latexText page

Extinction coefficients
from aerosol measurements

Christoph Gnendigera, Thorsten Schultzeb,
Kristian Börgera,c, Alexander Belta, Lukas Arnolda,c,111e-mail: l.arnold@fz-
juelich.de<EMAIL_ADDRESS>
aInstitute for Advanced Simulation (IAS),
Forschungszentrum Jülich,
D-52428 Jülich, Germany
bChair of Communication Systems (NTS),
University of Duisburg‐Essen,
D-47057 Duisburg, Germany
cChair of Computational Civil Engineering (CCE),
University of Wuppertal,
D-42119 Wuppertal, Germany

###### Abstract

In this contribution, we develop a model based on classical electrodynamics
that describes light extinction in the presence of arbitrary aerosols. We do
this by combining aerosol and light-intensity measurements performed with the
well-proven measuring systems ELPI+ and MIREX, respectively. The developed
model is particularly simple and depends on only a few input parameters,
namely on densities and refractive indices of the constituting aerosol
particles. As proof of principle, the model is in first applications used to
determine extinction coefficients as well as mass-specific extinction for an
infrared light source with a peak wave length of
$\lambda\\!=\\!\qty{0.88}{\text{\textmu}}$. In doing so, detailed studies
concentrate on two aerosols exemplary for characteristic values of the input
parameters: a non-absorbing paraffin aerosol in a bench-scale setup and soot
from a flaming $n$-heptane fire in a room-scale setup (test fire TF5 according
to standard EN54). As main results, we find numerical values for mass-specific
extinction that are first of all different in the two considered cases.
Moreover, obtained results differ in part more than a factor of three from
literature values typically used in practical applications. Based on the
developed model, we explicitly address and assess underlying reasons for the
deviations found. Finally, we propose a simple way how future light-extinction
studies can be performed comparatively easily by means of the ELPI+-system or
measuring devices that work in a similar way.

###### Contents

  1. 1 Introduction
  2. 2 Measuring and modelling light interaction with an aerosol
    1. 2.1 Aerosol measurements with ELPI+
    2. 2.2 Transition measurements with MIREX
    3. 2.3 Light extinction in classical electrodynamics
  3. 3 Application of the model
    1. 3.1 Extinction coefficient $\alpha_{\rm V}$
    2. 3.2 Mass-specific extinction coefficient $\alpha_{\rm M}$
    3. 3.3 Light extinction from ELPI+-measurements
  4. 4 Conclusions and outlook
  5. A Appendix

##  1 Introduction

Visibility in case of fire is an essential component in the performance-based
safety design of buildings and therefore an indispensable aspect in the field
of civil safety applications and research. Accordingly, a multitude of
experiments investigating the human capacity to identify safety-relevant
objects in the presence of soot has been conducted in the past, the first
experiments dating back to the early 70s of the last century [1, 2, 3, 4, 5].
While the application field of the gained knowledge was initially limited to
manual engineering procedures, it is nowadays also frequently applied in the
field of computational fluid dynamics (CFD), for instance in the widely used
Fire Dynamics Simulator FDS [6]. A reliable and comprehensive description of
visibility in the presence of an aerosol has therefore become of central
importance for the safety assessment of buildings and other places where
people are present.

The key quantity to describe visibility in the context of environmental
conditions is the extinction coefficient which quantifies properties of an
aerosol to attenuate parts of a light beam from its original path.
Conceptually, light extinction can be measured in two different ways: In a
first approach, the focus is put on the light beam and the total amount of
light-intensity reduction compared to the case without aerosol is measured.
Corresponding experiments are often called ‘direct’ visibility measurements
and are usually evaluated using the Beer–Lambert–Bouguer law [7]. In contrast,
in a second category of experiments, the focus is put on the aerosol instead
and, for instance, quantities like number and mass densities, sizes, or even
fractal properties of the constituting aerosol particles are measured. Using
models that typically rely on Rayleigh or Mie computations and extensions
thereof, visibility reduction is then deduced from the measured particle
properties. Since in the latter approach visibility conditions are derived
only in a downstream step and in the framework of the applied model, these
measurements are sometimes denoted as ‘indirect’ ones.

