Commit: e97c3b402ad223c16266d30c315a2ab67e7d2578 - myprojects-hg-reborn (hg)

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• Revision: e97c3b402ad223c16266d30c315a2ab67e7d2578 (tree)
• Zeit: 2007-10-01 17:17:24
• Autor: iselllo
• Commiter: iselllo

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Yannis added a new intro to the paper + some minor modifications.

Ändern Zusammenfassung

• modified: latex-documents/jas1/mydraft.tex

Diff

diff -r ef89800fd9ba -r e97c3b402ad2 latex-documents/jas1/mydraft.tex
--- a/latex-documents/jas1/mydraft.tex	Mon Oct 01 08:14:36 2007 +0000
+++ b/latex-documents/jas1/mydraft.tex	Mon Oct 01 08:17:24 2007 +0000
@@ -1,11 +1,10 @@
 % Template article for preprint document class `elsart'
 % SP 2006/04/26
 
-% %\documentclass{elsart}
+ \documentclass{elsart}
 
 % Use the option doublespacing or reviewcopy to obtain double line spacing
-
- \documentclass[]{elsart}
+% \documentclass[doublespacing]{elsart}
 
 % if you use PostScript figures in your article
 % use the graphics package for simple commands
@@ -16,7 +15,7 @@
 % \usepackage{epsfig}
 
 \usepackage{graphicx}
-
+\usepackage{url} 
 \usepackage{natbib}
 %\newcommand{\ol}{\overline}
 \renewcommand{\i}{\int}
@@ -122,7 +121,7 @@
 % \linenumbers
 \begin{document}
 
-%   
+%
 % \begin{linenumbers}
 \begin{frontmatter}
 
@@ -142,247 +141,214 @@
 % \address{Address\thanksref{label3}}
 % \thanks[label3]{}
 
-\title{One-dimensional Dynamics of Diesel Exhaust Particles}
+\title{One-dimensional dynamics of diesel exhaust particles}
 
 % use optional labels to link authors explicitly to addresses:
 
 
- \author{{L. Isella\corauthref{ise}}$^{\rm a}$}
+ \author{{L. Isella\corauthref{ise}}$^{\rm a}$},
 \ead{[e-mail]}
-\author{, B. Giechaskiel$^{\rm a}$}
-\author{, Y. Drossinos$^{\rm a,b}$}
+\author{B. Giechaskiel$^{\rm a}$},
+\author{Y. Drossinos$^{\rm a,b}$}
  \address{$^{\rm a}$European Commission, Joint Research Centre, I-21020 Ispra (VA), Italy}
  \corauth[ise]{Corresponding author. Tel.: [phone]}
- \address{$^{\rm b}$School of Mechanical and Systems Engineering, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom}
+ \address{$^{\rm b}$School of Mechanical and Systems Engineering, Newcastle University,
+ Newcastle upon Tyne NE1 7RU, United Kingdom}
 
-
-
+%\date{today, \textrm{OneDSep20\_yd1.tex}}
 
 %  \author{}
 
 %  \address{}
 
 \begin{abstract}
-% [few lines by Makis sketching the experiment].
-We study the one-dimensional dynamics of a diesel Euro-3 car exhaust solid particles along the transfer line (a cylindrical duct)  leading the exhaust fumes from the tailpipe to the dilution tunnel.
-For moderate Reynolds numbers and under steady-state conditions, we map the system dynamics into the dynamics of aerosol particles in an ageing chamber.
-We account for agglomeration, diffusional and thermophoretic losses but we do not include particle fragmentation in this study. Inertial  deposition is  neglected because the observed sub-micron aggregates have a low Stokes number.
-The time scales of the aerosol processes are estimated and thermophoretic deposition is found to be non-negligible at high engine loads, despite the dynamics being generally dominated by agglomeration.
- The fractal dimension of the aggregates, determined self-consistently in this study, exerts a significant influence on the evolution of the number-size distribution. When the  distribution remains ``frozen" inside the dilution tunnel, e.g. at high dilution ratios, the exhaust-particle residence time  in the transfer line  affects the measurement at the standard sampling point (dilution tunnel outlet).
-We apply the methodology in this study to the exhaust  solid particle measurements  conducted at VELA-2 laboratories at the JRC  Italy.
-
-
-
-
-
+We study, experimentally and theoretically, the dynamics of solid
+particles emitted from a diesel Euro-3 light-duty vehicle along the
+transfer line that conducts the exhaust fumes from the tailpipe to
+the dilution tunnel. Particle agglomeration, and diffusional and
+thermophoretic transport are modelled, whereas aggregate
+fragmentation and inertial effects are neglected. For turbulent, but
+moderate, Reynolds numbers and under steady-state conditions we map
+the average, one-dimensional dynamics into the dynamics of aerosol
+particles in an ageing chamber. The aggregate fractal dimension,
+determined self-consistently to vary from 2 to 2.2 depending on
+engine load, influences significantly the evolution of the
+number distribution along the transfer line. The relative
+importance of aerosol processes is estimated by defining appropriate
+characteristic time scales. For the experiments analyzed
+agglomeration is the dominant process, suggesting that the particle
+residence time in the transfer line might affect solid-particle
+measurements at the dilution tunnel exit. Thermophoretic losses are
+calculated non-negligible only at high engine loads.
 \end{abstract}
 
 \begin{keyword}
 % keywords here, in the form: keyword \sep keyword
-agglomeration, deposition, fractal, Smoluchowski, diesel particles
+Agglomeration, Fractal dimension, Smoluchowski equation, Diesel particles
 % PACS codes here, in the form: \PACS code \sep code
-\PACS 
+\PACS
 \end{keyword}
 \end{frontmatter}
 
 % main text
 
  \section{Introduction}
-Exhaust fine particles raise growing environmental concerns ranging from adverse health effect to climate change.
-The current European regulations for light-duty diesel engine particle emissions focus on particulate mass, but research  calls for new metrics, such as e.g. the number-size distribution or the active surface of the emitted particles \citep{burtscher review}.
-Car emissions (both in mass and number) are often measured using a constant volume sampler (CVS), which takes the exhaust fumes from the tailpipe to a dilution tunnel, where turbulent mixing with clean dilution air takes place.
-The exhaust dynamics along the experimental manifold is in general sensitive to the sampling conditions and care is needed to obtain repeatable and reproducible results \citep{burtscher review}.
-Understanding the diesel particle dynamics along the experimental manifold has a fundamental importance both for the planning and interpretation of the experiments and for the careful design of new instrumentation.
-% However, the study of the particle dynamics along the experimental manyfold is a very demanding task. The exhaust soot particle processes occur within a turbulent flow.
-In this study we investigate the dynamics of solid (soot) particles. The measurement of solid particles, through a procedure of hot sampling of the aerosol removing the volatiles which may coat particles, is recommended by the particle measurement protocol since it generally leads to more repeatable and reproducible results. 
-One distinctive feature of diesel exhaust aggregates is that they collide giving rise to complex, fractal-like structures.
-In this work, we do not include the possibility of aggregate breakage. 
-The fractal-like nature of the aggregates has profound implications on the  dynamics, but 
-the experimental mesurement of the fractal dimension usually requires  sophisticated techniques (see e.g. \cite{fractal experimental} and references therein) which may not be available to every laboratory.
- In the present study we devise a method for the algorithmic determination of the fractal dimension which relies both on the statistical properties  and on the evolution of the particle-size distribution.
-We limit the investigation of the dynamics of a Euro-3 diesel exhaust particles to the trasfer line connecting the car tailpipe to the dilution tunnel inlet.
-The rationale behind this choice is that, at least for high dilution ratios in the dilution tunnel, we found that most of the  evolution of the number-size distribution occurs  before the aggregates enter the  tunnel. In this situation the aerosol processes taking place in the transfer line heavily affect the outcome of a measurement conducted at the standard sampling point, typically located at the outlet of the dilution tunnel.
-Due to the simplicity of the transfer line geometry (a pipe), we formulate an approximated one-dimensional description of the aerosol dynamics, whose limits of applicability are discussed.
-The methodology is applied to  interpret a series of measurements conducted on a Euro-3 vehicle in the VELA-2 laboratories at  JRC, Italy, using a standard CVS system.
-\begin{table}
-% \begin{center}
-\begin{tabular}{|c|r@{}lr@{}lr@{}lr@{}l||r|}
-	\hline
-\multicolumn{10}{|c|}
-	{\rule[-3mm]{0mm}{8mm} \textbf{Statistics of the number-size distribution}} \\
-  Speed
-        & \multicolumn{2}{|c|}{Position}
-	& \multicolumn{2}{|c|}{$N$ [cm$^{-3}$] (fit)}
-	& \multicolumn{2}{|c|}{$N$  [cm$^{-3}$] (reconstructed)}
-	& \multicolumn{2}{|c|}{$\mu$ [nm]}
-	& $\sigma$ [nm] \\ \hline \hline
-50 Km/h   
-	 & \multicolumn{2}{|c|}{\rm Inlet}     
-         & \multicolumn{2}{|c|}{$7.6\cdot 10^7$}           
-	 &\multicolumn{2}{|c|}{$7.9\cdot 10^7$}  
-          &\multicolumn{2}{|c|}{$62$} 
-& $1.75$
-\\
-50 Km/h   
-	 & \multicolumn{2}{|c|}{\rm Outlet}     
-         & \multicolumn{2}{|c|}{$4.5\cdot 10^{7}$}           
-	 &\multicolumn{2}{|c|}{}  
-          &\multicolumn{2}{|c|}{$ 87.3$} 
-& $1.63$
-\\
+Diesel exhaust fine particles are extensively studied, both experimentally and theoretically, because they have been associated with various effects, ranging from adverse health effect to climate change. Current European regulations for light-duty diesel engine particle emissions are based on total emitted particulate mass. The exhaust gases of vehicles are diluted in a (full) dilution tunnel from where a sample is drawn and collected on a filter. Recent research results suggest, however, that different metrics, such as number distribution or active surface, may quantify particles effect more precisely, especially their possible health-related [Ober..].
+Moreover concerns are raised whether the results from the dilution tunnel are representative of the reality. \cite{makis2004} and \cite{ronkko} showed that tailpipe measurements were representative of vehicle emissions by comparing the size distributions a few meters behind a Euro 3 vehicle (with chasing experiments) with those measured with the same sampling system in the laboratory. However, \cite{maricq tailpipe} showed that there are differences between tailpipe and dilution tunnel in mass and number. Although they emphasized mainly the differences in the nucleation mode of the size distribution, the figures in their study show that there are also differences in the accumulation mode (at figure 10 in their paper the size distribution at the tunnel is more narrow, at bigger diameters and with lower concentration). \cite{vogt} presented that the size distributions at the tailpipe and the dilution tunnel were different with similar tendency as the one described by  \cite{maricq tailpipe}. They attributed the differences to agglomeration. More recently \cite{casati} found even bigger differences between the tailpipe and the dilution tunnel size distributions. The studies mentioned before measured total particles (volatile and non-volatile part) as their sampling systems used dilution air at ambient temperature, so many phenomena could explain the differences they observed (e.g. nucleation, condensation, agglomeration, deposition etc). There is a drawback with their approach: by favoring procedures like nucleation and condensation at the tailpipe measurements, the effect of the sampling system cannot be isolated from the procedures that take place actually in the transfer tube. Simple calculations show that no nucleation and condensation takes place in the transfer tube and in order to study the procedures that take place the sampling conditions must be such that only the accumulation (soot) mode is measured. In this direction the “PMP” protocol is more appropriate.
+The Particulate Measurement Programme (PMP), an international collaborative programme, has been established to develop a measurement protocol to measure repeatably (intra-laboratory) and reproducibly (inter-laboratory) particle number emissions to replace, or complement, the existing mass-based system for regulatory purposes. According to PMP the particle number system should consist of a first stage hot dilution and then a heated section at 300-400°C for the removal of volatiles, avoiding thus the uncertainty of the volatiles which affects the repeatability (and reproducibility) of the measurements. However, in order to minimize required changes to the current type approval facilities, the system will be sampling from the full dilution tunnel. The results of the light duty phase of the PMP, which were recently published \citep{andersson1}, showed that the number method has better sensitivity than the mass method and can be used for legislative purposes. In fact a new number limit of $5\cdot10^{11}$cm$^{-3}$ is already proposed \citep{good}. Although there is general guidline for the sampling point at the dilution tunnel (10-20 diameters downstream the mixing point), there are no specifications for the transfer tube. Thus there are labs with different lengths. For example, \cite{leon1} and \cite{vogt} have $6$m insulated tubes, \cite{maricq coagulation} have $8$m ($10$cm inner diameter), the lab of the present study has $9$m heated line.
+It is, thus, essential to analyze and model particle dynamics in the transfer tube from the tailpipe to the dilution tunnel, to ensure that measurements in different laboratories are comparable and to suggest the most appropriate experimental geometry for regulatory measurements. The transfer tube is considered the most critical part as, at least for high dilution ratios, in the dilution tunnel experimental measurements and theoretical calculation suggest that the size distribution freezes. 
+In this paper the particle number (and mass) distributions are studies both experimentally and theoretically. Procedures like agglomeration, thermophoresis, diffusion and deposition are studied and their relative importance is discusses for a Euro 3 vehicle at different steady state tests. The fractal structure of the particles is also taken into account.
 
-120 Km/h   
-	 & \multicolumn{2}{|c|}{\rm Inlet}     
-         & \multicolumn{2}{|c|}{$1.4\cdot 10^8$}           
-	 &\multicolumn{2}{|c|}{$1.38\cdot10^8$}  
-          &\multicolumn{2}{|c|}{$67.7$} 
-& $1.79$
-\\
-120 Km/h   
-	 & \multicolumn{2}{|c|}{\rm Outlet}     
-         & \multicolumn{2}{|c|}{$8.3\cdot 10^{7}$}           
-	 &\multicolumn{2}{|c|}{}  
-          &\multicolumn{2}{|c|}{$ 88.5$} 
-& $1.69$
-\\
 
- \hline
-% \vspace*{0.cm}
-\end{tabular}
-  \caption{Characteristic time scales for the aerosol processes considered in the two sets of experiments and residence time of the aggregates in the transfer line.}
- \label{table statistics}
-% \end{center}
-\end{table}
- 
 \section{Experiments}
+
+% Experimental set up
+
+% Particle size distributions, in particular that they
+% freeze at the CVS inlet.
+
 \subsection{Experimental set-up and particle sampling}
-The vehicle used in this study was a common rail DI FIAT Stilo JTD (Euro 3) with common market fuel with a sulfur content below $50$ppm and with lubrificant [{it find out!}].
-The vehicle was coupled to the dilution tunnel transfer line by a metal-to-metal join during testing to avoid the possibility of exhaust contamination by the high-temperature breakdown of elastomer coupling elements. The exhaust was transported to the dilution tunnel through a $9$m long heated at $70\,^{\circ}\mathrm{C}$ corrugated stainless steel tube. This tube, anaconda, was introduced along the dilution tunnel axis, near an orifice plate that ensured rapid mixing with the dilution air (position d). The results at the dilution tunnel will be presented elsewhere and will not be further discussed.
-Aerosol samples for particle number measurements were drawn with a modified Fine Particle Sampler (FPS) (Dekati Ltd.) with as short as possible not heated sampling lines. The same length probes were used at all different sampling positions at the dilution tunnel.
-The schematic of the sampling system is shown in Fig.~\ref{schematic experiment}.
-\begin{figure}
-\begin{center}
-\includegraphics[width=0.8\columnwidth, height=6cm]{JRC_CVS.jpg}
-\end{center}
-\caption{Schematic of the sampling system used in the experiments conducted in the VELA-2 laboratories at JRC, Italy.}\label{schematic experiment}
-\end{figure} 
-With FPS, dilution is carried out in two phases. The first dilution phase is conducted as close to a sampling point as possible, i.e., in a perforated tube diluter. This method assists reproducible results and prevents losses. The first, i.e., primary dilution can be hot or cold. With hot dilution, condensation of volatile species can be prevented, where as with cold dilution, the nucleation of volatile species can be maximized. The second dilution phase is an ejector type diluter, located downstream the primary dilution. The ejector diluter acts as a pump sucking a known amount of a diluted sample from the primary dilution. Simultaneously, a secondary dilution is carried out. In the FPS, critical parameters of the dilution are controlled and simultaneously monitored. The dilution ratio of the measurement can be calculated reliably, thus the measured concentration of particles can be converted to an actual particle concentration. 
-In this study most measurements were close to the PMP protocol specifications (hot dilution). The dilution ratios at FPS were between $30:1$ and $40:1$ (probe diluter dilution ratio $1.5:1$ to $2:1$ and ejector dilutor dilution ratio $17:1$ to $20:1$). The dilution ratio was further checked with $CO_2$ concentrations when FPS was measuring at the tailpipe. For $50$ Km/h there was no difference, but for $120$ Km/h there was (the DR was overestimated $20\%$). This factor was taken into account at the calculations. The temperature was set to $400\,^{\circ}\mathrm{C}$ for the probe heater and $150\,^{\circ}\mathrm{C}$ for the (ejector) dilution air in order to evaporate volatile particles and reduce the partial pressures of the gas phase species to prevent recondensation at the diluter exit. These temperatures led to diluted aerosol temperatures at the exit of the diluter $\sim 120\,^{\circ}\mathrm{C}$. 
-Downstream FPS an evaporation tube (ET) at $400\,^{\circ}\mathrm{C}$ was connected in order to evaporate all volatiles and semi-volatile compounds. The residence time in this tube was estimated $0.2$s (needs to be confirmed). 
-Immediately downstream of the evaporation tube there was an ejector diluter in order to minimize the diffusion losses, cool the hot diluted exhaust gas and reduce the particle number concentration below $10^5$ cm$^{-3}$ in order to be within the detection limit of the instruments (CPCs). The dilution ratio was constant $10.5:1$ for the specific overpressure (1 bar) in this set up. An extra ejector diluter was used in some measurements (e.g. tailpipe and end anaconda) where the particle number concentrations were higher. The dilution ratio of this diluter was calculated $13.5:1$. The calculations of the dilution ratios were done with particles upstream and downstream the diluters as no calibration gas or flowmeters were available at that period. The ejector dilutors were cleaned only at the beginning of the measurement campaign. 
-Some measurements were conducted without any heating at FPS and the ET in order to measure volatile particles also (as nucleation mode or condensed material on the soot particles). These measurements are noted as “cold” dilution measurements because ambient temperature air was used for the dilution. The particles measured with cold dilution are called “wet” particles in order to distinguish them from the normal measurements (PMP protocol or “hot” dilution that give as a result non-volatile or “dry” particles).
+The vehicle used in this study was a common rail DI FIAT Stilo JTD (Euro 3) fueled with common market fuel (sulfur $<50$ ppm).
+The schematic of the experimental setup and sampling system is given in Fig.~\ref{schematic experiment}.
+The vehicle was coupled to the dilution tunnel transfer line by a metal-to-metal join during testing to avoid the possibility of exhaust contamination by the high-temperature breakdown of elastomer coupling elements (position 1). The exhaust was transported to the dilution tunnel through a $9$m-long (inner diameter $10$cm) heated at $70\,^{\circ}\mathrm{C}$ corrugated stainless steel tube (the end of the tube is considered position 2). This tube was introduced along the (full) dilution tunnel axis, near an orifice plate that ensured rapid mixing with the dilution air. Particle samples were drawn at the beginning of the transfer tube (position 1) and at the end (position 2) before entering the dilution tunnel (Fig.~\ref{schematic experiment}). Samples were also drawn from the normal legislated position at the dilution tunnel $10$ tunnel diameters after the mixing point. 
+Aerosol samples for particle number measurements were drawn with a modified Fine Particle Sampler (FPS) (Dekati Ltd.) with as short as possible not heated sampling lines. The same length probes were used at the two different sampling positions. With FPS, dilution is carried out in two phases. The first dilution phase is conducted as close to a sampling point as possible, i.e., in a perforated tube diluter. This method assists reproducible results and prevents losses. The second dilution phase is an ejector type diluter, located downstream the primary dilution. The ejector diluter acts as a pump sucking a known amount of a diluted sample from the primary dilution. Simultaneously, a secondary dilution is carried out. In the FPS, critical parameters of the dilution are controlled and simultaneously monitored. The dilution ratio of the measurement can be calculated reliably, thus the measured concentration of particles can be converted to an actual particle concentration. 
+In this study the measurements were according to the PMP protocol specifications (hot dilution). The dilution ratios at FPS were between $30:1$ and $40:1$ (probe diluter dilution ratio $1.5:1$ to $2:1$ and ejector dilutor dilution ratio $17:1$ to $20:1$). The temperature was set to $400\,^{\circ}\mathrm{C}$ for the probe heater and $150\,^{\circ}\mathrm{C}$ for the (ejector) dilution air in order to evaporate volatile particles and reduce the partial pressures of the gas phase species to prevent recondensation at the diluter exit. These temperatures led to diluted aerosol temperatures at the exit of the diluter $\sim 120\,^{\circ}\mathrm{C}$. 
+Downstream FPS an evaporation tube at $400\,^{\circ}\mathrm{C}$ was connected in order to evaporate all volatiles and semi-volatile compounds. The residence time in this tube was estimated $0.15$s. Immediately downstream of the evaporation tube there were two ejector diluters \citep{makis2004} in order to minimize the diffusion losses, cool the hot diluted exhaust gas and reduce the particle number concentration below $10^5$cm$^{-3}$ in order to be within the detection limit of the instruments (CPCs). The dilution ratio was constant $\sim 150$ for the specific overpressure ($1$bar) in this set up. The estimations of the dilution ratios were done with particles upstream and downstream the diluters as no calibration gases or flowmeters were available at that period. The ejector dilutors were cleaned only at the beginning of the measurement campaign.
 
