport/dispersion of
aerosols (Csanady, 1980). Certainly, instantaneous evolution times for particular
aerosol/air samples (i.e., the time of aerosol transportation to the monitoring point) vary
significantly due to violent turbulent fluctuations – Fig. 6.2. However, the developed
approach is based on averaging evolution time, and thus determining averaged
evolutionary changes of aerosol nano-particles (which is analogous to considering
turbulent diffusion as opposed to instantaneous acts of stochastic transportation of
particular air portions (Csanady, 1980)). This is the reason for using average values of
wind speed and direction (Fig. 6.2), i.e., wind components v⊥ and v||, and considering
their differences for the two different selected sets of scans.
105
Fig. 6.2. The time dependencies of (a) the wind direction (i.e. the angle between the wind
direction and the road), and (b) wind speed. Both the curves have been plotted by means of the
Freedman super smoother (Venables & Ripley, 2000). The measurements of the wind direction
and wind speed were taken by an automatic weather station at the rate of one measurement per
minute and three measurements per minute, respectively. The sets of 11 and 20 scan are denoted
by the dashed and two dotted vertical lines, respectively.
The corresponding average (over 11 and 20 scans) particle size distributions are
presented in Fig. 6.3a,b. Not much can be seen from these distributions, apart from the
strong maximum between ~ 10 nm and ~ 12 nm in both the figures, and a weak ~ 50 nm
mode in Fig. 6.3a. No conclusions about possible interaction of modes, and/or their
evolution can be made directly from these figures. Nevertheless, even if distinct features
of a size distribution are effectively smoothed out by strong turbulent mixing (as in
106
Figs. 6.3a,b), as shown below, the statistical analysis will clearly reveal a number of
particle modes and their mutual interactions.
Fig. 6.3. The average particle size distributions of concentrations in 100 channels on 25
November 2002 at the same distance of 40 m from the centre of the road, but at significantly
different normal and parallel one hour average wind components: (a) v⊥ = 1.65 ± 0.16 ms-1, v|| =
1.6 ± 0.2 ms-1 (for the set of 11 scans); (b) v⊥ = 0.8 ± 0.1 ms-1, v|| = 2.4 ± 0.3 ms-1 (for the second
set of 20 scans). The meteorological and traffic parameters, as well as their standard deviations,
are presented in Table 6.1.
We re-define particle modes as groups of particles in neighbouring channels,
such that particle concentrations in these channels tend to fluctuate in correlation with
each other. In particular, this suggests that such groups of particles are likely to have the
same nature and/or come from the same source (i.e., are likely to be generated by the
same mechanism). Usually, this new definition does not contradict the conventional
mode definition as distinct maxima of a particle size distribution. We will see below
that in several cases it results in the same modes as those that were directly observed
107
under more favourable meteorological conditions (Chapter 5 and [A3]). However, the
new definition makes more physical sense, emphasises the actual nature of the aerosol
particles, and will allow mode analysis in the case of strong turbulent mixing (Fig.
6.3a,b). If two or more modes overlap, i.e. the corresponding particles come from
different sources / physical processes, then correlations between particle concentrations
in neighbouring channels should typically reduce in the overlap region. This effect
enhances the appearance of the correlation maximums corresponding to separate
particle modes, which makes the analysis of evolution and transformation of different
types of aerosol particles much more obvious. As a result, it will be possible to clearly
see and identify particle modes even if there are no any distinct features of the
conventional size distributions (e.g., due to strong turbulent mixing – Figs. 6.3a,b).
To determine particle modes (as maximums of the correlation coefficients) from
multiple scans of size distribution in the absence of obvious features of this distribution
(Figs. 6.3a,b), correlations between particle concentrations in neighbouring channels of
the size distribution were determined using the following steps.
Step 1. Normalise particle concentrations in each channel in every scan to the
total number concentration in this scan. Thus we eliminate trivial correlations between
channel concentrations, caused by changing total number concentration.
Step 2. Select an interval of n channels out of the overall N channels in a scan
(in our case, N = 100).
Step 3. Choose two different channels from the n-channel interval, and
determine the simple correlation coefficient (Larsen and Marx, 1986) between particle
concentrations in these two channels over the M scans (in our case M = 11 or 20). In
other words, simple correlation is determined between two columns of concentrations in
two chosen channels, the number of elements in each of the columns being equal to the
number of scans M.
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Step 4. The procedure is repeated m = n!/[2!(n – 2)!] times for all possible
different pairs of channels in the n-channel interval.
Step 5. The average correlation coefficient for the considered n-channel interval
is determined.
Step 6. If n is odd, then steps 2 – 5 are repeated for all N – (n – 1) different n-
channel intervals, giving N – (n – 1) different values for average correlation coefficient.
At the same time, each n-channel interval can be characterised by an average
particle diameter. For example, if n is odd, then this will be the diameter corresponding
to the middle channel of the n-channel interval. As a result, we obtain a dependence of
the moving average of the correlation coefficient versus average particle diameter. The
number n is chosen as a compromise between mode resolution and statistical errors
(standard deviation of the moving average), both increasing with decreasing n.
The procedure was applied to the two sets of M = 11 and M = 20 scans (see
above) with N = 100 and n = 7 (m = 21). The resultant dependencies of the moving
average correlation coefficients are presented in Fig. 6.4a,b. It can clearly be seen that
despite the absence of noticeable features on the average particle size distributions
(Figs. 6.3a,b), the dependencies in Figs. 6.4a,b display a number of distinct and
pronounced maxima and minima with the level of confidence of the corresponding
correlations (Larsen & Marx, 1986) of no less than 95% for all points on the curves (see
the Appendix to this Chapter for the procedure of calculating the levels of confidence).
The corresponding error curves (representing the errors of the moving average
correlation coefficients) are presented in Figs. 6.5a,b. These clearly demonstrate that all
of the obtained maxima, except for those at ~ 14.6 nm and ~ 31 nm in Fig. 6.4a and ~
31 nm and ~ 35 nm in Fig. 6.4b, are statistically significant. A correlation maximum
was assumed to be statistically significant if the lower limit of the error band at this
109
maximum is higher than the upper limits of the error band at the two neighbouring
minima (Figs. 6.5a,b).
Fig. 6.4. The dependencies of the moving average of the correlation coefficient, R, between the
particle concentrations in different channels on particle diameter. The averaging is carried out in
each of the different intervals of 7 neighbouring channels, thus giving 94 average values of the
correlation coefficients, i.e., 94 points in the presented graphs. The curves are obtained by
simply connecting the neighbouring points. The moving average correlation coefficients are
calculated over (a) the first set of 11 scans (scans from 1 to 11); (b) the second set of 20 scans
(scans from 19 to 38).
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Fig. 6.5. The error curves determining the errors of the moving average correlation coefficients
represented in Figs. 6.4a,b. The two curves in each of the figures (a) and (b) give the limits of
the range of possible statistical errors for every considered particle diameter. This demonstrates
that all the modes are statistically significant, except for the 14.6 nm and 31 nm modes in Figs.
6.4a, and 31 nm and 35 nm modes in Figs. 6.4b.
The obtained maxima correspond to groups of particles of similar size with
strong mutual correlations, which means that concentrations of these particles tend to
increase/decrease simultaneously in strong correlation with each other. For example,
this is the case for particles within the size range from ≈ 10 nm to ≈ 15 nm in Fig. 6.4b.
These particles appear to be in strong correlation with each other, and thus can be
regarded as a mode.
Thus particle modes can be re-defined as groups of particles of similar size, that
are characterised by pronounced maxima of mutual correlations. This definition is more
111
general and physically meaningful, than the previous definition of a mode as a group of
particles corresponding to a maximum on a particle size distribution.
It is important that such a large number of different particle modes (Figs. 6.4a,b)
in a single size distribution has never been obtained previously (even at more favourable
atmospheric conditions with weaker turbulent mixing (Zhu et al, 2002, Chapter 5 and
[A3]). This clearly demonstrates the effectiveness of the proposed method that provides
a new physical insight into the processes of aerosol evolution.
6.3. Maximum of the total number concentration
In Chapter 5 and [A3], for the first time, a maximum of the total number
concentration was observed at an optimal distance from the road. This means that the
total number concentration may increase with increasing distance from the road. For
example, in Chapter 5, this maximum was observed at the distance of ~ 80 m from the
road, and the total number concentration increased within the interval from ~40 m to ~
80 m from the road (for the normal component of wind ≈ 1.15 m/s). It is important that
the experiment described above in Section 6.2 clearly confirms those previous results.
Indeed, the total number concentrations calculated for the particle size
distributions presented in Figs. 6.3a,b were determined as N1 ≈ (23 ± 3)×104
particles/cm3 and N2 ≈ (40 ± 3)×104 particles/cm3, respectively. The background
particle concentration was taken to be N0 ~ 2,400 particles/cm3 (measured for the
similar conditions in (Chapter 3 and [A1]). However, the direct comparison of N1 and
N2 is not possible, since the wind conditions were different for the two sets of scans
corresponding to Figs. 6.3a,b. Therefore, the following procedure was used. Using the
calculated value of N1 and the software package CALINE4 (adjusted for predicting
aerosol dispersion near a busy road (Chapter 3 and [A1])), the model emission factor Em
= 229 from the average fleet on the road (Chapter 3 and [A1]) was determined for the
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meteorological and traffic parameters corresponding to the first set of 11 scans at the
considered distance 45 m from the centre of the road.
scans 1 to 11 scans 19 to 38
wind direction, degrees to the road
43 ± 20 18 ± 19
wind speed, ms-1 2.3 ± 0.2 2.6 ± 0.9
temperature, °C 28.2 ± 0.3 27.2 ± 0.3
humidity, % 29 ± 1 36 ± 1
solar radiation, Wm-2
860 ± 40 560 ± 80
trucks, hour-1 810 ± 140 790 ± 140
cars, hour-1 3800 ± 400 4800 ± 500
Table 6.1. Meteorological and traffic parameters.
The traffic flow for both the sets of scans is regarded to be approximately the
same. This is because the emission factors for cars are about two orders of magnitude
less than those for heavy trucks (Chapter 4 and [A2]), and increase of the number of
cars by ~ 1000 for the second set of scans (see Table 6.1) is approximately cancelled by
the simultaneous reduction of the number of trucks by ≈ 20 (Table 6.1). Therefore, the
effective traffic flow can be regarded as approximately the same. Then, using the
determined emission factor from the average fleet on the road Em and the CALINE4
model, the total number concentration was calculated for the wind parameters
corresponding to the second set of scans: 2N′ ≈ 19.6×104 particles/cm3. This is the
concentration that one would have expected for the second set of scans if the Gaussian
plume approximation were correct.
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In other words, we determine the model emission factor corresponding to the
evolution time t1, which is the time of transport of the aerosol from the road to the point
of observation for the first set of scans. Then, using this model emission factor (which
should not change significantly between the two sets of scans), we determine the
corresponding total number concentration at a different evolution time t2 ≈ t1 + 25 s,
which is the time of transport of the aerosol from the road to the point of observation for
the second set of scans.
However, we have: (N2 – N0)/( 2N′ – N0) ≈ 2.19. Thus, the Gaussian plume
approximation is not valid, and an additional strong influx of particles from some
“internal” sources exists during the evolution of the aerosol within the ~ 25 s between
the two sets of scans (see section 2). In other words, due to the presence of these
“internal” sources, the number of particles in the aerosol increases 2.19 times within the
time interval of ~ 25 s. This means that the total number concentration may increase
with increasing distance from the road, if the dispersion is not sufficient to suppress the
above increase of the number of particles. This is another confirmation of the
experimental observation of a maximum of the total number concentration at an optimal
distance from the road in (Chapter 5 and [A3]).
6.4. Failure of the conventional mechanisms of the aerosol evolution
Interpretation of the results obtained in sections 2 and 3 may present a challenge
if we intend to use only the conventional mechanisms of aerosol evolution, such as
particle formation, condensation, evaporation, coagulation, and dispersion.
For example, the observed increase of the number of particles during the
evolution of the aerosol within ~ 25 s cannot be explained by particle formation. Indeed,
if this were the case, the number of smaller particles within the range < 8 nm would
have increased much more significantly than the number of the larger particles with the
diameters > 8 nm. At the same time, the comparison of Figs. 6.3a,b suggests the
114
opposite: the number of particles within the range between ~ 8 nm and ~ 30 nm
increases, whereas the number of particles with the diameters less than ~ 8 nm is
practically the same for both size distributions. In addition, it is difficult to imagine that
particles of ~ 10 – 20 nm diameter may be formed or significantly increase their
diameter due to nucleation processes within ~ 25 s near a busy road with relatively low
levels of gaseous pollutant concentrations (Fig. 6.3a,b). Therefore, there should be some
other “internal” source of particles as the aerosol is transported away from the road.
Strong differences can be seen in the mode structure for the two sets of scans in
Figs. 6.4a and 6.4b. In the range between ~ 6 nm and ~ 14 nm, and between ~ 35 nm
and ~ 80 nm, maxima in Fig. 6.4a correspond to minima in Fig. 6.4b. In addition, Fig.
6.4a displays three sharp minima of correlations at ~ 7 nm, ~ 10 nm, and ~ 70 nm. In
Fig. 6.4b, these minima disappear, turning into maxima. A very strong and broad
maximum of correlations appears between 9.5 nm and 15.5 nm (Fig. 6.4b). All this
strongly suggests the presence of fast and strong processes of evolution and
transformation of aerosol particles. For example, substantial variations of the correlation
coefficient (e.g., at ~ 10 nm or ~ 70 nm) must be linked to an inflow or outflow of
particular particles into or out of the corresponding range of diameters. This means that
within the time interval of ~ 25 s (the difference in the evolution time between the sets
of the scans – see section 2) particles in the ranges from ~ 6 nm to ~ 14 nm and from ~
35 nm to ~ 80 nm experience substantial variations of their properties and size.
It is hardly possible to expect that coagulation may be responsible for the
observed mode evolution. Indeed, according to (Jacobson and Seinfeld, 2004),
coagulation is highly inefficient at the considered particle size and concentrations (Figs.
6.3a,b) and cannot account for the drastic evolution of the particle modes (Figs. 6.4a,b).
In addition, coagulation hypothesis is in obvious contradiction with the observed
115
increase of the number of particles in the aerosol within the time interval ~ 25 s between
the two sets of scans.
Condensation/evaporation alone would be likely to shift the mode pattern in Fig.
6.4a to the right or to the left without significant changes of the mode structure
(assuming similar chemistry for the particles). However, this contradicts the results
presented in Fig. 6.4b. In addition to significant variations of the overall pattern of
minima/maxima, Fig. 6.4b displays a significantly larger shift (by more than ~ 10 nm)
of the modes in the range between ~ 35 nm and ~ 80 nm, than in the range between ~ 6
nm and ~ 14 nm (by ~ 2 – 3 nm). This would be difficult to explain using the
condensation/evaporation mechanism alone. The appearance of a strong and broad
maximum at ~ 12 nm (Fig. 6.4b) with surprisingly small errors (Fig. 6.5b) would also
be difficult to explain by condensation/evaporation.
Even more importantly, the analysis of correlations between the 5.7 nm mode
and other modes in Fig. 6.4a, and between the 7.9 nm mode and other modes in Fig.
6.4b suggests that the 7.9 nm mode in Fig. 6.4b is unlikely to result from shifting the 5.7
nm mode to the right, since these modes are characterised by strongly different
correlations with other modes (for more detail see (Chapter 7 and [A5]). In addition, the
same analysis shows that the 7.9 nm mode in Fig. 6.4b is likely to result from the 8.8
nm mode in Fig. 6.4a, shifting to the left (towards smaller particle size). At the same
time, the 5.7 nm mode simply disappears in Fig. 6.4b. This can be explained by an
assumption that the 5.7 nm mode is likely to consist of liquid volatile particles that
effectively evaporate within the evolutional time interval of 25 s between the two sets of
scans corresponding to Figs. 6.4a,b.
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Fig. 6.6. Particle size distributions obtained on 20 November 2002 at the indicated distances
from the centre of the road (reproduced from Chapter 5). The normal one hour average wind
component v⊥ = 1.15 m/s. The other meteorological and traffic parameters are presented in
Table 5.1.
From this point of view, it is interesting that 8.8 nm particles can be obtained
when a 7.9 nm (presumably solid – see below) particle coagulates with a liquid 5.7 nm
particle. Therefore, it is possible that the 8.8 nm mode in Fig. 6.4a may have resulted
from coagulation of the 5.7 nm and 7.9 nm modes near the exhaust pipe where the
concentration of particles is sufficient for this process to occur (Jacobson, 1999,
Jacobson and Seinfeld, 2004).
As the aerosol is transported away from the road, the concentration of volatile
compounds in the air is reduced, and the fluid particles (and the volatile coating of solid
particles) start evaporating. This results in shifting the 8.8 nm mode (Fig. 6.4a) to the
117
left into the 7.9 nm mode (Fig. 6.4b). The strong minimum of correlations between the
5.7 nm and 8.8 nm modes in Fig. 6.4a can be explained by the presence of both solid
and liquid particles with different (opposite) correlations.
Similarly, the conventional mechanisms of particle transformation can hardly
explain a number of previously observed features of the particle size distribution near
the same road on at least four other days of measurements (Chapter 5 and [A3]). For
example, Figs. 6.6 and 6.7 reproduced from (Chapter 5 and [A3]) clearly demonstrate
the appearance of a strong 30 nm mode at some distance from the road (curves 1 and 6
in Figs. 6.6a and 6.7b, respectively), its subsequent shift to the left into the 20 nm mode
(curves 2 and 3 in Fig. 6.6a, and curve 7 in Fig. 6.7b), and then into the 12 nm mode
(curves 3 and 4 in Fig. 6.6) with the simultaneous increase of the 7 nm mode (curves 2 –
5 in Fig. 6.6, and 7 in Fig. 6.7b). The correspondence of these curves to the
experimental points is demonstrated in (Chapter 5 and [A3]). The levels of confidence
for the observed modes and analysis of the experimental errors are also discussed in
(Chapter 5 and [A3]). As a result, the coagulation model obviously runs into severe
problems with explaining the discussed pattern of evolution (in addition to the serious
doubts about its effectiveness (Jacobson, 1999, Pihiola et al, 2003, Jacobson and
Seinfeld, 2004, Zhang & Wexler, 2004, Zhang et al, 2004)).
Fig. 6.6a clearly suggests that the 12 nm mode substantially increases with
increasing distance from the road from 45 m to 82 m. One might think that this could be
caused by particle formation from gasses due to heterogeneous nucleation of particles,
e.g., from the 7 nm mode. However, if this were the case, the 7 nm mode would have
been a source for the 12 nm sink mode. This is in contradiction with curves 4 and 5 in
Fig. 6.6b, which display an increase of the source 7 nm mode with the simultaneous
decrease of the sink 12 nm mode. Obviously, a sink mode cannot decrease with
increasing its source (unless there appears an additional mechanism depleting the sink
118
mode, which in this case is unlikely). Therefore, the 7 nm mode cannot be a source for
the 12 nm mode, and increase of the 12 nm mode displayed by curves 1 – 4 in Figs.
6.6a,b cannot be explained by heterogeneous nucleation. On the other hand,
exceptionally large average correlation coefficients and noticeably reduced statistical
errors for the 12 nm (Figs. 6.4b, 6.5b) suggest that the particles in this mode should
come from the same source (must be caused by the same mechanism).
Comparison of curves 2 – 5 in Figs. 6.6a,b also suggests that typical
concentration of ~ 7 nm particles is almost independent of distance from the road
(staying at ~ 20000 cm-3), and results in a distinct maximum on curve 5. Note that at the
same time concentrations in other channels are significantly reduced by dispersion. This
strongly suggests that there is a substantial influx of particles into the 7 nm mode due to
some physical/chemical mechanisms. If these were only nucleation processes, why
would this influx increase when the neighbouring 12 nm mode starts decreasing
substantially (curve 5 in Fig. 6.6b)? Why would the maximum of the 7 nm mode (curve
5) coincide with almost two times drop in the 12 nm mode within the distance of 25 m
(compare curves 4 and 5 in Fig. 6.6a), whereas the Gaussian plume approximation
(Csanady, 1980, Gramotnev et al, 2003) suggests only 1.2 times decrease within the
same distance? Why would the influx into the 7 nm mode practically cease or at least
significantly reduce only after other modes have been seriously depleted (see the 7 nm
mode strongly decreasing on curves 6 and 7 in Fig. 6.6b)?
119
Fig. 6.7. Particle size distributions obtained on 23 December 2002 at the indicated distances
from the centre of the road (reproduced from Chapter 5). The normal one hour average wind
component v⊥ = 2.05 m/s. The other meteorological parameters are approximately the same as
for Fig. 6, and are presented in Table 5.1.
In addition, as mentioned in section 3, the experimental measurements clearly
demonstrate that the process of shifting the 30 nm mode to the left (i.e. into the 20 nm
and 12 nm modes) coincides with a substantial increase of the total number
concentration of nano-particles within the interval of distances from ~ 45 m to ~ 82 m
from the road – see Fig 6.7a from (Chapter 5 and [A3]) and section 3 above.
It is thus clear that the conventional mechanisms, such as particle formation,
condensation, evaporation, coagulation, and dispersion, cannot fully explain all the
experimentally observed effects of aerosol evolution and mode transformation near a
busy road. The discussed contradictions suggest that a new mechanism of aerosol
120
evolution is required. This mechanism will be based on thermal fragmentation of nano-
particles.
6.5. Fragmentation model of aerosol evolution
The possibility of fragmentation of soot aggregates has previously been
discussed in (Mikhailov E. F., et al, 1996, Harris & Maricq, 2002, Kostoglou &
Konstandopoulos, 2003). For example, Mikhailov et al (1996) demonstrated
experimentally that large soot aggregates (with ~ 2 μm2 – 5 μm2 projection) produced
under special laboratory conditions can experience fragmentation at ~ 100oC and
between 300oC and 500oC. Harris and Maricq (2002) showed that allowing for
fragmentation of relatively large soot nano-particles (of mobility diameter ~ 100 nm)
slightly improved the fit of the theoretical curves to the experimental data points for
particle size distributions obtained under special laboratory conditions. Kostoglou and
Konstandopoulos (2003) suggested an oxidation mechanism for fragmentation of large
soot aggregates, also under special laboratory conditions, and again with relatively
minor effect of fragmentation on particle size distribution.
In this section, fragmentation of much smaller particles will be considered and
used for the explanation of the strong changes in the size distribution modes near a busy
road during field experiments. The main idea of the fragmentation model is based on the
existence of small solid ~ 7 nm carbon/graphite primary particles. Larger carbon
particles (~ 20 – 30 nm) were readily observed in the products of combustion of
different materials (Colbeck et al, 1997), and in the vehicle exhaust (Meszaros, 1999,
Wentzel et al, 2003). However, several researchers also reported the existence of a large
number of smaller solid particles around ~ 7 nm mobility diameter. For example,
Abdul-Khalek et al (1998) reported the experimental observation of a strong mode of
particles from the diesel exhaust within the considered range. The analysis of these
121
particles by means of a catalytic stripper suggested that a large number of them are solid
(Abdul-Khalek et al, 1998), and the authors assumed these particles to be metallic ash
formed from oil and fuel additives. On the other hand, Bagley et al,(1996) suggested
that this mode may consist of primarily carbonaceous particles. More recent
investigation using a thermal desorption particle beam mass spectrometer demonstrated
the existence of solid nuclei particles in the diesel exhaust of ~ 3 – 7 nm (Sakurai et al,
2003). This was explained by evaporation of a thick volatile layer from the surface of
the solid nuclei particles (Sakurai et al, 2003). However, an alternative explanation
might be fragmentation of nano-particles. Yet another very careful recent experiment on
separation of volatile and solid particles in the diesel exhaust by means of a hot dilution
system also demonstrated the existence of a large number of solid (presumably,
carbon/graphite) particles with the mobility diameters between 6 nm – 10 nm (Fiertz
and Burtscher, 2003).
It can be expected that these small carbon/graphite particles are mainly of non-
spherical shape. For example, these may be graphite scale-like particles (Wentzel et al,
2003) with free sp2 bonds at the edges of the scale. These particles are formed in the
exhaust pipe at high temperatures. They can then coagulate by means of the covalent
bonds, and/or various types of volatile molecules or their fragments may engage the free
bonds, forming a kind of a frill. Thus the primary ~ 7 nm particles may, to a significant
extent, consist of volatile compounds. Nevertheless, it is argued that they can be
regarded as “solid”, because the volatile molecules (or their fragments) are bonded to
the carbon/graphite core by means of strong covalent bonds. More volatile molecules
can then deposit onto the primary ~ 7 nm particles (e.g., forming the 8.8 nm mode – Fig.
6.4a). However, they interact with the “frilled” primary particles by means of weak van
der Waals forces, rather than covalent bonds. Therefore, these additional condensed
molecules can also evaporate from the particle surface.
122
If the primary, ~ 7 nm particles, are non-spherical, then their coagulation near
the exhaust pipe by means of the condensed volatile compounds may result in particles
with mobility diameters (Rogak et al, 1993, Cleary et al, 1990, Okada and
Heintzenberg, 2003) that are approximately multiples of 7 (see the modes at ~ 7.9 nm,
10 – 15 nm, 20 nm, etc. in Fig. 6.4b). The suggested pattern of the evolution of the
aerosol can then be represented by the following five stages (Fig. 6.8). First, small non-
spherical carbon/graphite particles are formed inside the exhaust pipe by means of
homogeneous and heterogeneous nucleation and coagulation by means of strong
covalent bonds (stage 1). The graphite scale-like particles have free covalent bonds
around their perimeters. As mentioned above, volatile molecules and/or their fragments
engage these bonds forming the primary ~ 7 nm non-spherical (frilled) particles (stage
2). This also is likely to occur at high temperature inside the exhaust pipe. Immediately
after their emission, the exhaust gasses with the primary ~ 7 nm particles effectively
mix with the ambient air and rapidly cool down. As a result, effective condensation of
volatile molecules onto the primary nano-particles occurs (stage 3), with subsequent
coagulation of these particles by means of these volatile molecules (stage 4). As the
aerosol propagates away from the road, the concentration of volatile compounds
gradually reduces to that in the ambient air. As a result, the volatile molecules start
evaporating from the surface of the particles, and bonding may be significantly
weakened (stages 5). Eventually, the thermal energy kT may become sufficient for
fragmentation of the particles, and they break apart (stage 6).