Historically, there has been considerable restraint and criticism regarding
the ’indirect’ approach. Objects of concern typically comprise the alleged
presence of large experimental uncertainties [8, 9] or the used models for
light interaction of the aerosol particles [8, 10, 9]. Accordingly, many
statements can be found in the literature claiming that ‘direct’ visibility
measurements are the only reliable, or at least the preferable way to
determine optical properties of aerosols. An interesting quantity in this
respect is the mass-specific extinction coefficient. Since it combines, by
definition, light extinction with the overall mass density of the present
aerosol, mass-specific extinction can not be obtained through ‘direct’
visibility measurements alone but has always to be combined with other
experimental results like gravimetric ones. In the literature, the existence
of a (universal) mass-specific extinction coefficient has so far been
investigated empirically, based on the hypothesis that post-flame smoke
produced by overventilated fires behaves almost identically and is more or
less independent of the burned fuel (see for instance the review in [11] and
references therein). Although functional dependences on different influencing
factors like the wave length of the incident light [11] or the ventilation
state of the fire [9] are qualitatively known, these aspects are often
neglected for the sake of simplicity and, accordingly, the same constant value
for mass-specific extinction is used in practical applications. FDS, for
instance, uses by default the constant value proposed in [12] which itself is
the review result of various small-scale experiments that have been conducted
under well-controlled conditions. Given this fact, it is of major importance
to know if universal mass-specific extinction does exist at all and, if so, to
know its actual value in real-scale applications. Only then, it is possible to
draw reliable conclusions that neither overestimate visibility conditions
(which would be detrimental for safety aspects) nor underestimate them (which
could result in unnecessary high expenses for safety measures).

In the present contribution, we therefore put the determination of (mass-
specific) extinction on a firm basis by combining ‘direct’ and ‘indirect’
visibility measurements. More precisely, we combine aerosol measurements
performed with the cascade impactor ELPI+ [13] and light-intensity reduction
measurements performed with the extinction measuring equipment MIREX [14]. The
goal of the comparison is to develop a practical model for the interaction of
a light beam with aerosol particles that

  * •

allows to reliably predict the total amount of light-intensity reduction
in the presence of an arbitrary aerosol,

  * •

is at the same time compatible with known value ranges
of the entering model parameters,

  * •

has expected predictive uncertainties
comparable with ‘direct’ visibility measurements, and

  * •

is preferably as simple as possible.

In order to achieve this, the structure of the paper is as follows: In Secs.
2.1 and 2.2, we summarize key aspects of ELPI+- and MIREX-measurements in
general and identify possible output of the two systems allowing to describe
light extinction in a combined framework. Building on these findings, in Sec.
2.3 we develop a first-principle model that is based on classical
electrodynamics and that fulfils the above requirements. In doing so, we
explicitly assess all relevant assumptions made in the formulation of the
model and discuss possible modifications that may (or may not) improve the
predictive power in practical applications. In Sec. 3, we then apply the model
and consider two different aerosols, namely paraffin and soot from flaming
$n$-heptane fires which are exemplary for characteristic differences of the
input variables. For the two cases, we determine value combinations of the
model parameters which allow to bring ELPI+- and MIREX-measurements into the
best possible agreement. This is done by explicitly computing extinction
coefficients in Sec. 3.1 and mass-specific extinction in Sec. 3.2. Before
concluding, in Sec. 3.3 we finally propose a simple methodology based on the
developed model that allows to comparatively easily determine light extinction
in future applications of the ELPI+-device.

##  2 Measuring and modelling light interaction with an aerosol

###  2.1 Aerosol measurements with ELPI+

The ELPI+-device is a cascade impactor which counts and classifies aerosol
particles according to their aerodynamic and gravimetric properties [13]. Its
working principle is as follows: In a first step, a sample of the aerosol
under investigation is aspirated through a nozzle with an opening size of
$\qty{10}{\text{\textmu}}$ and the contained aerosol particles are charged in
a corona charger. For each electrically charged particle, the total amount of
collected charge depends on the surface of the particle and therefore
indirectly on its size and shape. After charging, the aerosol sample is formed
to a jet that successively passes $14$ different impactor plates. Before
passing each of the plates, the particles have to make a sharp turn, which for
larger particles is not possible due to larger inertia. Accordingly, these
particles are collected in the respective impactor plate and the arising
electrical current $I(t)$ is measured. At the same time, smaller particles
remain in the jet flow and move to the next impactor stage. Since the opening
size of the device is decreasing from stage to stage, the speed of the jet
flow is successively increasing, allowing also smaller particles to be
collected. In this way, particles are collected and classified from larger to
smaller size. The opening size of the lowest impactor stage is
$\qty{0.006}{\text{\textmu}}$.