 \subsection{Instrumentation and PM Sampling}
-Two CPCs (TSI Inc. 3025A, Dekati prototype), one SMPS (TSI Inc. 3936) and one DMM (Dekati Ltd.) were measuring total particle number concentration, particle number size distribution and mass concentration respectively downstream the ejector dilutor.
-Mass samples were drawn downstream the FPS with a flowrate of $30$lpm. A long metal tube was used to cool the hot ($\sim 120°C$) diluted exhaust gas.
-Mass samples were also drawn from the CVS at the normal sampling position at a constant flowrate of 25 lpm at normal conditions ($0\,^{\circ}\mathrm{C}$ and $1$ bar). Mass samples were collected on a single $47$mm Quarzo filters to permit chemical analysis with MEXA. The  results of the chemical analysis are not available yet as the MEXA unit had to be sent back to the manufacturer for calibration. 
-For some tests normal T60A20 filters were used at CVS. After their weight was determined the Non Volatile Fraction (NVF) content was determined indirectly with the following procedure: The previously loaded and weighted filters were heated in a furnace under continuous flux of $N_2$. Afterwards filters remained in the conditioning room for another $24$h and then heated filters weights were taken. NVF emissions of the vehicle were calculated as the difference between tare and heated filter weights. It should be noted that even when a blank filter follows this procedure losses some of its mass ($\sim30\mu$g for T60A20 filters) due to its volatile content. This weight loss was also taken into account, but it didn’t affect the results as the weight of the filters at CVS was $>1500\mu$g.
+One Condensation Particle Counter (CPC, TSI Inc. 3025A) and one Scanning Mobility Particle Sizer (SMPS, TSI Inc. 3936) were measuring total particle number concentration and particle number size distribution downstream the sampling system. The CPC is capable of measuring the number concentration of submicrometer airborne particles that are larger than $3$nm in diameter. The particles are detected and counted by a simple optical detector after a supersaturated vapor condenses onto the particles, causing them to grow into larger droplets. The range of particle concentration detection extends from less than $0.01$ particle/cm$^33$ to $9.99 \cdot 10^4$ particles/cm$^3$.
+In SMPS particles are classified with an Electrostatic Classifier ($103$ bins) and their concentration is measured with a 3010 CPC. The SMPS system also uses a personal computer and custom software to control individual instruments and perform data reduction. Losses in the SMPS were taken into account with the manufacturer’s new software. In this study sheath/sample flowrates used were $3/0.3$. With these flowrates it was possible to measure particles in the range of $15-640$nm and it was ensured that the activity of the neutralizer of the SMPS (model 3077) was adequate for the charging of the particles. The sample flowrate was checked with a bubble flowmeter (BUCK calibrator M-5). The upscan and downscan times for the SMPS were 90/30 s respectively. 
+Mass samples were also drawn from the CVS at the normal sampling position at a constant flowrate of $25$lpm at normal conditions ($0\,^{\circ}\mathrm{C}$ and $1$bar). Mass samples were collected on a single $47$mm Quarzo filters to permit chemical analysis with a Sunset PM analyser. For some tests T60A20 filters were used at CVS. After their weight was determined, the Non Volatile Fraction (NVF) content was determined indirectly with the following procedure: The previously loaded and weighted filters were heated in a furnace under continuous flux of N2. Afterwards filters remained in the conditioning room for another 24 h and then heated filters weights were taken. NVF emissions of the vehicle were calculated as the difference between tare and heated filter weights. It should be noted that even when a blank filter follows this procedure losses some of it’s mass ($\sim30\mu$g for T60A20 filters) due to its volatile content. This weight loss was also taken into account, but it didn’t affect the results as the weight of the filters at CVS was $>1500\mu$g.
 \subsection{Steady-state tests: stability and repeatability}
-Steady-state tests at $50$ and $120$Km/h were run. The steady-state tests lasted for $20$ minutes. In these $20$ minutes two CVS flowrates were tested: $6$ and $12$ m$^3$/min.
-% The detailed procedure will be given in the results section.
-The stability of the measurements (for DMM, CPC1=TSI, CPC2=DEKATI and total concentration of SMPS) can be checked by estimating the  CoV, defined as the standard deviation divided by the mean. Usually it was better than $5\%$ with the exception of some measurements at the mixing point (between) where it was around $10\%$. These values are within the common experimental uncertainties.
-For SMPS the size dependent stability was also checked.
-% Figure 7 shows a size distribution and the mean variation at 50 and 120 km/h. 
-At the size range of $40-200$nm the CoV is $<10\%$ and for $60-120$nm is $<5\%$. In these size ranges the concentration is high. Outside this range the concentration of particles is very low leading to high CoVs.
-The repeatability of the measurements was checked by comparing the results at the same sampling position of two different days (beginning and end of measurement campaign).
-%  Figure 8 shows the results at the end of anaconda for 120 km/h where the FPS was used with exactly the same settings and temperatures. Red lines are the results of the first period and green lines for the second after a -10\% correction. The size distributions are exactly the same (mean, stdev) indicating no change of the size distribution. The -10\% difference of the concentration can be attributed to the dirtiness of the sampling device (FPS and ejector) or uncertainties in the ejector dilution air pressure setting.
- The second repeatition was conducted after the sampling system had measured at the tailpipe. When the sampling system gets dirty its dilution ratio increases and this cannot be taken into account from the instruments dilution ratio indication. The results presented in this report have been corrected for this $-10\%$ for the second period of measurements (after the measurements at the tailpipe).
-\section{Aggregate structure} 
+Steady state tests at $50$ and $120$km/h of $20$ minutes duration were run. Only the last $10$ minutes of each test were taken into account to be sure that the vehicle had stabilized \citep{makis2007}. The measured exhaust gases flowrates were $1$ and $2$m$^3$/min respectively leading to Re number in the transfer tube (taking into account the high exhaust temperature) of $\sim 8000-13000$. The stability of the measurements was checked with the CoV (=stdev/mean) of each test. Usually it was better than 5\%. For SMPS the size dependent stability was also checked. At the size range of $60-120$nm the stability was very good (<5\%), but outside this range it was higher due to the low particle number concentration measured. The peak of the size distributions was also quite stable during the measurements (CoV $<2\%$ or $<2$nm)
+The repeatability of the measurements was checked by comparing the results at the same sampling position (end of transfer tube) of two different days (beginning and end of measurement campaign with other measurements in between). The two tests had a difference of $10\%$ probably due to the dilution ration change of the system as it got dirty during the measurements. When the sampling system get’s dirty its dilution ratio increases and this cannot be taken into account from the instruments dilution ratio indication. 
+It should be also emphasised that due to the high temperature and pressure fluctuations at the tailpipe the value given by the instrument might differ from the actual one. At the end of the transfer tube the temperature and the pressure fluctuations are lower so the values given by the instrument should be close to the actual ones. For this reason the size distributions measured at the end of the transfer tube were considered the “accurate” ones and from the tailpipe size distributions the concentration was fitted in order to have mass balance (mean and standard deviation were kept the same) as it will be explained later. However it was found that the differences between theoretical and experimental size distributions were within $10\%$.
+Other uncertainties of the experimental data are associated with the measurement instruments. Generally the CPC was $10\%$ higher than the total concentration given by SMPS. However, the measurement at the tailpipe at $50$km/h showed a $25\%$ difference between the two instruments, which was taken into account by correcting the SMPS size distribution by this factor. Although there is no clear explanation for this difference the theoretical analysis later also showed that the CPC results were the correct ones. Finally it should be mentioned that depending on the SMPS parameters (or the company of SMPS) small differences at the size distributions can be obtained (mean, standard deviation and total concentration). Although this was not investigated, the effect of different size distribution parameters is discussed in the Discussion section of the paper.
+\section{Aggregate structure}
+
 \subsection{Effective density and fractal dimension}
+
 Aggregate morphology has a large effect on the transport and thermal properties of
- aerosol particles. Solid diesel-exhaust carbonaceous particles do not coalesce perfectly
- but give rise to complex, quasi-fractal structures \citep{friedlander book}.
-The morphology of large aggregates consisting of a number $n_{\rm{agg}}$ of smaller primary spherules of size $d_0$ is often characterised by their fractal dimension $\df$ defined by:
+aerosol particles. The morphology of large aggregates consisting of a number $n_{\rm{agg}}$ of smaller
+primary particles of effective
+diameter $d_0$ is often characterised by their fractal dimension $\df$ defined by:
 \beq\label{rel_nagg}
 n_{\rm{agg}}=a\lro\f{\dagg}{d_0}\rro^{\df},
 \eeq
 where $\dagg$ is the characteristic size of the aggregate.
-In the experiments, we measured the aggregate number-size distribution using an SMPS. In the SMPS, the aerosol particles are classifed by an electrostatic method, which is based on
-the electrical mobility of electrically charged particles. Electrical mobility is defined as the velocity
-that an electrically charged aerosol particle  acquires in an electric field of unit strength.
-If the aerosol
-particle is non-spherical, the SMPS returns a mobility equivalent diameter that indicates the diameter of
+The aggregate number distribution was measured with an SMPS; thus,
+particles were classified according to their electrical mobility.
+For non-spherical particles the SMPS estimates a mobility equivalent diameter that indicates the diameter of
 a sphere with the same electrical mobility as the actual non-spherical particle.
-  We then choose  this diameter as the characteristic size of an aggregate $\dagg$.
-The prefactor $a$, weakly dependent on $\df$ \citep{naumann code}, is often taken as a constant equal to one, as in the present study. The exact value of $a$  is not crucial, since it can be absorbed in a re-definition of $d_0$,  the only important parameter being the ratio $a/d_0^{d_f}$.
-Here we do not allow for a time evolution of the fractal dimension (see at this regard \citet{evolving fractal 1}). 
+We chose this diameter as the characteristic size of an aggregate $\dagg$, in agreement
+with previous works \citep{maricq experimental}, where
+the effective density is expressed as a function of the
+mobility diameter instead of the aggregate radius
+of gyration. The value of the prefactor $a$, weakly dependent on $\df$ \citep{naumann code},
+is not crucial, since it can be absorbed in $d_0$,
+the only important parameter being the ratio $a/d_0^{d_{\rm f}}$.
+We, thus, take it to be unity.
 The mass of an aggregate made up of $\nagg$ primary particles can then be expressed as:
-\beq\label{mass aggregate}
-\magg=\nagg \f{\pi}{6}d_0^3=\reff \vagg,
+\beq
+\label{mass aggregate}
+\magg=\nagg \f{\pi}{6}d_0^3 \equiv \reff \vagg,
 \eeq
-where $\vagg=\pi\dagg^3/6$ is the volume of the aggregate and $\reff$ is the effective density given by: 
+where $\vagg=\pi\dagg^3/6$ is the volume of the aggregate and $\reff$ is the effective density given by:
 \beq\label{eff dens}
-\rho_{\rm{eff}}=\rho_0\lro \f{\dagg}{d_0} \rro^{\lro{\df-3}\rro},
+\rho_{\rm{eff}}=\rho_0 \lro \f{\dagg}{d_0} \rro^{\lro{\df-3}\rro},
 \eeq
 where $\rho_0$ is the primary particle density.
 % The effective density in Eq.~(\ref{eff dens}) is useful to determine the mass of an aggregate once its
 %  fractal dimension and mobility equivalent size are known.
-A large body of experimental data (see e.g. \cite{maricq experimental} ) suggests an empirical expression for the effective density of light-duty  diesel engine  exhaust particles:
+A large body of experimental data (see, e.g., \cite{maricq experimental, park fractal, virtanen fractal} ) indicate that the power-law in Eq.~(\ref{eff dens}) breaks down at aggregates sizes $\dagg\simeq 50$nm, below which the effective density is approximately constant with value $\reff\simeq 1$g/cm$^3$.
+Therefore we choose  the following expression to interpolate the power-law behavior at large aggregate sizes to the constant value of $\reff$ at for small aggregates:
 \beq\label{rho eff revised}
 \reff=\min\lro\rho_0,\rho_0\lro \f{\dagg}{d_0} \rro^{\lro{\df-3}\rro}\rro,
 \eeq
 with $\rho_0\simeq1$g/cm$^3$, $d_0\simeq50$nm, and a fractal dimension  in the interval $[1.8,2.8]$.
-This reflects the fact that aggregates acquire a real fractal nature only when  above a  size threshold.
-
+Equation (\ref{rho eff revised}) implies that the primary particle diameter $d_0$
+is the aggregate diameter beyond which aggregates become fractal-like.
+We tested numerically the robustness of the results we obtain against small ($\sim 10-20\%$) variations of $\rho_0$ and $d_0$. 
+% Aggregates of a given effective diameter may be considered fractal objects
+% beyonf a threshold.
 