The possibility of such thermal fragmentation becomes clear from the fact that
volatile molecules may evaporate (break away) from the surface of a particle (when the
thermal energy kT becomes of the order of the van der Waals energy of interaction
between the molecule and the particle). Since the molecules can break away from a
123
nano-particle, the bonds between particles (due to these molecules) can also eventually
break, resulting in particle fragmentation.
Fig. 6.8. Stages of the evolution of nano-particle aerosol near a busy road, according to the
fragmentation model. Stage 1: Coagulation (first type) of carbon nano-particles and formation
of the particle frill by means of covalent bonding in the exhaust pipe (i.e. at high temperature
and lack of oxygen) – formation of the original 7 nm particles. Stage 2: Heterogeneous
nucleation, i.e. further adsorption of volatile/organic/water molecules onto the surface of the 7
nm particles (due to van der Waals interaction) near the exhaust pipe where the concentration of
such molecules is large. Stage 3: Coagulation (second type) of the original 7 nm particles with
the adsorbed molecules providing bonding between the particles – formation of larger particles.
Stage 4: Evaporation of volatile/organic/water molecules in the ambient air where the
concentration of such molecules is much lower than near the exhaust pipe. Stage 5: Weakening
bonds between the 7 nm particles due to evaporation of the bonding molecules. Stage 6:
Fragmentation by means of breaking the primary 7 nm and 12 nm particles away from larger
particles.
It is important that the fragmentation is more likely to occur by means of
breaking smaller (~ 7 nm and/or ~ 12 nm) particles away from larger composite
124
particles. This is because smaller particles are likely to have smaller area of bonding. As
a result, the broad and strong maximum of correlations within the range from ~ 10 nm
to ~ 15 nm in Fig. 6.4b can be caused by fragmentation of larger particles with
diameters > 15 nm. Small errors of the moving average correlation coefficient in the
region of this maximum (Fig. 6.5b) strongly suggest that despite the breadth of this
mode, all its particles are likely to come from the same source – fragmentation of larger
particles. At the same time, it is also important to understand that fragmentation into
larger particles is also possible, and the corresponding evidence will be presented in
Chapters 9 and 11.
The shift of the modes within the range from ~ 35 nm to ~ 80 nm in Fig. 6.4b to
the left (compared to Fig. 6.4a) is likely due to fragmentation (mainly by breaking away
the ~ 7 nm and ~ 12 nm particles). On the contrary, the 98 nm mode does not change its
position noticeably. Thus, it might largely consist of a different type of particles that do
not experience thermal fragmentation at the considered temperatures (though another
possibility may be insufficient resolution of the size distribution at large particle
diameters). The modes at ~ 20 nm, ~ 26 nm, and ~ 31 nm (Figs. 6.4a,b) do not change
their positions significantly, because they are likely to be transformed one into another
by means of fragmentation. Note that the direct observation of the average particle size
distribution does not allow the resolution of, for example, the 26 nm and 31 nm modes –
they are normally seen as one mode at ~ 30 nm (Figs. 6.6 and 6.7). It is only the
developed statistical approach (see above) that reveals these modes and their mutual
transformation (the same is also true for the modes at Dp > 30 nm – compare Figs.
6.4a,b with Figs. 6.6 and 6.7).
The difficulties with interpretation of the main features of evolution of particle
size distributions in Figs. 6.6 and 6.7 are also eliminated if we assume the fragmentation
model of evolution. Indeed, as fragmentation goes on, a significant number of 7 nm and
125
12 nm particles are expected to be released. As a result, a significant increase of
concentration of particles in the 7 nm and 12 nm modes can be expected (see Figs. 6.6
and 6.7). As seen from Fig. 6.7b, the broad plateau of curves 4 and 5 (representing the
same size distribution) within the range 40 nm < Dp < 80 nm is strongly reduced by
fragmentation, resulting in curve 6 raising above the plateau of curve 5 within the range
15 nm < Dp < 40 nm. Thus the discussed 30 nm mode appears (curve 6).
Further fragmentation of particles from the 30 nm mode results in the 20 nm and
7 nm modes, and/or 12 nm mode, which is clearly shown by curves 2 – 5 in Figs. 6.6a,b
and curve 7 in Fig. 6.7b. This is also the reason for broadening the maximum of curve 4
in Fig. 6.6b. The strong influx of particles occurs into the 7 nm mode, caused by
fragmentation of 30 nm, 20 nm, and 12 nm modes. Despite dispersion, it results in
almost constant concentration of these particles within a significant range of distances
(curves 1 – 5 in Fig. 6.6 and curves 5 – 7 in Fig. 6.7b).
When the 20 nm and 30 nm modes have disintegrated, the particle influx into the
12 nm mode ceases, and this mode starts decaying with distance from the road due to
dispersion and further fragmentation – see Fig. 6.6b. This is why the 12 nm mode
significantly decreases before the 7 nm mode starts doing the same (curve 5 in Fig.
6.6b).
If we assume that the time that it takes for the volatile compounds to evaporate
does not depend on wind (if all other meteorological parameters are approximately the
same), then the distance at which the 30 nm mode appears in the size distribution must
increase approximately proportionally to increasing normal wind component. This is the
reason for the 30 nm mode appearing at ≈ 89 m from the road in Fig. 6.7b (curve 6),
compared to ≈ 45 m in Fig. 6.6a (curve 1).
The effects of nano-particle fragmentation can also be seen when the number of
heavy-duty trucks on the road is minimal – see Fig. 6.8 from (Chapter 5 and [A3]) for
126
the weekend measurements. In this case, the height of the 12 nm mode stays the same,
whereas the 20 nm mode disappears with increasing distance from 45 m to 70 m.
The observed maximum of the total number concentration at an optimal distance
from the road (Chapter 5 and [A3]) and strong increase of the total number
concentration within the evolution time of 25 s between the two sets of scans (section 3)
can also be naturally understood in the fragmentation model. Obviously, fragmentation
of the 12 nm, 20 nm, 30 nm, etc. modes should result in increasing number of smaller ~
7 nm and ~ 12 nm particles, which causes a maximum (Chapter 5 and [A3]) or strong
increase (section 3) of the total number concentration.
It is also clear that the fragmentation model could give alternative explanations
of several previously observed features of the size distribution. For example, the
observed dependencies from (Sakurai et al, 2003) could easily be re-interpreted by
means of fragmentation of larger particles into the primary 7 nm particles at
temperatures < 100 oC. In our opinion, such a re-interpretation seems more plausible
than the assumption that the 12 and 30 nm particles are ~ 95% volatile consisting of
thick shells of lubricating oil (of significantly different thicknesses) around small core
solid particles (Sakurai et al, 2003). The smaller size of the core particles of ~ 2.5 nm –
5 nm obtained at 200 oC (Sakurai et al, 2003) might be related to further reduction of
the size of primary 7 nm particles due to stripping the frill molecules off the
carbon/graphite nuclei at the temperatures > 100 oC (Sakurai et al, 2003).
6.6. Conclusions.
As a result of this work, a new method of statistical analysis of particle modes in
combustion aerosols has been developed. This method has been shown to provide a
unique insight into physical and chemical processes of evolution and transformation of
aerosol nano-particles. In particular, a new more accurate definition of modes of particle
size distribution has been introduced. A number of different modes has been revealed
127
from the experimental particle size distributions in the presence of strong turbulent
mixing. Highly unusual features of evolution of these modes and their mutual
transformation have been demonstrated and discussed.
As a result, a new mechanism of mode transformation, based on thermal
fragmentation of nano-particles, has been suggested and discussed. A comprehensive
complex pattern of aerosol evolution near a busy road has been proposed and discussed,
involving several different processes, such as particle formation, condensation,
coagulation, evaporation and, finally, thermal fragmentation. It has been demonstrated
that the presented model appears to be highly successful in explaining the major
experimental features of mode evolution. Interestingly, it can also account for the
previous experimental results on diesel nano-particles investigated by means of the
thermodesorption technique (Sakurai et al, 2003).
Appendix for Chapter 6
Here, we present an alternative approach to obtaining graphs in Figs. 6.4a,b
from Figs. 6.3a,b. Though not essential for understanding Figs. 6.4a,b, this approach
gives an interesting statistical insight into the described procedure of calculating moving
average correlation coefficient, and leads to the determination of the levels of
confidence for the correlations represented by Figs. 6.4a,b.
The alternative method of calculating moving average correlation coefficients
can be described as follows.
Step 1: As in Section 6.2, we consider an n-channel (n > 2) interval from the
overall N channels in a scan. Suppose that we again have M scans.
Step 2: Construct a column with the corresponding M different concentrations,
for example, in the first channel of the n-channel interval.
128
Step 3: Extend periodically this column, so that the resultant (extended) column
contains n – 1 sections identical to the original column with the M elements. Thus we
form a column with (n – 1)M elements.
Step 4: The remaining (n – 1) channels in the considered n-channel interval also
correspond to (n – 1) columns of M different particle concentrations corresponding to
the M scans. Then the second column with (n – 1)M elements is constructed of these (n
– 1) columns (with M elements each) by placing them one on top of another.
Step 5: A simple correlation coefficient between the two extended data columns
with (n – 1)M elements is then determined (Larsen & Marx, 1986).
Step 6: Steps 1 – 4 are repeated for the remaining n – 1 channels in the n-
channel interval, and the average correlation coefficient is again determined for the n-
channel interval.
Step 7: Steps 1 – 5 are repeated for all different n-channel interval, and the
dependence of the moving average correlation coefficient versus average particle
diameter is again obtained.
It can be shown that in terms of determining moving average correlation
coefficients, this statistical procedure is equivalent to that described in Section 6.2.
Therefore, Figs. 6.4a,b can equally be obtained either by means of the first procedure
described in Section 6.2, or the above second approach with the extension of the data
columns. However, the second approach provides significantly higher levels of
confidence of the obtained correlations due to larger number of elements in the data
columns (Larsen & Marx, 1986, Venable & Ripley, 2000). Thus the correct levels of
confidence should be determined using the second procedure. At the same time, one
should be careful when using the second procedure for the determination of statistical
errors (errors of the mean) for the obtained moving average correlation coefficients. If
these errors are calculated directly as the errors of the mean in the above procedure, they
129
appear to be smaller than those resulting from the first approach. This is because in the
second procedure we obtain only a fraction of the overall error of the moving average
correlation coefficient, and the correct estimate of the errors will require methods based
on analysis of variance (see Section 11 in (Larsen & Marx, 1986)). Thus, it is easier to
determine the errors of the average correlation coefficients, using the first approach
(Figs. 6.5a,b).
130
CHAPTER 7
MODELLING OF AEROSOL DISPERSION FROM A BUSY ROAD IN THE PRESENCE OF NANO-PARTICLE FRAGMENTATION
([A5, A19, A22, A26])
7.1. Introduction
In the previous Chapter, we have discussed and justified the new mechanism of
aerosol evolution based on intensive fragmentation of nanoparticles. It was suggested
that at least in some cases this mechanism may play a dominant role in the processes of
dispersion and transformation of nano-particle combustion aerosols, resulting in a
serious impact on human exposure and reliable forecast of aerosol pollution from busy
roads and road networks. This, in turn, may have significant implications for the
environmentally friendly urban design and minimise the effect of transport emissions on
human health. The presence of such implications can already be seen from the observed
maximum of the total number concentration at an optimal distance from the road (see
Chapters 5 and 6 and [A3, A4]). In accordance with the proposed physical interpretation
of this maximum (Chapter 6 and [A4]), it results from fragmentation of larger aerosol
particles as they are transported away from the road. Therefore, the observed maximum
of the total number concentration is mainly related to generation of small particles at
distances from the road of ~ 100 m, or so. This distance is usually of the order of, or
larger than typical setbacks for residential developments near busy roads, which may
clearly result in a significant exposure for the residents to the most harmful
nanoparticles from transport emissions.
Previously, the main interest of aerosol scientists has been focused on the study
of decay of the total number concentration of particles with distance from a busy road
(Benson, 1992, Shi et al, 1999, Hitchins et al, 2000, Zhu et al, 2002, [A1]). In
131
particular, exponential (Hitchins et al, 2000, Zhu et al, 2002) and power (Chapter 3 and
[A1]) decay laws were used for the description of the total number concentration of fine
particles as a function of distance from the road. This can approximately be done within
the range of particles > 30 nm, for which the Gaussian plume approximation (Csanady,
1980) is approximately correct (Chapter 3 and [A1]). However, this approximation
clearly fails in the range < 30 nm and, as has been suggested, this is largely due to not
taking into account intensive particle fragmentation.
Unfortunately, there are no current models for aerosol dispersion that take
particle fragmentation into account. None of the existing models can predict increase of
the total number concentrations near busy roads, and/or determine conditions at which
such increase may exist (resulting in a significant increase of exposure).
Therefore, the aim of this Chapter is to develop a simple semi-analytical model
of dispersion of fine particle aerosols from a busy road in the presence of fragmentation
of nano-particles. Rate equations for particle concentrations will be presented for the
cases when the fragmentation process is switched on abruptly and slowly (linearly) at
some distance from the road. In particular, it is demonstrated that the total number
concentration may be characterized by a significant maximum at an optimal distance
from the road. Simple analytical existence conditions of such a maximum are derived.
Comparison of the model with the available experimental results will be carried out, and
the applicability conditions will be derived and discussed; the typical fragmentation rate
coefficient will be determined to equal ≈ 0.086 s-1 with the estimated error of ~ 30%.
7.2. Dispersion as a chemical reaction
Consider a continuous ground line source of nano-particles, i.e. a road with the
wind direction at an angle θ to the road (Fig. 7.1). Initially we will assume that these
particles do not experience fragmentation/transformation in the atmosphere, i.e., we
132
assume the Gaussian plume approximation (Csanady, 1980). If the wind is normal to the
road, then the self-similarity theory (Csanady, 1980) suggests that the concentration c of
nano-particles is given by a power function of distance from the source (road):
c = C0 x–μ, (7.1)
where x is the dimensionless distance from the road in metres divided by 1 metre, C0 is
a constant depending on the strength of the source, and μ is a parameter that is close to 1
(it depends on wind speed and x; μ → 1 when x → + ∞ (Csanady, 1980)). The
numerical and experimental analysis (Chapter 3) suggested that this power dependence
is also valid for arbitrary wind direction, with the parameter μ depending on both the
wind components. In this case, the rate of changing concentration from the road (due to
dispersion of the Gaussian plume) is
dc/dx = – (μ/x)c (7.2)
(this follows from differentiation of Eq. (7.1)). This equation is valid in the stationary
frame K with the coordinates (x,y,z). Since the source is continuous and steady-state, the
concentration c is time-independent, and varies only with distance from the road (see
Eqs. (7.1) and (7.2)).
Fig. 7.1. The geometry of the problem: a road, point of observation, and wind direction.
133
Let the slender plume approximation (Csanady, 1980) be satisfied, i.e. the
normal component Ux of the wind velocity is sufficiently large, so that the aerosol
transport perpendicular to the road (along the x-axis) due to turbulent diffusion is
negligible compared to that due to the average wind (for a detailed discussion of the
applicability conditions for the model see Section 7.6). Consider another frame K′ with
coordinates (x′,y′,z′), moving away from the road with the velocity Ux, i.e., x = x′ + Uxt.
In the frame K′, the normal wind component is zero (on average), and the concentration
depends on time but not the x′-coordinate. Transformation of Eq. (7.2) to the K′ frame at
x′ = 0 (i.e., at the origin of the K′ frame) gives:
dc/dt = – kd*c, (7.3)
where kd* = μ/t.
Eq. (7.3) has the form of an equation describing a first-order chemical reaction
with the reaction rate coefficient kd*. Obviously, in the K frame, the reaction rate
coefficient for dispersion is kd = kd*/Ux = μ/x – see Eq. (7.2).
As has been demonstrated experimentally ((Csanady, 1980) and Chapter 3), μ
can approximately be regarded as a constant depending on atmospheric conditions. The
values of the constants μ and C0 are determined numerically by using, for example, the
software package CALINE4 (Benson, 1992) adapted for the analysis of dispersion of
fine particle aerosols in the absence of particle transformation (Chapter 3). Typically,
the coefficient μ ranges between ≈ 0.7 and ≈ 1.
7.3. Fragmentation of particles
In this section, fragmentation of nano-particles will be analysed together with
dispersion with the aim of modelling the observed maximum of the total number
concentration at an optimal distance from the road (Chapters 5 and 6).
134
Consider dispersion of particles A, each of which decomposes into n (different)
particles B1, B2, …, Bn (n ≥ 2):
A ⎯→⎯k B1 + B2 + … + Bn. (7.4)
According to the previous section, the variations of concentrations of A and
B1,…,n can be described by the rate equations:
d[A]/dx = – kd(x)[A] – k(x)[A], (7.5)
⎪⎩
⎪⎨
⎧
−=
−=
],)[(])[(d/][d...
],)[(])[(d/][d 11
ndn
d
BxkAxkxB
BxkAxkxB (7.6)
where the letters in the square brackets denote concentrations of the corresponding
particles, kd(x) = μ/x (see Eq. (7.2)), and k(x) is the x-dependent rate coefficient for the
reaction of fragmentation of A into B1,…,n. In other words, each of the rates of changing
concentrations [A], [B1], [B2], …, [Bn] (left-hand sides of Eqs. (7.5), (7.6)) is simply
equal to the sum of two independent increments due to dispersion (the terms with kd(x);
always negative), and due to fragmentation (the terms with k(x); negative for
fragmenting particles and positive for resulting particles).
Note that, unlike fragmentation, coagulation depends on particle concentrations,
and unless this concentrations are very large, it is highly inefficient (Jacobson, 1999,
Shi, et al, 1999, Jacobson and Seinfeld, 2004, Pihiola et al, 2003, Zhang & Wexler,
2004, Zhang et al, 2004). Thus, the reverse process of coagulation is neglected in Eqs.
(7.5) and (7.6). If required, coagulation can be considered by means of methods similar
to those used for analysis of dispersion of reactive gasses (Fraigneau, et al, 1995).
Since fragmentation is expected to switch on at some distance from the road
(where the bonds between the particles are weakened by evaporation of bonding volatile
molecules (Chapter 6), the coefficient k(x) is generally a function of distance from the
135
road x. Here, we will assume that this coefficient depends linearly on distance within
the interval (x1, x2):
⎪⎪⎩
⎪⎪⎨
⎧
>
≤≤−−
<
=
,for,
,for,
,for,0
)(
20
2112
10
1
xxk
xxxxxxxk
xx
xk (7.7)
where x1 is the distances from the road at which fragmentation starts, and x2 is the
distance at which the rate coefficient for fragmentation reaches the steady-state value k0,
i.e. the fragmentation process is completely “switched on” (Fig. 7.2a).
We use here a linear function approximation for the rate coefficient k(x), since
this allows simple analytical solution of the problem, results in lucid physical
interpretation of the results, and clearly demonstrates the general tendencies of the
solution (including the conditions for a maximum of the total number concentration –
see Section 7.4). In addition, linear function can also be used as an approximation of
other dependencies (e.g., an exponential dependence of k(x)). At the same time,
generalization to the case of a general monotonic (e.g., exponential) function k(x) does
not present major difficulties, but will in general require numerical methods of solution
of the rate equations.
Fig. 7.2. The considered linear (a) and step-wise (b) dependencies for the fragmentation rate
coefficient k(x) as a function of distance from the road, given by Eqs. (7.13), and (7.7),
respectively.
136
Here, we will mainly be interested in the total number concentration for all
particles. Therefore, Eqs. (7.6) are added together to give:
d[A]/dx = – kd(x)[A] – k(x)[A], (7.8)
d[B]/dx = nk(x)[A] – kd(x)[B], (7.9)
where [B] is the overall concentration of the fragmentation products: [B] = [B1] + [B2] +
… + [Bn].
Assuming that the concentration of the B particles before the fragmentation
process (i.e. at x ≤ x1) is zero, the solution to Eqs. (7.8) and (7.9) is
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
>+−−
≤≤−
−−
<
=
μ−
μ−
μ−
,for)],2
(exp[
,for],)(2
)(exp[
,for,
][
221
00
2112
210
0
10
xxxxxkxC
xxxxx
xxkxC
xxxC
A (7.10)
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
>+−−−
≤≤−
−−−
<
=
μ−
μ−
,for)]},2
(exp[1{
,for]},)(2
)(exp[1{
,for,0
][
221
00
2112
210
0
1
xxxxxkxnC
xxxxx
xxkxnC
xx
B (7.11)
In this case, the total number concentration [T] can be written as:
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
>+−−−−
≤≤−
−−−−
<
=
μ−
μ−
μ−
,for)]},2
(exp[)1({
,for]},)(2
)(exp[)1({
,for,
][
221
00
2112
210
0
10
xxxxxknnxC
xxxxx
xxknnxC
xxxC
T (7.12)
In the above derivation, n was assumed to be integer, corresponding to the
number of particles B that result from fragmentation of one particle A. However, if
originally we have particles A and A*, and only particles A experience fragmentation,
then Eqs. (7.10) and (7.11) do not change, while the extra term (1 – p)p-1C0x–μ should be
137
added to the total number concentration in Eq. (7.12) (where p = [A]/([A] + [A*])). This
term corresponds to the dispersion of particles A*. The resultant equation for [T] will be
exactly the same as Eq. (7.12), but generally with fractional n = p(n1 – 1) + 1, where n1
is the integer number of the B particles resulting from fragmentation of an A particle.
If the fragmentation process switches on abruptly at t = t0, i.e. at x = x0 ≡ x1 = x2,
k(x) is a step-wise function (Fig. 7.2b):
⎩⎨⎧
≥<
=,for,
,for,0)(
00
0
xxkxx
xk (7.13)
then the corresponding solutions for the rate equations (7.8) and (7.9) are obtained by a
simple limit transition x2 → x1 in Eqs. (7.10) – (7.12).
Eq. (7.13) means that all the particles reach the fragmentation stage at the same
time t0 = x0/Ux. It also assumes that the speed of transport of different particles from the
road is the same and equal to the average normal component of the wind Ux. This means
that the spread of particles (plume) in space due to turbulent diffusion is ignored (the
slender plume approximation (Csanady, 1980)). For more detailed analysis of the
applicability conditions for this assumption see Section 7.6.
The resultant typical total number concentrations as functions of distance from
the road are presented in Fig. 7.3a-d for different values of parameters μ, k0, and x1,2 at
the assumption that n = 2, i.e. each particle A disintegrates into two particles B1 and B2.
Figs. 7.3a-c are for the case when the fragmentation process switches on gradually
within several different intervals between x1 and x2 (Eq. (7.7)), whereas Fig. 7.3d
demonstrates the situation when the fragmentation process switches on abruptly at
several different distances x0 (Eq. (7.13)).
138
Fig. 7.3. Typical dependencies of the total number concentration of fine particles on distance
from a line source (busy road) in the presence of fragmentation of each of the original particles
into two. (a) – (c) Fragmentation process is “switched on” gradually, i.e., the rate coefficient
k(x) is a linear function within the interval between x1 and x2 (see Eq. (7.7)). (d) Fragmentation
is “switched on” abruptly at the distance x0 ≡ x1 = x2 from the road. C0 = 6×105 cm-3 for all the
subplots. The dotted curves in subplots (a) – (c) are identical.
(a) The effect of different values of x1,2 on the total number concentration: μ = 0.8, k0 = 0.04 m-1,
1) x1 = 0, x2 = 50 m, 2) x1 = 20 m, x2 = 70 m, 3) x1 = 40 m, x2 = 70 m, 4) x1 = 40 m, x2 = 90 m.
(b) The effect of k0 on the total number concentration: μ = 0.8, x1 = 40 m, x2 = 70 m, 1) k0 = 0,
2) k0 = 0.02 m-1, 3) k0 = 0.04 m-1, 4) k0 = 0.08 m-1. (c) The effect of μ on the total number
concentration: k0 = 0.04 m-1, x1 = 40 m, x2 = 70 m, 1) μ = 0.7, 2) μ = 0.8, 3) μ = 0.95. (d) The
effect of x0 on the total number concentration in the case of abruptly switched fragmentation
process: 1) x0 = 0, 2) x0 = 20 m, 3) x0 = 30 m, 4) x0 = 50 m, 5) x0 = 70 m; other parameters are
the same as in (a).
139
In particular, if the fragmentation process switches on (abruptly or gradually as a
linear function k(x)) immediately after the emission by the source, i.e., x1 = 0, then
strong dispersion in the immediate proximity to the source overpowers the tendency for
increasing total number concentration due to fragmentation. As a result, the
corresponding curves monotonically decrease with increasing distance from the road
(see curves 1 in Figs. 7.3a,d). However, if the distance at which the fragmentation
process switches on increases (x1 and x0 are non-zero), then an obvious ‘shoulder’
appears on the corresponding curves (see curves 2 in Figs. 7.3a,d). Further increase of
x1 and x0 may result in an appearance of a maximum of the total number concentration
at an optimal distance from the road (curves 3 – 5 in Figs. 7.3a,d).
Expectedly, decreasing slope of the linear dependence (7.7) results in a
reduction of the maximum of the total number concentration – compare curves 3 and 4
in Fig. 7.3a. Indeed, smaller slope of the dependence (7.7), results in slower
fragmentation, which leads to smaller maximum of the total number concentration.
Thus, in general, Figs. 7.3a,d suggest that the relative height of the maximum of
the total number concentration has a tendency to increase with increasing x1 and
decreasing Δx0 ≡ x2 – x1.
Similarly, the effect of variations of the steady-state rate coefficient k0 on the x-
dependencies of the total number concentration is demonstrated by Fig. 7.3b. Increasing
k0 at given values of x1 and x2 initially results in a shoulder on the curve (see curve 2 in
Fig. 7.3b), and then in a clear maximum that increases with increasing k0 (curve 3, 4 in
Fig. 7.3b). If k(x) is given by Eq. (7.13), then the corresponding dependencies of the
total number concentration on x at different values of k0 are very similar to those in Fig.
7.3b, but display derivative discontinuities similar to those in Fig. 7.3d.