As an output of an individual ELPI+-measurement one obtains in total $14$
time-dependent electrical currents $I_{i}(t)$ with a time resolution of one
second. Using conversion factors, measured currents can be converted into
physical quantities like number densities $\rho_{\text{N},i}$ and mass
densities $\rho_{\text{M},i}$ of the collected aerosol part [13]:

$\displaystyle\rho_{\text{N},i}(t)$ $\displaystyle=\frac{N_{i}(t)}{V_{\rm
u}}\,=X_{N_{i}}\cdot
I_{i}(t)\,,\phantom{\rho_{\text{N},i}(t)\cdot\rho_{\text{part},i}\cdot}\qquad
i\in\\{1,\dots,14\\}\,,$ (1a) $\displaystyle\rho_{\text{M},i}(t)$
$\displaystyle=\frac{M_{i}(t)}{V_{\rm
u}}=\rho_{\text{N},i}(t)\cdot\left(\pi\frac{d_{\text{S},i}^{3}}{6}\right)\cdot\rho_{\text{part},i}\,,\qquad
i\in\\{1,\dots,14\\}\,.$ (1b)

By convention, the ordering of the channel labels is such that $i\\!=\\!1$
denotes the ELPI+-channel collecting the smallest particles, whereas
$i\\!=\\!14$ corresponds to the channel collecting the largest particles. In
Eqs. (1), $N_{i}$ and $M_{i}$ denote absolute particle numbers and particle
masses per unit volume $V_{\rm u}\\!=\\!\qty{1}{\cubed}$ of the measured
aerosol and the conversion factors $X_{N_{i}}$ consider charging and
collection inefficiencies of the ELPI+-device. The latter are calibrated in
[15] and provided in Appendix A1 for the sake of completeness. Explicit
definitions of $d_{\text{S},i}$ and $\rho_{\text{part},i}$ are given in the
next paragraph.

A crucial point in the evaluation of ELPI+-data is that each of the $14$
measured currents $I_{i}$ (and therefore each of the derived densities) can be
assigned a dedicated particle size which is directly related to aerodynamic
and gravimetric properties of the collected particles. For that, each
(irregularly shaped) aerosol particle is internally converted into a virtual
equivalent particle which is spherical and which has the same settling
velocity $v_{s}$ as the actual aerosol particle. While the settling velocity
of the aerosol particle is indirectly measured by the ELPI+-device, it is for
the virtual equivalent particle given by Stoke’s law, i. e.

$\displaystyle v_{s}=\frac{g}{18\,\mu_{\rm air}}(\rho-\rho_{\rm air})\cdot
d^{2}\,.$ (2)

Here, $g\\!\approx\\!\qty{9.81}{\per\squared}$ is the standard acceleration
due to gravity, $\mu_{\rm
air}\\!\approx\\!1.8\\!\cdot\\!10^{-5}\,\unit{\per\per}$ and $\rho_{\rm
air}\\!\approx\\!1.2\,\unit{\per\cubed}$ are the dynamic viscosity and mass
density of air at a given temperature, respectively. The three quantities are
related to measuring conditions of the experiments and are approximately the
same for each individual ELPI+-measurement.222Of course, $g$, $\mu_{\rm air}$,
and $\rho_{\rm air}$ are affected by various boundary conditions of the
experiments like ambient temperature or height above sea level. Compared to
other sources of uncertainty, however, these factors can be regarded
negligible, as will be shown below.  In contrast, $\rho$ and $d$ denote the
mass density and the diameter of the equivalent particle, respectively. The
two quantities therefore strongly depend on the measured aerosol. For a given
settling velocity, there are two common ways to define the diameter of the
equivalent sphere, see also Fig. 1:

Figure 1:  Possible definitions of the equivalent diameter. The figure is
adopted from the ELPI+-manual [13].