-\subsection{Particle-size distribution}
+\subsection{Particle number distribution}
 
-Ambient aerosol number-size distributions are typically multi-lognormal distributions \citep{whitby classical}.
-The SMPS used in the experiments binned the number-size distribution into $103$ logarithmically-spaced channels. 
-Since the experiments showed only a single-modal distribution, the  measured particle number in a unit volume, as a function of the aggregate mobility diameter, $dN(\dagg)$  was fitted according to:
+%Ambient aerosol number-size distributions are typically multi-lognormal distributions \citep{whitby classical}.
+The SMPS used in the experiments binned the number distribution into $103$ logarithmically-spaced channels.
+Since the experiments showed only a single-modal distribution, the
+measured particle number per unit volume, as a function of the aggregate mobility diameter, $dN(\dagg)$
+was fitted to a single lognormal distribution according to:
 \beq\label{lognormal}
 dN=
 \f{N}{\sqrt{2\pi}\log\sigma}\exp\lsq-\f{(\log \dagg-\log\mu)^2}{2\log^2\sigma}\rsq d\log\dagg,
 \eeq
-where $N$ is the total particle number in a unit aerosol volume and  $\mu$ and $\sigma$ are the geometric mean and standard deviation of the distribution, respectively.
- We choose $N$, $\mu$ and $\sigma$ as fit parameters of the lognormal distribution. The nonlinear 
+where $N$ is the total particle number per unit volume,
+and  $\mu$ and $\sigma$ are the geometric mean and standard deviation of the distribution, respectively.
+We choose $N$, $\mu$ and $\sigma$ as the fitting parameters of the lognormal distribution. The nonlinear
 least-square optimisation used for the fit relies on  Levenberg-Marquardt algorithm
- \citep{numerical recipes} as implemented in the Minpack library of  R statistical software \citep{minpack}.
-Once the number-size distribution in Eq.~(\ref{lognormal}) is known, the corresponding mass-size distribution can be easily obtained:
+\citep{numerical recipes} as implemented in the Minpack library of R statistical software \citep{minpack}.
+
+Once the number distribution in Eq.~(\ref{lognormal}) is known,
+the corresponding mass distribution can be easily obtained
 \beq\label{mass size}
 dM=\reff\f{\pi}{6}\dagg^3dN.
-\eeq 
+\eeq
 
 
-\begin{figure}
-\begin{center}
-\includegraphics[width=0.5\columnwidth, height=6cm]{mass-sizes.pdf}
-\end{center}
-\caption{Cumulative mass distribution function  for the particle-size distributions measured at the inlet for a representative $d_f=2.3$, $d_0=50$nm and $\rho_0=1$g/cm$^3$. Most of the total mass is due to aggregates with $\dagg$ larger than $100$nm. This proves true for any $d_f$ in the range $[1.8,2.8]$.}\label{mass-size distr}
-\end{figure} 
-The total particle mass in a unit volume, $M_{\rm{tot}}$, is obtained from the integration of Eq.~(\ref{mass size}), after choosing the fractal dimension $d_f$.
-Although we generally evaluate it numerically, if we express the effective density as in Eq.~(\ref{eff dens}), i.e. we neglect its deviation from a power-law for aggregate sizes below $d_0$, we deduce the approximate expression:
+
+The total particle mass per unit volume, $M_{\rm{tot}}$, is obtained from the integration of Eq.~(\ref{mass size}), after choosing the fractal dimension $d_{\rm f}$.
+Although we generally evaluate it numerically, if we express the effective density as in Eq.~(\ref{eff dens}),
+i.e. we neglect its deviation from a power-law for aggregate sizes below $d_0$, we obtain the approximate expression:
 \beq\label{total mass}
 % M=\f{N\pi}{6d_0^{(\df-3)}}\exp\lsq\df\ln\mu_{\rm geo}+\f{\df^2}{2}\ln^2\sigma_{\rm geo}\rsq.
-M_{\rm tot}=\i_0^{\infty}dM\simeq N\rho_0\f{\pi}{6d_0^{(d_f-3)}}\exp\lro d_f\ln\mu+\f{d_f^2}{2}\ln^2\sigma\rro.
+M_{\rm tot}=\i_0^{\infty}dM \simeq N\rho_0\f{\pi}{6d_0^{(d_{\rm f}-3)}}\exp\lro d_{\rm f}\ln\mu+\f{d_{\rm f}^2}{2}\ln^2\sigma\rro.
 \eeq
-It turns out that most of the mass is found at aggreagate sizes above $100$nm, irrespective of the fractal dimension, so this estimate  is not very sensitive to the behavior of $\reff$  for aggregate diameters  smaller than $d_0$. This is shown in Fig.~\ref{mass-size distr} plotting, for a characteristic fractal dimension $d_f=2.3$, the cumulative mass distribution function (CDF) for the initial state measured at the inlet of the transfer line in both the experimental cases investigated, namely car speed $120$ and $50$Km/h. A similar conclusion  holds in general for $d_f$ in the range from $1.8$ to $2.8$. 
-In the case of diesel-engine nanoparticles, we expect the smallest generated particles to be of the size of a few nanometers, so we limit the applicability of Eq.~(\ref{rho eff revised}) down to a monomer size $d_{\rm mon}=2$nm.
-Again, the precise value of $d_{\rm mon}$, in this order of magnitude, is irrelevant for the mass balances and for the aerosol processes considered in this work. 
+It turns out that most of the mass is found at aggregate sizes above $100$nm, irrespective of the fractal dimension: hence,
+the approximate expression for the total mass, Eq. (\ref{total mass}), is not very sensitive to the behavior
+of $\reff$ for aggregate diameters smaller than $d_0$.
+This is shown in Fig.~\ref{mass-size distr} where the normalized cumulative mass distribution function (CDF)
+at the transfer line inlet is plotted for both experiments, vehicle speed $50$ and $100$Km/h.
+The normalized mass CDF in Fig.~\ref{mass-size distr} was calculated for a characteristic fractal dimension $d_{\rm f}=2.3$, but
+a similar conclusion holds in general for $d_{\rm f}$ in the range from $1.8$ to $2.8$.
+In the case of diesel-engine nanoparticles, the smallest generated particles are of the order of a few nanometers,
+so we limit the applicability of Eq.~(\ref{rho eff revised}) to a monomer size $d_{\rm mon}=2$nm.
+Again, the precise value of $d_{\rm mon}$, as long as it is of the same order of magnitude, is irrelevant for the mass balance
+and for the aerosol processes considered in this work.
+
 % \subsection{Application to the experimental number-size distributions}
 
-The statistics described in the previous section can be applied to the experimentally measured number-size distributions.
-Figure~\ref{experimental number-size} shows the experimental results, together with their statistical uncertainty, at the inlet and outlet of the trasfer line for the two steady-state velocities $50$ and $120$Km/h.
-For any fractal dimension in the range $[1.8,2.8]$, the calculation of the total particle mass, for steady-state velocity $50$Km/h, leads to an excess mass around $10\%$ at the outlet with respect to the inlet.
-This is most likely due to the presence of strong pressure fluctuations when sampling at the inlet of the transfer line which can affect the determination of the sampling system dilution ratio.
-In Section~\ref{numerical method} we show how to deal with that experimental uncertainty.
-For car speed $120$Km/h, instead, we find about $10\%$ extra particle mass at the inlet. This experimental result is not problematic and, as we show in Section~\ref{results}, can be almost entirely explained by taking into account thermophoretic deposition.
- \begin{figure}
-\includegraphics[width=0.5\columnwidth]{experimental-distr-with-error-bars_50.pdf}
-\includegraphics[width=0.5\columnwidth]{experimental-distr-with-error-bars.pdf}
-\caption{Left: experimental number-size distributions, together with statistical uncertainty, at inlet and outlet, for car speed equal to $120$Km/h. Right: as on the left, but now the car speed is $120$Km/h.}\label{experimental number-size}
-\end{figure}
+A comparison of the lognormal fit of the experimental distributions, along with the experimental statistical uncertainty
+estimated from repeatability experiments, is shown in Figure~\ref{experimental number-size}.
+The experimental number size distributions at the inlet and outlet of the transfer line for the two steady-state velocities $50$ and $120$Km/h
+are shown. The experimental data and the lognormal fits are practically indistinguishable.
+
+%In addition to the analysis of the particle size distribution, the total particle mass was calculated,
+%both via numerical integration and via the approximate expression. For the experiments at the steady-state
+%velocity $50$Km/h we find that, for any fractal dimension in the range $[1.8,2.8]$,
+%the calculated total particle mass at the outlet is $10\%$ greater than the mass at the inlet.
+%This discrepancy is most likely due to strong pressure fluctuations during sampling
+%at the transfer line inlet: these fluctuations affect the determination of the sampling system dilution ratio,
+%rendering the corresponding measurements less reliable than the measurements at the outlet.
+%In Section~\ref{numerical method} we show that the proposed algorithm
+%for the determination of the average fractal dimension
+%can be used to reconstruct the initial state, and, hence, how to deal with this experimental uncertainty.
+%For car speed $120$Km/h, instead, the total calculated particle mass at the inlet is about $10\%$ larger
+%than at the outlet. We show in Section~\ref{results} that the extra inlet particle mass
+%can be almost entirely explained by taking into account thermophoretic deposition.
+
 
 
 
 \section{Aerosol dynamics}
+
 \subsection{One-dimensional treatment}
-The dynamics of solid aerosol particle (e.g. transport and
+The dynamics of solid aerosol particles (e.g. transport and
 agglomeration) in the transfer line is described by the
-General Dynamic Equation (GDE) for the particle-size distribution
+General Dynamic Equation (GDE) for the number distribution
 (gas-to-particle conversion processes are not considered because
-we are only interested in the dynamics of solid particles, as described
-in Section 2).  As the main interest of our work is the effect of the
+we only model dynamics of solid particles).
+As the main interest of our work is the effect of
 aggregate structure on particle dynamics, and the development of a
 simplified approach to estimate the main aerosol process
 within the transfer line we make a number of approximations to simplify the GDE.
@@ -392,26 +358,22 @@
 turbulent GDE an extremely complex task.
 
 The effect of particle inertia on transport and deposition
-can be estimated by considering the appropriate
-dimensionless number, the particle
-Stokes number. The particle response time is given by:
-\beq\label{tau_part}
-\tau_p=\f{\rho_0\dagg^2}{18\nu\rho_f},
-\eeq
-where $\rho_{f}$ is the fluid density and $\nu$ is the fluid kinematic viscosity.
-If we consider $\dagg\simeq 70$nm a typical aggregate size, see the corresponding
-figure of the experimental distributions, the characteristic particle response time
-becomes $\tau_p\sim 10^{-8}$s. The characteristic fluid time scale in a turbulent flow
-is the Kolmogorov time scale \citep{pope turbulence}:
-\beq
-\tau_f=\sqrt{\f{\nu}{\varepsilon}},
-\eeq
-where $\varepsilon$ is the turbulent energy dissipation rate. Computational Fluid Dynamics
-calculations of the fluid flow in the transfer line estimate the
-Kolmogorov time-scale to be $\tau_f\sim 10^{-3}$s. Therefore, the
-corresponding Stokes number becomes ${\rm St}=\tau_p/\tau_f\sim 10^{-5}$.
-Particle inertia is, thus, not important and particles may be considered fluid points,
-namely, they follow the fluid streamlines.
+can be estimated by considering the dimensionless particle
+Stokes number, the ratio of the particle response time
+to a characteristic fluid time scale. The particle response time
+is $\tau_{\rm p} = \rho_0\dagg^2/(18\nu\rho_{\rm f})$, where $\rho_{\rm f}$ is the fluid density
+and $\nu$ is the fluid kinematic viscosity.
+If we consider $\dagg\simeq 70$nm a typical aggregate size, cf. Fig. \ref{experimental number-size},
+the characteristic particle response time
+becomes $\tau_{\rm p}\sim 10^{-8}$s. The characteristic fluid time scale in a turbulent flow
+is the Kolmogorov time scale \citep{pope turbulence}, $\tau_{\rm f}=\sqrt{\nu/\varepsilon}$,
+where $\varepsilon$ is the turbulent energy dissipation rate. We performed
+Computational Fluid Dynamics calculations (relying on the $k-\epsilon$ turbulence module as implemented in the finite-element software \cite{comsol}) of the fluid flow in the transfer line
+that estimate the Kolmogorov time-scale to be $\tau_{\rm f}\sim 10^{-3}$s. The
+corresponding Stokes number becomes ${\rm St}~=~\tau_{\rm p}/\tau_{\rm f}\sim 10^{-5}$.
+Hence, particle inertia is not important and, in the absence of
+diffusion, particles may be considered fluid points,
+in that they follow the fluid streamlines.
 
 % The first, and foremost, approximation is neglect of
 % turbulent fluctuations (that break the cylindrical-tube azimuthal
@@ -428,80 +390,87 @@
 The effects of instantaneous particle concentration inhomogeneities, solely induced by turbulent
 flow fluctuations, on agglomeration  are neglected.
 In essence, this implies that the carrier flow is not  highly turbulent otherwise,
- even in the case of a simple duct geometry, a
- two-dimensional treatment becomes unavoidable.
- Accordingly, the flow field is described by its mean
-value having only an axial component, $\vec U=(U_z,U_r)=(U_m,0)$, with
- $U_m$ the mean axial velocity along the transfer line.
-Furthermore, the experiments were performed with a
-constant flow field and under steady-state conditions.
-Axial turbulent  diffusion is easily shown to be negligible but radial diffusion and thermophoresis may become significant.
-They are accounted for by introducing  thermophoretic and diffusional velocities directed towards the duct walls.
+even in the case of a simple duct geometry, a
+two-dimensional treatment becomes unavoidable.
+Our Computational Fluid Dynamics calculations estimate the
+average contribution of fluid velocity fluctuations to the
+total fluid kinetic energy to be of the order of 1\%.
+Accordingly, the leading order approximation of
+the flow field becomes the mean flow with
+only an axial component, $\vec U=(U_z,U_r)=(U_m,0)$,
+ $U_m$ being the mean axial velocity along the transfer line.
+Axial turbulent diffusion is easily shown to be negligible (we estimated a Peclet number ${\rm Pe\simeq 80}$ along the transfer line), but radial
+diffusion and thermophoresis may become significant.
+They are accounted for by introducing
+thermophoretic and diffusional velocities directed towards the duct walls.
+As the Stokes number is small turbophoresis may be neglected,
+and temperature fluctuations, induced by fluid flow fluctuations,
+are also neglected.
 Since the emitted particles are in the nanoparticle
 regime sedimentation is not significant.
-% Although  but other external processes,
-% like .
-% Turbulent
-% diffusion and correlation of velocity fluctuations
-% with particle concentration fluctuations are, consequently, neglected.
-
-
-% : particle concentration
-% is considered radially uniform. 
-
-
-
+Furthermore, the experiments were performed with a
+constant flow field and under steady-state conditions.
 
 Under these assumptions,
 the full GDE may be approximated by a one-dimensional, steady-state
-equation. Therefore, simple mass balance considerations convert the
-GDE (a population balance equation) into an effective one-dimensional equation for
+equation. Therefore, a simple balance on the number concentration,
+or equivalently an average over the pipe cross section, converts the
+GDE into an effective one-dimensional equation for
 $n_q$, the mean (flux-averaged) axial aggregate concentration
-(aggregate number per unit volume) of characteristic size $\dagg=d_q$, to
+(aggregate number per unit volume) of characteristic
+aggregate size $d_q$, to
 \beq
 \label{1D-advection-intermediate}
-U_m \, \f{\p n_q}{\p z}=-\f{2(v_d+v_{th})}{R}n_q + \omega_q(z),
+U_m \, \f{d n_q}{d z}=-\f{2(v_{\rm dif}+v_{\rm th})}{\rm R}n_q + \omega_q(z),
 \eeq
 
-where $z$ is the axial coordinate and $R$ is the radius of the transfer line.
+where $z$ is the axial coordinate and ${\rm R}$ is the radius of the transfer line.
 The first term on the right-hand side of Eq.~(\ref{1D-advection-intermediate})
 describes changes of the size distribution due to the external processes considered:
-diffusional and thermophoretic deposition. The corresponding deposition
-velocities are denoted $v_d$ and $v_{th}$. The last term $\omega_q$
+turbulent diffusional and thermophoretic deposition. The corresponding deposition
+velocities are denoted $v_{\rm dif}$ and $v_{\rm th}$. The last term $\omega_q$
 describes particle aggregation, the only internal particle processes taken into account.
 It is worth pointing out that the aggregation term has no effect on
 mass balance since aggregation conserves the total mass; it only affects
-particle number by reducing the concentration.
+particle number by reducing its concentration.
 A similar argument leads to a one-dimensional steady-state equation for the
-steady-state, axially averaged fluid temperature (see, for example,
-Ref.~\cite{yannis 1d deposition}).
+steady-state, axially averaged fluid temperature [see, for example,
+Ref.~\cite{yannis 1d deposition}]
+\beq
+\label{1D-temperature}
+\frac{d T_m}{dz} = - \frac{2 \textrm{Nu}}{\rm{R} \textrm{Re Pr}} \, (T_w -T_m)  ,
+\eeq
+where the mean temperature is $T_m$, the wall temperature $T_w$, \textrm{Re} is
+the flow Reynolds number, $\textrm{Nu}$ the (turbulent) Nusselt number,
+and $\textrm{Pr}$ the (turbulent) Prandtl number.
 
 According to Eq.~(\ref{1D-advection-intermediate}) agglomeration becomes
 a function of the aggregate position within the tube. Under steady-state conditions
-the axial position of an aggregate is trivially related to its residence time
+the axial position of an aggregate is related to its residence time
 in the tube, denoted by
 $\tau$ to be distinguished from the physical time $t$, via $z=U_m\tau$. Agglomeration, thus, becomes
 a function of the residence time of the aggregate along the
 duct. In other words, aggregates
 at different axial penetrations along the duct spent
-a longer time inside the pipe, and they are thus more affected by agglomeration.
+a longer time inside the pipe, and they are, thus,
+ more affected by agglomeration.
 By making the $\tau$-dependence explicit in Eq.~(\ref{1D-advection-intermediate}) one obtains:
 \beq\label{smolu-tau}
-\f{\p n_q(\tau)}{\p\tau}=-\f{2(v_d+v_{th})}{R}n_q(\tau)+\omega_q(\tau),
+\f{d n_q(\tau)}{d\tau}=-\f{2(v_{\rm dif}+v_{\rm th})}{\rm R}n_q(\tau)+\omega_q(\tau),
 \eeq
 namely, a Smoluchowski equation in the particle residence time along the duct, $\tau$,
 with additional linear terms for particle losses.
 We have, thus, mapped the one-dimensional dynamics of aerosol particles convected inside
 the transfer line into the dynamics of particles ageing in a chamber.
 