The effect of different values of μ on the dependencies of the total number
concentration is demonstrated by Fig. 7.3c, where typical curves are presented for three
140
different values of μ = 0.7, 0.8, and 0.95. In general, increasing μ results in decreasing
maximum of the total number concentration (Fig. 7.3c). This is because μ usually
decreases with increasing normal component of the wind (as can be seen from the
analysis using the CALINE4 model). In this case, if Δx0 is constant, increasing normal
component of the wind results in faster (in time) switching on the fragmentation
process, which causes an increased maximum (Fig. 7.3c).
7.4. Existence conditions for the maximum of the total number concentration
The above consideration suggests that the predicted maximum of the total
number concentration depends on several parameters, namely, x1, x2, k0, μ, and the slope
of the linear dependence k(x). Firstly, assume that the fragmentation process switches
on abruptly (see Eq. (7.13)). Then the total number concentration is
⎪⎩
⎪⎨⎧
>−−−−
<=+
μ−
μ−
.for)]},(exp[)1({
,for,][][
0000
00
xxxxknnxC
xxxCBA (7.14)
Taking the derivative of Eq. (7.14) at x > x0 and equating it to zero gives
)(10xk
nn +μμ− = exp[k0(x – x0)], (7.15)
which should be satisfied at the maximum of the total number concentration.
Using the graphical method of solving Eq. (7.15), we can derive that this
equation has a solution only if
M1 ≡ μ−1
00nxk > 1, (7.16a)
or
M1 ≡ k0x0/μ > 1 (7.16b)
for n = 2, i.e. when fragmentation of each of the particles A occurs into two B particles.
141
Conditions (7.16a,b) determine the existence of a maximum of the total number
concentration in the case of a step-wise function k(x) (Eq. (7.13)). The number M1 in
this case is a fundamental number that determines the behaviour of the total number
concentration as a function of distance from the road. It also reflects the tendency of
increasing maximum of the total number concentration with increasing k0 and x0, and
decreasing μ.
For example, for curves 1 – 5 in Fig. 7.3d, M1 = 0, 1, 1.5, 2.5, and 3.5,
respectively. It can be seen that if condition (7.16b) is not satisfied (curves 1 and 2 in
Fig. 7.3d), the maximum of the total number concentration is absent, though curve 2
(with M1 = 1) displays a ‘shoulder’ near x = x0.
If the function k(x) is given by Eq. (7.7), i.e. by a linear function of x within the
interval x1 ≤ x ≤ x2, then conditions (7.16a,b) are still approximately applicable for the
determination of existence of the maximum of the total number concentration, if Δx0 ≡
x2 – x1 is noticeably smaller than Δxc. Here, Δxc is the distance within which particles A
decay into the B particles, i.e., the typical concentration of A decreases e times. In this
case, x0 in conditions (7.16a,b) should be replaced by x0 = x1 + Δx0/2.
If, on the contrary, Δx0 is large, so that the number of A particles decreases e
times within the distance that is smaller than Δx0 (i.e., Δxc < Δx0) then conditions
(7.16a,b) are not applicable, since the value of k0 does not matter in this case for the
process of fragmentation (particles A decay before k(x) reaches k0).
In this case, the typical distance within which the concentration of particles A
drops e times is
Δxc ~ kc-1, (7.17)
where kc is the average value of k(x) within the interval Δxc. On the other hand, since
k(x) is a linear function with the slope α, we have: kc = αΔxc/2. Substituting this value
142
of kc into Eq. (7.17), gives: Δxc ~ 2/(αΔxc), or Δxc ~ α/2 . As a result, the distance
from the road, at which the fragmentation process can be regarded as switched on, is xf
~ x1 + Δxc/2 ~ x1 + α2/1 . Substituting xf into condition (7.16a,b) instead of x0, and
taking into account that in this case k0 ~ kc = αΔxc/2 ~ 2/α , gives:
2)1(2/12/12 >−
μ+α≡ nxM , (7.18a)
or for n = 2 (fragmentation of one A particle into two B particles):
22/12/12 >
μ+α≡ xM . (7.18b)
Note that 2 in the right-hand side of these conditions does not follow directly from
the above derivations. It has been used, because the numerical analysis has
demonstrated that conditions (7.18a,b) are more accurate with the factor 2 .
If Δxc is noticeably larger than Δx0, then conditions (7.16a,b) and the number M1
should be used as the approximate existence conditions for the maximum of the total
number concentration. In this case, one should use x0 ≈ x1 + Δx0/2. On the contrary, if
Δxc is noticeably smaller than Δx0, then conditions (7.18a,b) and the second number M2
should be used instead. In fact, the numerical analysis shows that M1 can be used if Δx0
< Δxc/3, while M2 should be used if Δx0 ≳ Δxc. It is also important to understand that
conditions (7.16a,b) do not formally follow from conditions (7.18a,b) in the limit case
of Δxc ≈ Δx0. This is because they have been derived at different physical and
mathematical conditions. Therefore, the first number M1 depends on k0, whereas the
second number M2 depends on α – the slope of the linear function k(x).
It can also be seen from the above derivations that the inequality Δx0 > Δxc can
be expressed in terms of the slope α and k0:
143
α < k02/2. (7.19)
In this form this inequality is more convenient to use, since these are α and k0 that might
be known in practice.
As an illustration of condition (7.18b), we present the values of the number M2
for the curves in Figs. 7.3a-c. For the curves in Fig. 7.3a, M2 is equal to: (1) 0.625, (2)
1.125, (3) 1.916, (4) 1.625. Therefore, condition (7.18b) is satisfied for curves 3 and 4
in Fig. 7.3a, and these curves indeed display maximums of the total number
concentration. At the same time, condition (7.18b) is not satisfied for curves 1 and 2 in
Fig. 7.3a, and these curves do not display such a maximum (though curve 2 shows a
‘shoulder’ at ~ 70 m from the road). For curves in Fig. 7.3b, M2 equals: (1) 0.625, (2)
1.538, (3) 1.916, (4) 2.451. Curve 2 in this figure is at the border-line for condition
(7.18b), and only a small maximum can be seen in this case. For curves in Fig. 7.3c, the
values of M2 are: (1) 2.190, (2) 1.916, (3) 1.614.
These examples demonstrate the direct relationship between the maximum of
the total number concentration and values of the second number M2: increasing M2
beyond the critical value 2 results in increasing relative strength of the maximum of
the total number concentration at an optimal distance from the road.
7.5. Comparison with the experimental results
One of the most interesting results in Chapter 5 was the experimental
observation of a significant maximum of the total number concentration at some
distance from a busy road. The measured total number concentrations and their errors at
9 different distances from the centre of the road (presented in Fig. 5.7 in Chapter 5) are
reproduced in Fig. 7.4a by the experimental points. As can be seen, a significant
maximum of the total number concentration is reached at the distance ≈ 82 m from the
road.
144
Fig. 7.4. The comparison of the theoretical curves with the experimental results (points
with the error bars) from (Chapter 5 and [A3]).
(a) Dotted curves are the x-dependencies of the total number concentration of particles
for the Gaussian plumes in the absence of fragmentation and dry deposition for the two different
values of C0 = 6.69×105 cm-3 (dotted curve 1) and C0 = n6.69×105 cm-3 ≈ 12×105 cm-3 (dotted
curve 2). The other parameters are as follows: μ = 0.87, background concentration: ~ 1000 cm-3,
θ = 41o (±40o), U = 2.05 m/s (±0.9 m/s), and temperature: 26.5 oC (±0.9 oC) [A3]. The solid
curve is obtained from Eq. (7.12), i.e. in the presence of fragmentation, with the parameters: x1
= 46 m, x2 = 71 m, n = 1.8, and k0 = 0.1, which gives M2 ≈ 2.9 > 2 .
(b) Demonstration of the effect of dry deposition of particles on the theoretical
dependence and its fit to the experimental points (including the additional point at x = 232 m,
discarded in [A3] as an outlier). The solid curve is identical to that in Fig. 7.2a, while the dotted
curve is the theoretical dependence taking into account dry deposition of the 7 nm particles at x
> xm ≈ 75 m. The parameter for the dotted curve are the same as for the solid curve, except for k0
= 0.075 m-1, and n = 1.9, resulting in M2 ≈ 2.6 > 2 .
145
Explanation of this maximum by means of particle formation is not feasible
because of several reasons discussed in detail in Chapter 6 and [A4]. One such reason is
that the concentration maximum is mainly related to increasing concentration of
particles within the range between ~ 7 nm and ~ 20 nm. At the same time, if particle
formation were responsible for this effect, concentration within the range ≤ 7 nm should
have increased especially strongly (Chapter 6 and [A4]).
The second possible explanation may be related to a thermal rise of the heated
exhaust gasses due to buoyancy. This is similar to having an elevated source that may
result in a concentration maximum at some distance from the road. To demonstrate the
impossibility of using this mechanism for the explanation of the observed maximum
(Fig. 7.4a) consider a strongly exaggerated case of the road elevated by 8 m above its
actual level. Then the numerical analysis of aerosol dispersion from the road by means
of the CALINE4 model ((Benson, 1992) and Chapter 3) with the stability class 1
(unstable) suggests that the concentration maximum should have been obtained at the
distance x ~ 45 m. In this case, the concentration at x = 25 m (the first point in Fig.
7.4a) should have been smaller (by ~ 10%) than that at x = 45 m (decreasing elevation
results in decreasing maximum and shifting it closer to the road). This is in obvious
contradiction with the experimental points in Fig. 7.4a. Thus, thermal rise has nothing to
do with the observed maximum of the total number concentration, and we are left only
with the third possible interpretation – fragmentation of nano-particles. The expectation
that the thermal rise does not have a noticeable effect on aerosol dispersion near a busy
road (highway) was confirmed experimentally by Zhu & Hinds (2005).
Using the atmospheric conditions mentioned above, and the actual shape of the
road of 27 m width as the inputs in the CALINE4 model (Benson, 1992) adjusted for
the analysis of a Gaussian plume of fine particles (Chapter 3 and [A1]), we obtain that μ
≈ 0.87. We also choose C0 = 6.69×105 cm-3, x1 = 46 m, x2 = 71 m, k0 = 0.1 m-1, and n =
146
1.8, which means that only 80% of particles A experience fragmentation into two (on
average) particles B. Substituting all these parameters into Eq. (7.12), results in the
dependence of concentration on distance from the centre of the road, given by the solid
curve in Fig. 7.4a. Dotted curve 1 in Fig. 7.4a gives the dependence of concentration of
particles A on distance in the absence of fragmentation, whereas dotted curve 2
represents the same dependence, but with 1.8 times larger concentrations (i.e. with 1.8
times larger constant C0). In this case, if n = 1.8 (as assumed above), the curve for the
total number concentration [T] (solid curve in Fig. 7.4a) must obviously split from
dotted curve 1 at the distance x1 where fragmentation starts switching on, and merge
with dotted curve 2 when fragmentation finishes – see Fig. 7.4a.
In particular, it can be seen that the solid curve in Fig. 7.4a fits well to the
experimental points up to ~ 100 m from the road (the statistical consideration is
presented below). However, at distances > 100 m from the road, the theoretical solid
curve starts deviating (with the tendency upwards) from the most of the experimental
points (Fig. 7.4a).
A possible reason for this is that the theoretical model presented in Sections 7.2
and 7.3 assumes that there are no particle losses during dispersion, apart from the
fragmentation process. However, it has been suggested that fragmentation should
mainly occur by means of breaking the primary 7 nm particles away from larger
particles from the 12 nm, 20 nm, and 30 nm modes (Chapter 6 and [A4]). Thus, we
should expect that the observed maximum of [T] should be caused by generation of the
7 nm particles due to fragmentation. On the other hand, it is known that 7 nm particles
may be effectively deposited on interfaces by means of dry deposition (Jacobson, 1999),
resulting in a noticeable loss of the overall number of such particles. This however has
not been taken into account in Eqs. (7.10) – (7.12).
147
Dry deposition occurs when particles are transported by turbulent diffusion to
the 0.01 cm – 0.1 cm laminar layer that exists near any surface (Jacobson, 1999). Then
the particles cross the layer by means of molecular diffusion and stick to the surface. It
is known that the smaller the particles, the larger their deposition coefficient due to
larger molecular diffusivity. For example, for 7 nm particles, the deposition speed is v ~
0.1 m/s (Jacobson, 1999), while deposition of larger particles will be neglected.
To estimate the contribution of dry deposition to the theoretical solid curve in
Fig. 7.4a, we assume that the 7 nm particles appear as a result of fragmentation at the
distance x ≈ xm ≈ 75 m (at which the maximum of the solid curve in Fig. 7.4a is
achieved), and their concentration is approximately equal to the concentration of
particles A at the same distance in the absence of fragmentation. This assumption is
reasonable, because we took n = 1.8, i.e., 80% (on average) of all particles A
disintegrate into two particles B. If all these B particles were from the 7 nm mode, then
their concentration should have been ~ 1.8 times larger than that of particles A at the
same distance in the absence of fragmentation. However, not all particles B should
necessarily belong to the 7 nm mode, because fragmentation may occur into one 7 nm
particle and another larger composite particle that may not experience further
fragmentation, for example, due to stronger bonding. Therefore, in this estimate, it is
reasonable to assume that the concentration of the 7 nm particles at x ≈ xm ≈ 75 m is
approximately equal to that of particles A at the same distance in the absence of
fragmentation.
Thus, if dry deposition is not taken into account, then Gaussian dispersion of the
7 nm particles at x > xm must be approximately given by dotted curve 1 in Fig. 7.4a.
The contribution of dry deposition is estimated from conservation of particles.
Consider an imaginary plane at the distance xm from the road, at which the
fragmentation process is assumed to have finished. This plane is parallel to the road,
148
perpendicular to the ground, has a length l in the direction parallel to the road (along
the y-axis), and is infinitely high (along the z-axis). If the transport due to turbulent
diffusion along the x-axis is neglected compared to the average wind transport (see
Section 7.6), then the flux of the 7 nm particles through this plane is
∫=+∞
0d),()( zzxcUxF mxm l , (7.20)
where z is the vertical distance from the ground, Ux is the average x-component of the
wind (the fact that the normal wind component may depend on z is not essential for this
consideration, and Ux is assumed z-independent), c(xm, z) is the z-dependent
concentration of the 7 nm particles (it is estimated by dotted curve 1 in Fig. 7.4a at x =
xm ≈ 75 m).
As the aerosol is transported away from the road beyond x = xm, the flux of the 7
nm particles into the ground, due to dry deposition, can be estimated as
∫ ′=′=x
xd
m
xdzxcvxF )0,()( l , (7.21)
where v ≈ 0.1 m/s is the speed of dry deposition of the 7 nm particles (Jacobson, 1999),
and c(x, z = 0) is the concentration of the 7 nm particles at the ground level, estimated in
the absence of dry deposition, i.e. given by dotted curve 1 in Fig. 7.4a. The actual
ground concentration of these particles is affected by dry deposition, and will be slightly
smaller. However, this is unlikely to be an overestimate, because, for example, dry
deposition of larger particles was not taken into account, and we have probably
underestimated the concentration of 7 nm particles (e.g., because they also exist before
the beginning of fragmentation).
At a given distance from the road (and fixed meteorological parameters), the
flux of particles through a vertical plane is proportional to the concentration at the
ground level. On the other hand, for a Gaussian plume in the absence of deposition, the
149
flux through the considered plane should be independent of distance from the road (due
to particle conservation). Therefore, the ratio of the concentration cd(x, z = 0) at a
distance x from the road, including the losses due to dry deposition, to that in the
Gaussian plume in the absence of the deposition, c(x, z = 0), is given as
)()(1
)0,()0,()(
m
dd
xFxF
zxczxcxf −===≡ . (7.22)
Thus, the dependence for the total number concentration [T] in the region x > xm
in Eq. (7.12) should be multiplied by the factor f(x). The resultant new dependence of
[T] on x takes into account the effect of dry deposition of 7 nm particles.
Using the above meteorological parameters and the constant C0 = 6.69×1011 m-3
(for dotted curve 1 in Fig. 7.4a) in the CALINE4 model, the vertical concentration
profile c(xm, z) for the 7 nm particles and the corresponding flux F(xm) are calculated.
Estimating the ground concentration of the 7 nm particles from dotted curve 1 in Fig.
7.4a, the flux due to dry deposition is also calculated: Fd(x) = (1 – μ)-1C0(x1 – μ – xm1 – μ)
(see Eq. (7.21)). As a result, the factor f(x) is determined from Eq. (7.22).
If dry deposition is taken into account (as described above), the best fit of the
theoretical curve to the experimental points (Venables and Ripley, 2000) is achieved if
the value of k0 is reduced to 0.075 m-1, and n is slightly increased to n = 1.9, i.e. 90% of
particles A experience fragmentation into two (on average) particles B. The resultant
theoretical (dotted) curve taking into account dry deposition is presented in Fig. 7.4b
together with the experimental points, together with the previously discarded point at x
= 232 m. For comparison, the previous theoretical (solid) curve calculated without dry
deposition (Fig. 7.4a) is also presented in Fig. 7.4b. It can be seen that the fit of the new
curve allowing for dry deposition of the 7 nm particles at x > xm is significantly better at
distances x > 100 m, compared to that without dry deposition.
150
The statistical analysis has shown that the residual standard error of the curve
with dry deposition is about ± 1500 cm-3 (Venables and Ripley, 2000). The value for the
fragmentation rate coefficient can thus be estimated as k0 ≈ 0.075 m-1 (with the error ~
30%). If we wish to represent the fragmentation reaction as a function of time, rather
than distance from the road, then the corresponding value of the fragmentation rate
coefficient is given by k0* = k0Ux ≈ 0.086 s-1 (assuming that at the time of concentration
measurements near the maximum Ux = 1.15 m/s (Chapter 5 and [A3])).
7.6. Applicability conditions
As mentioned in Section 3, if we have a step-wise variation of the fragmentation
rate coefficient from zero to k0 (Eq. (7.13)), this means that the fragmentation stage is
reached by all particles A simultaneously at a given distance from the road. However,
even if the time that takes for different particles to reach the fragmentation stage is the
same: t0 = x0/Ux, turbulent diffusion results in stochastic motion, and different particles
reach the fragmentation stage at different distances. Thus step-wise function (7.13) must
be spread by means of turbulent diffusion, i.e. particles reaching the fragmentation stage
should be found typically within the interval x0 – σx < x < x0 + σx, where σx is the
standard deviation of the plume along the x-axis (i.e., 2σx is the increase of the size of
the plume along the x-axis due to turbulent diffusion). Use of Eq. (7.13) can only be
justified if
σx << x0. (7.23)
To estimate σx, we consider the turbulent intensities i∥ and i⊥ (Csanady, 1980)
in the directions parallel and perpendicular to the wind, respectively. These intensities
are determined by the fluctuations u∥ and u⊥ of the wind components on the directions
parallel and normal to the average direction of the wind (Csanady, 1980):
151
i∥ = 2||
1 uU − , i⊥ = 21⊥
− uU , (7.24)
where U is the magnitude of the mean velocity of the wind (Fig. 7.1).
Suppose that the average direction of the wind makes an angle θ with respect to
the road (Fig. 7.1). It can be shown that the root-mean-square fluctuation of the wind
component onto the x-axis (normal to the road) is given as
θ+θ= ⊥2222
||2 cossin uuux , (7.25)
and the corresponding turbulent intensity
ix = θ+θ= ⊥− 2222
||21 cossin iiuU x . (7.26)
The standard deviation of the plume along the x-axis σx ≈ Dixxα (Csanady,
1980), where α ≈ 0.87, and D is a dimensional coefficient of the order of 1 with the
units m1–α. This relationship and inequality (7.23) give the required applicability
condition for using Eq. (7.13).
Usually, i� ~ 0.1, and i� ~ 0.25 – 0.55 for unstable atmospheric conditions
(Csanady, 1980). Assuming that i� ~ 0.4, θ = 41o, xm ≈ 75 m (these are the parameters
corresponding to Figs. 7.4a,b), and using Eq. (7.26), we have σx ~ 12.5 m. It can be seen
that in this case, the function k(x) can hardly be regarded as step-wise due to relatively
large value of σx (the turbulent spread of the step is ~ 25 m). The approximation of a
step-wise dependence of k(x) can thus be reasonable only for near-normal wind (i.e.,
when Ux is significantly larger than Uy) and fairly stable atmospheric conditions. For
example, if θ = 90o, and i∥ ~ 0.1, then σx ≈ 4 m << xm ≈ 75 m. In more common
situations, condition (7.23) is fairly difficult to satisfy.
Turbulent spread of the step-wise function (7.13) results in different particles A
reaching the fragmentation process at different distances from the road, which on
152
average can be approximated by a linearly increasing fragmentation rate coefficient
within the interval between x1 = x0 – σx and x2 = x0 + σx. This is another reason for using
linear dependence of k(x) – see Eq. (7.7).
Therefore, condition (7.23) is not essential for the applicability of the developed
model, since if this condition is not satisfied, the model can still be used, but with the
fragmentation rate coefficient given by Eq. (7.7) with x1 = x0 – σx and x2 = x0 + σx.
If the rate coefficient k(x) is approximated by a linearly increasing function
because of gradually switching fragmentation process (see Section 3 and Eq. (7.7)), then
turbulent diffusion will result in a further reduction of the typical slope of the x-
dependence of the effective rate coefficient. In this case, a new interval between x′1 = x1
– σx and x′2 = x2 + σx should rather be used.
For example, the above estimate of σx ≈ 12.5 m for the experimental results in
Figs. 7.4a,b suggests that the interval within which the effective (average)
fragmentation rate coefficient should change from 0 to k0 could be about 25 m, which is
in agreement with the difference between x2 and x1 used for plotting curves in Figs.
7.4a,b. This may suggest that in the considered experiment, the dependence of k(x) in
the absence of turbulent diffusion should be close to a step-wise function given by Eq.
(7.13). However, turbulent diffusion has spread this function within the interval of ≈ 25
m, effectively giving the linear dependence.
The approximation of the effect of turbulent diffusion by means of a linearly
changing fragmentation rate coefficient has its limitations. It can only be used if the
interval 2σx is not larger than the interval Δxc, within which the fragmentation process
finishes (here, Δxc is calculated in the same way as in Section 7.4, but for the effective
rate coefficient k(x) varying linearly between x′1 and x′2):
Δxc > 2σx. (7.27)
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If condition (7.27) is not satisfied, but turbulent diffusion is represented by a
linearly changing k(x), then fragmentation will formally result in the total depletion of
particles A within the interval Δxc that may be significantly smaller than the actual
interval within which this depletion occurs. This actual interval should not be smaller
than 2σx – the interval within which particles A that have just reached the fragmentation
process are dispersed by turbulent diffusion. More detailed analysis of situations when
condition (7.27) is not satisfied is beyond the scope of this chapter.
As mentioned above, for the experimental results presented in Figs. 7.4a,b, we
have 2σx ≈ 25 m, and Δxc is estimated to be ≈ 26 m. Therefore, the developed theory can
approximately be used (see Figs. 7.4a,b and Section 7.5).
The finite width of the road may also contribute to an additional spatial
dispersion (along the x-axis) of particles reaching the fragmentation stage. However,
this additional dispersion may not be as strong as one could expect it to be. Thermal
fragmentation may occur when volatile molecules responsible for bonds between the
particles evaporate (Chapter 6 and [A4]). On the road (in the mixing zone), the process
of evaporation of volatile compounds may be significantly impeded by large
concentrations of these compounds. Therefore, it is possible that all particles that cross
the kerb and leave the road may have approximately the same amount of volatile
molecules that would require approximately the same time to evaporate (irrespectively
of the width of the road). Thus finite width of the road may not lead to an additional
significant spread of the dependence of k(x) in space (along the x-axis). However, this
suggestion requires further confirmation by means of more detailed theoretical and
experimental investigation.
It is possible to indicate that the obtained applicability conditions for the model
are not over-restrictive. They are typically satisfied when the average wind direction
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makes a noticeable angle with respect to the road (e.g., in the considered experiment θ =
41o). The conventional slender plume approximation (Csanady, 1980) is typically a
sufficient (but not necessary) applicability condition for the model.
7.7. Conclusions
This Chapter developed a new model for a simple semi-analytical analysis of
dispersion of ultra-fine particle aerosols from a busy road in the presence of intensive
fragmentation of nano-particles (Chapter 6 and [A4]). The possibility of existence of a
maximum of the total number concentration at an optimal distance from the road,
caused by particle fragmentation, was derived theoretically. Simple analytical existence
conditions for this maximum were determined when the fragmentation process switches
on abruptly at some distance from the road, or linearly increases within some distance
near the road. As a result, two fundamental numbers representing these existing
conditions were suggested and discussed.
An agreement between the model and the previous experimental results (Chapter
5 and [A3]) was demonstrated. In particular, it was shown that taking into account dry
deposition of particles noticeably improves the fit of the theoretical curve to the
available experimental data. As a result, the value of the fragmentation rate coefficient
was found: k0* ≈ 0.086 s-1 (± 30%).
Main applicability conditions of the developed model were derived. The
important role of turbulent diffusion for the model was analysed. In particular, it was
suggested that increasing turbulence may eventually result in breaching the applicability
of the model. At the same time, the usual approximations (e.g., the slender plume
approximation (Csanady, 1980)) should normally be sufficient (but not always
necessary) for the applicability of the model.
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CHAPTER 8
MULTI-CHANNEL CANONICAL CORRELATION ANALYSIS OF COMBUSTION AEROSOLS: SOURCES OF PARTICLE MODES
([A6])
8.1. Introduction
In Chapters 6 and 7, a new mechanism of aerosol evolution based on intensive
particle fragmentation has been developed and used for the detailed interpretation of
evolution of particle modes in combustion aerosols near a busy road. As a result, a
complex evolution pattern has been suggested. This includes several stages such as
formation of solid and liquid nano-particles inside and in the vicinity of the vehicle
exhaust, their coagulation near the exhaust pipe (where particle concentrations are
sufficient for such a process to occur (Shi et al, 1999, Jacobson, 1999, Jacobson and
Seinfeld, 2004)), evaporation of volatile compounds from the surface of solid particles
as the aerosol is transported away from the road, loss of bonding between coagulated
nano-particles due to evaporation of bonding volatile molecules and, finally, thermal
fragmentation of nano-particles (Chapter 6). Numerous experimental evidence
supporting this evolution pattern have been presented, including substantial
transformation of particle modes and their shift towards smaller particle size (Chapters
5 and 6), observation and modelling of a maximum of the total number concentration at
an optimal distance from the road (Chapters 5, 6, and 7), direct observations and
statistical confirmation of generation of strong modes that would be expected as a result
of fragmentation of larger particles (Chapter 6), etc.
A new method of statistical analysis of particle modes in combustion aerosols
near a busy road has been developed (Chapter 6), based on the moving average of the
correlation coefficients between neighbouring channels of the particle size distribution.
This method allows determination and analysis of particle modes in the presence of
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strong turbulent mixing, even if these modes are not directly seen (as distinct
maximums) on the size distribution. As a result, particle modes have been re-defined as
groups of particles of similar dimensions, corresponding to distinct maximums of the
moving average correlation coefficient (Chapter 6).
The aim of this Chapter is to develop and use new statistical approaches for the
detailed analysis of modes of particle size distribution near a busy road, including their
possible sources, mutual transformations during the aerosol evolution, correlations with
atmospheric and meteorological parameters. This will be done by means of the
extension of the previously developed method based on the moving average technique
(Chapter 6) to the multi-variate canonical correlation analysis. In particular, modes
resulting primarily from heavy diesel trucks and petrol cars will be identified, and the
dependencies of particle modes on such parameters as temperature, humidity and solar
radiation will be analysed. Physical interpretation of the obtained results is presented on
the basis of the fragmentation model of aerosol evolution.
8.2. Experimental data and particle modes
The development of the new statistical methods and their application for the
analysis of particle modes and their evolution will be conducted on the basis of the
experimental data previously discussed in Chapter 6. The data were obtained during the
field campaign on 25 November 2003 near Gateway Motorway in Brisbane, Australia.
Measurements of particle size distribution were conducted, by means of a scanning
mobility particle sizer (SMPS-3936) and condensation particle counter (CPC-3025). 50
scans (previously discussed in Chapter 6) were taken during ~ 3 hours of measurements
at the distance of ≈ 40 m from the centre of the road, within the range of particle
diameters from 4.6 nm to 163 nm in 100 equal intervals (channels) of Δlog(Dp), where
Dp is the particle diameter in nanometres. The time for one full scan was 2.5 minutes,
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with a 1 min down-scan. The width of the Motorway was ≈ 27 m , and its elevation
above the surrounding area was ≈ 2 m.
Traffic conditions were recorded on a video camera (with short breaks for
changing batteries). The traffic on the road was subdivided into two groups: heavy-duty
trucks and cars (the car group including gasoline and diesel cars and light trucks).
Numbers of vehicles in each of these two groups were determined within 2.5 min
intervals by means of direct counting from the video tape. The beginning of each of the
2.5 min intervals for traffic counting was taken L/vn seconds earlier than the beginning
of the corresponding scans. Here, L = 40 m is the distance from the centre of the road,
and vn is the one-hour average normal component of the wind. This was done in order to
take into account average time delays associated with the aerosol transport from the
road to the point of monitoring, so that to determine the traffic conditions corresponding
to particular scans. The obtained results from traffic counting were then used for the
statistical analysis of canonical correlations between particle modes, traffic and
meteorological conditions (see below).
Wind speed, wind direction, temperature, humidity, and solar radiation were
measured every 20 seconds by a automatic weather station at the same distance from the
road. In Chapter 6, the two sets of scans (out of the overall 50 scans) from 1 to 11 and
from 19 to 38 were chosen, because these sets correspond to approximately constant
(but noticeably different) one-hour average normal wind components (the variations of
the average wind components within these sets are within the standard deviation of the
mean – see also Fig. 6.2). In this paper, when considering correlations between particle
modes and traffic and meteorological conditions, we will choose slightly different sets
of scans from 6 to 16 (11 scans) and from 28 to 43 (16 scans) – Table 8.1. This new
choice has been made because of the following two reasons. First, the new sets still
correspond to approximately constant (but distinctly different) one-hour average normal
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wind components – see Fig. 6.2 from Chapter 6 and Table 8.1. Second, the recorded
traffic conditions are available for each of the scans from the new sets, and this will
allow analysis of canonical correlations between particle modes and traffic conditions.
Part (a) Part (b).
Wind direction (degrees to the road) 50 ± 20 20 ± 20 Wind speed (ms-1) 2.4± 0.2 2.6 ± 0.8 Normal component of the wind velocity, v⊥ (ms-1)
1.7 ± 0.16 1.0 ± 0.2
Parallel component of the wind velocity, v|| (ms-1)
1.7 ± 0.2 2.3 ± 0.8
Solar radiation (Wm-2) 800 ± 40 530 ± 70 Humidity (%) 30 ± 2 36 ± 1 Temperature (°C) 28 ± 1 27 ± 1 Total number concentration (cm-1) 22.6×103 39×103
Heavy-duty trucks (hour-1) 800 ± 200 790 ± 140 Cars (hour-1) 3800 ± 400 4800 ± 500
Table 8.1. Average meteorological and traffic conditions together with their standard
deviations.
Note that significantly different average normal wind components for the two
selected sets (Table 8.1) correspond to significantly different times that it takes for the
aerosol to be transported from the road to the point of observation. The aerosol
transportation time for the first set of 11 scans is smaller than that for the second set by
≈ 16 s. As a result, the two sets correspond to two different stages of the aerosol
evolution.
The average size distributions are determined for each of the two sets of 11 and
16 scans – Fig. 8.1. These distributions are plotted using the moving average method.
The concentrations in every channel in each scan are normalised to the total number
concentration in this scan. Then we choose an interval of 5 neighbouring channels (out
of the total 100 channels) and average the normalised particle concentrations over these
5 channels in the scan and over all scans within a particular set of 11 or 16 scans. Thus
we obtain an average particle concentration, which gives us one point on the size
distribution. Then another interval of 5 neighbouring channels is chosen and the
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procedure is repeated, giving another point on the size distribution. Repeating this
procedure for all 96 different 5-channel intervals, we obtain 96 average concentrations.
Connecting all these points, gives the moving-average particle size distributions for the
11-scan and 16-scan sets. The errors associated with these distributions are obtained by
calculating the errors of the mean for each of the 96 average concentrations. These
errors and the determined size distributions are represented in Fig. 8.1 by the two bands
corresponding to the sets of 11 and 16 scans.
Fig. 8.1. Average particle size distributions and their errors of the mean for the first set of 11
scans (light band) and the second set of 16 scans (dark band), obtained using the moving
average technique. Before averaging, the concentrations in each channel in every scan were
normalised to the total number concentration in the corresponding scan.
The moving average technique for plotting average particle size distributions
seems to be more efficient and accurate than that used in Chapter 6. This is because it is
highly efficient in revealing main average features and tendencies of a particle size
distribution and results in easy and straightforward way of determination of the
corresponding errors of this distribution (Fig. 8.1). Some differences between the size
distributions in Fig. 8.1 and those obtained in Chapter 6 can be explained by the
different (more accurate) smoothing technique used in this Chapter and slightly
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different normal wind components for the considered sets, which correspond to slightly
different times of aerosol evolution, especially for the second set of 16 scans. For this
second set, the average normal wind component is ≈ 1.0 m/s (see Table 8.1), while for
the slightly shifted set considered in Chapter 6 it was ≈ 0.75 m/s (rounded up to 0.8 m/s
– Chapter 6).
Note that the 5-channel intervals for plotting the moving average size
distributions (Fig. 8.1) have been chosen as a compromise between the errors of the
mean (which increase with decreasing number of channels in the interval) and the
resolution of the features of the size distribution (which decreases with increasing
number of channels in the interval). In principle, an optimal number of channels within
a selected interval of neighbouring channels should be chosen separately for each
experimental data set.
Despite the use of the improved averaging and smoothing technique, direct
observation of the particle size distributions (Fig. 8.1) can still hardly be used for clear
identification of all particle modes and especially their mutual interaction, evolution,
major tendencies and correlations, and possible sources. Therefore, in this Chapter, the
detailed analysis of particle modes and determination of their possible sources were
conducted by means of the previously developed statistical method based on the moving
average correlation coefficient (Chapter 6). In this method, particle concentrations in
each of the 100 channels in every scan are normalized to the total number concentration
in the corresponding scan. Then we choose 7 neighbouring channels out of the 100
channels in a scan. The number of channels within the interval is again determined as a
compromise between the mode resolution and statistical errors of the resultant curves
(Chapter 6). Simple correlations between particle concentrations in all possible pairs of
different channels from the 7-channel interval are determined for the sets of 11 or 16
scans, and then the average correlation coefficient is calculated. Thus, considering all
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possible different 7-channel intervals, the moving average correlation coefficient is
plotted versus particle diameter (for more detailed description of this procedure see
Chapter 6).
Fig. 8.2. The dependencies of the moving average of the correlation coefficient, R, between the
particle concentrations in neighbouring channels on particle diameter; the bands represent the
associated statistical errors (similar to Figs. 6.4 and 6.5). The moving average correlation
coefficients are calculated over (a) the first set of 11 scans (scans from 6 to 16); (b) the second
set of 16 scans (scans from 28 to 43).
The resultant dependencies of the moving average correlation coefficient on
particle diameter, together with their errors of the mean are presented in Figs. 8.2a,b for
the two sets of 11 and 16 scans, respectively. As discussed in (Chapter 6), it is
reasonable to redefine particle modes as groups of particles with similar diameters,
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corresponding to maxima of the moving average correlation coefficient. The reason for
such a definition is that particle concentrations in different channels corresponding to
these maxima tend to increase/decrease in strong (maximal) correlation with each other,
and this is an indication that these groups of particles are likely to come from the same
source and/or have the same physical/chemical nature. Therefore, these particle groups
can naturally be called “modes”. In our opinion, this is physically more reasonable than
defining particle modes as maxima of the size distributions (Fig. 8.1). At the same time,
it is important to note that modes as maxima of the moving average correlation
coefficient often (but not always) include modes as maxima of the particle size
distributions (compare Fig. 8.1 with Figs. 8.2a,b).
The dependencies in Figs. 8.2a,b appear to be somewhat different from those
considered in Figs. 6.4a,b from Chapter 6, though the major features and modes remain
the same. These differences are again due to the different choice of the sets of scans,
and the resultant differences in the one hour average normal wind components, which is
equivalent to slightly different stages of the aerosol evolution. For example, previously
obtained ~ 12 nm mode (Fig. 6.4b) was noticeably stronger and had much smaller
statistical errors of the correlation coefficient, than the same mode in Fig. 8.2b. The
existence of the ~ 12 nm mode in Fig. 6.4b was explained by means of intensive thermal
fragmentation of larger particles, resulting in generation of a large number of particles
within the range ~ 10 – 13 nm (Chapter 6). However, Fig. 8.2b corresponds to the larger
(by ≈ 0.25 m/s) one hour average normal wind component (Table 8.1). Therefore, Fig.
8.2b corresponds to ≈ 13 s earlier stage of aerosol evolution than Fig. 6.4b. In other
words, the ~ 12 nm mode in Fig. 8.2b did not have sufficient time to properly develop,
and thus it is characterized by smaller moving average correlation coefficient and larger
errors of the mean. Fig. 8.2b is thus intermediate between Fig. 6.4a and Fig. 6.4b for the
two sets of scans considered in (Chapter 6).
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Similar situation occurs for ~ 6 – 8 nm particles. Thermal fragmentation results
in an inflow of a large number of solid particles into this range (Chapter 6). This causes
large moving average correlation coefficients (Fig. 6.4b), because these particles have
the same nature (they mainly come from the process of fragmentation), and their
concentrations in different channels tend to increase/decrease in strong correlation with
each other. However, Fig. 8.2b corresponds to ~ 13 s earlier stage of evolution, and the
number of ~ 6 – 8 nm particles resulting from fragmentation is still relatively small.
This is the reason for significantly lower correlation coefficients within this range in
Fig. 8.2b, compared to Fig. 6.4b.
The correlation maximums at ~ 6 – 7 nm diameter in Figs. 8.2a and 6.4a are
unrelated to the above interpretation, because, as it will be demonstrated in the next
section, particles causing these maximums are mostly volatile. They rapidly evaporate
during the aerosol evolution, and thus do not exist in Figs. 8.2b and 6.4b. In particular,
the evaporation process results in shifting the maximum from ~ 6 – 7 nm in Fig. 8.2a to
~ 5 – 6 nm in Fig. 6.4a, because Fig. 6.4a corresponds to ~ 3 – 5 s later stage of the
evolution, compared to Fig. 8.2a. For more justification of this conclusion see below.
8.3. Moving average approach to the canonical correlation analysis
In this section, the developed moving average approach is extended to the case
of canonical correlation analysis of particle modes and their dependence of traffic and
meteorological parameters.
Canonical correlation analysis determines correlations between two groups of
variables, when the variables in each of the groups depend on each other (Dillon and
Goldstein, 1984, Johnson and Wichern, 2002). A significant advantage of this method is
that it can determine the effect of each of the mutually dependent parameters (e.g.,
traffic and/or meteorological parameters) from a group A on some variable or a group
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of variables (e.g., particle concentrations in different modes or channels) from another
B.
Canonical correlation analysis gives three major output parameters: correlation
coefficients between specially determined linear combinations of variables from group
A with linear combinations of variables from group B, canonical weights which are the
coefficients in these linear combinations, and canonical loadings which are the simple
correlation coefficients between the a variable from one of the groups and the
considered linear combination containing this variable (Dillon and Goldstein, 1984,
Johnson and Wichern, 2002). The canonical correlation coefficient R measure the
strength of association between the two groups of variables.
For example, suppose that group A contains only one variable (e.g.,
concentration in one of the particle modes) and group B contains traffic and
meteorological parameters (see below). Then R2 determines the strength of correlation
between the mode concentration and the mentioned traffic and meteorological
parameters. If R2 = 1, then the mode concentration depends only on the parameters from
group B. If however R2 < 1, then the mode concentration depends on some other factors
that are not included in group B. That is, usually, the larger the coefficient R, the larger
the contribution of the considered parameters to variations of the mode concentration.
Canonical weights determine the contributions of the corresponding variables from one
of the groups of variables to the variance (i.e., variation) of the linear combination of
the other group of variables (Dillon and Goldstein, 1984, Johnson and Wichern, 2002).
In the above example, if the canonical weight, i.e., the coefficient in front of a variable
from group B, is large and positive, then increasing/decreasing this variable results in a
significant increase/decrease of the mode concentration. Finally, canonical loadings
may be used for the assessment of stability of the obtained correlations and
relationships. For example, if the canonical loading corresponding to a variable from
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group B and the corresponding canonical weight have different signs, then the
correlation is not stable, i.e., the obtained weight cannot be used for the above
interpretation. For the detailed mathematical description of the canonical correlation
analysis see (Dillon and Goldstein, 1984, Johnson and Wichern, 2002).
The canonical correlation coefficient R is actually the correlation coefficient for
the first canonical variates, and it should have been denoted as R1 (Dillon and
Goldstein, 1984, Johnson and Wichern, 2002). However, the correlation coefficient for
the second canonical variates R2 << R1 for all cases considered below. Therefore,
correlations for the second and higher variates can be neglected (Dillon and Goldstein,
1984, Johnson and Wichern, 2002), and for the sake of simplicity, we omit index 1 in
the correlation coefficient for the first variates, denoting it simply by R.
In order to use the canonical correlation analysis for the investigation of particle
modes and their sources, the counted numbers of heavy-duty trucks and light cars
corresponding to each of the scans from the two selected sets of scans (as discussed in
Section 8.2) are included in group B of variables. For each of the scans, we determine
the average temperature, humidity, and solar radiation and also include them in group
B. Group A includes average particle concentrations in different channels. Usually,
group A will be assumed to contain only one concentration, for example, in a particular
channel under investigation.
To apply the moving average approach to the canonical correlation analysis, we
choose an interval of 7 neighbouring channels out of the overall 100 channels in one
scan. As usual, the number of channels within one interval is chosen so that to achieve a
reasonable compromise between the statistical errors (e.g., the scatter of the resultant
points on the graph) and the sufficient resolution of the major features (particle modes)
on the resultant distributions – see also Section 8.2 and (Chapter 6). Consider, for
example, the first set of 11 scans from 6 to 16 (the analysis for the second set of 16
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scans is the same). Each of the selected 7 neighbouring channels corresponds to a
column of 11 different concentrations from each of the 11 scans. Taking the average
over the 7 channels, we obtain one column with 11 different average concentrations.
The concentrations in this average column are taken as the average concentrations for
the central channel of the considered 7-channel interval.
Each scan corresponds to the determined numbers of heavy-duty trucks and light
cars (see Section 8.2), and particular (averaged over the scan time) meteorological
parameters: temperature, humidity and solar radiation. We take one channel out of the
selected 7-channel interval. This channel corresponds to a column of 11 different
concentrations corresponding to 11 different scans. Canonical correlations between this
column of concentrations (one variable in group A), and the numbers of trucks and cars
and the corresponding meteorological parameters (group B) are calculated. As a result
we determine the canonical correlation coefficient R, canonical weights and loadings
(for example, for the numbers of trucks and cars). The procedure is then repeated for the
remaining 6 channels from the selected 7-channel interval. Taking the average over the
7 channels, we can obtain the average canonical correlation coefficient, canonical
weights and loadings for the considered 7 channel interval.
This procedure is then repeated for all 94 possible different 7-channel intervals
(out of the 100 channels in each scan), and thus 94 different canonical correlation
coefficients, weights and loadings are found. As a result, we obtain moving average
canonical correlation coefficients, weights and loadings as functions of particle diameter
that is taken as the diameter for the middle channel for each of the 7-channel intervals.
Similar dependencies of moving average correlation coefficient, weights and
loadings can also be obtained by means of an alternative statistical procedure. Instead of
calculating the average concentrations for each scan within each of the 7-channel
intervals, we rearrange these 7 columns with 11 different concentrations into one
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column by placing the 7 columns one on top of another (Chapter 6). The order in which
these 7 columns are placed on top of each other does not matter. The resultant big
column will have 77 different concentration values for the sets of 11 scans. The column
with 11 different numbers of trucks on the road (corresponding to each of the 11 scans)
is then periodically repeated 7 times, resulting in another extended column with 77
numbers of trucks in it (each of the 7 “sub-columns” of this column being identical).
The same is done for the corresponding columns with the car numbers on the road,
average (over the scan time) temperature, humidity, and solar radiation. Then the
canonical correlations between the concentration column with the 77 elements (group A
with just one variable) and the other 5 similar columns for the truck and car numbers,
temperature, humidity and solar radiation (group B with 5 variables) are calculated,
resulting in the corresponding correlation coefficient, canonical weights and loadings
for all the 5 variables from group B.
The same procedure is repeated for all other 94 different 7-channel intervals.
Thus, we obtain the dependencies of the corresponding moving average canonical
correlation coefficient, weights and loadings on particle diameter. This diameter is again
taken to equal the diameter corresponding to the central channel of the considered 7-
channel interval. The same procedure is repeated for the other selected set of 16 scans
(however, in this case, each of the extended columns contains 112 elements rather than
77).
As mentioned above, the resultant dependencies of the canonical correlation
coefficient, weights and loadings on particle diameter are very similar (though not
identical) for both the procedures. The two procedures have some complementary
advantages. For example, the direct determination of levels of confidence of canonical
correlations in the first procedure is difficult, whereas this can easily be done in the
second procedure by means of the usual methods in the canonical correlation analysis
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(Dillon and Goldstein, 1984, Johnson and Wichern, 2002). On the other hand, errors of
weights are difficult to find in the second procedure, whereas the first approach
immediately provides them as the errors of the mean.
The use of canonical correlation analysis solely for descriptive purposes requires
no assumptions about the data distribution used. However, for testing levels of
confidence of the analysis results the data should meet the requirements of normality
and homogeneity of variance (Dillon & Goldstein, 1984). Therefore, all the extended
data columns for concentrations, traffic and meteorological parameters were checked
for normality using quantile-normal plots (Dillon & Goldstein, 1984), demonstrating
high correlations with quantiles (sorted values) of normal distribution. For example, the
resultant correlation coefficients for both the sets of scans (between 6 and 16 and
between 28 and 43) were within the range ≈ 0.96 ± 0.02 with the exception of a few
channels with Dp < 6 nm in the first set of 11 scans. For these few channels, the
correlation coefficients with the quantiles of the normal distribution were within the
range ≈ 0.72 ± 0.9. As a result, it was concluded that the data transformation to the
normal distribution is not required (Dillon & Goldstein, 1984).
All the data was standardised to enable quantitative conclusions and
comparisons of the canonical weights and loadings (Dillon & Goldstein, 1984, Johnson
& Wichern, 2002).
It is also important to note that similar approach is applicable not only for the
canonical correlation analysis, but also for the determination of simple moving average
correlation coefficients, for example, between the average particle concentration in a
particular channel and the number of trucks, or cars, etc. However, canonical correlation
analysis gives a much more comprehensive and reliable analysis of mutual relationships
between multiple variables in real-world situations (Dillon and Goldstein, 1984,
Johnson and Wichern, 2002), like those considered in this thesis.
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8.4. Sources of particle modes.
The dependencies of the moving average simple correlation coefficients between
the average particle concentrations in different channels on the one hand, and the
numbers of trucks and cars for the considered two sets of scans on the other, are
presented in Figs. 8.3a,b. The dotted horizontal lines represent the 80% level of
confidence, and the solid horizontal lines correspond to the 95% level of confidence.
Fig. 8.3. The dependencies of the moving average correlation coefficient, R, between channel
concentrations and numbers of trucks (curve 1) and cars (curve 2) on particle diameter. The
moving average correlation coefficients are calculated for (a) the first set of 11 scans (scans
from 6 to 16); (b) the second set of 16 scans (scans from 28 to 43).
One of the main aspects of Figs. 8.3a,b is that particles with different diameters
are characterised by substantially different simple correlations with numbers of cars and
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trucks on the road. This is an indication that the statistical approach may be highly
effective in the determination of sources of different types of particles in combustion
aerosols. That is, it should be possible to determine which particles originate
predominantly from cars, and which from trucks. The second important aspect of Figs.
8.3a,b is that the correlations drastically change with changing evolution time (compare
Figs. 8.3a and 8.3b). This is a clear indication of rapid evolution processes (e.g.,
evaporation and thermal fragmentation of nano-particles (Chapter 6) that occur as the
aerosol is transported from the road (recall that the evolution time difference between
the set of 11 scans (Fig. 8.3a) and the set of 16 scans (Fig. 8.3b) is ≈ 16 s – see Section
8.2). Thus the proposed statistical approaches can also be used for the investigation of
these processes.
For example, the curves in Fig. 8.3a suggest that at earlier stages of aerosol
evolution (~ 24 s after the emission) there is a possibility of an association between the
~ 14, 25, 40, 90 nm particles and cars (curve 2 in Fig. 8.3a), and between the ~ 6, 55
nm, ≳ 110 nm particles and trucks (curve 1 in Fig. 8.3a). However, 16 s later (Fig.
8.3b), these associations have drastically changed. For example, strong positive
correlations with trucks for the ~ 6 nm and ~ 120 nm particles for the first set of scans
(earlier stage of evolution – Fig. 8.3a) change to negative correlations for the second set
of scans (≈ 16 s later stage of evolution). This suggests that the maximum of
correlations at ~ 5 nm in Fig. 8.2b can hardly result from shifting the maximum at ~ 6 –
7 nm in Figs. 8.2a to the left, because this maximum is strongly related to trucks (solid
curve in Fig. 8.3a), whereas the maximum at ~ 5 nm in Fig. 8.2b is related to cars (Fig.
8.3b). The most logical explanation may be that the maximum of the solid curve in Fig.
8.2a at ~ 6 – 7 nm is likely to be volatile, and its particles simply evaporate within the
16 s of evolution to the stage corresponding to Fig. 8.3b. This is also in agreement with
the transformation of the dependence in Fig. 8.2a into the dependence in Fig. 6.4a
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within ~ 3 s of aerosol evolution (see the last paragraph of Section 8.2). This is also the
reason for increased correlations of particles with < 6 nm diameters with cars (Fig.
8.3b). More justification that the 6 – 7 nm mode in Fig. 8.2a is volatile is provided
below by the canonical correlation analysis.
Strong variations of correlations of large particles with > 110 nm diameter
within the same 16 s of evolution between Figs. 8.3a and 8.3b is likely to be related to
thermal fragmentation of these particles, which noticeably change the concentration of
these particles and their mutual correlations (see also below).
At the same time, using just simple correlations for the determination of possible
sources of different types of particles and their mutual transformation/evolution may not
be reliable. Indeed, only small sections of the curves in Figs. 8.3a,b lie beyond the 95%
level of confidence. The other problem is that simple correlations establish relationships
only between two variables (e.g., particle concentration and number of trucks). At the
same time, these correlations may be significantly affected by some other parameters
(e.g., number of cars and meteorological parameters), variation of which may introduce
significant errors into the resultant simple correlation coefficients. Therefore, simple
correlation analysis is not reliable and accurate for a real-world problem with numerous
independent (and dependent) variables and external factors, as in the considered case of
aerosol dispersion and evolution near a busy road.
Therefore, canonical correlation analysis should be used instead. As mentioned
in Section 8.3, the canonical correlation analysis establishes correlations between
groups of independent or dependent parameters, which gives more accurate and reliable
predictions with significantly higher levels of confidence. Thus, though Figs. 8.3a,b
may suggest some interesting tendencies and results, these must surely be verified by
the canonical correlation analysis involving the determination of correlations between
particle concentrations on the one hand (group A), and traffic and meteorological
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parameters on the other (group B). Here, we will mainly use the second procedure
described in Section 8.3 with extended columns of the particle concentrations, and the
first procedure will only be used for estimating statistical errors of canonical weights.
Five parameters affecting particle concentration are considered. These are
numbers of trucks and cars, corresponding to each scan, and average (over the
considered scan) temperature, humidity and solar radiation. Wind speed and direction
are not included, because we choose the sets of scans for which these parameters are
approximately constant (see above). This was done because increasing number of
parameters involved (in this case up to 7) results in reduction of levels of confidence of
the obtained results (Dillon & Goldstein, 1984). Therefore, the sets with constant
average wind were chosen instead, in order to determine correlations with other
parameters. Same signs of canonical weights and loadings for almost all particles (see
below) will confirm the validity of this approach.
For example, the procedures described in Section 8.3 were used for the
determination of the dependencies of the moving average canonical correlation
coefficient on particle diameter for the set of 11 scans (thick solid curve in Fig. 8.4a)
and for the set of 16 scans (thick solid curve in Fig. 8.4b).