-The agglomeration term $\omega_q$ is given by:
+The collision term $\omega_q$ is given by:
 \beq\label{omega q as in books}
 \omega_q=\f{1}{2}\sum_{i+j=q}\mathcal{K}_{ij}n_in_j+n_q\sum_i\mathcal{K}_{iq}n_i,
 \eeq
 where $\mathcal{K}_{ij}$ is the collision kernel between aggregates of solid volume $\nu_i$ and $\nu_j$.
-For low mass particles the gravitational kernel does not contribute
-significantly. For aggregates smaller than $1\mu$m, such as in the
-experiments, and $Re\simeq 8000-16000$, Brownian motion is the leading mechanism
+The gravitational kernel does not contribute
+significantly for collisions of nanoparticles. For aggregates smaller than $1\mu$m, such as in the
+experiments, and $\textrm{Re}\simeq 8000-16000$, Brownian motion is the leading mechanism
 for particle collisions \citep{pandis book}.
 We, thus, consider the Brownian collision kernel for collisions of aggregates of solid volume $\nu_i$ and $\nu_j$
 in the continuum regime. The effect of aggregate structure is
@@ -509,18 +478,17 @@
 on the aggregate fractal dimension \citep{continuum formula},
 \beq
 \label{cont kernel}
-\mathcal{K}_{ij}^{\rm cont}=\f{2k_BT}{3\mu_{\rm{air}}}\lro \nu_i^{1/\df}+\nu_j^{1/\df}\rro\lro
+\mathcal{K}_{ij}^{\rm cont}=\f{2k_B T_m}{3\mu_{\rm{air}}}\lro \nu_i^{1/\df}+\nu_j^{1/\df}\rro\lro
 \f{C_i}{\nu_i^{1/\df}}+ \f{C_j}{\nu_j^{1/\df}}  \rro,
 \eeq
-where $k_B$ is the Boltzmann constant, $T$ the temperature, $\mu_{\rm{air}}$ the air dynamic
-viscosity, $C_i=1+\Kn_i(1.17+0.53\exp(-0.78/\Kn_i))$
-the Cunningham slip factor, and $\Kn_i$ is the Knudsen number,
-that ratio of the air mean free path
-to the particle diameter, calculated for a particle in the $i$-th bin.
+where $k_B$ is the Boltzmann constant, $T_m$ the mean temperature, $\mu_{\rm{air}}$ the air dynamic
+viscosity, $C_i=1+\Kn_i[1.17+0.53\exp(-0.78/\Kn_i)]$
+the Cunningham slip factor, and $\Kn_i$the Knudsen number,
+the ratio of the air mean free path
+to the particle diameter, calculated for a $\nu_i$ particle.
 For a characteristic particle size $d_q~\sim ~70$nm, and an exhaust temperature $T~\sim ~380$K,
-the particle Knudsen number is in between the continuum and the
-free molecular regime, i.e., in the transition regime.
-Accordingly, we consider a modification of the continuum Brownian kernel
+the particle Knudsen number is in the transition regime, between the continuum and the
+free molecular regime. Accordingly, we consider a modification of the continuum Brownian kernel
 via the Fuchs interpolation factor $\beta$ \citep{pandis book},
 a quantity considered herein independent of the aggregate fractal dimension. The Fuchs factor
 interpolates the kernel from the continuum to the free molecular regime.
@@ -529,65 +497,81 @@
 \label{fuchs_kernel}
 \mathcal{K}_{ij}=\mathcal{K}_{ij}^{\rm cont}\beta \ .
 \eeq
+The collision kernel depends explicitly on the aggregate fractal dimension.
+Here we do not allow for a time evolution of the aggregate
+fractal dimension. The bivariate distribution was
+studied extensively \citet{evolving fractal 1}) where they
+found that the fractal dimension eventually settles to an average
+fractal dimension as long as agglomeration progresses.
+This is surely the case of the experiments we model in this study, since the
+typical size of the aggregates is well above the few nanometers typical of the smallest  diesel combustion-generated nanoparticles.
 
 For low Stokes number the effect of homogeneous turbulence
-on particle collisions can be modelled by the Saffman and Turner kernel \citep{pandis book}.
-The ratio of the continuum to the turbulent kernels may be easily estimated
+on particle collisions can be modelled by the Saffman and Turner kernel \citep{saffman}.
+The ratio of the continuum to the turbulent kernel may be easily estimated
 for typical experimental conditions to be:
 \beq
-\f{\mathcal{K^{TS}}}{\mathcal{K}^{\rm cont}}=\f{3\mu_{\rm air}(\pi\varepsilon/120\nu)^{1/2}\dagg^3}{k_BT}\sim 10^{-4},
+\f{\mathcal{K^{TS}}}{\mathcal{K}^{\rm cont}}=\f{3\mu_{\rm air}(\pi\varepsilon/120\nu)^{1/2}\dagg^3}{k_B T_m}\sim 10^{-4},
 \eeq
-i.e. agglomeration is dominated by Brownian collisions and the role of turbulence
+namely, the collision kernel is dominated by Brownian collisions, and the role of turbulence
 in particle collisions is negligible, in agreement with usual estimates \citep{pandis book}.
 
 The kernel in Eq.~(\ref{fuchs_kernel}) and the thermophoretic and diffusional deposition velocities  are
-$\tau$-dependent. Indeed the temperature $T$ varies along the transfer line as a function of the position $z=U_m\tau$
-and so do, as a consequence,  air viscosity,  Knudsen number and  particle diffusion coefficient.
-In the effective one-dimensional transport equation (\ref{smolu-tau}), we are neglecting the effect of turbulent
+$\tau$-dependent. Indeed the mean temperature $T_m$ varies along the transfer line as a function of the position $z=U_m\tau$,
+Eq. (\ref{1D-temperature}), and so do, as a consequence,  air viscosity,  Knudsen number and  particle diffusion coefficient.
+In the effective one-dimensional transport equation (\ref{smolu-tau}), we neglect the effect of turbulent
 collisions which could introduce a radial dependence in the agglomeration process.
-The thermophoretic and diffusional velocities [see Eqs.~\ref{expression vt}-\ref{expression vd}], are estimated by means of one-dimensional correlations.
-We resort to a correlation also for the calculation of the mean one-dimensional temperature along the anaconda \citep{yannis 1d deposition}.
-The details about all the correlations used in this study can be found in appendix \ref{appendix corr}.
+The thermophoretic and diffusional velocities, as well as the one-dimensional
+mean temperature, are estimated by means of one-dimensional correlations.
+% Details are summarized in appendix \ref{appendix corr}, see, also, Ref.~\citep{yannis 1d deposition}.
+The thermophoretic velocity is calculated as:
+ \beq
+ \vth= -\kt\;\f{\nu}{T}\f{T_w-T}{2{\rm R}/\nusselt},
+ \eeq
+
+
 
 The relative importance of the aerosol processes in
 Eq.~(\ref{smolu-tau}) is most conveniently estimated by
 recasting it in a dimensionless form.
-Convection fixes the total residence time of the aggregates in the transfer line, $\tau_{\rm res}=L/U_m$, where $L$ is the total length of the transfer line.
- We introduce appropriate
-characteristic time scale for agglomeration, $\tau_{\rm ag}$, thermophoretic
+Convection fixes the total residence time of the aggregates in the transfer line, $\tau_{\rm res}=L/U_m$,
+where $L$ is the total length of the transfer line.
+We introduce appropriate characteristic time scale for agglomeration, $\tau_{\rm ag}$, thermophoretic
 deposition, $\tau_{\textrm{th}}$, and diffusional deposition,
 $\tau_{\textrm{dif}}$, as follows
 \begin{align}
 \label{dimensionless times}
-\tau_{\rm ag} &\equiv  \lro N_0\f{2k_BT_0}{3\mu_{\rm{air}}}\beta(\mu_0,T_0)\rro^{-1},
-  &\tau_{\rm{dif}} \equiv &
- \gdif^{-1}(\mu_0,T_0) ,&\tau_{\rm{th}} \equiv &  \gth^{-1}(\mu_0^,T_0),
+\tau_{\rm ag}^{-1} &\equiv  N_0 \, \f{2k_BT_0}{3\mu_{\rm{air}}} \, \beta(\mu_0,T_0) ,
+  &\tau_{\rm{dif}}^{-1} \equiv &
+ \gdif(\mu_0,T_0) ,&\tau_{\rm{th}}^{-1} \equiv &  \gth(\mu_0, T_0),
 \end{align}
 
-
 % \begin{align}
 % \label{dimensionless times}
 % \nonumber   \tau_{\rm ag} &\equiv  \lro N_0\f{2k_BT_0}{3\mu_{\rm{air}}}\beta(\mu_0,T_0)\rro^{-1},
 %  &\tau_{\rm{dif}} \equiv &
 % \gdif^{-1}(\mu_0,T_0) ,&\tau_{\rm{th}} \equiv &  \gth^{-1}(\mu_0^,T_0),
 % \end{align}
-where $N_0$, $\mu_0$ and $T_0$
-are the total number of aggregates, arithmetic mean of the number-size distribution, and the
-average temperature at the inlet, respectively. The thermophoretic and
+where $N_0$ are the total number of aggregates per unit volume,
+$\mu_0$ the arithmetic mean of the number distribution, and
+$T_0$ the mean temperature at the inlet. The thermophoretic and
 diffusional time scales are determined from the corresponding deposition
-velocities via $\gamma_{{\rm th(dif)}}=2v_{{\rm th(dif)} }/R$.
+velocities via $\gamma_{{\rm th(dif)}}=2v_{{\rm th(dif)} }/{\rm R}$.
 The agglomeration time scale, $\tau_{\rm ag}$, is a
-generalization, through Fuchs interpolation factor, of
-the agglomeration characteristic time used in \cite{pratsinis preserving adimensional}.
- Its inverse value is obtained multiplying the initial total aggregate concentration, $N_0$, by   the collision kernel evaluated for a typical temperature and aggregate size, a quantity we call $\mathcal{K}$. 
-We illustrate the role of $\tau_{\rm ag}$ by considering the case of an initially monodispersed aerosol interacting with a constant kernel $\mathcal{K}$. 
-One can prove that \citep{friedlander book}  the total aerosol concentration decreases according to:
+generalization, through the Fuchs interpolation factor, of
+the characteristic agglomeration time used in \cite{pratsinis preserving adimensional}.
+Its inverse is obtained by multiplying the initial total aggregate concentration, $N_0$,
+times the collision kernel evaluated at a typical temperature and aggregate size,
+a quantity we call $\mathcal{K}_0$.
+The role of $\tau_{\rm ag}$ becomes apparent
+by considering an initially monodispersed aerosol interacting with a constant kernel $\mathcal{K}_0$.
+Then, the total aerosol concentration decreases according to \citep{friedlander book}
 \beq\label{explanation tau agglomeration}
-N(\tau)=\f{N_0}{1+\mathcal{K}N_0t/2},
-\eeq 
-which shows the role of the $\tau_{\rm ag}=1/(\mathcal{K}N_0)$ as the appropriate time scale for agglomeration.
+N(\tau)=\f{N_0}{1+\mathcal{K}_0 N_0 t/2} .
+\eeq
+Hence, $\tau_{\rm ag}=1/(\mathcal{K}_0 N_0)$ is the appropriate time scale for agglomeration.
 
-Equation~(\ref{smolu-tau}) expressed in terms of following dimensionless
+Equation~(\ref{smolu-tau}) expressed in terms of the dimensionless
 variables
 \begin{align}
 \label{dimensionless variables}
@@ -597,7 +581,7 @@
 becomes
 \beq
 \label{smolu-dimensionless}
-\f{\p \tilde n_q}{\p\tilde \tau}= \f{\tau_{\rm res}}{\tau_{\rm ag}}\lro\f{1}{2}\s_{i,j}^{N_b}\tilde{\mathcal{K}}_{ij}\tilde n_i\tilde n_j-\s_i^{N_b}\tilde{\mathcal{K}}_{iq}\tilde n_i\tilde n_q\rro 
+\f{d \tilde n_q}{d\tilde \tau}= \f{\tau_{\rm res}}{\tau_{\rm ag}}\lro\f{1}{2}\s_{i,j}^{N_b}\tilde{\mathcal{K}}_{ij}\tilde n_i\tilde n_j-\s_i^{N_b}\tilde{\mathcal{K}}_{iq}\tilde n_i\tilde n_q\rro
 -\f{\tau_{\rm res}}{\tau_{\rm{dif}}} \tilde n_q -\f{\tau_{\rm res}}{\tau_{\rm{th}}} \tilde n_q.
 \eeq
 % Equation~(\ref{smolu-dimensionless}) explicitly shows that there are  independent time-scales
@@ -605,196 +589,382 @@
 % processes included in this study. Furthermore, it shows that there are two important
 % combinations the may be used to estimate the relative importance of each process,
 % a procedure that we follow in the Results and Discussion Section.
-Equation~(\ref{smolu-dimensionless}) explicitely shows that there are four independent time scales that govern the system dynamics, three corresponding to  the aerosol 
-processes included in this study and one given by convection. Furthermore, it shows that there are three important
-combinations the may be used to estimate the relative importance of each process,
+Equation~(\ref{smolu-dimensionless}) explicitly shows the four independent time scales
+that govern the system dynamics, three corresponding to aerosol
+processes and one determined by convection. Furthermore, it shows that there are three important
+combinations the may be used to estimate the relative importance of the processes,
 a procedure that we follow in Section \ref{results}.
 
-
+\subsection{Numerical method}
+\label{numerical method}
 
-\subsection{Numerical method and determination of the fractal dimension} \label{numerical method}
-The dynamics of particles on a nanosize scale dispersed throughout the fluid is governed by the aerosol general dynamic equation (GDE) \citep{friedlander book}.
-Its formulation for a discrete size distribution  amounts to  a population balance equation for each particle size. Such a formulation  quickly becomes computationally unpractical when the number of primaries in an aggregate reaches  $n_{\rm{agg}}\sim 1000$.
-Here we follow a sectional approach: the initial number-size distribution is split into $N_b$ bins and  particle properties are averaged on each bin \citep{garrick section, jacobson sectional}.
-Despite relying heavily on the lognormality of the number-size distribution for mass balances and statistics, the sectional method does not require any assumption about its functional form. 
-Equation~(\ref{smolu-tau}) is solved on a logarithmically-spaced bin structure.
-Each bin is labelled according to the average size of the aggregate it contains.
-After choosing an interval $[d_{\rm mon},d_{\rm{max}}]$, where $d_{\rm{max}}$ is the maximum aggregate size included in the simulation, we fix the grid in aggregate diameters:
+The formulation of the GDE for a discrete size distribution,
+a population balance equation for every particle size,
+quickly becomes computationally impractical when the number of primary
+particles in an aggregate reaches $n_{\rm{agg}}\sim 1000$.
+Herein, we follow a sectional approach: the initial number distribution is
+split into $N_b$ bins and particle properties are averaged on each
+bin \citep{jacobson sectional,garrick section}.
+%Even though the experimental size distributions
+%are accurately represented by a lognormal distribution we chose a numercial
+%scheme to solve the GDE, the sectional method, that does not require any assumption about its functional form.
+Equation~(\ref{smolu-tau}) is solved on a logarithmically-spaced bin structure, where
+each bin is labelled according to the average aggregate size.
+For a diameter interval $[d_{\rm mon},d_{\rm{max}}]$, where $d_{\rm{max}}$ is the maximum
+aggregate size included in the simulation, we fix the grid in aggregate diameters
 \beq
 d_q=\exp\lsq  \f{(q-1)\ln(d_{\rm{max}}/d_{\rm mon})}{N_b}\rsq d_{\rm mon} ,\;\;\;\; q=1\dots N_b+1.
-\eeq  
-The $q$-th bin is given by the interval $[d_q,d_{q+1}]$ and its population $n_q$ can be determined from the numerical integration of the lognormal distribution in Eq.~(\ref{lognormal}).
-We  identify a characteristic size in each bin $\overline{d}_q=(d_q+d_{q+1})/2$ and a corresponding aggregate mass $\overline{m}_q=\reff\pi\overline{d}_q^3/6$ and aggregate volume of solid $\nu_q=\overline{m}_q/\rho_0$.
+\eeq
+The $q$-th bin is the interval $[d_q,d_{q+1}]$ and the number concentration $n_q$
+is obtained by integrating numerically the lognormal distribution in Eq.~(\ref{lognormal}).
+We  identify a characteristic size in each bin $\overline{d}_q=(d_q+d_{q+1})/2$,
+an aggregate mass $\overline{m}_q=\reff\pi\overline{d}_q^3/6$, and
+an aggregate solid volume $\nu_q=\overline{m}_q/\rho_0$.
 
-The  $\omega_q$ term in Eq.~(\ref{smolu-tau}) accounts for the effect of particle collisions.
-We now re-cast it in a suitable form for the application of the sectional method on a non-uniform grid:
+The  collision term $\omega_q$ in Eq.~(\ref{smolu-tau})
+may be rewritten into a form suitable for the application of the sectional method on a non-uniform grid
 \beq\label{omega_q}
 \omega_q=\f{1}{2}\s_{i,j}^{N_b}\chi_{ijq}\mathcal{K}_{ij}n_in_j-\s_i^{N_b}\mathcal{K}_{iq}n_in_q,
 \eeq
-where   $\chi_{ijq}$ is the splitting operator (introduced by  \citet{jacobson sectional} and here implemented as in   \citet{garrick section}) which accounts for the non-uniformity of the volume grid and re-distributes the newly created aggregates into different bins.
-Equation~(\ref{smolu-tau}) becomes a system of first-order nonlinear coupled  ODE's  we solve  using the LSODA algorithm \citep{lsoda1,lsoda2}.
-We tested the procedure by comparing the numerical  solution to analytical expressions when the latter are available.
-The accuracy of the solution is checked by ensuring its independency on the number of bins used to discretize the initial number-size distribution.
-In the absence of diffusional and thermophoretic losses, Eq.~(\ref{smolu-tau}) conserves the total aerosol particle mass in a unit volume. This is also tested in our numerical simulations  to make sure that the employed grid contains all the significant particle sizes for the system in exam.
-Finally, the use of the splitting operator $\chi_{ijk}$ has been tested by comparing the results obtained with a uniform grid and $6000$ bins to those we get using $100$ logarithmically spaced bins.
-They turn out to be equivalent, but the numerical advantage of such a reduction in the number of bins, i.e. equations to solve, is obvious.
+where $\chi_{ijq}$ is the splitting operator (introduced by  \citet{jacobson sectional}, here implemented as
+in \citet{garrick section}) that accounts for the non-uniformity of the volume grid
+by re-distributing newly created aggregates into different bins.
+Equation~(\ref{smolu-tau}) becomes a system of first-order nonlinear coupled
+ODE's that we solve using the LSODA algorithm \citep{lsoda1}.
+We tested the numerical algorithm
+by comparing the numerical solution to analytical expressions, when available,
+e.g. for a constant collision kernel.
+The accuracy of the solution was checked by ensuring numerical
+results were independent of the number of bins used to discretize the
+initial number distribution.
+In the absence of particle deposition, Eq.~(\ref{smolu-tau}) conserves
+total particle mass concentration. Mass conservation was tested
+to ensure that the chosen grid contained all relevant particle sizes.
+Finally, the splitting operator $\chi_{ijk}$ was tested by comparing results obtained in
+a uniform grid of $6000$ bins and on a  grid of $100$ logarithmically spaced bins.
+Numerical results were identical, but the computational advantage
+of the reduction in the number of bins,
+i.e. equations to solve, was noticeable.
 