In particular, Figs. 8.4a,b clearly demonstrate that the canonical correlation
analysis substantially improves the levels of confidence of the obtained results. Indeed,
in this case, almost the entire curves lie beyond the lines corresponding to the 95% level
of confidence. This is one of the important indications that the obtained results may be
reliable (for the discussion of stability of the obtained correlations see the analysis of
canonical weights and loadings). Only within a few relatively small regions the level of
confidence of the obtained correlations is below 95% (Figs. 8.4a,b), and this is an
indication that within these regions the statistical analysis is not reliable.
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Fig. 8.4. The dependencies of the moving average canonical correlation coefficients on particle
diameter. Group A: channel particle concentration (one variable). Solid thick curve: five
variables in group B – temperature, humidity, solar radiation, number of trucks, and number of
cars. The other three curves correspond to group B with just four variables: (1) without the
number of trucks, (2) without the number of cars, (3) without the solar radiation. (a) The
dependencies for the first set of 11 scans (scans from 6 to 16). (b) The dependencies for the
second set of scans (scans from 28 to 43). Straight horizontal lines indicate the 95% levels of
confidence when group B contains all five variables (solid lines), and for groups B with just
four variables (dash-and-dot lines).
Note that Figs. 8.4a,b demonstrate again that the obtained canonical correlations
strongly depend on particle diameter. Numerous strong maximums of the correlation
coefficient may provide important information about the nature and evolution of nano-
particles in a combustion aerosol. For example, if one of the parameters is removed
174
from group B, and the corresponding moving average correlation coefficient
significantly decreases for some particular particle diameters, then this is a strong
indication that this missing parameter has a significant impact on the particle
concentration. For example, curves 1 in Figs. 8.4a,b correspond to group B without the
number of trucks on the road. Therefore, not including truck number in group B has a
significant effect on correlations within the particle diameters ~ 6 – 7 nm and ~ 11 nm
in Fig. 8.4a. Therefore, it is possible to conclude that the corresponding particle modes
in this set of scans (Fig. 8.2a), i.e., ~ 24 s after emission, are likely to be more
associated with trucks rather than cars. At the same time, particles with the diameters
between ~ 20 nm and ~ 28 nm, and between ~ 60 nm and ~ 110 nm are more related to
cars than to trucks – see curve 2 in Fig. 8.4a (although particles around ~ 90 nm also
show relation to trucks). The effect of solar radiation is most noticeable for particles
between ~ 9 nm and ~ 50 nm (curve 3 in Fig. 8.4a).
For the second set of 16 scans (~ 16 s later), the situation significantly changes.
The most noticeable aspect is the strong dependence of the particles between ~ 5 nm
and ~ 10 nm on number of cars, which was completely different for the set of 11 scans.
This suggests that something has happened within the ~ 16 s of evolution between the
two sets of scans, which has led to the drastic increase of the correlation of small
particles with cars (see also for other significant differences between Figs. 8.4a and
8.4b).
This analysis, however, gives only qualitative information about the importance
of the contribution of different parameters to variations of particle concentrations in
different channels. In order to obtain more specific quantitative information, including
signs of correlations, we need to consider canonical weights (see Section 8.3). The
dependencies of the moving average canonical weights and loadings for trucks and cars
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on particle diameter for the sets of 11 and 16 scans are presented in Figs. 8.5a,b and
Figs. 8.6a,b.
Fig. 8.5. Moving average canonical weights (solid curves) and loadings (dotted curves) for
trucks in group B with the five variables: temperature, humidity, solar radiation, number of
trucks, and number of cars. (a) The dependencies for the first set of 11 scans (from 6 to 16). (b)
The dependencies for the second set of scans (from 28 to 43). The dashed horizontal lines
indicate the 95% level of confidence for the loading curves. The horizontal solid line
corresponds to zero. The typical errors of the curves for the moving average canonical weights
(solid curves) are ~ ± 0.1.
It can be seen that the presented dependencies of canonical weights for trucks
and cars on particle diameter are rather similar to the dependencies of the simple
correlation coefficients in Figs. 8.3a,b. However, the canonical weights and loadings in
Figs. 8.5a,b and 8.6a,b provide more reliable information about ranges of particles
where the determined tendencies can be trusted. For example, for the first set (evolution
time of ~ 24 s) for particles with the diameters < 8 nm, both canonical weights and
loadings for trucks (solid and dotted curves in Fig. 8.5a) have the same (positive) signs.
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This is an indication that the corresponding correlations are likely to be stable (Dillon
and Goldstein, 1984, Johnson and Wichern, 2002), and increasing number of trucks
results in increasing number of particles within the range < 8 nm.
Fig. 8.6. Moving average canonical weights (solid curves) and loadings (dotted curves) for cars
in group B with the five variables: temperature, humidity, solar radiation, number of trucks, and
number of cars. (a) The dependencies for the first set of 11 scans (from 6 to 16). (b) The
dependencies for the second set of scans (from 28 to 43). The dashed horizontal lines indicate
the 95% level of confidence for the loading curves. The solid horizontal line corresponds to
zero. The typical errors of the canonical weights (solid curves) are ~ ± 0.1.
On the contrary, the weights and loadings for cars (thin solid and dashed curves
in Fig. 8.6a) both have negative signs within the range of particle diameters between ~ 5
nm and ~ 8 nm. Therefore, the corresponding correlations are again likely to be stable,
and increasing number of cars results in decreasing particle concentration within the
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indicated range. This could be explained by the fact that increasing number of cars on
the road should automatically results in decreasing number of trucks (due to limited
traffic flow), thus causing number of the considered particles to drop, because their
main source is heavy trucks.
Another reason for the opposite signs of traffic correlations for modes related to
cars and trucks is due to the normalisation procedure. If, for example, we consider a
mode that is predominantly related to cars, and the number of trucks changes, then there
will be additional negative correlations (due to normalisation) of the considered mode
with the trucks, because increasing number of trucks will result in increasing total
number concentrations. This, in turn, will lead to decreasing relative concentration in a
mode that is predominantly related to cars. It is important to note that this effect is
rather beneficial because it allows better distinguishing between modes coming from
different sources (different types of vehicles).
Thus, according to the statistical evidence obtained from three different methods
(see Figs. 8.3a, 8.4a, and 8.5a), the ~ 6 – 7 nm mode in Fig. 8.2a (i.e., for the first set of
measurements) is very likely to be associated with heavy-duty trucks on the road. The
comparison of Figs. 8.5a and 8.5b also shows that the large positive maximums of
canonical weights and loadings for trucks at ~ 5 – 7 nm (Fig. 8.5a) have drastically
dropped into strong negative minimums in just 16 s of evolution (Fig. 8.5b). These
maximums in Fig. 8.5a do not seem to move to the right, but rather disappear in Fig.
8.5b (shift to the left beyond the detectable range). This is a fairly clear indication that
these particles are mostly volatile, and they simply evaporate within the 16 s of
evolution between Figs. 8.5a and 8.5b. Thus, as has been mentioned in (Chapter 6) and
during the discussion of Fig. 8.3a,b, the ~ 6 – 7 nm mode in Fig. 8.2a and ~ 5 – 6 nm
mode in Fig. 6.4a are likely to be of the same nature and formed of liquid particles that
are related to heavy-duty diesel trucks on the road. These volatile particles disappear in
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Figs. 6.4b and 8.2b due to their evaporation in the ambient air, leaving instead solid
particles resulting from fragmentation of larger particles, and from shrinking 8 – 10 nm
particles caused by evaporation of the volatile shell (Chapter 6).
The ~ 10 – 14 nm modes in the first set of scans (Fig. 8.2a) seem to be more
associated with cars rather than with trucks (Figs. 8.5a and 8.6a). Indeed, the canonical
weights and loadings for cars are both positive (Fig. 8.6a), while for trucks they are both
negative (Fig. 8.5a). This suggests that decreasing number of trucks results in increasing
particle concentration in these channels, and vice versa for cars.
For the same first set of scans, the canonical weights and loadings for trucks
within the range between ~ 15 nm and ~ 25 nm provide mainly inconclusive results,
because in this case weights and loadings either have opposite signs (indicating
instability of the correlations), or are small in magnitude (also indicating possible
instabilities) – Fig. 8.5a. On the other hand, the canonical weights and loadings for cars
are both positive for particles within the range 22 – 30 nm (Fig. 8.6a), which suggests
that the 25 nm mode is associated with cars (see also Fig. 8.3a). For particles between ~
30 nm and ~ 35 nm, the level of confidence of canonical correlations (Fig. 8.4a) is
insufficient for reliable conclusions. At the same time, for the ~ 40 nm mode (Fig. 8.2a),
the truck weights and loadings are both negative (Fig. 8.5a), while the weights and
loadings for cars are both positive (Fig. 8.6a). Therefore, this mode has a tendency to be
related to cars rather than to trucks. This is also confirmed by the simple correlations in
Fig. 8.3a. There is a strong indication that the 50 – 60 nm mode is associated with
trucks (Fig. 8.5a). However, this conclusion may be put under some question by the
insufficiently large canonical correlation coefficient within this range of particle
diameters (see Fig. 8.4a). Particles within the range between ~ 60 nm and ~ 100 nm
provide inconclusive results, because they are negatively correlated with both trucks
and cars (Fig. 8.5a). Therefore, it is possible to expect that there are some other more
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dominant factors/processes affecting particle concentrations within this range, which
may overpower the effect of increasing traffic flow and result in non-linear
dependencies causing correlation instabilities (Fig. 8.6a). At the same time, there is a
clear and strong association of particles with larger diameters ≳ 120 nm with diesel
trucks – the corresponding weights and loadings are large and positive (Fig. 8.5a), while
the weights and loading for cars are both negative (Fig. 8.6a).
The described pattern of correlations is likely to be typical for the stage
preceding particle fragmentation, because this is the stage when volatile liquid shells
have not yet evaporated from the solid particles, and bonding between coagulated
particles (caused by the volatile molecules) is still strong (Chapter 6). The next 16 s of
aerosol evolution result in substantial changes of the correlation pattern – Figs. 8.5b and
8.6b. For example, the major difference between Figs. 6a and 6b is the disappearance of
the volatile ~ 6 – 7 nm mode and the substantial alterations of weights and loadings
within the range > 70 nm.
Fig. 8.6b also demonstrates a few interesting results. First, the smallest
registered particles at ~ 5 nm are clearly associated with cars – see the positive weights
and loadings for cars (Fig. 8.6b) and negative weights and loadings for trucks (Fig.
8.5b). The second region is between ~ 7 nm and ~ 9 nm, where canonical weights and
loadings for cars are large and negative (Fig. 8.6b), and the weights and loadings for
trucks are positive (Fig. 8.5b). This is an indication that these particles are more
associated with trucks. Curve 2 in Fig. 8.4b and simple correlations in Fig. 8.3b confirm
this conclusion.
The third range where some conclusions could be made is between ~ 17 nm and
~ 23 nm. Particles from this range tend to be associated with both cars and trucks – see
the corresponding weights and loadings in Fig. 8.5b and 8.6b. The last two ranges
within which more or less certain conclusions can be made are ~ 40 – 100 nm
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(association with trucks) and ~ 70 – 110 nm (association with cars) – see Figs. 8.5b and
8.6b. Note, however, that the association with trucks seems to be significantly stronger
for these ranges (compare curves in Fig. 8.5b and Fig. 8.6b).
In other regions the presented canonical correlation analysis does not give
consistent results and reasonable conclusions. This means that in those regions, particles
and their modes cannot be associated reliably and predominantly with cars or trucks, but
rather come from both these sources, or turbulent fluctuations and additional processes
(like thermal fragmentation) mask the correlations. For example, the ~ 10 – 13 nm
particles in Figs. 8.5b and 8.6b tend to be negatively correlated with both cars and
trucks. Surprisingly, this could be explained by the fragmentation mechanism of
evolution of nano-particle aerosols. Indeed, as has been mentioned in discussion of Fig.
8.2b and Fig. 6.4b (see also Chapter 6), particles from this range at the considered
evolution time and conditions are likely to result from thermal fragmentation of larger
particles. Fragmentation may only occur when volatile bonding molecules effectively
evaporate, resulting in weakening bonds between the coagulated particles, eventually
leading to their fragmentation (Chapter 6). Evaporation of volatile molecules may only
occur if the concentration of these molecules in the ambient air is below the
concentration in the saturated vapour. On the other hand, increasing traffic flow (i.e.,
numbers of cars and trucks on the road) results in increasing concentration of volatile
compounds in the air. As a result, increasing traffic flow may cause decreasing rate of
fragmentation, and thus decreasing number of particles generated by fragmentation.
Because ~ 10 – 13 nm particles in the second set of scans are assumed to be generated
by fragmentation of larger particles (Chapter 6), their concentration may decrease with
increasing numbers of cars and trucks, leading to negative correlations with both (Figs.
8.5b and 8.6b). The presented interpretation is also likely to be the key reason for
observing condensation, rather than fragmentation, at the same distances from the road
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in (Zhang, et al, 2005) – in that paper the traffic flow was ~ 5 times larger than in our
experiments.
The discussed mechanism of partial suppression of fragmentation by increased
traffic flow should affect not only the ~ 13 nm mode, but also the ~ 7 nm mode, which
also results from fragmentation of larger particles (Chapter 6). As a result, particles with
~ 6 – 7 nm diameter in Figs. 8.5b and 8.6b are also characterised by strong negative
correlations with both cars and trucks on the road. A similar situation possibly occurs
for particles within the range between ~ 30 nm and ~ 40 nm in Figs. 8.5b and 8.6b. This
may also be the reason for exceptionally large negative correlations with number of
cars, and insufficiently strong positive correlation with number of trucks in the range
between ~ 8 nm and ~ 10 nm (Fig. 8.6b).
Further confirmation of the determined particle sources and correlations can be
obtained by means of canonical correlation analysis of concentrations in different
modes. This is done by including particle concentrations in different particle modes
(Fig. 8.2a) into group A of variables. Concentration in a mode is taken as the average
concentration within the 7-channel interval with the middle channel centred at the
considered mode. This middle channel will naturally be used for identification of the
considered modes.
Table 8.2 gives examples of the canonical correlation coefficients, weights and
loadings for the first canonical variates. Particle modes included into the A group of
variables are shown in the first column of Table 8.2. Canonical weights for modes are
the coefficients in front of mode concentrations in the first canonical variate for group
A; weights for traffic are the coefficients in front of numbers of trucks and cars on the
road in the first canonical variate for group B. The canonical correlation coefficient R
(the second column of Table 8.2) is the correlation coefficient for the first canonical
variates (Dillon and Goldstein, 1984, Johnson and Wichern, 2002). The correlation
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coefficient for the second canonical variates R2 << R for all cases considered below and
correlations for the second variates can be neglected.
Particle modes,
nm
R Level of conf.
Weights for modes
Loadings for modes
Traffic Weights for traffic
Loadings for traffic
6, 53, 136
0.56 99% 0.58 0.07 0.47
0.99 0.15 0.46
TrucksCars
0.93 -0.22
0.98 -0.39
26, 31
0.35 95% 1.14 -0.27
0.95 0.28
TrucksCars
-0.14 0.96
-0.30 0.99
14, 26, 40, 88
0.66 99% 0.87 0.67 0.55 0.32
0.69 0.34 -0.20 0.18
TrucksCars
-0.58 0.71
-0.71 0.82
Table 8.2. The results of the canonical correlation analysis for different particle modes
in the first set of 11 scans (Fig. 8.2a). Group A: the considered particle modes (the first column
of Table 8.2). Group B: numbers of trucks and cars (the traffic column).
This results yet again confirm that particles from the 6 nm mode are associated
with trucks. This is because the corresponding canonical weights and loadings for trucks
are large and both positive, while for cars they are also large but negative (Table 8.2).
Therefore, increasing number of trucks results in a significant increase of particle
concentration in this mode, and vice versa for cars (the correlation is strong and stable).
Similarly, the 136 nm mode is also associated with trucks, which is in agreement
with Figs. 8.3a, 8.4a, and 8.5a. The corresponding weights and loadings are large and
positive (the correlation is thus sufficiently stable and strong).
The 53 nm mode also has the same tendency towards the association with
trucks, and this is in agreement with the comments about the same association derived
from Fig. 8.5a. However, its weight and loading are not very large (Table 8.2), and there
might be questions about its stability and strength. This is probably related to the
insufficient level of confidence for the canonical correlations of this mode, displayed by
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Fig. 8.4a. Therefore, the conclusions about this mode may not be as certain as for the 6
nm and 136 nm modes.
The 26 nm and 31 nm modes represent two components of essentially the same
~ 30 nm mode (Chapter 6) – see also Figs. 8.2a and 6.4a. However, it is interesting that
the canonical correlation analysis suggests that these components could come from
different sources. Indeed, the canonical weight for the 26 nm mode is positive with
respect to the car weight and negative with respect to the truck weight (Table 8.2).
Therefore, we can conclude that the 26 nm mode mainly comes from cars. Since the
corresponding canonical loadings have the same signs, the considered correlation is
stable. Similarly, the 31 nm mode mainly comes from heavy trucks, which is
demonstrated by the signs of the corresponding canonical weights in Table 8.2.
However, this correlation is unstable, since the weight and loading for this mode have
different signs. Therefore, more extensive experimental data will be required to confirm
this conclusion.
As can be seen from the third row of Table 8.2, all the remaining 14, 40 and 88
nm modes are related to cars (although the 40 nm mode shows instability of correlation
because of the different sign of its canonical loading). This conclusion is again
confirmed by the simple moving average correlation coefficient (Fig. 8.3a). The
association of the 14 nm and 40 nm modes with cars is also in agreement with Fig. 8.5a
(see above). However, Figs. 8.4a and 6a provide inconclusive results about the 88 nm
mode, which makes the conclusions about its association with cars more questionable,
and more experimental data and further analysis is needed.
8.5. Meteorological parameters.
As has been mentioned, the canonical correlation analysis (Figs. 8.4 – 8.6) has
been conducted for group B of 5 variables including numbers of cars and trucks on the
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road, and average (over the scan time) meteorological parameters, such as temperature,
humidity and solar radiation. Therefore, the conducted analysis has automatically
provided canonical weights and loadings for temperature, humidity and solar radiation
for all particle diameters.
However, it has been shown that for both the sets of 11 and 16 scans,
correlations of particle modes with humidity are highly unstable. Signs of canonical
weights and loadings do not normally coincide with each other, with typical values for
loadings less than 0.2 (which corresponds to the level of confidence < 95%). Stable
correlation between the particle concentrations and humidity for the first set of scans
could only be revealed within the range ~ 7 – 19 nm (with negative weights and
loadings) and within the range ~ 70 – 110 nm (with positive weights and loadings). The
situation is not better for the second set of 16 scans: only ~ 40 nm particles show stable
(negative) correlation with humidity. Therefore, it is hardly possible to talk about
reliable and useful relationships between the observed particle modes and humidity.
One of the reasons of this instability is related to the significant dependence of
humidity on traffic. For example, it has been experimentally estimated that traffic-
induced fluctuations of humidity at the distance of ~ 40 m from the road may be of the
order of a few percent. If this results in non-linear dependencies, then the canonical
correlations may become unstable.
For better understanding of these relationships, canonical correlations between
temperature/humidity and traffic have been calculated for both the sets of scans (Table
8.3). In particular, one can see strong positive association between humidity and the
number of cars for both the sets. This is because the corresponding canonical weights
and loadings for cars are large and both positive, while for trucks they are negative
(Table 8.3). Therefore, increasing number of cars results in a significant increase of
humidity. At the same time, the correlations between traffic and temperature are
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unstable, because the weight and loading for temperature have different signs – see
Table 8.3.
Temperature and humidity Trucks and cars Set R Level
of conf.
weights loadings weights loadings
1 0.73 99% 0.32 (temp) 1,12 (hum)
-0.24 (temp) 0.96 (hum)
-0.09 (trucks) 0.98 (cars)
-0.27 (trucks) 0.996 (cars)
2 0.64 99% 0.34 (temp) 1.22 (hum)
-0.54 (temp) 0.97 (hum)
-0.36 (trucks) 0.89 (cars)
-0.47 (trucks) 0.94 (cars)
Table 8.3. The results of the canonical correlation analysis between temperature and
humidity (group A) and the number of trucks and cars (group B).
Thus, it is indeed possible that correlations between humidity and particle
concentrations may be significantly affected by the additional dependence of humidity
on traffic conditions on the road. On the contrary, temperature does not display a
consistent dependence on traffic conditions. Therefore, it is possible to expect that
correlations between temperature and particle concentrations should be much more
stable and thus can be used for the analysis of temperature-related processes of aerosol
evolution.
Correlations of particle modes with temperature are illustrated by Figs. 8.7a,b by
the dependencies of the moving average canonical weights and loadings for
temperature. These dependencies show very strong and stable correlations with same
signs of canonical weights and loadings in wide ranges of particle diameters (Fig.
8.7a,b). An important feature of these dependencies are the drastic differences between
correlations for the two considered sets of scans. This means that just 16 s of evolution
(between Figs. 8.7a and 8.7b) have resulted in a significant alteration of the
temperature-related correlations. This is a clear indication of strong and fast processes
of evolution of combustion aerosols near a busy road. For example, particles in the
broad range from ~ 8 nm to ~ 20 nm (the ~ 13 nm mode) for the second set of scans are
characterised by strong and stable positive correlations with temperature (Fig. 8.7b).
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This means that increasing temperature results in increasing number of particles within
this range. On the contrary, particles within the ranges ~ 20 – 80 nm and > 100 nm are
negatively correlated with temperature (Fig. 8.7b), i.e. increasing temperature results in
decreasing concentration of these particles. This behaviour is again in a good qualitative
agreement with the fragmentation mechanism of aerosol evolution. Indeed, particles
within the ~ 8 – 20 nm range are expected to be generated by fragmentation of larger
particles. At the same time, thermal fragmentation should strongly depend on
temperature – its rate rapidly (exponentially) increasing with increasing temperature
(Chapter 10). This is the reason for strong negative correlations for fragmenting larger
particles (their concentration decreases with increasing temperature) and strong positive
correlations with particles resulting from fragmentation (their concentration increases
with increasing temperature).
An interesting feature is the strong and stable negative correlations with
temperature for very small particles with ~ 5 – 6 nm for both the sets of scans (i.e., at
both considered stages of aerosol evolution). This behaviour is unclear at this stage and
will require further investigation. It could be interpreted by the presence of a large
number of small volatile particles coming from diesels (see the discussion of Figs. 8.3a
and 8.5a). The number of these particles could increase with decreasing temperature due
to slower evaporation. However, the same is not applicable to Fig. 8.7b, where the
volatile particles have already evaporated (see the discussion of Figs. 8.6a,b). Similarly,
further investigation is needed for understanding of temperature dependence of particle
correlations in the range ~ 80 – 100 nm (Fig. 8.7b).
Temperature correlations for the first set are less understandable at this stage and
will also need further investigation. One of the possibilities is that these correlations (at
least partly) may still be related to correlations between temperature and traffic
conditions, though this is not entirely confirmed by the weights and loadings for solar
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radiation (Figs. 8.8a,b). Indeed, weights and loading for solar radiation tend to have
some similar features to those observed on the dependencies for temperature (compare
curves in Figs. 8.7a and 8.8a). These similarities may suggest the existence of processes
that are similarly related to temperature and solar radiation, and it is obvious that solar
radiation does not depend on traffic conditions.
Fig. 8.7. Moving average canonical weights (solid curves) and loadings (dotted curves) for
temperature in group B with the five variables: temperature, humidity, solar radiation, number
of trucks, and number of cars. (a) The dependencies for the first set of 11 scans (from 6 to 16).
(b) The dependencies for the second set of scans (from 28 to 43). Straight dashed horizontal
lines indicate the 95% level of confidence for the loading curves. The dash-and-dot horizontal
line corresponds to zero.
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Fig. 8.8. Moving average canonical weights (solid curves) and loadings (dotted curves) for solar
radiation in group B with the five variables: temperature, humidity, solar radiation, number of
trucks, and number of cars. (a) The dependencies for the first set of 11 scans (scans from 6 to
16). (b) The dependencies for the second set of scans (scans from 28 to 43). Straight dashed
horizontal lines indicate the 95% level of confidence for the loading curves. The dash-and-dot
horizontal line corresponds to zero.
An interesting aspect that can be drawn from the comparison of Figs. 8.7a and
8.8a is that for small particles (< 8 nm), the effect of solar radiation is opposite to that of
temperature. This may be explained again by the fact that particles in this range in the
first set of scans are mostly volatile. Such particles are expected to be formed by means
of condensation of volatile compounds onto very small core particles (beyond the
detectable range). The core particles could be formed by means of the processes of
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nucleation in the immediate proximity of the exhaust pipe (Zhang & Wexler, 2004,
Zhang et al, 2004, Pohjola, et al, 2003). The nucleation processes may be induced by
solar radiation (Shi, et al, 2001, Kulmala, et al, 2004). Therefore, the number of core
particles should have positive correlations with solar radiation. If the volatile particles
from the range < 8 nm grow from the core particles by means of condensation, then they
must also have positive correlations with solar radiation (Fig. 8.8a). On the contrary,
increasing temperature should result in increasing evaporation or decreasing
condensation processes, resulting in negative correlations of the same particles with
temperature (Fig. 8.7a).
The situation with the second set of scans is more understandable, if we use the
fragmentation model. Again, positive stable correlations with solar radiation for the
particles within the range ~ 8 – 20 nm (Fig. 8.8b) are similar to those in Fig. 8.7b.
Larger particles are predominantly characterised by negative (ranges ~ 20 – 30 nm and
> 100 nm) or unstable (inconclusive) correlations with solar radiation. Both these results
are in agreement with fragmentation of larger particles into smaller particles from the
range ~ 8 – 20 nm. This process may be expected to intensify (e.g., due to heating
and/or radiative effects) with increasing solar radiation, resulting in increased
concentration of smaller particles (positive correlations) and decreased concentration of
larger particles (negative correlations).
8.6. Conclusions.
In this Chapter, a new statistical method of analysis of nano-particle aerosols,
based on the moving average approach and canonical correlation analysis has been
developed and applied for the investigation of particle modes and their possible sources
in combustion aerosols near busy roads. This method was demonstrated to provide an
important new physical insight into the processes of evolution of combustion aerosols
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near busy roads. It enables detailed investigation of contribution of different
environmental and meteorological factors and processes on evolution of combustion
aerosols. Several particle modes were identified to originate predominantly from petrol
cars and heavy-diesel trucks. Levels of confidence and the associated statistical errors
were determined, demonstrating reliability and accuracy of the developed new
approach. The obtained results will be important for the development of effective
practical measures for reduction of human exposure to nano-particle aerosols and
development of new strategies for reduction of the impact of modern transport on our
environment.
The developed new statistical approach will also be important for further
development of our understanding of fundamental physical and chemical effects in
combustion aerosols. For example, it has led to further confirmation of the discovered
new major mechanisms of aerosol evolution – fragmentation of nano-particles.
It is also important to note that the developed approach is directly applicable not
only to the considered combustion aerosols near busy roads, but also to any other type
of anthropogenic and natural aerosols, for which the determination of possible sources
of different particle modes and investigation of evolution processes is important.
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CHAPTER 9
CORRELATIONS BETWEEN PARTICLE MODES: FRAGMENTATION THEOREM ([A7])
9.1. Introduction
As was indicated in Chapter 8, the development of new statistical methods of analysis
of combustion aerosols resulting from transport emissions near busy roads is one of the
important directions of current research in the aerosol science. This is because such methods
are expected to provide new insights into the physical and chemical processes of evolution of
airborne nano-particles, their possible sources, mutual relationships, and eventually lead to
the development of effective means for elimination or reduction of their impact of human
health and environment. For example, these methods will be important for the investigation
of fundamental processes and nature of nano-particle aerosols, including particle formation,
condensation/evaporation, coagulation, and thermal fragmentation in the real-world
environment with strong stochastic fluctuations of atmospheric and environmental
parameters (Zhang and Wexler, 2004, Zhang et al, 2004, Zhu et al, 2002, Shi et al, 1999, Shi
et al, 2001, Pohjola et al, 2003, Kulmala et al, 2004) – see also Chapters 6 and 8.
Therefore, the new statistical methods based on the moving average concentration of
particles in different channels of the size distribution (Chapters 5 and 8), and moving average
correlation coefficients, canonical weights and loadings (Chapters 6 and 8) were developed
for the determination and investigation of particle modes in combustion aerosols. These
approaches provided an excellent new physical insight into the evolution processes and
possible sources of nano-particle modes (Chapters 6 and 8).
However, these techniques mostly provide information about how particle
concentrations in neighbouring channels correlate with each other (e.g., forming a mode), or
how groups of particles with similar diameters react to variations of traffic and/or
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meteorological conditions (Chapters 6 and 8). It is more difficult to use these methods for the
direct investigation of correlations between particles with significantly different diameters
(i.e., between different channels/modes in the particle size distribution). At the same time,
such an analysis is expected to provide new important evidence of physical processes and
particle transformations during evolution of combustion aerosols.
For example, because thermal fragmentation of nano-particles was demonstrated to
play a major role at particular stages of aerosol evolution (Chapters 6 and 8), this process
should have a substantial effect on mutual correlations between particle modes. For example,
if fragmentation results in mutual transformation of some particle modes, then these modes
may be characterised by strong negative correlations (anti-correlations). This is because
increasing concentration in the fragmenting mode (e.g., due to turbulent fluctuations of
evolution time) may result in decreasing concentration in the modes resulting from
fragmentation.
Therefore, in this Chapter, we extend the developed moving average approach to the
analysis of correlations and mutual transformations/interactions between different particle
modes in combustion aerosols near busy roads. As a result, a new moving average cross-
correlation approach will be developed. Strong anti-correlations between several particle
modes will be demonstrated and investigated. A unique anti-symmetric correlations between
some particular modes will be demonstrated and investigated. The interpretation of these
anti-symmetric correlations will be conducted on the basis of the derived fragmentation
theorem, which may be regarded as one of the most convincing confirmations of the
fragmentation mechanism of evolution of combustion aerosols.
9.2. Moving average approach for particle modes.
In this Chapter we again analyse the set of monitoring data near Gateway Motorway
in the Brisbane area, Australia, which was considered in Chapters 6 and 8. The detailed
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description of the experimental procedure, the obtained monitoring data, and the
corresponding meteorological and traffic parameters are presented in Section 8.2.
The extension of the moving average approach described in Chapters 6 and 8 to the
case of correlations between particle modes and/or different channels from the size
distribution can be done as follows. We again consider multiple scans of the particle size
distribution. For example, these can be the sets of 16 scans (from 28 to 43) corresponding to
Figs. 8.2a,b from Chapter 8, or 20 scans (from 19 to 38) corresponding to Figs. 6.4a,b from
Chapter 6.
We again choose a 7-channel interval (which we will call ‘primary interval’)
corresponding to a particular mode, i.e., the central channel of this interval corresponds to
one of the maximums in Figs. 8.2a,b or 6.4a,b. We also choose another arbitrary interval
(‘secondary interval’) of 7 neighbouring channels from the overall 100 channels in the
particle size distribution. Each of the channels from any of the chosen (primary or
secondary) 7-channel intervals will correspond to a column of different particle
concentrations corresponding to each of the multiple scans. For example, for a set of 11
scans, each channel will correspond to a column of 11 different particle concentrations.
Every value of concentration in every column is normalised to the total number
concentration in the respective scan. This eliminates trivial correlations associated with
turbulent fluctuations of the total number concentration.
We select a channel from the primary 7-channel interval and calculate the simple
correlation coefficient between the concentration columns corresponding to this channel and
one of the channels from the secondary 7-channel interval. This procedure is then repeated
for all possible different pairs of channels – one from the primary interval and the other from
the secondary interval. As a result, we obtain 49 simple correlation coefficients, and their
average is calculated. Each 7-channel interval is identified by the particle diameter
corresponding to its central channel. The obtained average simple correlation coefficient is
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taken as the correlation coefficient between the two central channels from the two selected 7-
channel intervals. Then we take a different secondary interval, and repeat the whole
procedure (this is done 94 times for all possible choices of the secondary 7-channel interval
and the fixed primary interval; one of the secondary intervals coincides with the primary
interval). As a result, we obtain 94 average correlation coefficients between the selected
channel (i.e., the central channel of the selected primary interval) and all other
channels/modes. Thus we obtain a dependence of the moving average correlation coefficient
between the selected channel/mode and all other channels on particle diameter (each such
diameter is associated with the central channel of a secondary interval). These dependences
are obviously different for different primary intervals.
The statistical errors of the obtained dependencies of the moving average correlation
coefficients can be obtained by calculating the errors of the mean correlation coefficients for
each of the 94 pairs of the primary and secondary 7-channel intervals. As a result, we obtain
two error curves surrounding the main dependence, similar to how it was done for Figs.
8.2a,b and 6.4a,b.
9.3. Numerical results and their discussion
For the numerical analysis in this section, we will choose three sets of scans: 16 scans
(scans from 1 to 16), 16 scans (from 28 to 43), and 20 scans (from 19 to 38). The first set of
16 scans includes the two sets from 1 to 11 and from 6 to 16 scans, considered in Chapter 8.
We use here one joint set, instead of the two separate considered in Chapter 8, because they
both correspond to approximately the same wind, and the difference between them in terms
of the evolution time was only ~ 3 s (Chapter 8). Therefore, within the limits of errors, it is
difficult to distinguish between these two sets, and we join them in one larger set of 16 scans.
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Fig. 9.1. The dependencies of the moving average correlation coefficients for the primary intervals
centred at the following channels: 126 nm (solid curve 1), 136 nm (2 channels above 126 nm –
dashed curve 1), 113 nm (2 channels below 126 nm – dotted curve 1), 13.6 nm (solid curve 2), 13.1
nm (1 channel below 13.6 nm – dotted curve 2), and 14.1 nm (1 channel above 13.6 nm – dashed
curve 2) for the set of 16 scans from 1 to 16. The average particle diameters on the horizontal axis
correspond to the central channels of the secondary intervals. The 95% levels of confidence for the
considered simple correlations are shown by the horizontal solid lines. Average normal wind
component ≈ 1.65 m/s; evolution time ~ 24 s. The dash-and-dot horizontal line corresponds to zero
correlation coefficient.
For the first set of 16 scans (from 1 to 16), the dependence of the correlation
coefficients for the 126 nm channel (i.e., the central channel of the primary interval
corresponds to the diameter 126 nm) on particle diameter is given by the solid curve 1 in Fig.
9.1. Slight variations of the position of the primary interval does not result in drastic
alterations of the obtained dependence. For example, if we choose the 136 nm and 113 nm
channels as the central channels of the primary 7-channel interval, variations of the resultant
correlations coefficients are relatively small (dashed and dotted curves 1 in Fig. 9.1). This is
expected, because variations of position of the primary interval within the same mode are not
anticipated to produce substantial variations of correlations with other channels. Similar
196
situation occurs for the curves corresponding to the primary intervals centred at the 13.1 nm,
13.6 nm, and 14.1 nm channels (dotted, solid and dashed curves 2 in Fig. 9.1). Therefore,
this approach can thus be regarded as a method for the determination of particle modes – a
mode will thus be an interval of neighbouring channels within which correlations with other
modes/channels, obtained by means of the described procedure, are approximately the same.
Curves 1 in Fig. 9.1 suggest that particles with diameters around 126 nm (the 126
mode in Fig. 8.2a) are positively correlated with the modes ~ 55 nm and ~ 6 nm. Note that
this is in agreement with the solid curve in Fig. 8.3a from Chapter 8, which demonstrates that
all these three modes are associated with trucks on the road. Therefore, the positive
correlations between these three modes on the curves 1 in Fig. 9.1 are likely due to their
mutual source – diesel trucks (Chapter 8).
Negative correlations shown by curves 1 in Fig. 9.1 for ~ 10 – 30 nm particles can be
explained by stronger association of these particles with cars (Chapter 8). Therefore, due to
limited overall traffic capacity of the road, and normalisation-induced negative correlations
(see Section 8.4), it may be expected that, for example, increasing relative concentrations of
particles associated with trucks (e.g., 126 nm mode) may result in a tendency of decreasing
relative concentrations in the channels more associated with cars, leading to negative
correlations with the 126 nm mode (curves 1 in Fig. 9.1).
As the aerosol is transported from the road, particles experience rapid processes of
transformation/evolution. These processes drastically change the correlation pattern for the
particle modes within just 16 s – see Figs. 8.2 – 8.7. Similar significant alterations are also
expected for the moving average simple correlations between the modes. For example, the
dependencies of the moving average correlation coefficient for the 126 nm mode (the
primary 7-channel interval is centred at the 126 nm channel) and for the 13.6 nm mode (the
primary interval is centred at the 13.6 nm channel) for the second set of 16 scans (from 28 to
43) are presented in Fig. 9.2 by solid curves 1 and 2, respectively.
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The dashed and dotted curves 1 are plotted for the primary interval centred at the 136
and 113 nm channels, respectively. These curves again suggest (as expected) relative
insensitivity, i.e., stability of the resultant dependencies with respect to variations of position
of the primary interval within the limits of one mode. The examples of the resultant error
curves (obtained by the approach described above in Section 9.2) are represented by the two
dashed curves around solid curve 2 in Fig. 9.2. These error curves clearly demonstrate that
the corresponding errors of the moving average correlation coefficients are typically
insignificant and may not be considered.
Fig. 9.2. Moving average simple correlation coefficient between each of the following four channels
(modes): 126 nm (solid curve 1), 136 nm (dashed curve 1), 113 nm (dotted curve 1), and 13.6 nm
(solid curve 2) and all other channels in the particle size distribution for the second set of 16 scans
(from 28 to 43). Dashed curves 2 represent an example of the error lines for the dependence given by
solid curve 2. The average particle diameter for each of the secondary intervals is taken as the
diameter corresponding to the central channel of the interval. The average normal wind component ≈
1 m/s; evolution time ~ 40 s. The horizontal straight lines indicate the 95% level of confidence of the
considered simple correlations.
The 13.6 nm and 126 nm modes are thus negatively correlated (anti-correlated) with
each other. The most interesting aspect of this figure is that the dependencies for the primary
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intervals centred at the 126 nm and 13.6 nm channels (solid curves 1 and 2, respectively) are
almost symmetric of each other with respect to the zero line (anti-symmetric correlations) –
Fig. 9.2. Moreover, all the correlation curves for the channels 113 nm, 126 nm, and 136 nm
(curves 1 in Fig. 9.2) have almost exactly the same positions of their maximums and
minimums, corresponding to the same positions of the respective minimums and maximums
of the correlation curve for the 13.6 nm mode (Fig. 9.2).
This is a non-trivial result. It means that the correlation coefficient between the 126
nm channel and, for example, 40 nm channel is the same in magnitude, but different in sign
to the correlation coefficient between 13.6 nm channel and the same 40 nm channel. The
same is correct for any other channel, but not just the 40 nm channel. This clearly
demonstrates that all the channels are related by some mutual physical process. It is
important that this process was not present or was suppressed/masked for the first set of 16
scans (from 1 to 16), because the correlations in Fig. 9.1 are not anti-symmetric. The average
normal wind component for the first set of 16 scans is 1.65 m/s, whereas for the second set of
scans (from 28 to 43) it is ≈ 1 m/s (Chapter 8). This means that the anti-symmetric
correlation pattern (Fig. 9.2) has developed within just ~ 16 s of aerosol evolution.
An explanation of anti-correlations in Fig. 9.2 should be different from that for Fig.
9.1. Because significant variations of correlations between the modes have occurred, anti-
correlations in Fig. 9.2 cannot be explained by traffic effects, similar to how it was done for
Fig. 9.1, whereas average traffic conditions have not changed noticeably between these two
sets of scans. Therefore, these are rather evolutionary processes within the 16 s of the time
difference between Figs. 9.1 and 9.2 that have resulted in anti-symmetric correlations in Fig.
9.2.
The fact that the 13.6 nm mode is negatively correlated with all larger modes
suggests that there may be a process of formation of the 13 nm mode from the larger
particles, or, vice versa, larger particles are formed by the particles from the 13 nm mode.
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This is because the observed anti-correlations between the ~ 13 nm particles and particles
with diameters > 30 nm (Fig. 9.1), and especially particles with diameters ~ 126 nm clearly
suggest that, for example, if concentration in the 13 nm mode increases, then the
concentration in the 126 nm mode decreases, and vice versa. Therefore, one of this modes
may be the source for the other, and the process of transformation should be characterised by
the typical relaxation time that is of the order of the time of evolution between the two sets of
scans, which is ≈ 16 s.
It seems to be possible to immediately dismiss such processes of particle evolution as
condensation/evaporation, homogeneous and heterogeneous nucleation, deposition and
dispersion as possible mechanisms for the observed strong anti-correlations and anti-
symmetric correlations (Fig. 9.2). It difficult to imagine (from the physical nature of these
processes) how they could result in such anti-correlations on the considered scale of particle
diameters. The only option out of the conventional mechanisms of aerosol evolution would
be coagulation. However, as has been shown by Shi, et. al. (1999), Jacobson (1999), and
Jacobson & Seinfeld (2004), coagulation at the considered levels of particle concentrations
will take at least hours (rather than 16 s) to result in noticeable changes in particle
distribution and correlation pattern. Therefore, coagulation should also be excluded from the
list of possible mechanisms.
As a result, the only suitable explanation of the observed anti-symmetric correlations
(Fig. 9.2) is thermal fragmentation of nano-particles (Chapters 6 and 8). In accordance with
the obtained results (Fig. 9.2), it can be assumed that the ~ 13 nm particles are generated due
to thermal fragmentation of larger nano-particles (e.g., the ~ 126 nm mode). As a result, it is
expected that if particles with the diameters of ~ 13 nm break away from, for example, 126
nm particles, then particle concentration in the ~ 126 nm mode decreases, while the
concentration in the ~ 13 nm mode increases, resulting in the observed anti-correlations.
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However, Fig. 9.2 also suggests that particles not only from ~ 126 nm mode, but also
within the large range (> 30 nm) experience similar anti-correlations with the ~ 13 nm mode.
This suggests that all these particles experience fragmentation resulting in breaking away ~
13 nm particles. For example, the ~ 66 nm particles have strong negative correlations with
the ~ 13 nm particles, and strong positive correlations with the ~ 126 nm particles (Fig.
9.2).This could be explained by the fact that particle concentration for the considered second
set of 16 scans decreases monotonically with increasing particle diameter –see the dark band
in Fig. 8.1. This means that during the process of fragmentation, the inflow of the particles
into, for example, the 66 nm mode due to fragmentation of the larger particles is less than the
outflow of the particles from the 66 nm mode due to its own fragmentation. Thus
fragmentation causes an overall shift of the dark band at larger particle diameters in Fig. 8.1
to the left. Therefore, fragmentation results in a simultaneous decrease of particle
concentrations in all the channels in the range > 30 nm, thus giving positive correlations
between all the modes from this range and the 126 nm mode – Fig. 9.2.
Fig. 9.3 shows the moving average correlation curves for 40, 50, 68 and 126 nm
particles from the range of particles experiencing anti-correlations with the ~ 13 nm particles
(Fig. 9.2). Curve 1 in Fig. 9.3 is identical to solid curve 1 in Fig. 9.2. The most significant
result of Fig. 9.3 is that all the presented curves demonstrate almost identical negative
correlations between the considered channels and the ~ 13 nm particles. This is a clear
indication that all the particles within the considered range > 30 nm have approximately the
same probability to fragment with the release of ~ 13 nm particles. The fragmentation
mechanism works in approximately the same way for the larger particles in a large range of
particle diameters (between ~ 30 nm and ~ 136 nm). In other words, the fraction of particles
participating in the process of fragmentation is approximately the same for all the channels in
the mentioned range. This suggests approximately the same values of the fragmentation rate
coefficient for particles in the considered range.
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Fig. 9.3. Moving average simple correlation coefficient between each of the following four channels
(modes): 126 nm (curve 1), 68 nm (curve 2), 50 nm (curve 3), and 40 nm (curve 4) and all other
channels in the particle size distribution for the second set of scans from 28 to 43. Average normal
wind component ≈ 1 m/s; evolution time ~ 40 s. The horizontal straight lines indicate the 80%
(dashed straight lines) and 95% (solid straight lines) levels of confidence of the considered simple
correlations.
Another interesting aspect of Fig. 9.3 is that decreasing average particle diameter for
the primary 7-channel interval from ~ 126 nm to ~ 40 nm results in a steady and significant
reduction of the correlation coefficients with smaller particles of ≲ 8 nm (compare curves 1 –
4).
This might be because of the tendency that smaller particles may fragment by means
of the release of smaller (~ 7 nm) primary particles (Chapter 6). If this is true, then one
should expect that further decrease of diameter of fragmenting particles below ~ 40 nm
should significantly increase anti-correlations (negative correlations) with the ~ 7 nm mode.
This is indeed the case and is demonstrated by Fig. 9.4, where correlations for the 7 nm and
32 nm modes (channels) are considered. We again obtain the two dependencies that tend to
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be approximately symmetric with respect to the zero line (Fig. 9.4). Because of the same
reasons as for Fig. 9.2, the obtained anti-correlations between the 32 nm and 7 nm modes can
be explained by fragmentation of nano-particles by means of breaking away primary ~ 7 nm
particles (Chapter 6).
Fig. 9.4. Moving average simple correlation coefficient between each of the following two channels
(modes): 32.2 nm (curve 1) and 7.37 nm (curve 2) and all other channels in the particle size
distribution for the second set of 16 scans from 28 to 43. Average normal wind component ≈ 1 m/s;
evolution time ~ 40 s. The horizontal straight lines indicate the 80% (dashed straight lines) and 95%
(solid straight lines) levels of confidence of the considered simple correlations.
As a result, Figs. 9.2 – 9.4 suggest the following possible interpretation. There are
two main types of primary particles: ~ 7 nm and ~ 13 nm. They can possibly be related to
different types of vehicles (needs to be confirmed by further analysis). These particles form
different types of larger particles. This is expected, because coagulation may occur only in
the immediate proximity to the exhaust pipe or inside it, where particle concentration is
sufficient for coagulation to occur (Jacobson, 1999, Jacobson & Sienfeld, 2004). As a result,
large composite particles are primarily formed of one type of primary particles – either ~ 7
nm or ~ 13 nm (particles from one vehicle are highly unlikely to get to the exhaust pipe of
another vehicle in sufficient concentrations). Larger particles with diameters closer to ~ 100
203
nm are mainly formed of larger primary particles, and their fragmentation results mainly in
breaking away particles from the ~ 13 nm mode. This leads to anti-correlations shown in
Figs. 9.2 and 9.3. However, at smaller diameters (~ 20 – 40 nm) there is a large number of
particles formed by the ~ 7 nm primary particles (Chapter 6). As a result, their fragmentation
leads to the anti-correlations with the ~ 7 nm mode (Fig. 9.4).
Fig. 9.5. Moving average simple correlation coefficient between each of the two channels (modes):
126 nm (solid curve) and 13.6 nm (dashed curve) and all other channels in the particle size
distribution for the third set of scans from 19 to 38. Average normal wind component ≈ 0.75 m/s;
evolution time ~ 53 s. The horizontal solid straight lines indicate the 95% levels of confidence of the
considered simple correlations.
The anti-symmetry of correlations between different particle modes is approximately
preserved for the third set of 20 scans (from 19 to 38) that corresponds to the largest
evolution time of ~ 53 s (Fig. 9.5). This is because the fragmentation process continues at
this stage of the aerosol evolution. However, the anti-symmetry has started to break down,
for example, in the region below ~ 8 nm diameter (Fig. 9.5). This is possibly because anti-
symmetry requires very specific conditions that may possibly be achieved only at some
particular stage of aerosol evolution (see the next section). Probabilistic time delays (Chapter
10) with fragmentation of intermediate particle modes resulting from fragmentation of even
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larger particles may also play a noticeable role in the formation and breach of the anti-
symmetric pattern of correlations. However, this requires further investigation.
Certainly, fragmentation mechanism does not necessarily imply that all of the larger
particles should necessarily fragment into primary particles. Some of larger particles may
have completely different origins and physical/chemical nature. The presence of such
different particles may also result in breaching anti-symmetric pattern of correlations
between the modes in the particle size distribution (for more detail see the next section).
9.4. Fragmentation Theorem
In Section 9.3, we have attempted a qualitative interpretation of the obtained anti-
symmetric pattern of mutual correlations between the modes in the particle size distribution.
Here, we will present a more rigorous mathematical derivation and justification of conditions
when such a pattern may occur. As a result, we will obtain further confirmation of the
fragmentation model of evolution of combustion aerosols.
In order to explain the observed anti-symmetric correlations between concentrations
in different channels of the size distribution (Fig. 9.2), consider again N scans of the particle
size distribution and simple correlations between two columns of particle concentrations
corresponding to two selected channels. Each such column contains N different
concentrations that are normalised to the total number concentration in the respective scan.
The sufficient and necessary condition for the anti-symmetry of the dependencies of the
moving average correlation coefficients for two different k-th and m-th channels/modes (e.g.,
126 nm channel and 13.6 nm channel in Fig. 9.2) can be written in the form:
Rkn = – Rmn, (9.1)
where Rkn is the correlation coefficient between the k-th channel (with the larger particle
diameter) and an arbitrary n-th channel of the particle size distribution, and Rmn is the
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correlation coefficient between the m-th channel (with the smaller particle diameter) and the
same n-th channel.
Using the equation for the simple correlation coefficient between the two data
columns (Larsen and Marx, 1986), we re-write Eq. (9.1) as follows:
∑∑Δ
Δ
∑Δ
Δ−=∑∑Δ
Δ
∑Δ
Δi
jnj
ni
jmj
mi
ij
nj
ni
jkj
ki
y
y
y
y
y
y
y
y2222
, (9.2)
where yki, ymi, and yni are the particle concentrations in the k-th, m-th, and an arbitrary n-th
channels of the particle size distribution, respectively, sums are taken over the N different
scans, Δyki, Δymi, and Δyni are the concentration fluctuations in the i-th scan around the mean
values ky , my , and ny calculated over the N scans.
The fragmentation theorem establishes conditions at which Eqs. (9.1) and (9.2) are
satisfied, i.e., the dependencies of the moving average correlation coefficients for two
different channels/modes are anti-symmetric (Fig. 9.2).
Conditions of the fragmentation theorem:
1. Particles taking part in fragmentation consist of primary particles that can break away
from larger composite particles. Primary particles belong to the sink mode.
2. Fragmentation occurs by means of breaking away a primary particle – one primary
particle breaks away during each act of fragmentation (Fig. 9.6). Each act of
fragmentation adds one primary particle to the sink mode and results in a transition of
the fragmenting particle to the next intermediate mode with smaller diameter (Fig.
9.6). Only the last act of fragmentation of a particle from the smallest intermediate
mode results in two primary particles being added to the sink mode.
3. Fragmentation by means of breaking away non-primary particles or more than one
primary particles at a time are assumed not to be possible.
4. The number of fragmenting intermediate particle modes can be finite (when there is a
maximal diameter of fragmenting particle, i.e. a maximal number of primary particles
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in a fragmenting particle), or infinite (when there is no maximal diameter of
fragmenting particles).
5. Fluctuations of particle concentrations in different scans in at least two different k-th
and m-th channels are caused predominantly by the process of fragmentation (for
more detailed explanation of this condition see below).
6. Particle concentrations in the k-th and m-th channels are negatively correlated with
each other: Rkm < 0 (i.e., if the concentration in the k-th channel increases, then the
concentration in the m-th channel decreases, and vice versa).
7. Fragmentation rates for particles from all the intermediate modes are equal.
Fragmentation theorem: If conditions 1 – 7 are satisfied, then the dependencies of the
moving average correlation coefficients for the k-th and m-th channels are anti-symmetric of
each other.
Fig. 9.6. Fragmentation scheme for particles in the intermediate modes into the sink mode of primary
particles by means of breaking away primary particles – one per each act of fragmentation. Each
intermediate mode is obtained from the previous mode by means of breaking away one primary
particle that is automatically added to the sink mode.
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For example, in accordance with Fig. 9.2, it is possible to expect that for the second
set of scans considered in Section 9.3, the particle diameter for the k-th mode is equal to 126
nm, whereas the m-th mode corresponds to the sink mode with the primary particles of 13.6
nm diameter. Fragmentation occurs by means of breaking the ~ 13.6 nm particles away from
the larger particles from the source mode or intermediate modes.