-In the experiments, the uncertainty associated with the determination of the dilution ratio at the inlet of the transfer line, connected directly to the car tailpipe, is  larger than the one at the outlet.
-This affects the determination of the total concentration $N_{\rm in}$ at the inlet,  but not the estimates of the geometric mean and standard deviation.
-After choosing a fractal dimension $d_f$ we first use Eq.~(\ref{total mass}) to determine the total mass at the outlet $M_{\rm out}$.
-The same procedure is then applied at the inlet. Of course, the uncertainty about $N_{\rm in}$  leads to the same uncertainty in the total mass at the inlet, $M_{\rm in}$.
-As a first approximation, one can reconstruct $N_{\rm in}$ by imposing $M_{\rm in}=M_{\rm out}$ and adjusting the total particle concentration at the inlet accordingly.
- It is also possible to account for some extra mass at the inlet if particle losses along the transfer line are not negligible.
-The fractal dimension (which is not experimentally determined) affects both the mass balances and the particle dynamics.
-We determine it self-consistently until a good agreement (i.e. minimization of the least-square or $L_2$-norm distance) is reached between the mass-size distribution 
-returned by our numerical simulations and its value determined from the fit of the corresponding  experimental number-size distribution (see Eq.~(\ref{mass-size distr})).
-The mass-size distribution is favoured over  the number-size distribution  since the methodology developed in this study relies on mass balances.
-The algorithm we follow is sketched   in Fig.~\ref{diagram for d_f}.
- \begin{figure}
-\includegraphics[width=\columnwidth]{diagram.pdf}
-\caption{Algorithm used to determine the fractal dimension of the aggregates.}\label{diagram for d_f}
-\end{figure}
+\subsection{Determination of the aggregate fractal dimension}
+\label{DetermineFractalDimension}
+
+The algorithm we use for the self-consistent calculation
+of the average fractal dimension is based on the strong
+dependence of the system dynamics, through the collision kernel, on the fractal dimension.
+In fact, the kernel in Eq.~(\ref{fuchs_kernel}) exhibits a $1/d_{\rm f}$ power-law dependency on the fractal dimension, leading to enhanced collsions between aggregates with the same volume of solids but lower $d_{\rm f}$.
+As in \cite{maricq coagulation}, we treat both the total aggregate
+number concentration and the fractal dimension as fitting parameters.
+However, in our experiments, the uncertainty of the overall aggregate concentration
+is probably greater at the transfer-line inlet than at
+the outlet. This is probably due to strong pressure fluctuations
+during sampling at the inlet that affect the determination of the
+sampling-system dilution ratio. The geometric mean and standard deviation of the
+experimental distribution, however, may be considered weakly dependent on
+the dilution-ratio uncertainty. Consequently, we base out algorithm on the assumption
+that outlet measurements are more accurate than inlet measurements.
+
+Accordingly, an aggregate fractal dimension is chosen, and then
+the total particle mass per unit  volume at  the inlet, $M_{\rm in}$, and at the outlet,  $M_{\rm out}$,
+are calculated via Eq.~(\ref{total mass}). Unlike Ref.~\cite{maricq coagulation}, we explicitly
+impose, as a first approximation, mass conservation from inlet to
+outlet and reconstruct $N_{\rm in}$ such that $M_{\rm in} = M_{\rm out}$.
+This reconstruction uses the chosen fractal dimension, and the experimentally determined
+$\mu$ and $\sigma$ at the inlet to determine $N_{\textrm{in}}$.
+If the dynamics shows non-negligible particle losses, we account for the
+additional mass concentration at the inlet.
+We then propagate the reconstructed inlet number distribution for
+the appropriate residence time, thus obtaining the outlet number distribution
+and the mass distribution (calculated via the effective density [see Eq.~(\ref{mass size})]).
+The experimental mass distribution at the outlet is similarly calculated
+using the lognormal fit of the experimental number distribution.
+Since we rely explicitly on mass-balance considerations to fit $N_{\rm in}$,
+we iterate the procedure described above for different values of $d_{\rm f}$
+until we minimize the least-square (or $L_2$-norm) distance
+between the experimental and simulated mass distributions at the outlet.
+Another natural choice would be the minimization
+the least-square distance between the experimental and calculated number distributions.
+The two minimizations are not equivalent in general and a comparison
+between them is shown in Section \ref{results}.
+The algorithm we follow is sketched in Fig.~\ref{diagram for d_f}.
+
+%  after choosing the  fractal dimension, outlet particle distribution is calculated propagating
+%  the inlet distribution for the appropriate
+%  residence time. Due to the experimental uncertainties, we consider
+% A similar methodology was used in Ref.~
 
 
-% \subsection{Statistical Treatment of Particle Distributions }
-\section{Results}\label{results}
-We investigated the dynamics of diesel-exhaust aggregates emitted by a Euro-3 vehicle at constant speeds $50$ and $120$Km/h.
-Table~\ref{table time scales} shows a the relevant time scales.
-At $120$Km/h the dynamics is dominated by agglomeration and thermophoretic deposition, whereas diffusional losses are not significant. 
-The exhaust velocity inside the transfer line is  $4.3-4.5$m/s,  the line being about $9$m long, which leads to a total aggregate residence time $\tau_{\rm res}~\simeq 2$s.
-The exhaust mean temperature at the inlet can be estimated as about $200\,^{\circ}\mathrm{C}$. 
-Thermophoretic losses are estimated to account for about $7\%$ of the total mass. 
-% In \cite{lee_deposition}, a correlation similar to the one from \cite{yannis 1d deposition}, i.e. the one employed in this study, was used to estimate thermophoretic deposition in a tailpipe.
-% There it was found to be insufficient but the experimental conditions (decreasing  instead of constant wall temperature) were, as pointed out by the authors, outside the realm of validity of the correlation used.
-% It is simply worth pointing out that \cite{lee_deposition} also found thermophoretic deposition to be non-negligible for inlet temperatures and number-size distributions comparable with those measured at the VELA-2 laboratories.
- 
-The experiments at car speed equal to $50$Km/h  show some similarities in the role of agglomeration on particle dynamics.
- The total residence time of aerosol aggregates inside the transfer line is  $\tau_{\rm res}\simeq4-4.5$s, due to the lower exhaust flow rate, and is again comparable to the characteristic time for agglomeration.
-The exhaust initial temperature is about $60\,^{\circ}\mathrm{C}$ lower than in the case of the $120$Km/h.
-This significantly alters the role played by thermophoresis, which is mirrored by a new value of $\tau_{\rm th}~\sim 10^2$s and in this case thermophoretic losses amount to roughly $1\%$ of the total particle mass. 
-% With a characteristic time  $\tau_{\rm ag}~\sim 4$s, agglomeration remains by far the dominating process.
-Figure~\ref{mass-size fig} shows the  numerical results in this study.
- For the experiments at car speed $120$Km/h (top row), we estimated a fractal dimension $\df=2.2$. One observes the combined effects of agglomeration and losses in decreasing particle concentration and increasing the mean of  the number-size distribution (left).
-On the right we also show the mass-size distribution [see Eq.~(\ref{mass-size fig})] as obtained from the numerics and from fitting    the experimental number-size distribution at outlet.
-For car speed equal to $50$Km/h, we determine the fractal dimension to be $\df=2$ and we show the quantities of interest in the bottom row of Fig.~\ref{mass-size fig} as for the $120$Km/h case.
-\begin{table}
-\begin{center}
-\begin{tabular}{|c|r@{}lr@{}lr@{}l||r|}
-	\hline
-\multicolumn{8}{|c|}
-	{\rule[-3mm]{0mm}{8mm} \textbf{Time scales for the two sets of experiments}} \\
- Car Speed
-	& \multicolumn{2}{|c|}{Agglomeration}
-	& \multicolumn{2}{|c|}{Diffusion}
-	& \multicolumn{2}{|c|}{Thermophoresis}
-	& Residence Time \\ \hline \hline
-50 Km/h    & \multicolumn{2}{|c|}{$\tau_{\rm ag}\simeq 4$s}               & \multicolumn{2}{|c|}{$\tau_{\rm dif}\simeq 10^3$s}                 &\multicolumn{2}{|c|}{$\tau_{\rm th}\simeq 10^2$s}   & $\tau_{\rm res}\simeq 4$s \\
-120 Km/h    & \multicolumn{2}{|c|}{$\tau_{\rm ag}\simeq 2$s}               & \multicolumn{2}{|c|}{$\tau_{\rm dif}\simeq  10^3$s}                 &\multicolumn{2}{|c|}{$\tau_{\rm th}\simeq 13$s}   & $\tau_{\rm res}\simeq 2$s \\
- \hline
-% \vspace*{0.cm}
-\end{tabular}
-  \caption{Characteristic time scales for the aerosol processes considered in the two sets of experiments and residence time of the aggregates in the transfer line.}
- \label{table time scales}
-\end{center}
-\end{table}
-\begin{figure}
-\includegraphics[width=0.5\columnwidth, height=6cm]{2D-distribution.pdf}
- \includegraphics[width=0.5\columnwidth, height=6cm]{2D-mass_fitting.pdf}
-%  \includegraphics[width=6.cm, height=6cm]{errors_120.pdf}
-%  \includegraphics[width=10.cm, height=6cm]{evolution_3d_120.pdf}
- \includegraphics[width=0.5\columnwidth, height=6cm]{2D-distribution_50.pdf}
- \includegraphics[width=0.5\columnwidth, height=6cm]{2D-mass_fitting_50.pdf}
-\caption{Top: results for for fractal dimension $\df=2.2$ and car speed $120$Km/h. Top left: fit of the measured number-size distribution  at the inlet (dashed line) together with experimental (crosses) and numerical (continuous line) findings  at the outlet. 
-Top right: mass-size distribution derived from the corresponding fitted number-size distributions both at the inlet (dashed line) and  at the outlet (crosses) together with the output of our simulations at the outlet (continuous line).
-Bottom: as on top, but results for $\df=2$ and car speed $50$Km/h. }\label{mass-size fig}
-\end{figure}
 
 
-The dependence of the  distance (in a least-square sense) on $\df$,  between the calculated and fitted  mass-size  distributions at outlet,  is shown in Fig.~\ref{mass-size residuals}. 
-On the left, we plot the square root of the  sum of the residuals of the mass-size distribution as a function, normalized by the total mass,  of the fractal dimension $\df$.
+% The proposed algorithm for the self-consistent calculation
+% of the average fractal dimension is based on the strong
+% dependence of the collision kernel, and consequently
+% on the propagation of the number distribution
+% along the transfer line, on the fractal dimension.
+% Specifically, an inlet fractal dimension is chosen,
+% the inlet distribution is propagated for the appropriate
+% residence time, and the calculated outlet particle
+% size distribution is compared to the experimental distribution.
+% If they differ, a new initial fractal dimension is chosen till the
+% difference of the calculated and experimental outlet distributions
+% is minimized. A similar procedure was used by Maricq to
+% determine the fractal dimension and the total particle number (CHECK).
+
+% However, for the experiments at steady-state velocity $50$Km/h we found that
+% the calculated total particle mass at the outlet
+% was $5\%$ greater than the mass at the inlet
+% for a typical fractal dimension of $d_f = 2.0$.
+% For a vehicle speed of $120$Km/h, instead, the total calculated particle mass
+% at the inlet was about $10\%$ larger than at the outlet.
+% We attribute this discrepancy to the experimental uncertainty associated with the
+% determination of the dilution ratio at the inlet of the transfer line.
+% This uncertainty is greater at the inlet than at
+% the outlet, as the transfer line is directly connected
+% to the tail pipe. It is most likely due to strong pressure fluctuations
+% during sampling at the transfer line inlet: these fluctuations
+% affect the determination of the sampling system dilution ratio,
+% rendering the corresponding measurements less reliable than the
+% measurements at the outlet. In Section~\ref{results} we
+% show that the additional inlet particle mass can be
+% almost entirely explained by taking into account thermophoretic
+% deposition. The greater outlet mass is dealt with by
+% reconstructing the initial state such that the total inlet
+% particle mass is equal (or greater in case of significant
+% deposition losses) than the outlet mass.
+% It is, thus, necessary not only to determine
+% the fractal dimension self consistently but the initial state
+% has to be reconstructed, a reconstruction based on the calculation
+% of the particle mass distribution.
+
+% The algorithm is modified accordingly. We choose a fractal
+% dimension $d_f$ and then we use Eq.~(\ref{total mass}) to calculate the
+% total mass at the outlet $M_{\rm out}$. The same procedure is then
+% applied at the inlet.
+
+% As a first approximation, we reconstruct
+% $N_{\rm in}$ by requiring $M_{\rm in}=M_{\rm out}$ and adjusting the total
+% particle concentration at the inlet accordingly.
+% We consider that the
+%  uncertainty in the sampling dilution ratio
+% affects only the determination of the total
+% concentration $N_{\rm in}$ at the inlet,
+% but not the estimates of
+% the geometric mean and standard deviation:
+% hence the reconstructed initial state retains the
+% experimental $\mu$ and $\sigma$.
+% Of course, the uncertainty of $N_{\rm in}$
+% leads to the same uncertainty in the total mass at the inlet,
+% $M_{\rm in}$. It is also possible
+% to account for additional mass at the inlet if particle losses along
+% the transfer line are not negligible.
+%  Consequently we determine the (average) fractal dimensions
+% self-consistently until a
+% good agreement (i.e. minimization of the least-square or $L_2$-norm
+% distance) is reached between the mass-size distribution returned by
+% our numerical simulations and its value determined from the fit of
+% the corresponding  experimental number-size distribution (see
+% Eq.~(\ref{mass-size distr})).
+% As the fractal dimension affects both mass balance and
+% particle dynamics the minimization procedure could be used
+% either on the particle size distribution or the mass-size distribution.
+% The mass-size distribution is favoured
+% over the number-size distribution  since the methodology developed
+% in this study relies on mass balances (??) and also
+% the figure in the next section (Rewrite!!!). A sketch of the
+% algorithm we follow is shown in Fig.~\ref{diagram for d_f}.
+
+
+
+\section{Results}
+\label{results}
+
+We compare experimental and calculated number distributions
+for two sets of experiments performed at constant speeds $50$ and $120$Km/h
+of a Euro-3 light-duty vehicle. Our results are summarized
+in Table~\ref{table statistics} that shows the statistics of the number
+distributions and the calculated fractal dimensions.
+The difference between  $N_{\rm in}$, as calculated from the experimental data (fit)
+and as reconstructed according to mass-balance considerations (rec), amounts to only
+a few percents, and it is within experimental uncertainties. The same, of course,
+is true for the total inlet mass concentration. Note, however,
+that the fitted inlet mass concentration is smaller than
+the outlet mass concentrations (experiments at $50$Km/h).
+For the experiments at car speed $120$Km/h, we
+estimated a fractal dimension $\df=2.2$, whereas
+$d_{\rm f} = 2$ the $50$Km/h experiments.
+
+Mass losses are almost entirely due to thermophoretic deposition. This is illustrated in
+Table~\ref{table time scales} where the relevant time scales are presented.
+At steady-state speed $120$Km/h the dynamics is dominated by agglomeration and
+thermophoretic deposition, whereas diffusional losses are not
+significant. The exhaust velocity inside the $9$m long transfer line is
+$4.3-4.5$m/s, leading to a total aggregate residence time $\tau_{\rm res}~\simeq 2$s.
+The mean exhaust temperature at the inlet can be estimated to be
+$200\,^{\circ}\mathrm{C}$. Thermophoretic losses are estimated to
+account for about $7\%$ of the total mass.
+% In \cite{lee_deposition}, a correlation similar to the one from \cite{yannis 1d deposition}, i.e.
+%the one employed in this study, was used to estimate thermophoretic deposition in a tailpipe.
+% There it was found to be insufficient but the experimental conditions (decreasing  instead of constant wall temperature)
+%were, as pointed out by the authors, outside the realm of validity of the correlation used.
+% It is simply worth pointing out that \cite{lee_deposition} also found thermophoretic deposition to be non-negligible for inlet t
+%emperatures and number-size distributions comparable with those measured at the VELA-2 laboratories.
+The experiments at $50$Km/h show similarities in the role of agglomeration on particle dynamics.
+The aggregate residence time is  $\tau_{\rm res}\simeq4-4.5$s, due to the lower exhaust flow rate:
+as before it is comparable to the characteristic time for agglomeration.
+The exhaust initial temperature is about $60\,^{\circ}\mathrm{C}$ lower than the $120$Km/h experiments.
+This significantly alters the role played by thermophoresis,
+a change that is mirrored in a new value of $\tau_{\rm th}~\sim 10^2$s:
+in these experiments thermophoretic losses amount to roughly $1\%$ of the total particle mass.
+% With a characteristic time  $\tau_{\rm ag}~\sim 4$s, agglomeration remains by far the dominating process.
+
+The evolution of the number and mass distributions is plotted in Fig.~\ref{mass-size fig}
+for both experiments at $120$Km/h (top row) and $50$Km/h (bottom row).
+The combined effects of agglomeration and losses in decreasing particle concentration and increasing
+the mean of  the number-size distribution (left) are easily noted (the same effect is shown in
+Table~\ref{table statistics}). On the right we also show the
+mass distributions [see Eq.~(\ref{mass-size fig})] as obtained from the simulations and the fit of
+the experimental outlet number distribution.
+
+The difference between the calculated and fitted
+mass distributions at the outlet, defined as the
+distance (in a least-square sense, or equivalently the $L_2$ norm),
+is shown in Fig.~\ref{mass-size residuals}.
+On the left we plot (in arbitrary units) the square root of the
+sum of the residuals of the mass distribution
+as a function of the fractal dimension $\df$.
 This function exhibits a clear minimum justifying its use for the  self-consistent determination of the fractal dimension.
-On the right, we carry out the same calculations replacing the mass-size with the number-size distribution.
-We notice that in particular for car speed $50$Km/h, the resulting function is much flatter and the determination of the minimum is less obvious. This is somehow expected, since the number-size distribution does not distinguish between small and large sizes, thus weighting equally every aggregate size.
-\begin{figure}
-\includegraphics[width=0.5\columnwidth, height=6cm]{errors_mass.pdf}
-\includegraphics[width=0.5\columnwidth, height=6cm]{errors_number.pdf}
-\caption{Left: square root of the sum of the residuals of the mass-size distribution, normalized by the total mass, as a function of the fractal dimension, for $\df$ in $[1.8,2.8]$, normalized by the minimum of value assumed by the sum. Right: as on the left, but now the residuals of the number-size distribution are normalized by the total aggregate number.  }\label{mass-size residuals}
-\end{figure}
+On the right, the same calculations for the mass distribution are shown.
+We notice that the distance function is much flatter and the determination of
+the minimum is less obvious, especially for the $50$Km/h experiments.
+This is somehow expected, since the number distribution does not
+distinguish between small and large aggregates, thus weighting equally every aggregate size.
+Therefore, even though the reconstructed inlet number distribution does not differ
+significantly from the experimental distribution,
+the use of the number or mass distribution to determine the fractal dimension is not equivalent.
+We suggest the use of the mass distribution as the preferable metric for the determination of $d_{\rm f}$,
+regardless of the specific experimental uncertainties,
+since the effective density, and therefore $d_{\rm f}$, is intimately bound to
+the mass rather than to the number concentration of the aggregates.
 
 % Some experimental studies relate the aggregate fractal dimension to the engine load \citep{virtanen fractal, skillas fractal, park fractal}.
 % Unlike in this study, there the fractal dimension was found to be a decreasing function of the engine load.
 % However, the presence of the a diesel oxidation catalist (DOC), as in our experiments, can increase the sulphur content of the exhaust particles at high temperature \citep{olfert-fractal}  thus giving rise  to more compact aggregate structures, i.e. with a higher fractal dimension [{\it does it make sense when we do hot sampling???}].
 
 
-\section{Discussion}
-We can appreciate the importance of the modeling  of the whole distribution, rather than a representative particle size, by investigating the 
- dynamics of the system  according to Eq.~(\ref{explanation tau agglomeration}).
-This amounts to approximating the initial number-size distribution with a monodisperse aerosol interacting with a constant collision kernel evaluated for a single aggregate size  while neglecting deposition.
- Such a treatment turns out to be  a rather crude approximation, which underestimates the decrease of the particle concentration by several tens of percent. One can see that in Fig.~(\ref{comparison total number}) where  $N(\tau)$ is calculated both from the output of the numerical simulations and according to Eq.~(\ref{explanation tau agglomeration}). This shows the importance of including  all the aggregate sizes and the deposition mechanisms in the study of aerosol dynamics. 
-\begin{figure}
-\includegraphics[width=0.5\columnwidth, height=6cm]{Total_particle_number_vs_time_comparison_monodisperse_approx.pdf}
- \includegraphics[width=0.5\columnwidth, height=6cm]{Total_particle_number_vs_time_comparison_monodisperse_approx_50.pdf}
-\caption{Left: for car speed $120$Km/h, $\tau$-evolution of the total aggregate concentration obtained from the numerics and the analytical expression in Eq.~(\ref{explanation tau agglomeration}). Right: as on left, but now the car speed is $50$Km/h. In both cases, Eq.~(\ref{explanation tau agglomeration}) underestimates the decrease in the number concentration.  }\label{comparison total number}
-\end{figure}
-
-\begin{figure}
-\includegraphics[width=0.5\columnwidth, height=6cm]{2D-distribution_short_anaconda_50.pdf}
- \includegraphics[width=0.5\columnwidth, height=6cm]{2D-distribution_short_anaconda.pdf}
-\caption{bla bla  }\label{comparison anaconda length}
-\end{figure}
-
-The precise determination of $\df$ is affected by experimental uncertainties (the experimental reading of the exhaust air flow rates assumes  air to be in normal conditions always inside the duct, whereas it is generally hotter  and  with non-constant density due to the decrease in  temperature).
-As a sensitivity study, we repeat the least-square minimization of the mass-size distribution errors to determine the fractal dimension $d_f$ while fictitiously varying the residence time $\tau_{\rm res}$ (in this test we exaggerate the experimental uncertainty) of the aerosol particles inside the transfer line. We allow $d_f$ to vary in $[1.8,2.8]$ in discrete steps of $0.05$ and, for each chosen $\tau_{\rm res}$, we determine the best fractal dimension in a least-square sense.
-We notice that, while for the $120$Km/h case $d_f$ varies linearly with the residence time, it rather looks like a sum of step functions for car speed $50$Km/h. Such a behavior could be smoothed out by using a more refined grid on $d_f$, but this was not the purpose of the exercise, which was aimed at understanding the sensitivity of the fitted $d_f$ on the residence time.
-This is indirectly an estimate of the effect of the fractal dimension on the system dynamics: particle-size distributions  with different fractal dimensions would need a different residence time to undergo a similar evolution.
-% For instance, in Fig~(\ref{df_fitting}) we show the values of the fitted $d_f$ as a function of the residence time.
-% The  uncertainty on the residence time is due to the non-constant value of the air density along the anaconda, which affects the determination of $U$. 
 
 
 
-\begin{figure}
-\includegraphics[width=0.5\columnwidth, height=6cm]{df_fitting_120.pdf}
-\includegraphics[width=0.5\columnwidth, height=6cm]{df_fitting_50.pdf}
-\caption{Test on the sensitivity of the fitted fractal dimension on the exhaust particles residence time in the transfer line. The experimental uncertanty on $\tau_{\rm res}$ has been exeggerated in this test. Left: estimate of $d_f$ as a function of the residence time for car speed $120$Km/h. The fitted $d_f$ depends linearly on the residence time. Right: as on the left, but now the car speed is $50$Km/h. The apparent jumps of $d_f$ are due to the finite resolution of the set of fractal dimensions which could be chosen by the least-square minimization.   }\label{df_fitting}
-\end{figure}
-
-
-\begin{figure}
-\includegraphics[width=0.5\columnwidth, height=6cm]{thermo_losses_50.pdf}
-\includegraphics[width=0.5\columnwidth, height=6cm]{thermo_losses.pdf}
-\caption{bla bla thermphoresis  }\label{thermo_losses}
-\end{figure}
 
 
- We can conclude that the dynamics inside the duct is mainly dominated by agglomeration, which is sensitive to the residence time of aggregates inside the transfer line.
-At high dilution ratios, the number-size distribution remains frozen inside the dilution tunnel and its measurement at the standard sampling point is thus significantly affected by the length of the transfer line. A difference in this  length between two experimental set-ups could hinder the reproducibility of the measurements.
-We also notice that the same conclusion does not necessarily hold true for a Euro-4 or Euro-5 vehicle.
-This is due to the depence on $N_0$ of characteristic  agglomeration time (see Eq.~(\ref{dimensionless times})) $\tau_{\rm ag}\sim 1/N_0$. Modern diesel-particulate-filter equipped cars have significantly lower number emissions, which translates into a time-scale for agglomeration which can easily be one or two orders of magnitude above the one estimated here. In that case, the role of agglomeration in the transfer line is much more limited. 
-Furthermore, for high exhaust temperature, thermophoresis is not negligible at all at should taken into account. This also raises important issues about the possibility of a later accidental resuspension of deposited material into the carrier flow.
+
+\section{Discussion}
+\label{sec:discussion}
+
+The algorithm described in the previous sections provides
+an efficient and simple procedure for the interpretation of
+experimental results under different conditions. Accordingly, we
+performed a number of sensitivity studies corresponding either
+to simpler modelling approaches or to different experimental geometries.
+
+The importance of the modeling the complete number
+distribution, rather than a representative particle size, is
+appreciated by investigating
+the dynamics of particles as given by Eq.~(\ref{explanation tau agglomeration}).
+The initial number distribution is approximated by a monodisperse aerosol interacting
+with a constant collision kernel evaluated for a single aggregate size,
+neglecting deposition. Such a treatment is a rather crude approximation that
+underestimates the decrease of the particle concentration by several tens of percent.
+Figure~(\ref{comparison total number}) presents the total aggregate number
+concentration $N(\tau)$ calculated with the numerical simulations presented
+herein and according to Eq.~(\ref{explanation tau agglomeration}).
+The significant difference of the two curves,
+Eq.~(\ref{explanation tau agglomeration}) underestimates the decrease in the number concentration
+for both experiments, shows the importance of including
+all aggregate sizes and the dominant deposition mechanisms in a proper
+description of aerosol dynamics in the transfer line.
+
+
+
+Furthermore, Fig.~(\ref{comparison total number}) shows  the effect of the length of the transfer line (i.e. the residence time, since $U_m$ is not affected by the length of the duct) on the total aggregate concentration.
+A simulated decrease from $9$ to $5$m  amounts to   almost halving  $\tau_{\rm res}$ for both car speeds.
+This is investigated as not all experimental geometries used to measure
+number and mass concentrations have transfer lines of equal length.
+ The role of agglomeration on particle concentration is also reflected by the outlet number distributions for different lengths  of the
+transfer line. Figure~\ref{comparison anaconda length} shows that
+a significant variation of its length 
+would clearly lead to a different output number distribution.
+This would severely affect solid-particle measurements at the
+sampling point located at the outlet of the dilution tunnel, especially if the
+distribution remains frozen in the dilution tunnel.
+Finally. Fig.~\ref{comparison mu and sigma} shows the influence of the transfer line length on the geometric mean and standard deviation of the number distribution.
+
+We should remark, however, that the same conclusion might not necessarily hold for
+more recent Euro-4 or Euro-5 vehicle. This is due to the dependence
+of characteristic agglomeration time (see Eq.~(\ref{dimensionless
+times})) $\tau_{\rm ag}\sim 1/N_0$ on the total number of
+aggregate concentration at the inlet, $N_0$. Modern diesel-particulate-filter
+equipped vehicles emit significantly lower number of particles: consequently, the
+corresponding agglomeration time scale may easily be
+one or two orders of magnitude above the estimate presented here. In that
+case the role of agglomeration in the transfer line is much more
+limited, as convection in the transfer line would be the dominant
+aerosol process.
+
+Finally, we estimated the role of thermophoresis by varying
+the wall temperature of the transfer line. For variations
+of the order of $\sim 30-40\,^{\circ}\mathrm{C}$, we predict a
+moderate (few percent) variation of the thermophoretic mass
+deposition. In general, for high exhaust temperature, thermophoresis
+is not negligible and it should taken into account. This also
+raises important issues about the possibility of a later accidental
+resuspension of deposited material into the carrier flow.
+
+
+% The determination of $\df$ depends sensitively on the transfer-line
+% residence time, itself dependent on experimental uncertainties, e.g., the experimental reading of
+% the exhaust-air flow rates assumes air to be in normal conditions inside the duct,
+% whereas it is hotter with non-constant, temperature dependent density.
+% As a sensitivity study we, thus, repeated the least-square minimization of the mass distribution
+% residuals by artificially varying the residence time $\tau_{\rm res}$ (admittedly an exaggeration of
+% experimental uncertainties).
+We finally estimate the effect of the fractal dimension on the aerosol dynamics in the duct.
+We chose a fractal dimension
+$d_{\rm f}$ in the interval $[1.8,2.8]$ and, after propagating the reconstructed inlet number distribution, we determine an artificial residence time  which minimizes the least-square difference between  the simulated and the fitted mass distributions at the outlet.
+As expected, since a low fractal dimension enhances the collision rate between aggregates with the same volume of solids, the artificial residence time turns out to be a monotonic function of the fractal dimension. 
+We note that for the $120$Km/h experiments $\tau_{\rm res}$ varies linearly with the fractal dimension,
+whereas for the $50$Km/h experiments it becomes a sum of step functions.
+Such a behavior could be smoothed by using a more refined grid on $\tau_{\rm res}$, i.e. saving more often the mass distribution along the evolution,
+but this was not the purpose of the exercise that aimed at understanding the sensitivity of the dynamics on the fractal dimension.
+% Furthermore, this sensitivity study is indirectly an estimate of the effect of the fractal
+% dimension on the system dynamics: number distributions with different fractal dimensions would
+% need a different residence time to exhibit a similar evolution.
+% For instance, in Fig~(\ref{df_fitting}) we show the values of the fitted $d_f$ as a function of the residence time.
+% The  uncertainty on the residence time is due to the non-constant value of the air density along the anaconda, which affects the determination of $U$.
+
 
 \section{Conclusion}
+\label{sec:conclusion}
+
 We developed a method for the study of one-dimensional aerosol dynamics in a turbulent flow.
-For low Reynolds numbers and under steady-state conditions, we find ourself solving the GDE for aerosol particles in an ageing chamber with additional terms for particle losses.
- The fractal dimension is a priori an unknown parameter but is determined self-consistently from the output of the dynamics.
-Statistical considerations on the total aerosol mass per unit volume are used to reconstruct the initial total particle concentration.
+For low Reynolds numbers and under steady-state conditions, we map the dynamics into a  GDE for aerosol particles in an ageing chamber with additional terms for particle losses.
 Although the turbulent nature of the flow is neglected, we estimate the magnitude of the errors introduced by this approximation.
-The fractal dimension of the agglomerates is treated as a constant in this study, but this is not an inherent limitation of the method developed here.
+Statistical considerations on the total aerosol mass per unit volume are employed to reconstruct the initial total particle concentration.
+The fractal dimension is a priori an unknown parameter but is determined self-consistently from the output of the dynamics
+and we explain why we favour the mass over the number distribution for its determination. The sensitivity of the method on the particle total residence time in the transfer line is discussed.
+We identify the time-scales of the relevant aerosol processes and show the influence of the length of the transfer line on the output number distribution.
+Finally, we link the results to the case of the more up-to-date Euro-4 and Euro-5 vehicles.
 
 \section*{Acknowledgements}
-The research conducted here has been supported by the European Commission. The authors thank Dr. Panagiota Dilara 
-for [\emph{to be written}].
-
-% \begin{figure}
-% \includegraphics[width=8cm, height=6cm]{2D-distribution.pdf}
-% \caption{Evolution of the number-size distribution with chosen fractal dimension $2.25$ for car speed $120$Km/h. }\label{2D-120km}
-% \end{figure}
+The authors thank P. Dilara for helpful discussions and
+G. de Santi for his support.
 
 % The Appendices part is started with the command \appendix;
 % appendix sections are then done as normal sections
- \appendix 
-
- \section{One-dimensional correlations} \label{appendix corr}
-In this appendix we list for completeness the 1D correlations employed to model  losses and average temperature along the transfer line.
+%  \begin{appendix}
+% \section{Depositional velocities}
+% \label{appendix corr}
+% \section{One-dimensional correlations} \label{appendix corr}
+%In this appendix we list for completeness the 1D correlations employed to model  losses and average temperature along the transfer line.
 %  \subsection{Air properties}
 %  Air kinematic viscosity, in  units of m$^2$/s,  is given as a function of temperature by the following empirical expression [{\it I took the following 2 expressions from some fittings on the web; I tested they are accurate wrt what you find in books, but how should I reference them???}]:
 %  \begin{align}\label{air kinematic viscosity}
@@ -810,61 +980,169 @@
 %  \eeq
 %  where $p$ is the atmospheric pressure in units of bars.
 %  The dynamic air viscosity, $\mu_{\rm air}$, is of course given by $\rho_{\rm air}\nu_{\rm air}$.
-
-
-
 % \label{}
-\subsection{Thermophoretic and diffusional losses coefficients}
-The thermophoretic velocity can be expressed as in \cite{yannis 1d deposition}:
-\beq
-\vth= -\kt\;\f{\nu}{T}\f{T_w-T}{2R/\nusselt},
-\eeq
-where  $R$ is the duct radius, $\nusselt$ is the Nusselt number and $\kt$ is Talbot coefficient.
-Further correlations are needed to obtain the last two quantities.
-In the case of a pipe, one can estimate the Nusselt number using Dittus-Boelter correlation \citep{book heat mass transfer}:
-\beq
-\nusselt=0.023\re^{0.8}\pr^{0.4},
-\eeq
-where $T_w$ is the transfer line wall temperature, $\re$  the Reynolds number and $\pr$  the Prandtl number, typically quite insensitive to temperature.
- More sophisticated correlations, e.g. Glieninski formula \citep{book heat mass transfer}, lead fundamentally to the same estimate for $\nusselt$ when the Reynolds number ranges from $8000$ to $19000$, as in the case of the VELA-2 experiements.
-Talbot coefficient can be estimated according to \citep{talbot}:
-\beq
-\kt=2C_sC_i\f{\ra+C_t\Kn_i}{(1+3C_m\Kn_i)(1+2\ra+2C_t\Kn_i)},
-\eeq 
-where $C_s$, $C_m$ and $C_t$ are three constants equal to $1.17$, $1.14$ and $2.18$, respectively, $\ra=\kair/\kp$, $\kp$ being the carbonaceous particle thermal conductivity. 
-Then the coefficient for thermophoretic losses is  given by \citep{yannis 1d deposition}:
-\beq\label{expression vt}
-\gth=\f{2\vdif}{R}.
-\eeq
+%\subsection{Thermophoretic and diffusional losses coefficients}
 
-The diffusion velocity is:
-\beq\label{expression vd}
-\vdif=\sh\f{\mathcal{D}_q}{2R},
-\eeq
-where $\sh$ is the Sherwood number which is calculated through the following correlation \citep{sherwood number}:
-\beq\label{sherwood}
-\sh=0.042\re f^{0.5}\sc^{1/3},
-\eeq
-where $\sc=\nu_{\rm air}/\mathcal{D}_q$ is the Schmidt number and $f$ is Fanning friction factor which in turn can be obtained from \citep{pope turbulence}:
-\beq
-f=0.0791\re^{-0.25},
-\eeq
-which is practically equivalent to other correlations widespread in literature like Churchill expression \citep{churchill}.
-Similarly to Eq.~(\ref{expression vd}), the coefficient for diffusional losses is then given by $\gdif=2\vdif/R$.
-\subsection{Temperature correlation}
-The duct walls are kept at a constant temperature $T_w=70\,^{\circ}\mathrm{C}$ and $T$ (average temperature along the pipe cross-section) evolves according to \citep{yannis 1d deposition}:
-\beq
-T=T_w+(T_0-T_w)\exp\lro\f{-2\nusselt x}{R \re\pr}\rro,
-\eeq
-where $T_0$ is the average exhaust temperature at the pipe inlet.
-As mentioned before, $x=U\tau$ so linking the temperature along the transfer line to the particle residence time is straightforward. 
+% For completeness we summarize the 1D correlations employed to model losses along the transfer line.
 