Proving the fragmentation theorem is equivalent to verifying the validity of the
sufficient and necessary conditions (1) and/or (2) under the conditions of the theorem 1 – 7.
In accordance with Fig. 9.6, we can write the equations for the concentration
fluctuations Δypi in an arbitrary p-th intermediate mode:
Δypi = ypi – py = addpi
outpi
inpi yyy Δ+Δ+Δ , for p ≠ 1; (9.3)
Δy1i = y1i – 1y = – outi
p
outpi yy 2
32Δ−∑ Δ
∞
= + add
iy1Δ =
= iniy1Δ + add
iy1Δ . (9.4)
Here, outpiyΔ is the variation of the concentration in the p-th intermediate mode in the i-th
scan, caused by fragmentation of particles from this mode. Therefore, this variation results in
particle outflow from the p-th intermediate mode, caused by fragmentation. Simultaneously,
this variation results in an inflow of particles into the sink mode (see Eq. (9.4)), because
fragmentation occurs through the release of primary particles. If p ≠ 1, then inpiyΔ is the
variation of the concentration in the p-th intermediate mode due to the inflow of particles
from the (p + 1)-th intermediate mode. This inflow is caused by fragmentation of the (p – 1)-
th intermediate mode. On the other hand, iniy1Δ are the variations of the concentration in the
sink mode, caused by the inflow of primary particles into the sink mode due to fragmentation
of all the intermediate modes, and outixΔ . The variations add
piyΔ (p = 1, 2, 3, …) are the
additional fluctuations of the concentrations in the sink and intermediate modes, caused by
factors/processes other than the process of fragmentation. There is a factor of 2 in front of
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outiy2Δ in Eq. (9.4), because the intermediate mode with p = 2 fragments into two primary
particles. There is no term with outiy1Δ in Eq. (9.4), because the sink mode (with p = 1) is
assumed to consist of primary particles that do not experience further fragmentation. Note
also that in Eqs. (9.3) and (9.4) the index p indicate all possible modes, i.e., p may also be
equal to m or k.
Condition 5 of the theorem states that the k-th and m-th modes (for which the anti-
symmetry of the correlation coefficients is expected) are caused predominantly by the
process of fragmentation. That is, the effect of any other mechanisms on fluctuations of the
normalised particle concentrations in these modes is negligible:
addkiyΔ = add
miyΔ = 0. (9.5)
Physically, fragmentation-induced fluctuations of particle concentrations occur
because of turbulent fluctuations. Turbulent fluctuations result in fluctuations of evolution
time for the aerosol, because they lead to stochastic variations of the transport time from the
road. As a result, each scan corresponds to slightly different evolution times, i.e., slightly
different stages of the aerosol evolution. For example, smaller evolution time always results
in smaller concentration of primary particles in the sink mode, and normally (but not always
– see below) in larger concentration of fragmenting particles. Thus we usually have anti-
correlations between the sink and the intermediate modes (some exclusions may apply – see
below).
If the fragmentation rate is the same for all the modes (condition 7), then ypif = αi pfy
for all p > 1, where ypif is the concentration of particles in the p-th mode in the i-th scan,
which take part in fragmentation, αi is some coefficient that is the same for all p > 1 (because
of the same fragmentation rate), but are different for different scans. In this case, Eq. (9.4)
for the k-th and m-th modes (for which the anti-symmetry of correlation coefficients is
investigated) can be reduced as
209
Δyki = yki – ky = (αi – 1) kfy – (αi – 1) fky )1( + , (9.6)
Δymi = ymi – my = (αi – 1) mfy – (αi – 1) fmy )1( + . (9.7)
Note again, that Eq. (9.5) is assumed to be satisfied for the k-th and m-th modes. Therefore,
the effect of any other mechanism (apart from fragmentation) of concentration fluctuations is
negligible, and therefore, concentrations of particles that do not take part in fragmentation
cancel out from the right-hand sides of Eqs. (9.6) and (9.7).
Eqs. (9.6) and (9.7) suggest that condition 6 of the fragmentation theorem is satisfied
if Δyki and Δymi have opposite signs, i.e., for example, fky )1( + < kfy , and fmy )1( + > mfy . If
this is the case, then substituting Eqs. (9.6) and (9.7) into condition (9.2), reduces it to an
obvious equality.
If m = 1, then according to condition 5 of the theorem, 01 =Δ addiy , and Eq. (9.4) gives
Δy1i = (1 – αi) ⎟⎟⎠
⎞⎜⎜⎝
⎛+∑
∞
=f
ppf yy 2
1 = (1 – αi)(Nf + fy2 ). (9.8)
Substituting Eqs. (9.6) and (9.8) into condition (9.2) again gives an equality, if fky )1( + <
kfy , i.e., Δyki and Δy1i have opposite signs (condition 6 of the theorem).
According to condition 4, fragmentation theorem is also correct if the number of
intermediate modes is finite. If this is the case, then there exists a maximal diameter of
particles that take part in fragmentation. In this case, the k-th mode may correspond either to
any of the intermediate modes, or the mode with the maximal diameter, as long as the other
conditions of the theorem are satisfied for this mode. In particular, if N is the number of
modes and k = N, then Eq. (9.6) gives
ΔyNi = yNi – Ny = (αi – 1) Nfy , (9.9)
i.e., there is no influx of particles into this mode due to fragmentation. This does not change
the above proof of the fragmentation theorem, which means that the k-th mode can indeed be
210
either one of the intermediate modes, or the mode with maximal diameter (as long as such a
diameter exists and other conditions of the fragmentation theorem are satisfied).
Note also that if k = N (mode with the maximal diameter) and m = 1 (sink mode),
then condition 6 is satisfied automatically (see Eqs. (9.8) and (9.9)).
The fragmentation theorem provides yet another mathematical confirmation of the
fragmentation mechanism of evolution of nano-particle combustion aerosols near a busy
road. It demonstrates that the anti-symmetry of the moving average correlation coefficients
for the 13.6 nm mode and 126 nm mode in Fig. 9.2 can be explained by fragmentation of
nano-particles. Though the presented proof did not take into account the moving average
procedure, it can be understood that this procedure does not breach the anti-symmetry of the
correlation dependencies. Moving averaging only smoothes out the resultant dependencies
without changing their anti-symmetry.
If n = k, then substitution of Eqs. (9.8) and (9.9) into Eq. (9.2), gives the equality 1 ≡
1. Nevertheless, Fig. 9.2 gives less than unity correlation coefficients for the 13.6 nm mode
and 126 nm mode. This is because the moving average procedure smoothers out sharp
maxima and minima of the corresponding dependencies.
Condition (9.5) is essential for the anti-symmetry of the correlation coefficient
dependencies. If this condition is not satisfied, and the additional fluctuations of the
normalised particle concentrations are noticeable, then the common factor (1 – αi) in Eqs.
(9.6) – (9.9) cannot be isolated, and substitution these fluctuations into condition (9.2) does
not result in equality. That is, the necessary and sufficient condition for anti-symmetry is not
satisfied.
The fragmentation theorem may also apply to the process of evaporation that is very
similar to fragmentation. Indeed, evaporation also occurs due to breaking small particles (this
time, separate molecules) from the larger particles (e.g., evaporating droplets). However, the
dependencies in Fig. 9.2 cannot be explained by evaporation because of the following
211
reasons. First, if it were evaporation mechanism that is responsible for the evolution and anti-
symmetry of correlation dependencies (Figs. 9.2 and 9.5), then the 13.6 nm mode should
have been positively correlated with all modes with larger particle diameters. This follows
from the fact that the corresponding particle size distributions (Fig. 6.3b and dark band in
Fig. 8.1) are monotonically decreasing functions of particle diameter within the range ≥ 13
nm. It follows from here that decreasing particle diameter (caused by evaporation) would
have resulted in a uniform shift of this size distribution towards smaller particle diameters
(i.e., to the left), which should lead to a simultaneous decrease of particle concentrations in
all channels in the range ≥ 13.6 nm (see also the discussion before Fig. 9.3). This would have
resulted in positive correlations between the 13.6 nm mode and all other larger modes
(because the concentration in all these modes, including the 13 nm mode, would have
reduced), which is a contradiction to Figs. 9.2 – 9.5 and condition (6) of the fragmentation
theorem.
Secondly, evaporation mechanisms cannot explain the increase of the total number
concentration (Chapters 6 and 7) that was observed using the same sets of data. Neither can
it explain the increase in the concentration of particles with ~ 13 – 30 nm diameters (Fig.
6.3b and dark band in Fig. 8.1).
Therefore, the fragmentation theorem presented in this section suggests that the
observed anti-symmetric pattern of moving average correlation coefficients (Fig. 9.2) is very
likely to occur as a result of particle fragmentation. It establishes conditions for observation
of such anti-symmetry, and thus provides an additional physical insight into the physical
processes during the evolution of combustion aerosols. It may be important for the
determination of possible sources and behaviour of nano-particle aerosols and particle
modes. It is demonstrated that this theorem extends beyond the fragmentation mechanism,
and can be applicable to other types of evaporation/degradation processes, such as
212
evaporation of nano-particles, degradation of polymers, large biological molecules, polymer
networks, etc. (see also Chapter 10).
213
CHAPTER 10
PROBABILISTIC TIME DELAYS DURING MULTIPLE
STOCHASTIC DEGRADATION/EVAPORATION PROCESSES ([A8,
A9, A24, A26, A27])
10.1 Introduction
Multiple stochastic degradation/evaporation processes play an essential role in a
range of physical and chemical phenomena. These include degradation of double-stranded
polymers (Metanomski et al, 1993, Mark et al, 1990, Allen & Edge, 1992 ), fractals and
polymer multi-chains (Guan et al, 1999, Wickman & Korley, 1998), double-stranded DNA
(Lindahl, 1993), dye fading (Allen, 1992), interaction of UV and ionizing radiation with
large molecules (Sinha & Hader, 2002), reversible self-arrangement and
degradation/fragmentation of polymer networks (Sijbesma et al, 1997, Sijbesma & Meijer,
2003) and particle aggregates (Wickman & Korley, 1998, Mezzenga et al, 2003, Fan et al,
2004), thermal fragmentation of nano-particles due to evaporation of bonding molecules
(Chapters 6 and 7, and A4, A5), and even economic and social degradation caused, for
example, by stochastic financial processes (Buchanan, 2002, Mantegna & Stanley, 1995,
Mantegna & Stanley, 1996).
In this Chapter, we demonstrate that stochastic decay of multiple bonds between
elements of a system (e.g., particles/molecules) may lead to significant probabilistic delays
with further degradation of the intermediate substances/structures. A strong impact of such
delays on the life-time and strength of the intermediate modes due to accumulation of the
degradation/fragmentation products in these modes is demonstrated.
214
10.2 Time delays
The analysis is conducted on the example of the thermal fragmentation of composite
aerosol nano-particles (Chapters 6 and 7, and [A4, A5]), though the obtained results will be
directly applicable to any other type of multiple stochastic degradation processes. Aerosol
nano-particles often form aggregates of primary particles that are bonded together by
means of volatile molecules (Chapters 6 and [A4]). These molecules represent multiple
bonds between the particles. Stochastic evaporation of the bonding molecules results in
weakening of the interactions between the primary particles, and they may break away
from the composite particles (Chapters 6 and [A4]), similar to evaporation of molecules
from a droplet of fluid. This happens when the binding energy between the primary
particles becomes sufficiently close to the thermal energy ~ kT. Similar situation occurs
with stochastic decay of multiple covalent and non-covalent bonds in polymer-like
structures (Metanomski et al, 1993, Mark et al, 1990, Allen & Edge, 1992, Guan, 1999,
Lindahl, 1993, Sijbesma et al, 2003).
Consider a composite particle consisting of three primary particles – Fig. 10.1a.
Thermal fragmentation of such composite particles (3-particles) occurs by means of
breaking away one of the primary particles, resulting in one primary particle and one
composite 2-particle (consisting of two primary particles). However, further fragmentation
of the intermediate 2-particles may be significantly delayed compared to that of the 3-
particles. To understand this, we assume that fragmentation may occur only if there is one
bonding molecule left between the particles (for more than one molecules, the binding
energy is too large for fragmentation to occur with reasonable probability). Let one
bonding molecule be left between particles 1 and 2 (one bond) and two molecules between
particles 2 and 3 (two bonds) – Fig. 10.1a. We will say that the 3-particle is in the 1-2 state
(in accordance with the numbers of bonding molecules). We do not distinguish between
the i-k and k-i states, since they are equivalent.
215
Fig. 10.1. a) The 1-2 state of a 3-particle consisting of three primary particles that are bonded by
one and two volatile molecules. b) Two possible ways of evolution of the 3-particle from the 1-2
state: 1-2 1-1 and 1-2 0-2. The transition 1-2 0-2 corresponds to fragmentation of the 3-
particle.
Further evolution of the 3-particle may occur along two paths: 1-2 1-1 or 1-2
0-2 (Fig. 10.1b). During the process 1-2 1-1, the 3-particle does not fragment, but just
loses one of the bonding molecules (one of the two bonds between particles 2 and 3).
Further evaporation of either of the molecules will lead to fragmentation of the 3-particle:
1-1 0-1. The 0-1 state corresponds to one free primary particle and one 2-particle. The
2-particle fragments with no delays, i.e., the corresponding rate coefficient is “switched
on” immediately after the 3-particle fragments. This is because the resultant 2-particle in
the 0-1 state contains only one bonding molecule. Thus, no fragmentation delays occur
along the path 1-2 1-1 0-1 0-0.
If however evolution takes the path: 1-2 0-2 0-1 0-0, then fragmentation of
the 2-particles in the state 0-2 is delayed. This is because the 2-particle from the 0-2 state
cannot fragment until one of the two bonding molecules evaporates. The fragmentation
delay will thus be equal to the time of evaporation of one of the two bonding molecules
(bonds).
To determine the average delay between fragmentation of the 3-particles and 2-
particles, we need to determine the corresponding delays and numbers of 3-particles going
216
through different evaporation paths, e.g., 0-3 0-0, 0-4 0-0, etc. Suppose that we have
N0 3-particles in the initial 4-4 state. After losing one bonding molecule, all 3-particles will
pass through the 3-4 state (Fig. 10.2). A 3-particle in the 3-4 state has two options for
further evolution: to go into the 2-4 state (by losing one of the three bonding molecules), or
the 3-3 state (by losing one of the four bonds), etc. Thus we obtain a random graph
representing evaporation of bonding molecules in the considered 3-particles (Fig. 10.2).
Note that the initial 4-4 state also corresponds to three monomers with quadruple hydrogen
bonds considered in (Sijbesma et al, 1997, Sijbesma & Meijer, 2003).
Fig. 10.2. The random graph representation of the processes of stochastic evaporation of
bonding molecules. The 4-4 state is the initial state for the 3-particles, and 0-0 is the final state (in
which all three primary particles are free as a result of fragmentation). The arrows (edges of the
graph) indicate the direction of possible ways of particle evolution. N0 is the initial number of the
3-particles in the 4-4 state.
This random graph can be extended to the case with an arbitrary n-n or n-m initial
state. However, increasing n,m > 4 does not change the average time delays substantially,
because the probability for a particle to go through a 0-n state decreases with increasing n.
217
To determine the average fragmentation delays, we denote the number of 3-particles
passing through a particular i-k state (a vertex of the random graph – Fig. 10.2) as Nik.
Then, due to particle conservation, the sum of all numbers Nik for the vertices between any
two neighbouring dotted lines in Fig. 10.2 is equal to N0. This gives five equations relating
11 unknowns:
N33 + N24 = N23 + N14 = N22 + N13 + N04 = N12 + N03 = N11 + N02 = N0. (10.1)
The remaining 6 equations are obtained from the consideration of probabilities for
evaporation of bonding molecules in each of the i-k states of the 3-particle. Each bonding
molecule interacts with two bonded particles and other neighbouring bonding molecules by
means of van der Waals forces. Therefore, it is reasonable to expect that the energy that
should be given to one bonding molecule in order to remove it from between the particles
(binding energy of evaporation) increases with increasing number of bonding molecules,
because of the additional interaction between these molecules. This is analogous to a
possibility of interactions between the neighbouring bonds, e.g., due to overlap of the
electron wave functions in polymer molecules. As a result, the time of evaporation of one
bonding molecule (one bond) may increase by a factor of αk, where k is the number of
bonding molecules between the particles (α1 = 1).
For example, transformation of the 1-2 state into the 1-1 state occurs when one of the
two molecules evaporates. Let the time for evaporation of a single bond between two
particles be τ1. Then the time for evaporation of either of the two bonds (one of the two
molecules) is τ2 = α2τ1/2. Thus the probability for the process 1-2 1-1 is 2/α2 times
larger than for the process 1-2 0-2. Therefore, the number of particles going through the
graph edge 1-2 1-1 (which is N11) is 2/α2 times larger than the number of particles going
through the edge 1-2 0-2 (which is N02 – N03). This gives the first of the following six
equations:
N11 = 2(N02 – N03)/α2, (N12 – N22) = 3(N03 – N04)/α3,
218
(N13 – N23 + N22) = 4N04/α4, N22 = 3α2(N13 – N14 + N04)/(2α3); (10.2)
N23 – N33 = 2α2N14/α4; N33 = 4α3N24/(3α4).
The other five equations are obtained from the similar consideration of transitions from the
1-3, 1-4, 2-3, 2-4, and 3-4 states, respectively; α3 and α4 are the factors increasing the
average time for evaporation of one of the three and one of the four molecules,
respectively. Here, we have also assumed (for simplicity) that evaporation of each of the
three (four) molecules from between the bonded particles is equally probable.
The fragmentation delays for the 2-particles originating from different 0-i states (Fig.
10.2), and the corresponding numbers of the 3-particles are given in the table.
Vertex of the graph 0-1 0-2 0-3 0-4
Fragmentation delay,
Δτ1,2,3,4
0 α2τ1/2 α2τ1/2 + α3τ1/3 α2τ1/2 + α3τ1/3 + α4τ1/4
Number of 3-particles N11 N02 – N03 N03 – N04 N04
Table 10.1. Fragmentation delays for different types of 2-particles and the numbers of such
particles.
Thus the average fragmentation delay for the intermediate 2-particles
Δτ = [Δτ1N11 + Δτ2(N02 – N03) + Δτ3(N03 – N04) + Δτ4N04]/N0. (10.3)
10.3. Evolution time and kinetics of degradation
The average time that takes for the 3-particle to evolve from the initial 4-4 state to
one of the 1-k states preceding fragmentation can also be calculated from the graph in Fig.
10.2. Different paths on the graph may lead to the same 1-k state. For example, three
different paths lead to the 1-3 state: (1) 4-4 3-4 2-4 1-4 1-3, (2) 4-4 3-4
219
2-4 2-3 1-3, and (3) 4-4 3-4 3-3 2-3 1-3. The evolution time is
determined for each of the different paths by means of calculating time for each of the
possible transitions.
For example, the time for the transition 4-4 3-4 is equal to the time of evaporation
of one of the 8 bonding molecules: τ44-34 = α4τ1/8. One may think that the time for the
process 3-4 2-4 will then be α3τ/3. However, this is not correct, because two
simultaneous processes may originate from the state 3-4: 3-4 3-3 and 3-4 2-4.
Therefore, the time for a particle to leave the state 3-4 will be
τ34-33,24 = [4/(α4τ) + 3/(α3τ)]-1 = τ(α3α4)/(4α3 + 3α4). (10.4)
Since this is the time for the particles to leave the 3-4 state, the time for a particle to go
from the 3-4 state to the 2-4 state should not exceed the time τ34-33,24. This is because at
time t > τ34-33,24, there will be no particles in the state 3-4, and thus no particles can be
transferred from this state to the 2-4 state. This suggests that the time for the process 3-4
2-4 will be equal to τ34-33,24.
This conclusion can also be confirmed mathematically by solving the following
model problem. Consider particles A with the concentration [A], which can be transformed
by two different processes into particles B1 or B2. The rate equations for these processes
are:
d[A]/dt = – (k1 + k2)[A];
d[B1]/dt = k1[A]; d[B2]/dt = k2[A], (10.5)
where k1 and k2 are the rate coefficients for the transformations A B1, and A B2,
respectively. In other words, the typical times for these transformations are τ1 = k1-1 and τ2
= k2-1. Solutions to Eqs. (10.5) are
[A] = Cexp{– t(k1 + k2)}; [ ])}(exp{1][ 2121
11 kkt
kkCk
B +−−+
= ;
220
[ ])}(exp{1][ 2121
22 kkt
kkCk
B +−−+
= , (10.6)
where C is a constant and [B1,2] are the concentrations of particles B1,2, respectively.
From these solutions, one can see that the time for the particles to leave the A state
(e.g., the 3-4 state) is (k1 + k2)-1, as well as the times for the particles to be transformed into
the states B1 and B2 (e.g., the 3-3 and 2-4 states). This is equivalent to using Eq. (10.4) for
the determination of transition time from the 3-4 state of the 3-particle. In other words, the
time that it takes for a 3-particle to go from the 3-4 state to the 2-4 state is not equal to the
time of evaporation of one of the three molecules, but rather to the time of evaporation of
one of the 7 molecules, which is given by Eq. (10.4):
τ34-24 = τ34-33 = τ34-33,24 = τ(α3α4)/(4α3 + 3α4). (10.7)
Similarly, the time for a particle to go from the state 2-4 to the state 1-4 is
τ24-14 = τ(α4α2)/(4α2 + 2α4). (10.8)
The total time for the process 4-4 1-4 is
τ14 = τ44-34 + τ34-24 + τ24-14 =
= α4τ/8 + τ(α3α4)/(4α3 + 3α4) + τ(α4α2)/(4α2 + 2α4). (10.9)
The time that it takes for the particle to reach the 1-3 state is calculated similarly.
However, in this case, a particle can reach the 1-3 state by means of three different routes:
1) 4-4 3-4 3-3 2-3 1-3; 2) 4-4 3-4 2-4 2-3 1-3; or 3) 4-4 3-4
2-4 1-4 1-3. If α2 = α3 = α4 = 1 (i.e., there is no increase in the binding energy per
one molecule due to interactions of these molecules with each other), then the times for a
particle to go through any of these routes is the same and approximately equal to τ/8 + τ/7
+ τ/6 + τ/5, which is thus the time for the particle to reach the 1-3 state. The term
approximately is used, because the actual time will be slightly larger than that determined
by this equation, because direct averaging of times corresponding to different routes in the
221
graph (Fig. 10.2) gives slightly (by a few percent) smaller estimate that the actual time
obtained from the direct solution of the kinetic equations.
If α2 ≠ α3 ≠ α4 ≠ 1, then these different routes will result in different times:
1) τ*13 = τα4/8 + τα3α4/(4α3 + 3α4) + τα3/6 + τα2α3/(3α2 + 2α3); (10.10a)
2) τ**13 = τα4/8 + τα3α4/(4α3 + 3α4) +
+ τ(α4α2)/(4α2 + 2α4) + τα2α3/(3α2 + 2α3); (10.10b)
3) τ***13 = τα4/8 + τα3α4/(4α3 + 3α4) +
+ τ(α4α2)/(4α2 + 2α4) + τα1α4/(α4 + 4α1), (10.10c)
where the number of asterisks indicate the first, second, or third route undertaken by
particles reaching the 1-3 state. If all the alphas are equal to one, these equations are
reduced, as expected, to τ/8 + τ/7 + τ/6 + τ/5. If however, this is not the case, then it is
reasonable to take an average of all these times, taking into account the number of particles
taking each of the three routes.
For route 1, the number of particles getting into the 2-3 state is equal to N33. Of
these particles, N*23-13 will reach the 1-3 state, and N*
23-22 will not. The asterisk is used to
distinguish particles taking route 1 from other particles that may also undergo the same
transitions. Thus,
N*23-13 + N*
23-22 = N33. (10.11)
In addition, from the probabilities for a particle to go into the 1-3 state or the 2-2 state, we
obtain:
N*23-22 = 3α2N*
23-13/(2α3). (10.12)
Solving these two equations, we obtain:
33
2
3
*2223
32
1
1 NN
αα
+=− ; 33
3
2
*1323
23
1
1 NN
αα
+=− , (10.13)
which determines the number of particles N*23-13 taking the first route and reaching the 1-3
state.
222
Similarly, for the second route
)(
32
1
13323
2
3
**2223 NNN −
αα
+=− ; )(
23
1
13323
3
2
**1323 NNN −
αα
+=− .
Thus the number of particles taking the second route and reaching the 1-3 state is N**23-13.
For the third route, the number of particles reaching the 1-3 state is equal to N14 –
N04.
Thus the average time for a particle to reach the 1-3 state is
τ13 ≈ [τ*13N*23-13 + τ**13N**
23-13 + τ***13(N14 – N04)]/N13. (10.14)
For the 1-2 state, we have 5 different routes of evolution:
1) 4-4 3-4 2-4 1-4 1-3 1-2;
2) 4-4 3-4 2-4 2-3 1-3 1-2;
3) 4-4 3-4 2-4 2-3 2-2 1-2;
4) 4-4 3-4 3-3 2-3 1-3 1-2;
5) 4-4 3-4 3-3 2-3 2-2 1-2; (10.15)
The corresponding times of evolution:
1) τ(1)12 = τα4/8 + τα3α4/(4α3 + 3α4) + τ(α4α2)/(4α2 + 2α4) +
+ τα1α4/(α4 + 4α1) + τα1α3/(α3 + 3α1); (10.16a)
2) τ(2)12 = τα4/8 + τα3α4/(4α3 + 3α4) + τ(α4α2)/(4α2 + 2α4) +
+ τα2α3/(3α2 + 2α3) + τα1α3/(α3 + 3α1); (10.16b)
3) τ(3)12 = τα4/8 + τα3α4/(4α3 + 3α4) + τ(α4α2)/(4α2 + 2α4) +
+ τα2α3/(3α2 + 2α3) + τα2/4; (10.16c)
4) τ(4)12 = τα4/8 + τα3α4/(4α3 + 3α4) + τα3/6 +
+ τα2α3/(3α2 + 2α3) + τα1α3/(α3 + 3α1); (10.16d)
5) τ(5)12 = τα4/8 + τα3α4/(4α3 + 3α4) + τα3/6 +
+ τα2α3/(3α2 + 2α3) + τα2/4. (10.16e)
223
Now we find the numbers of particles passing through a particular route.