+% The thermophoretic velocity can be expressed as in Ref.~\cite{yannis 1d deposition}:
+% \beq
+% \vth= -\kt\;\f{\nu}{T}\f{T_w-T}{2{\rm R}/\nusselt},
+% \eeq
+% where  ${\rm R}$ is the duct radius, $\nusselt$ is the Nusselt number and $\kt$ is Talbot coefficient.
+% Further correlations are needed to obtain the last two quantities.
+% In the case of a pipe, one can estimate the Nusselt number using Dittus-Boelter correlation \citep{book heat mass transfer}:
+% \beq
+% \nusselt=0.023\re^{0.8}\pr^{0.4},
+% \eeq
+% where $T_w$ is the transfer line wall temperature, $\re$  the Reynolds number and $\pr$  the Prandtl number, typically quite insensitive to temperature.
+%  More sophisticated correlations, e.g. Glieninski formula \citep{book heat mass transfer}, lead fundamentally to the same estimate for $\nusselt$ when the Reynolds number ranges from $8000$ to $19000$, as in the case of the VELA-2 experiements.
+% Talbot coefficient can be estimated according to \citep{talbot}:
+% \beq
+% \kt=2C_sC_i\f{\ra+C_t\Kn_i}{(1+3C_m\Kn_i)(1+2\ra+2C_t\Kn_i)},
+% \eeq
+% where $C_s$, $C_m$ and $C_t$ are three constants equal to $1.17$, $1.14$ and $2.18$, respectively, $\ra=\kair/\kp$, $\kp$ being the carbonaceous particle thermal conductivity.
+% Then the coefficient for thermophoretic losses is  given by \citep{yannis 1d deposition}:
+% \beq\label{expression vt}
+% \gth=\f{2\vdif}{\rm R}.
+% \eeq
+
+% The diffusion velocity is:
+% \beq\label{expression vd}
+% \vdif=\sh\f{\mathcal{D}_q}{2 \rm R},
+% \eeq
+% where $\sh$ is the Sherwood number which is calculated through the following correlation \citep{sherwood number}:
+% \beq\label{sherwood}
+% \sh=0.042\re f^{0.5}\sc^{1/3},
+% \eeq
+% where $\sc=\nu_{\rm air}/\mathcal{D}_q$ is the Schmidt number and $f$ is Fanning friction factor which in turn can be obtained from \citep{pope turbulence}:
+% \beq
+% f=0.0791\re^{-0.25},
+% \eeq
+% which is practically equivalent to other correlations widespread in literature like Churchill expression \citep{churchill}.
+% Similarly to Eq.~(\ref{expression vd}), the coefficient for diffusional losses is then given by $\gdif=2\vdif/{\rm R}$.
+
+%\subsection{Temperature correlation}
+%The duct walls are kept at a constant temperature $T_w=70\,^{\circ}\mathrm{C}$ and $T$ (average temperature along the pipe cross-section) evolves according to \citep{yannis 1d deposition}:
+%\beq
+%T=T_w+(T_0-T_w)\exp\lro\f{-2\nusselt x}{R \re\pr}\rro,
+%\eeq
+%where $T_0$ is the average exhaust temperature at the pipe inlet.
+%As mentioned before, $x=U\tau$ so linking the temperature along the transfer line to the particle residence time is straightforward.
+% \end{appendix}
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+
+\hv{Good}{2007}{good}
+Good P. (2007). {\it An Overview of European Emissions Legislation for LD vehicles}. Presentation at the International Conference on transport and Environment, Milan 19-21 March 2007, Italy, available at \url{http://transportenv07.jrc.it/proceedings_D.html}
+
+\hv{Hindmarsh}{1980}{lsoda1} Hindmarsh, A.C. (1980). LSODE and LSODI, two new initial value ordinary differential equation solvers. {\it ACM-SIGNUM newsletter, 15}, 10-11.
+
+\harvarditem{Housiadas \& Drossinos}{2005}{yannis 1d deposition} Housiadas C., \& Drossinos Y. (2005).  Thermophoretic Deposition in Tube Flow. {\it Aerosol Science and Technology, 39}, 304-318.
+
+% \hv{Hull}{2005}{hull derivatives} Hull, J.C. (2006). {\it Options, Futures and Other Derivatives}, New York: Prentice-Hall.
+
+\hv{Incropera  \& DeWitt}{1996}{book heat mass transfer} Incropera, F.P. \& DeWitt, D.P. (1996), {\it Fundamentals of Heat and Mass Transfer}. New York: John Wiley \& Sons.
+
+\hv{Jacobson  \& Turco}{1994}{jacobson sectional} Jacobson M.Z., \& Turco, R.P. (1994). Modeling Coagulation among particles of different composition and size (1994). {\it Journal of Aerosol Science, 28}, 1327-1338.
+
+% \hv{Kittelson}{1997}{kittelson review} Kittelson, D.B. (1997). Engines and nanoparticles: a review. {\it Journal of Aerosol Science, 29}, 575-588.
+
+\hv{Kakac {\it et al.}}{1987}{churchill} Kakac, S., Shah, R. K., and Aung,W. (Eds.) (1987). {\it Handbook of Single-Phase
+Convective Heat Transfer.} John Wiley \& Sons: New York.
+
+\hv{Kostoglou  \& Konstandopoulos }{2001}{evolving fractal 1} Kostoglou, M., \& Konstandopoulos, A.G. (2001). Evolution of aggregate size and fractal dimension during Brownian coagulation. {\it Journal of Aerosol Science, 32}, 1399-1420.
+
+% \hv{Kostoglou {\it et al.}}{2006}{evolving fractal 2} Kostoglou, M.,  Konstandopoulos, A.G., \& Friedlander, S.K. (2006). Bivariate population dynamics simulation of fractal aerosol aggregate coagulation and restructuring. {\it Jouarnal of Aerosol Science, 37}, 1102-1115.
+
+% \hv{Lee  {\it et al.}}{2006}{lee_deposition} Lee, B.U., Byun, D.S., Bae, G., Lee, J.(2006). Thermophoretic Deposition of ultrafine particles in a turbulent pipe flow: Simulation of ultrafine particle behaviour in an automobile exhaust pipe. {\it Journal of Aerosol Science, 37}, 1788-1796.
+
+\hv{Maricq {\it et al.}}{1999}{maricq tailpipe}
+Maricq, M., Chase, R., Podsiadlik, D., Vogt, R. (1999). Vehicle Exhaust particle Size Distributions: A Comparison of Tailpipe and Dilution Tunnel Measurements. SAE 1999-01-1461
+
+
+
+\hv{Maricq \& Xu}{2004}{maricq experimental} Maricq, M.M., \& Xu, N. (2004). The effective density and fractal dimension of soot particles from premixed flames and motor vehicle exhaust. {\it Journal of Aerosol Science, 35}, 1251-1274.
+
+\hv{Maricq }{2006}{maricq coagulation} Maricq, M.M., (2006). Coagulation dynamics of fractal-like soot aggregates. {\it Journal of Aerosol Science, 38}, 141-156.
+
+\hv{Miller \& Garrick}{2004}{garrick section} Miller, S.E., \& Garrick, S.C. (2004). Nanoparticle Coagulation in a Planar Jet. {\it Aerosol Science and Technology, 38}, 79-89.
+
+
+
+\hv{Mountain \& Mulholland}{1986}{continuum formula} Mountain, R. D., Mulholland, G.W., \& Baum, H. (1986). Simulation of aerosol agglomeration in the free molecular and continuum flow regimes.
+ {\it Journal of Colloid and Interface Science, 114}, 67–81.
+
+\hv{Naumann}{2003}{naumann code} Naumann, K.H. (2003). COSIMA--a computer program simulating the dynamics of fractal aerosols. {\it Journal of Aerosol Science, 34}, 1371-1397.
+
+% \hv{Olfert {\it et al.}}{2006}{olfert-fractal}  Olfert, J.S., Symonds, J.P.R., \& Collings, N. (2006). The effective density and fractal dimension of particles emitted from a light-duty diesel vehicle with a diesel oxidation catalyst. {\it Journal of Aerosol Science, 38}, 69-82.
+\hv{Ntziachristos {\it at al.}}{2004}{leon2}
+Ntziachristos, L., Giechaskiel, B., Pistikopoulos, P., Samaras, Z., Mathis, U., Mohr, M., Ristimäki, J., Keskinen, J., Mikkanen, P., Casati, R., Scheer, \& V., Vogt, R. (2004). Performance Evaluation of a Novel Sampling and Measurement System for Exhaust Particle Characterization. {\it SAE 2004-01-1439}. 
+\hv{Ntziachristos {\it at al.}}{2005}{leon1}
+Ntziachristos, L., Giechaskiel, B., Pistikopoulos, P., \&  Samaras, Z. (2005). Comparative Assessment of Two Different Sampling Systems for Particle Emission Type-Approval Measurements, {\it SAE Transactions 2005-01-0198}.
+
+ \hv{Park {\it et al.}} {2003}{park fractal} Park, K., Cao, F., Kittelson, D.B., \& McMurry, P.H. (2003). Relationship between Particle Mass and Mobility for Diesel Exhaust Particles. {\it Environmental Science and Technology, 37}, 577-583.
+%  \hv{Petzold}{1983}{lsoda2} Petzold, L.R. (1983). Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations. {\it SIAM Journal on Scientific Computing, 4}, 136-148.
+
+\hv{Pope}{2000}{pope turbulence} Pope, S.B. (2000). {\it Turbulent Flows}, Cambridge: Cambridge University Press.
+
+\hv{Press {\it et al.}}{1988}{numerical recipes} Press, W.H., Flannery, B.P., Teukolsky, S.A., \& Vetterling W.T. (1988). {\it Numerical Recipes}, New York: Cambridge University Press.
+
+
+\hv{R\"{o}nkk\"{o} {\it et al.}}{2006}{ronkko}
+R\"{o}nkk\"{o}, T., Virtanen, A., Vaaraslahti, K., Keskinen, J., Pirjola, L., \& Lappi, M. (2006). Effect of dilution conditions and driving parameters on nucleation mode particles in diesel exhaust: Laboratory and on-road study. {\it Atmospheric Environment, 40}, 2893-2901.
+
+\hv{Saffman \& Turner}{1956}{saffman}  Saffman, P.G.,  \& Turner, J.S. (1956).  On the collision of drops in turbulent clouds. {\it J. Fluid Mech., 1}, 16-30.
+
+\harvarditem{Seinfeld  \& Pandis}{1998}{pandis book} Seinfeld, J.H., \& Pandis, S.N. (1998). {\it Atmospheric chemistry and physics}. NewYork: Wiley.
+% \hv{Skillas G. {\it et al.}}{1998}{skillas fractal} Skillas, G., Kunzel, S., Burtscher, H., Baltensperger, U., \& Siegmann, K. (1998). High Fractal-Like Dimension of Diesel Exhaust Soot Agglomerates. {\it Journal of Aerosol Science, 29,} 411-419.
+\hv{Talbot {\it et al.}}{1980}{talbot} Talbot, L., Cheng, R.K., Schefer, R.W., \& Willis, D.R. (1980). Thermophoresis of particles in a heated boundary layer. {\it J. Fluid. Mechanics, 101}, 737-758.
+\hv{Van Gulijk C. {\it et al.}}{2004}{fractal experimental}  Van Gulijk, C., Marijnissen, J.C.M., Makkee, M., Moulijn, J.A. \&  Schmidt-Ott, A. (2004). Measuring diesel soot witha scanning mobility particle sizer
+and an electrical low-pressure impactor: performance. {\it Journal of Aerosol Science, 35}, 633-655.
+assessment witha model for fractal-like agglomerates
+\hv{Vemury  \& Pratsinis}{1994}{pratsinis preserving adimensional} Vemury, S., \& Pratsinis, S.E. (1994). Self-preserving size distributions of agglomerates. {\it Journal of Aerosol Science, 26}, 175-185.
+\hv{Virtanen {\it et al.}}{2004}{virtanen fractal} Virtanen, A.K.K., Ristimaki, J.M., Vaaraslahti, K.M., \& Keskinen,  J.(2004). Effect of Engine Load on Diesel Soot Particles. {\it Environmental Science and Technology, 38}, 2551-2556.
+\hv{Vogt \& Scheer}{2002}{vogt}
+Vogt, R., \& Scheer, V. (2002). {\it Particles in diesel vehicle exhaust: A comparison of laboratory and chasing experiments}. Proceedings of the 11th International Symposium Ttransport and Air Pollution, Graz 19-21.6.2002, pp 79-84, Univ.-Prof. Dr. R. Pischinger, Graz, ISBN 3-901351-59-0
+
+\hv{Voutsis {\it et al.}}{2005}{vouitsis} Vouitsis, E., Ntziachristos, L., \&  Samaras, Z. (2005). Modelling of diesel exhaust aerosol during laboratory sampling.{\it Atmospheric Environment, 39}, 1335-1345.
+% \hv{Whitby}{1978}{whitby classical} Whitby, K.T. (1978). The physical characteristics of sulfur aerosols. {\it Journal of Aerosol Science, 12}, 135-159.
+
+\hv{Williams {\it et al.}}{1980}{sherwood number} Williams, M.M.R., Loyalka, S.K., 1991. {\it Aerosol Science.
+Theory and Practice}. Pergamon Press: Oxford.
+
 
 
 
@@ -883,93 +1161,10 @@
 
 
 
-\hv{Burtscher}{2004}{burtscher review} Burtscher, H. (2004). Physical characterization of particulate emissions from diesel engines: a review. {\it Journal of Aerosol Science, 36}, 896-932.
-
-
-
-\hv{Drossinos \& Housiadas}{2006}{yannis book} Drossinos, Y., \& Housiadas, C. (2006). Aerosol Flows. {\it Multiphase Flow Handbook}. Boca Raton: Taylor \& Francis.
-
-
-\hv{Elzhov}{2005}{minpack} Elzhov, T.V. (2005). Minpack.lm: R interface for least squares optimization library. 
-~http:~//~cran.r-project.org~/~src~/~contrib/~Descriptions~/~minpack.lm.html. 
-
-\harvarditem{Friedlander}{2000}{friedlander book} Friedlander S.K. (2000). {\it Smoke, Dust and Haze}.    New York: Oxford University Press.
-
-
-\hv{Gelbard  \& Seinfeld}{1980}{gelbard-seinfeld} Gelbard, F., \& Seinfeld, J.H. (1980). Simulation of Multicomponent Aerosol Dynamics (1980). {\it Journal of Colloid and Interface Science, 78}, 485-501.
-
-
-\hv{Hindmarsh}{1980}{lsoda1} Hindmarsh, A.C. (1980). LSODE and LSODI, two new initial value ordinary differential equation solvers. {\it ACM-SIGNUM newsletter, 15}, 10-11.
-
-\harvarditem{Housiadas \& Drossinos}{2005}{yannis 1d deposition} Housiadas C., \& Drossinos Y. (2005).  Thermophoretic Deposition in Tube Flow. {\it Aerosol Science and Technology, 39}, 304-318. 
-
-\hv{Hull}{2005}{hull derivatives} Hull, J.C. (2006). {\it Options, Futures and Other Derivatives}, New York: Prentice-Hall.
-
-
-\hv{Incropera  \& DeWitt}{1996}{book heat mass transfer} Incropera, F.P. \& DeWitt, D.P. (1996), {\it Fundamentals of Heat and Mass Transfer}. New York: John Wiley \& Sons.
-
-\hv{Jacobson  \& Turco}{1994}{jacobson sectional} Jacobson M.Z., \& Turco, R.P. (1994). Modeling Coagulation among particles of different composition and size (1994). {\it Journal of Aerosol Science, 28}, 1327-1338.
-
-\hv{Kittelson}{1997}{kittelson review} Kittelson, D.B. (1997). Engines and nanoparticles: a review. {\it Journal of Aerosol Science, 29}, 575-588.
-
-
-
-
 