1) )(
31
10414
1
3
)1(1214 NNN −
αα
+=− ; (10.17a)
2) )(
23
13
1
11424
3
2
1
3
)2(12132324 NNN −
⎟⎟⎠
⎞⎜⎜⎝
⎛αα
+⎟⎟⎠
⎞⎜⎜⎝
⎛α
α+
=−−− ; (10.17b)
3) )(
32
1
11424
2
3
)3(12222324 NNN −
αα
+=−−− ; (10.17c)
4) )(
23
13
1
1240
3
2
1
3
)4(1213233334 NNN −
⎟⎟⎠
⎞⎜⎜⎝
⎛αα
+⎟⎟⎠
⎞⎜⎜⎝
⎛α
α+
=−−−− ; (10.17d)
5) )(
32
1
1240
2
3
)5(1222233334 NNN −
αα
+=−−−− . (10.18e)
Thus, the average time of evolution for the transition from the 4-4 state to the 1-2
state is
τ12 ≈ (τ(1)12N(1)
14-12 + τ(2)12N(2)
24-23-13-12 + τ(3)12N(3)
24-23-22-12 +
+ τ(4)12N(4)
34-33-23-13-12 + τ(5)12N(5)
34-33-23-22-12)/N12. (10.19)
The time of evolution to the state 1-1 is determined simply by adding the time δτ =
α1α2τ/(α2 + 2α1) for a particle to leave the 1-2 state to all the times corresponding to the
evolution to the 1-2 state. This is because the only way the particles can get to the 1-1 state
is through the 1-2 state.
The number of particles passing from the 1-2 state into the 1-1 state is given by
N12-11 = N12/[1 + α2/(2α1)].
Therefore, to get the number of particles getting to the 1-1 state by means of 5 different
routes (10.15), we multiply the numbers of particles determined by Eqs. (10.17) by the
additional factor
224
1
22
1
1
αα
+.
Then the average time of evolution for the transition from the 4-4 state to the 1-1 state is
τ11 ≈ [(τ(1)12 + δτ)
1
2
)1(1214
21
αα
+
−N + (τ(2)
12 + δτ)
1
2
)2(12132324
21
αα+
−−−N +
+ (τ(3)12 + δτ)
1
2
)3(12222324
21
αα+
−−−N + (τ(4)
12 + δτ)
1
2
)4(1213233334
21
αα+
−−−−N +
+ (τ(5)12 + δτ)
1
2
)5(1222233334
21
αα+
−−−−N]/N11. (10.20)
Note that this procedure is to an extent similar to the determination of the
communication-related random delays in large-scale computing systems (Gu & Niculescu,
2004).
For example, consider the case with α1 = α2 = α3 = α4 = 1, i.e., there is no increase of
the binding energy and no increase of the evaporation time if there are two or more binding
molecules between two particles. In this case, if N0 = 100, then N11 = 57.1, N02 = 42.9, N12
= 85.7, N03 = 14.3, N13 = 45.7, N22 = 51.4, N23 = 85.7, N04 = 2.9, N14 = 14.3, N33 = 57.1, N24
= 42.9 are the numbers of particles passing through the corresponding graph vertices (Fig.
10.2). Then the average fragmentation delay for the 2-particles determined from Eq. (10.3)
is Δτ ≈ 0.269τ1, where τ1 is the time of evaporation of a single bonding molecule. From the
experimental observations of fragmentation, τ1 ~ 10 s (the fragmentation process occurs on
this time scale (Chapters 6 and 7, and [A4, A5])), which gives Δτ ≈ 2.7 s.
The typical binding energy E0 for a single volatile molecule between the particles is
evaluated from the Maxwell-Boltzmann distribution. We determine the probability P0(E0)
for a molecule to have the energy larger than E0 (this is the probability of fragmentation at
225
a given moment of time) by integrating the Maxwell-Boltzmann distribution between E0
and ∞. The ratio of τ1 to the average time between the collisions in the air L/c (L is the
mean free path for the molecules in the air, and c is the mean speed) must be of the order
of 1/P0(E0), or τ1 ~ L/[cP0(E0)]. Assuming that T = 300 K, L ~ 300 nm, c ≈ 467 m/s, and τ1
~ 10 s (the fragmentation rate coefficient k0 = 0.1 s-1), we get: E0 ≈ 1.045×10-19 J.
If the additional van der Waals interaction between two or more bonding molecules
increases the binding energy of evaporation per one molecule by just 5% for two molecules
and by 7% for three and more molecules, then Δτ ≈ 24 s. Thus, increasing binding energy
of evaporation due to mutual interaction of bonding molecules by just 5% – 7% results in
almost 10 times increase in the fragmentation delay. This certainly highlights a very
significant impact of such an energy increase on stochastic degradation processes,
including thermal fragmentation of nano-particles and polymer degradation.
In this example, the average evolution times that take for a 3-particle in the initial 4-4
state to reach the fragmentation stage, i.e., the states 1-4, 1-3, 1-2, and 1-1 are τ14 ≈ 23.41 s,
τ13 ≈ 33.10 s, τ12 ≈ 41.73 s, and τ11 ≈ 48.18 s, respectively. The corresponding numbers of
particles experiencing fragmentation through each of these states (if N0 = 100) are N04 ≈
11.32, N03 – N04 = 32.79, N02 – N03 ≈ 35.38, and N01 – N02 ≈ 20.51. Thus the average
evolution time τav ~ 38 s with the dispersion of the mean δτA ~ 8 s. Therefore, it is
reasonable to approximate that the fragmentation rate coefficient for the 3-particles
“switches on” at t = τav – δτA ≈ 30 s and linearly increases to its steady-state value k0 = 0.1
s-1 within the time interval 2δτA ≈ 16 s.
If the fragmentation delays for the intermediate 2-particles are assumed to be zero,
then the solution of the kinetic equations results in curves 1 – 3 in Fig. 10.3. However, if
the fragmentation delays are determined using Table 10.1, then Δτ1 = 0, Δτ2 ≈ 17.60 s, Δτ3
≈ 38.86 s, Δτ4 ≈ 51.31 s, and the average delay Δτ ≈ 24.5 s. As a result, the average
evolution time to fragmentation of the resultant 2-particles is t = τav + Δτ ≈ 62.5 s with the
226
dispersion δτB ≈ 9.5 s. Thus, we approximate that the fragmentation of the intermediate 2-
particles starts at t = τav + Δτ – δτB ≈ 53 s with the linearly increasing rate coefficient from
zero to k0 within the time interval 2δτB ≈ 19 s. The solution of the kinetic equations with
these parameters gives curves 1, 4 and 5 in Fig. 10.3.
Fig. 10.3. The time dependencies of the concentrations of the 3-particles (curve 1), intermediate 2-
particles (curves 3 and 5), and primary particles (curves 2 and 4). Curves 2 and 3: zero
fragmentation delay for the 2-particles (Δτ = 0). Curves 4 and 5: fragmentation delay for the
intermediate 2-particles Δτ = 24 s.
In particular, it can be seen that fragmentation delays have a substantial impact on
the kinetics of fragmentation/degradation processes. In the presented example, the number
(concentration) of the intermediate 2-particles in the presence of the fragmentation delays
(curve 5) increases up to ~ 5 times compared to that in the absence of this delays (curve 3).
At the same time, the time-dependent number of the primary particles in the presence of
fragmentation delays (curve 4) may be less up to 2 times compared to that at zero delays
(curve 2).
227
If the dispersion is taken into account with the parameters m = 0.87 (normal
component of the wind speed is 1 m/s) and the fragmentation rate coefficients k = 0.1 s-1,
then for the same parameters (evolution times, fragmentation delays, etc.) as for Fig. 10.3,
the dependencies of particle concentrations on distance from the road are presented in Fig.
10.4. As a result, dispersion does not introduce principle changes in the obtained pattern of
mode evolution, and this approach may thus be used for mode modelling near busy roads.
Fig. 10.4. The dependencies of the concentrations of the 3-particles (curve 1), intermediate 2-
particles (curves 3 and 5), and primary particles (curves 2 and 4) on distance from the road in the
presence of dispersion and turbulent diffusion. Fragmentation of the 3-particles starts at ≈ 30 m
from the road or, equivalently, after t = τav – δτA ≈ 30 s of evolution. The initial numbers of 3-
particles, 2-particles and primary particles at this moment are the same and equal to 100. Curves 2
and 3: zero fragmentation delay for the 2-particles (Δτ = 0). Curves 4 and 5: fragmentation delay
for the intermediate 2-particles Δτ = 24 s. Other parameters are identical to those for Fig. 10.3.
In particular, it can be seen that fragmentation delays significantly affect
concentrations of the primary particles and especially intermediate 2-particles. As a result,
the intermediate modes should be much more pronounced with fragmentation delays. The
resultant typical concentrations in these modes may increase several times, compared to
228
the situation when the fragmentation delays are ignored. This demonstrates that
fragmentation delays are one of the major effects in the process of mode formation in
combustion aerosols near busy roads.
It can also be seen that the determined fragmentation delays also strongly depend on
temperature. For example, decreasing temperature by 20 C results in τ1 ≈ 62 s, and Δτ ≈
168 s, which may effectively lead to suppression of fragmentation near a particle source.
The developed theory of fragmentation delays can naturally be extended to more
complex systems consisting of four or more particles, and include the possibility of
restoration of bonds between the particles (e.g., condensation of bonding molecules). This
will be especially important for the analysis of particle fragmentation in the presence of
significant pressure of vapour of volatile compounds, reversible self-assembling polymer
structures and networks (Sijbesma et al, 1997, Sijbesma & Meijer, 2003), and DNA repair
(Lindahl, 1993, Sinha & Hader, 2002).
229
CHAPTER 11
MULTI-CHANNEL STATISTICAL ANALYSIS OF BACKGROUND
FINE PARTICLE AEROSOLS ([A10, A28])
In this chapter, the new statistical methods for the determination and investigation of
particle modes and their mutual correlations in fine and ultra-fine particle aerosols in the
presence of strong turbulent mixing (Chapters 6, 9 and [A5,A7,A23,A25]) are applied
for the analysis of urban background aerosols. In particular, several distinct modes will
be obtained from the background particle size distribution, and their possible sources
will be discussed. Anti-symmetrical correlation pattern similar to that obtained near a
busy road in the presence of particle fragmentation (Chapter 9) will also be
demonstrated for background aerosols. Therefore, fragmentation theorem (Chapter 9)
will be used for possible interpretation of the obtained results.
The experimental measurements were conducted near Gateway Motorway in the
Brisbane area, Australia, on the upwind side of the road, so that traffic emissions from
the road did not affect the obtained background data (Fig. 11.1). There were no
buildings within ~ 150 m from the measurement area that was practically a flat grass
field with isolated scattered bushes and trees. Further away from the road, there is a
small residential area with a parkland – see Fig. 11.1. No particular sources of air
pollutants are known within at least several kilometers upwind from the place of
measurements. The distance from the curb of the road was ~ 60 m (the place of the
measurement is marked by a cross on the map – Fig. 11.1). This distance was sufficient
for not registering any noticeable particle concentrations coming directly from the road
due to turbulent diffusion.
230
Concentrations of fine particles in the background aerosol were measured within
the range from 13 nm to 763 nm at the height h = 2 m above the ground during the
afternoon of 30 July 2002. 38 scans with 113 equal intervals of Δlog(Dp), where Dp is
particle diameter in nanometers, were taken by means of a Scanning Mobility Particle
Sizer (SMPS). All the concentration measurements were conducted simultaneously with
the measurements of the temperature, humidity, solar radiation, wind speed and wind
direction every 20 seconds at the same height above the ground of ≈ 2 m.
Fig. 11.1. The area of measurements near Gateway Motorway, Brisbane, Australia. The
indicated receptor point is at the distance ~ 60 m from the curb of the road. The scale of the map
and the direction to the North are as indicated (the distances on the axes are given in meters).
The cross indicates the receptor point. The insert presents a section of the map with the area of
measurements.
231
Meteorological parameters
scans 11 to 27 scans 22 to 38
wind direction, degrees to the road
46 ± 50 35 ± 40
wind speed, ms-1 1.7 ± 0.8 1.5 ± 0.8
temperature, °C 20.4 ± 0.9 18.7 ± 0.6
humidity, % 40 ± 4 46 ± 2
solar radiation, Wm-2
200 ± 100 30 ± 30
Table 11.1. Average meteorological parameters.
38 scans in total were taken, and two separate sets of 17 scans (from 11 to 27 and
from 22 to 38) were considered. The choice of these sets was conducted primarily on
the basis changing solar radiation. The monitoring was conducted during the time of
sunset, and the second set of scans corresponded to nearly zero solar radiation, while for
the first set of scans it was still significant (Table 11.1).
Fig. 11.2. Typical size distribution in the urban background aerosol
in the Brisbane area, Australia, together with the experimental points.
232
The average particle size distributions for both the sets of scans, plotted by means
of the moving average technique, are presented in Figs. 11.2 and 11.3. The average
values of the meteorological parameters for the two sets of scans are presented in Table
11.1 together with their standard deviations.
Fig. 11.3. The moving average size distributions for the two sets of scans: (1) scans from 11 to
27 (from 2:43pm to 4:33pm - higher solar radiation), and (2) scans from 22 to 38 (from 3:54pm
to 5:44pm - negligible solar radiation). Curve 1 is identical to that in Fig. 11.2. Light and dark
bands show the standard errors of the mean for both the curves.
Only two maximums can be seen on the resultant particle size distributions – at ~
20 nm and ~ 100 nm (Fig. 11.2 and curve 1 in Fig. 11.3). Otherwise, the size
distributions do not display any significant features and/or distinct particle modes (the
maximum at ~ 20 nm is fairly small and not pronounced, which make it difficult to
analyse and interpret). Nothing can be said about possible relationships between
particles in the obtained size distribution, and their mutual dependencies. No firm
conclusions can be made about the differences between the two size distributions before
the sunset (curve 1 in Fig. 11.3) and after the sunset (curve 2 in Fig. 11.3).
233
Therefore, the statistical methods of analysis of background aerosols similar to
those developed in Chapters 6, 8, and 9 are expected to be of a significant help. As was
demonstrated (Chapters 6, 8 and 9), the methods that provide highly effective
determination and analysis of possible particle modes and mutual correlations between
different channels in the size distribution is based on the determination of the moving
average correlation coefficient. Application of this method with 7-channel moving
interval (Chapters 6 and 8) to the two background data sets of 17 scans results in the
dependencies of the moving average simple correlation coefficient displayed in Figs.
11.4a,b.
Fig. 11.4. The moving average correlation coefficients as functions of particle diameter for the
two sets of 17 scans: (a) from 11 to 27 (before the sunset), and (b) from 22 to 38 (after the
sunset). Both the dependencies were plotted using 7-channel moving interval. The shadow
bands indicate the associated errors of the mean.
234
One of the main important features of Figs. 11.4a,b is that the demonstration of
the existence of strong and distinct particle modes in the background particle size
distributions. At least five distinct modes (corresponding to particle diameters ~ 21 nm,
~ 35 nm, ~ 57 nm, ~ 105 nm, and ~ 168 nm) can be seen in Fig. 11.4a, and at least four
modes are displayed by the curve in Fig. 11.4b (at ~ 30 nm, ~ 55 nm, ~ 80 nm , and ~
168 nm). The presented error curves clearly demonstrate that all of these modes are
statistically significant.
The existence of these modes demonstrates that despite the smoothness of the
original particle size distributions with no significant distinct features, there are,
nevertheless, distinct groups of particles forming these size distributions.
Concentrations of particles in different channels within each of these groups have strong
tendencies to increase/decrease in correlation with each other. As has been mentioned in
Chapter 6, this is an indication that these particles (from each mode in Figs. 11.4a,b) are
likely to have come from the same source (origin) and/or have the same
physical/chemical nature.
It is interesting to note that the three modes in Figs. 11.4a,b with smaller particle
diameters approximately correspond to the modes previously described in Chapters 6
and 8 for transport emissions from busy roads. Therefore, it is possible that these modes
originate from transport emissions from the road network in the urban environment. At
the same time, it is important to indicate that these modes (strong maximums on the
correlation curve in Fig. 11.4a,b) correspond to relatively low particle concentrations
(Figs. 11.2 and 11.3). In addition, it is possible to see that these modes transform
noticeably with decreasing solar radiation (compare Figs. 11.4a and 11.4b).
The modes with larger particle diameters, e.g., at ~ 105 and ~ 168 nm (Fig. 11.4a)
correspond to noticeably larger number concentration (Figs. 11.2, 11.3). It may be
expected that at least some of these modes may come from natural sources, e.g., marine
235
aerosols (Seinfeld and Pandis, 1998). However, this should probably require further
confirmation by means of application of the mentioned approach to the analysis of
background aerosols at significantly different wind conditions.
A very interesting and unexpected result was obtained as a result of the
application of the moving average approach to the analysis of mutual correlations
between different particle modes (Chapter 9) in the background size distributions
displayed in Figs. 11.2 and 11.3. For example, Fig. 11.5 presents the dependencies of
the moving average cross-correlation coefficients for the two primary 7-channel
intervals centered at 40 nm (dashed curves) and 113 nm (solid curves) for the two
considered sets of scans (for more detailed description of the statistical procedure used
see Chapter 9).
The maxima for both the peaks at 40 nm and 113 nm for the dashed and solid
curves in Fig. 11.5 are slightly larger than the corresponding moving average
correlation coefficients in Fig. 11.4b. This is because for the curves in Fig. 11.4,
correlations of each channel with itself were discarded from the averaging procedure,
which results in slightly lower curves (without effecting anything else).
It can be seen that after the sunset the correlations between the different channels
in the particle size distribution have substantially changed. The dependencies for the
considered channels became almost exactly anti-symmetric of each other (Fig. 11.5b).
This is a clear indication of significantly changing processes of particle evolution. The
anti-symmetry of the correlation dependencies exists only for the pair of 40 nm and 113
nm channels. Changing to any other channels quickly breaks the anti-symmetry. Recall
that this anti-symmetry means that the correlation coefficient between the 40 nm
channel and, for example, ~ 200 nm channel is exactly the same in magnitude as the
correlation coefficient between the 113 nm channel and the same ~ 200 nm channel, but
236
opposite in signs. This is a highly non-trivial result, especially taking into account the
high degree of anti-symmetry displayed by Fig. 11.5b.
Fig. 11.5. Moving average simple correlation coefficients between the 113 nm channel and all
other channels (solid curves), and between the 40 nm channel and all other channels (dashed
curves). (a) The dependencies for the first set of scans from 11 to 27 (before the sunset), and (b)
for the second set of scans from 22 to 38 (after the sunset). The solid horizontal lines correspond
to zero correlation coefficient. The average particle diameter for each of the secondary 7-
channel intervals is taken as the diameter corresponding to the central channel of the interval.
It is important that Fig. 11.5a does not display the same anti-symmetry. This is an
indication that before the sunset the conditions and the evolution processes in the
background aerosol are not appropriate for producing such an anti-symmetry.
237
Fragmentation theorem proved in Chapter 9 was previously used for the
explanation of anti-symmetric correlation patterns similar to those in Figs. 11.5b, 9.2,
9.4 and 9.5. This theorem suggests that such anti-symmetric patterns can be explained
by fragmentation of nano-particles. Therefore, it is highly tempting to suggest that the
anti-symmetric pattern in Fig. 11.5b is also related to particle fragmentation. In our
opinion, this is indeed possible, but such a statement will nevertheless need further
confirmation by future analysis and measurements.
Note also that the reason for choosing the 40 nm and 113 nm channels in the size
distribution is because only these channels correspond to the antisymmetric cross-
correlation pattern. This means that the conditions of the fragmentation theorem
(Chapter 9) happen to be satisfied only for these channels. Therefore, it is possible that
mechanisms other than fragmentation may be at least partly responsible for fluctuations
of particle concentrations in other channels.
The breach of the anti-symmetric correlation pattern before the sunset (Fig. 11.5a)
does not necessarily mean that fragmentation does not occur before the sunset. On the
contrary, it might be even stronger due to higher temperature and solar radiation.
However, the conditions of the fragmentation theorem may only be satisfied after the
sunset, which results in the anti-symmetric pattern in Fig. 11.5b.
It is not clear at this stage if the ~ 40 nm particles are the primary particles or
belong to one of the intermediate modes (see Chapter 9). Further research in this
direction (that is however outside the scope of this thesis) is necessary. One of the
possibilities is that the ~ 40 nm particles are yet another type of primary particles from
the vehicle exhaust (coming from the upwind road networks). Such larger primary
particles may be formed in the exhaust pipe by means coagulation of smaller (~ 7 nm
and/or ~ 12 nm) particles by means of strong covalent bonds, before these bonds are
engaged by volatile fragments (Chapter 6). Because the ~ 40 nm particles are relatively
238
large, their fragmentation may be impeded by larger average bonding area, and thus
larger bonding energy (Chapter 6). Therefore, it takes longer for these particles to
fragment, and we can observe this process within several kilometers from the source
(i.e., in the background aerosol).
Another option is that these are particles of different nature (e.g., marine aerosols)
that may also experience fragmentation. However, at this stage, this option seems to be
less clear and less feasible.
In conclusion to this Chapter, it is important to note that the observed interesting
and unusual effects in the background urban aerosol in the Brisbane metropolitan area
are only the first step towards a comprehensive understanding of the nature, sources and
evolution processes in such aerosols. The presented evidence suggests that the
fragmentation mechanism of aerosol evolution may also be important for the
consideration of background urban aerosols. Unfortunately, at this stage, we do not have
sufficient proof to suggest this with certainty. The anti-symmetric correlation pattern for
the ~ 40 nm and ~ 113 nm modes is only the first piece of evidence in this direction.
Additional research will be required to confirm or reject this suggestion.
Nevertheless, the obtained results are a very important step demonstrating the
power of the developed new statistical methods in terms of providing new physical
insights into the processes of evolution and transformation of background aerosols,
which may be important for our understanding of aerosol behaviour and their impact on
the environment not only on local, but also on global scales. They have also provided
the first piece of evidence that one of the major mechanisms of evolution of background
aerosols may be fragmentation of nano-particles.
239
CHAPTER 12
CONCLUSIONS
This thesis was focused at gaining better understanding of fundamental and
applied aspects of evolution and dispersion of combustion aerosols near busy roads in
the urban environment, and development of practical methods and techniques for the
identification and investigation of the main physical mechanisms of behavior of
airborne nano-particles from vehicle exhaust with the ultimate aim of accurate
prediction of human exposure and environmental impact of combustion aerosols.
The research was based on a natural combination of experimental monitoring of
combustion aerosols and air quality in the urban environment, theoretical and numerical
modelling of aerosol dispersion and mechanisms of evolution of nano-particles, and
development and application of new statistical methods of data analysis in the presence
of strong turbulent mixing and stochastic variations of atmospheric and environmental
parameters in real-world situations. One of the major fundamental achievements of this
thesis is the suggestion, experimental investigation, and theoretical modelling of a
radically new mechanism of aerosol evolution based on intensive thermal fragmentation
of nano-particles. The significance of this new mechanism may be far beyond evolution
and transformation of combustion aerosols near busy roads (which is demonstrated by
this thesis). It may also play a major role in shaping patterns of atmospheric aerosols on
the global and local scales. At the same time, it is important to understand that the
presented theory and models will certainly need further extensive experimental and
theoretical confirmation/validation under different conditions. For example, at this stage
little is known about the actual types of volatile compounds that may be responsible for
the fragmentation mechanisms of aerosol evolution. Direct observations of solid
240
primary nano-particles and laboratory investigations of their possible
interaction/fragmentation would also be of a significant benefit in future. Therefore, this
thesis presents the first major step towards the further important research in this area.
The thesis has also developed several specific methods and approaches for the
accurate prediction of aerosol pollution levels in the urban environment, and techniques
for the determination and analysis of vehicle emissions under field conditions. Several
unique and powerful statistical methods for the investigation of dispersion, evolution
processes, and possible sources of nano-particle aerosols have been successfully
developed and applied for the analysis of combustion and background aerosols. All
these findings and methods will be important for the development of workable and
scientifically-based national and international standards for nano-particle emissions,
accurate prediction of human exposure to combustion aerosols, determination and
reduction of the environmental impact of aerosols and other types of air pollutants in the
indoor and outdoor environments.
List of main results
1. Adjustment of the software package CALINE4 (from the California Transport) for
prediction and modelling of aerosol pollution from busy roads in the Gaussian
plume approximation.
2. Development of a new method for the determination of the average emission
factors per vehicle on a road, based on the experimental measurements of particle
concentration at one point near the road. Determination of the average emission
factors for Gateway Motorway, Brisbane, Australia.
3. Development of two new methods of determination of average emission factors
from different types of vehicles (cars, light trucks, heavy-duty diesels) on a road.
241
4. Detailed experimental investigation of evolution of particle modes near a busy
road. Discovery of strong variations in the size distribution, including a maximum
of the total number concentration at an “optimal” distance from the road.
5. Development of a new mechanism of evolution of combustion aerosols, based on
intensive thermal fragmentation of nano-particles. Its theoretical justification and
experimental confirmed by means of several independent sets of data.
6. Development of a new model of dispersion of combustion aerosols near busy
roads, based on intensive particle fragmentation. Determination of the typical
fragmentation rate coefficient and existence conditions for a maximum of the total
number concentration at an optimal distance from the road.
7. Development of three major new methods of statistical analysis of mode evolution
and experimental data in the presence of strong stochastic fluctuations of external
and meteorological parameters, based on the moving average approach, simple
and canonical correlation analyses.
8. Re-definition of particle modes as groups of particles of similar dimensions and
strong mutual correlations. Statistical identification of particle modes emitted by
cars and heavy diesels. In particular, determination of a volatile ~ 7 nm mode that
exists at earlier stages of aerosol evolution and is strongly related to heavy-duty
diesel trucks.
9. Detailed canonical correlation analysis of the relationships between particle
modes and external atmospheric conditions.
10. Derivation and proof of the fragmentation theorem that provides further strong
confirmation of the fragmentation mechanism of evolution of combustion
aerosols.
242
11. Development of a theory of probabilistic time delays during stochastic
degradation/evaporation processes and their relationship to formation and
evolution of particle modes in combustion aerosols.
12. Determination and analysis of distinct particle modes in urban background
aerosols. Discovery of unique anti-symmetric dependencies of the moving
average correlation coefficients between different background modes, which are
closely related to the derived fragmentation theorem.
243
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