 
 
-\hv{Kakac {\it et al.}}{1987}{churchill} Kakac, S., Shah, R. K., and Aung,W. (Eds.) (1987). {\it Handbook of Single-Phase
-Convective Heat Transfer.} John Wiley \& Sons: New York.
-
-
-\hv{Kostoglou  \& Konstandopoulos }{2001}{evolving fractal 1} Kostoglou, M., \& Konstandopoulos, A.G. (2001). Evolution of aggregate size and fractal dimension during Brownian coagulation. {\it Journal of Aerosol Science, 32}, 1399-1420.
-
-\hv{Kostoglou {\it et al.}}{2006}{evolving fractal 2} Kostoglou, M.,  Konstandopoulos, A.G., \& Friedlander, S.K. (2006). Bivariate population dynamics simulation of fractal aerosol aggregate coagulation and restructuring. {\it Jouarnal of Aerosol Science, 37}, 1102-1115.
-
-
-\hv{Lee  {\it et al.}}{2006}{lee_deposition} Lee, B.U., Byun, D.S., Bae, G., Lee, J.(2006). Thermophoretic Deposition of ultrafine particles in a turbulent pipe flow: Simulation of ultrafine particle behaviour in an automobile exhaust pipe. {\it Journal of Aerosol Science, 37}, 1788-1796.
-
-
-
-\hv{Maricq }{2006}{maricq coagulation} Maricq, M.M., (2006). Coagulation dynamics of fractal-like soot aggregates. {\it Journal of Aerosol Science, 38}, 141-156. 
-
-\hv{Miller \& Garrick}{2004}{garrick section} Miller, S.E., \& Garrick, S.C. (2004). Nanoparticle Coagulation in a Planar Jet. {\it Aerosol Science and Technology, 38}, 79-89.
-
-
-
-\hv{Maricq M.M. \& Xu N.}{2004}{maricq experimental} Maricq, M.M., \& Xu, N. (2004). The effective density and fractal dimension of soot particles from premixed flames and motor vehicle exhaust. {\it Journal of Aerosol Science, 35}, 1251-1274.
-
-\hv{Mountain \& Mulholland}{1986}{continuum formula} Mountain, R. D., Mulholland, G.W., \& Baum, H. (1986). Simulation of aerosol agglomeration in the free molecular and continuum flow regimes.
- {\it Journal of Colloid and Interface Science, 114}, 67–81.
-
-\hv{Naumann}{2003}{naumann code} Naumann, K.H. (2003). COSIMA--a computer program simulating the dynamics of fractal aerosols. {\it Journal of Aerosol Science, 34}, 1371-1397.
-
-\hv{Olfert {\it et al.}}{2006}{olfert-fractal}  Olfert, J.S., Symonds, J.P.R., \& Collings, N. (2006). The effective density and fractal dimension of particles emitted from a light-duty diesel vehicle with a diesel oxidation catalyst. {\it Journal of Aerosol Science, 38}, 69-82.
-
-
-\hv{Park {\it et al.}} {2003}{park fractal} Park, K., Cao, F., Kittelson, D.B., \& McMurry, P.H. (2003). Relationship between Particle Mass and Mobility for Diesel Exhaust Particles. {\it Environmental Science and Technology, 37}, 577-583.
-\hv{Petzold}{1983}{lsoda2} Petzold, L.R. (1983). Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations. {\it SIAM Journal on Scientific Computing, 4}, 136-148.
-
-\hv{Pope}{2000}{pope turbulence} Pope, S.B. (2000). {\it Turbulent Flows}, Cambridge: Cambridge University Press.
-
-\hv{Press {\it et al.}}{1988}{numerical recipes} Press, W.H., Flannery, B.P., Teukolsky, S.A., \& Vetterling W.T. (1988). {\it Numerical Recipes}, New York: Cambridge University Press.
-
-\harvarditem{Seinfeld  \& Pandis}{1998}{pandis book} Seinfeld, J.H., \& Pandis, S.N. (1998). {\it Atmospheric chemistry and physics}. NewYork: Wiley.
-\hv{Skillas G. {\it et al.}}{1998}{skillas fractal} Skillas, G., Kunzel, S., Burtscher, H., Baltensperger, U., \& Siegmann, K. (1998). High Fractal-Like Dimension of Diesel Exhaust Soot Agglomerates. {\it Journal of Aerosol Science, 29,} 411-419.
-\hv{Talbot {\it et al.}}{1980}{talbot} Talbot, L., Cheng, R.K., Schefer, R.W., \& Willis, D.R. (1980). Thermophoresis of particles in a heated boundary layer. {\it J. Fluid. Mechanics, 101}, 737-758.
-\hv{Van Gulijk C. {\it et al.}}{2004}{fractal experimental}  Van Gulijk, C., Marijnissen, J.C.M., Makkee, M., Moulijn, J.A. \&  Schmidt-Ott, A. (2004). Measuring diesel soot witha scanning mobility particle sizer
-and an electrical low-pressure impactor: performance. {\it Journal of Aerosol Science, 35}, 633-655.
-assessment witha model for fractal-like agglomerates
-\hv{Vemury  \& Pratsinis}{1994}{pratsinis preserving adimensional} Vemury, S., \& Pratsinis, S.E. (1994). Self-preserving size distributions of agglomerates. {\it Journal of Aerosol Science, 26}, 175-185.
-
-\hv{Virtanen {\it et al.}}{2004}{virtanen fractal} Virtanen, A.K.K., Ristimaki, J.M., Vaaraslahti, K.M., \& Keskinen,  J.(2004). Effect of Engine Load on Diesel Soot Particles. {\it Environmental Science and Technology, 38}, 2551-2556.
-
-\hv{Whitby}{1978}{whitby classical} Whitby, K.T. (1978). The physical characteristics of sulfur aerosols. {\it Journal of Aerosol Science, 12}, 135-159.
-
-\hv{Williams {\it et al.}}{1980}{sherwood number} Williams, M.M.R., Loyalka, S.K., 1991. {\it Aerosol Science.
-Theory and Practice}. Pergamon Press: Oxford.
-
-\hv{Xiong  \& Pratsinis}{1993}{xiong pratsinis} Xiong, Y., \& Pratsinis, S.E. (1993). Formation of agglomerate particles by coagulation and sintering-part I. A two-dimensional solution of the population balance equation. {\it Journal of Aerosol Science, 24}, 283-300.
+% \hv{Xiong  \& Pratsinis}{1993}{xiong pratsinis} Xiong, Y., \& Pratsinis, S.E. (1993). Formation of agglomerate particles by coagulation and sintering-part I. A two-dimensional solution of the population balance equation. {\it Journal of Aerosol Science, 24}, 283-300.
 
 % notes:
 % \bibitem{label} \note
@@ -986,5 +1181,218 @@
 
 \end{thebibliography}
 % \end{linenumbers}
+
+\clearpage
+\begin{table}
+ \begin{center}
+\begin{tabular}{|c|r@{}lr@{}lr@{}lr@{}lr@{}lr@{}lr@{}l||r|}
+    \hline
+%\multicolumn{16}{|c|}
+%    {\rule[-3mm]{0mm}{8mm} \textbf{Overview of the Numerical Results}} \\
+  Speed
+        & \multicolumn{2}{|c|}{Position}
+    & \multicolumn{2}{|c|}{$N$ [cm$^{-3}$]}
+    & \multicolumn{2}{|c|}{$N$  [cm$^{-3}$]}
+& \multicolumn{2}{|c|}{$M$ [g cm$^{-3}$] }
+    & \multicolumn{2}{|c|}{$M$  [g cm$^{-3}$]}
+    & \multicolumn{2}{|c|}{$\mu$ [nm]}
+    & \multicolumn{2}{|c|}{$\sigma$[nm]}
+    & $d_{\rm f}$ \\
+
+        & \multicolumn{2}{|c|}{}
+    & \multicolumn{2}{|c|}{fit}
+    & \multicolumn{2}{|c|}{rec}
+& \multicolumn{2}{|c|}{fit}
+    & \multicolumn{2}{|c|}{rec}
+    & \multicolumn{2}{|c|}{}
+    & \multicolumn{2}{|c|}{}
+    &  \\ \hline \hline
+50 Km/h
+     & \multicolumn{2}{|c|}{\rm Inlet}
+         & \multicolumn{2}{|c|}{$7.6\cdot 10^7$}
+     &\multicolumn{2}{|c|}{$7.9\cdot 10^7$}
+    & \multicolumn{2}{|c|}{$1.41\cdot 10^{-8}$}
+     &\multicolumn{2}{|c|}{$1.47\cdot 10^{-8}$}
+          &\multicolumn{2}{|c|}{$62$}
+&\multicolumn{2}{|c|}{$1.75$}
+& $2$
+\\
+50 Km/h
+     & \multicolumn{2}{|c|}{\rm Outlet}
+         & \multicolumn{2}{|c|}{$4.5\cdot 10^{7}$}
+     &\multicolumn{2}{|c|}{}
+    & \multicolumn{2}{|c|}{$1.45\cdot 10^{-8}$}
+     &\multicolumn{2}{|c|}{}
+          &\multicolumn{2}{|c|}{$ 87.3$}
+&\multicolumn{2}{|c|}{$ 1.63$}
+& $2$
+\\
+
+120 Km/h
+     & \multicolumn{2}{|c|}{\rm Inlet}
+         & \multicolumn{2}{|c|}{$1.4\cdot 10^8$}
+     &\multicolumn{2}{|c|}{$1.38\cdot10^{8}$}
+    & \multicolumn{2}{|c|}{$4.1\cdot 10^{-8}$}
+     &\multicolumn{2}{|c|}{$4\cdot10^{-8}$}
+          &\multicolumn{2}{|c|}{$67.7$}
+&\multicolumn{2}{|c|}{$1.79$}
+& $2.2$
+\\
+120 Km/h
+     & \multicolumn{2}{|c|}{\rm Outlet}
+         & \multicolumn{2}{|c|}{$8.3\cdot 10^{7}$}
+     &\multicolumn{2}{|c|}{}
+          &\multicolumn{2}{|c|}{$3.7\cdot 10^{-8}$}
+&\multicolumn{2}{|c|}{}
+          &\multicolumn{2}{|c|}{$ 88.5$}
+&\multicolumn{2}{|c|}{$1.69$}
+& $2.2$
+\\
+
+ \hline
+% \vspace*{0.cm}
+\end{tabular}
+  \caption{Parameters of the experimental and calculated number distributions.}
+ \label{table statistics}
+\end{center}
+\end{table}
+
+% \clearpage
+\begin{table}
+\begin{center}
+\begin{tabular}{|c|r@{}lr@{}lr@{}l||r|}
+    \hline
+%\multicolumn{8}{|c|}
+%    {\rule[-3mm]{0mm}{8mm} \textbf{Time scales for the two sets of experiments}} \\
+ Car Speed
+    & \multicolumn{2}{|c|}{Agglomeration}
+    & \multicolumn{2}{|c|}{Diffusion}
+    & \multicolumn{2}{|c|}{Thermophoresis}
+    & Residence Time \\ \hline \hline
+50 Km/h    & \multicolumn{2}{|c|}{$\tau_{\rm ag}\simeq 4$s}               & \multicolumn{2}{|c|}{$\tau_{\rm dif}\simeq 10^3$s}                 &\multicolumn{2}{|c|}{$\tau_{\rm th}\simeq 10^2$s}   & $\tau_{\rm res}\simeq 4$s \\
+120 Km/h    & \multicolumn{2}{|c|}{$\tau_{\rm ag}\simeq 2$s}               & \multicolumn{2}{|c|}{$\tau_{\rm dif}\simeq  10^3$s}                 &\multicolumn{2}{|c|}{$\tau_{\rm th}\simeq 13$s}   & $\tau_{\rm res}\simeq 2$s \\
+ \hline
+% \vspace*{0.cm}
+\end{tabular}
+  \caption{Characteristic time scales of the aerosol processes.}
+ \label{table time scales}
+\end{center}
+\end{table}
+
+
+\clearpage
+\begin{figure}
+\begin{center}
+\includegraphics[width=0.8\columnwidth, height=6cm]{JRC_CVS.jpg}
+\end{center}
+\caption{Schematic of the experimental set-up and sampling system at the VELA-2 laboratories.}\label{schematic experiment}
+\end{figure}
+
+\clearpage
+\begin{figure}
+ \begin{center}
+ \includegraphics[width=0.5\columnwidth, height=6cm]{mass-sizes.pdf}
+ \end{center}
+ \caption{Calculated cumulative mass distribution function at the inlet with $d_{\rm f}=2.3$, $d_0=50$nm and $\rho_0=1$g/cm$^3$.}
+\label{mass-size distr}
+ \end{figure}
+
+\clearpage
+\begin{figure}
+\includegraphics[width=0.5\columnwidth]{experimental-distr-with-error-bars_50.pdf}
+\includegraphics[width=0.5\columnwidth]{experimental-distr-with-error-bars.pdf}
+\caption{Left: Experimental number distributions,  with statistical uncertainty, at inlet and outlet,
+vehicle speed $50$Km/h. Right:  Vehicle speed $120$Km/h.}
+\label{experimental number-size}
+\end{figure}
+
+
+\clearpage
+\begin{figure}
+\includegraphics[width=\columnwidth]{diagram.pdf}
+\caption{Algorithm used to determine the fractal dimension of the aggregates.}
+\label{diagram for d_f}
+\end{figure}
+
+
+\clearpage
+\begin{figure}
+\includegraphics[width=0.5\columnwidth, height=6cm]{2D-distribution.pdf}
+ \includegraphics[width=0.5\columnwidth, height=6cm]{2D-mass_fitting.pdf}
+%  \includegraphics[width=6.cm, height=6cm]{errors_120.pdf}
+%  \includegraphics[width=10.cm, height=6cm]{evolution_3d_120.pdf}
+\includegraphics[width=0.5\columnwidth, height=6cm]{2D-distribution_50.pdf}
+\includegraphics[width=0.5\columnwidth, height=6cm]{2D-mass_fitting_50.pdf}
+\caption{Top row: Experiments at $120$Km/h and $d_{\rm f} = 2.2$.
+Top left: Reconstructed (solid line) and experimental (crosses) number distributions  at the inlet,
+  experimental (circles) and numerical (dashed line) number distributions  at the outlet.
+Top right: Mass distribution derived from the fitted number distributions at the inlet (dashed line)
+and  at the outlet (crosses),  simulations at the outlet (continuous line).
+Bottom row:  Experiments at $50$Km/h and $\df=2$.}
+\label{mass-size fig}
+\end{figure}
+
+\clearpage
+\begin{figure}
+\includegraphics[width=0.5\columnwidth, height=6cm]{errors_mass.pdf}
+\includegraphics[width=0.5\columnwidth, height=6cm]{errors_number.pdf}
+\caption{Left: Square root of the sum of the mass distribution residuals (arbitrary units) 
+as a function of the fractal dimension.
+Right: Square root of the sum of the number distribution residuals.}\label{mass-size residuals}
+\end{figure}
+
+
+\clearpage
+\begin{figure}
+\includegraphics[width=0.5\columnwidth, height=6cm]{Total_particle_number_vs_time_comparison_monodisperse_approx.pdf}
+\includegraphics[width=0.5\columnwidth, height=6cm]{Total_particle_number_vs_time_comparison_monodisperse_approx_50.pdf}
+\caption{Left: calculated $\tau$-evolution of the total aggregate concentration, numerical simulations (circles), 
+ the analytical expression Eq.~(\ref{explanation tau agglomeration}) (crosses), experiments at $120$Km/h.
+ The arrow indicates the  total particle concentration corresponding to a shorter transfer line
+ (i.e.  a shorter residence time), namely a $5$m-long line.
+Right: Experiments at $50$Km/h.}
+\label{comparison total number}
+\end{figure}
+
+
+\clearpage
+\begin{figure}
+\includegraphics[width=0.5\columnwidth, height=6cm]{2D-distribution_short_anaconda_50.pdf}
+\includegraphics[width=0.5\columnwidth, height=6cm]{2D-distribution_short_anaconda.pdf}
+\caption{Left: Reconstructed inlet number distribution (solid) and calculated outlet number
+distributions for a transfer line length of $5$m (crosses) and $9$m (dashed), vehicle speed $50$Km/h.
+Right: Vehicle speed $120$Km/h.}
+\label{comparison anaconda length}
+\end{figure}
+
+\clearpage
+\begin{figure}
+\includegraphics[width=0.5\columnwidth, height=6cm]{mu_geo.pdf}
+\includegraphics[width=0.5\columnwidth, height=6cm]{sig_geo.pdf}
+\caption{Left: Evolution of the geometric mean of the number distribution
+ as a function of the residence time and indication 
+of the outlet value for a $5$m-long transfer line for car speed $120$ (circles) and $50$Km/h (crosses).. 
+Right: The evolution of the geometric standard deviation is plotted.}
+\label{comparison mu and sigma}
+\end{figure}
+
+\clearpage
+\begin{figure}
+\includegraphics[width=0.5\columnwidth, height=6cm]{df_fitting_120.pdf}
+\includegraphics[width=0.5\columnwidth, height=6cm]{df_fitting_50.pdf}
+\caption{Determination of the artificial residence time $\tau_{\rm res}$ minimizing the least-square difference between the numerical and fitted mass distributions at the outlet for a set of choices of the fractal dimension.
+Left: Experiments at $120$Km/h. Right: Experiments at $50$Km/h.}
+\label{df_fitting}
+\end{figure}
+
+
+
+
+% \begin{figure}
+% \includegraphics[width=8cm, height=6cm]{2D-distribution.pdf}
+% \caption{Evolution of the number-size distribution with chosen fractal dimension $2.25$ for car speed $120$Km/h. }\label{2D-120km}
+% \end{figure}
+
+
+
 \end{document}
-