  * (1)

Stokes equivalent diameter $d_{\rm S}$
In this case, the diameter of the equivalent particle is defined such that the
latter has the same settling velocity and the same particle density $\rho_{\rm
part}$ as the real aerosol particle:

$\displaystyle v_{s}=\vcentcolon\frac{g}{18\,\mu_{\rm air}}(\rho_{\rm
part}-\rho_{\rm air})\cdot d_{\rm S}^{2}\,.$ (3a)

  * (2)

Aerodynamic equivalent diameter $d_{\rm aero}$
In this case, the diameter of the equivalent particle is defined such that the
latter has the same settling velocity as the real aerosol particle. However,
the equivalent particle is assumed to have unit density, i. e.
$\rho\\!=\\!\qty{1}{\per\cubed}$:

$\displaystyle v_{s}=\vcentcolon\frac{g}{18\,\mu_{\rm
air}}(\qty{1}{\per\cubed}-\rho_{\rm air})\cdot d_{\rm aero}^{2}\,.$ (3b)

It should be noted that both definitions introduce auxiliary objects that may
differ from the actual size of the initial aerosol particle. For instance,
setting equal both definitions in Eqs. (3) and neglecting the density of air,
it is straightforward to derive the following relation:333 Actually, the
functional dependence on the value of $\rho_{\rm part}$ is slightly more
complicated and even depends on the respective ELPI+-channel, see in
particular the ELPI+manual [13]. However, Eq. (4) serves as a useful
approximation for the present considerations.

$\displaystyle\frac{d_{\rm S}}{d_{\rm
aero}}\approx\sqrt{\frac{\qty{1}{\per\cubed}}{\rho_{\rm part}}}\,.$ (4)

Accordingly, in case the actual particle density $\rho_{\rm part}$ is larger
(smaller) than $\rho\\!=\\!\qty{1}{\per\cubed}$, aerodynamic diameters are
larger (smaller) than corresponding Stokes diameters. Since an exemplary
particle density of $\rho_{\rm part}\\!=\\!\qty{4}{\per\cubed}$ is assumed in
Fig. 1, for instance, the diameter of the aerodynamic equivalent sphere is
twice as large as the diameter of the corresponding Stokes sphere and
therefore much larger than the size of the real aerosol particle. In contrast,
the given Stokes equivalent diameter is compatible with the geometrical size
of the irregularly shaped aerosol particle since it considers one additional
physical property, namely the density $\rho_{\rm part}$ of the aerosol
particle. In order to get equivalent diameters that are close to the
geometries of the aerosol particles, therefore Definition (3a) should be used
instead of (3b). In Appendix A3 we further investigate differences between
both approaches and explicitly quantify the impact of the choice on obtained
numerical results. At this point it remains to be mentioned that, since
$\rho_{\rm part}$ is not measured by the ELPI+-device, its value has either to
be measured separately or to be taken from the literature.

Introducing Stokes equivalent spheres via Eq. (3a) allows to assign dedicated
particle diameters to each of the $14$ measured currents $I_{i}(t)$ of an
ELPI+-experiment, as mentioned before:

$\displaystyle I_{i}(t)\ \leftrightarrow\
d_{\text{S},i}(\rho_{\text{part},i})\,,\qquad i\in\\{1,\dots,14\\}\,.$ (5)

The quantities $d_{\text{S},i}$ and $\rho_{\text{part},i}$ denote the
geometric mean of the Stokes equivalent diameters and the particle density
associated with ELPI+-stage $i$, respectively. In Sec. 3, these combinations
of measured electrical currents and deduced diameters are used to compute
extinction coefficients of different aerosols.

#### Aerosol density $\rho_{\rm aerosol}$

An important quantity the total amount of light extinction depends on is the
overall mass density $\rho_{\rm aerosol}$ of the aerosol. One advantage of
ELPI+-measurements in general is that the quantity can be deduced from
obtained experimental data. This is done by summing up particle masses
measured by the different ELPI+-stages and normalizing to unit volume, as done
e. g. in Eq. (1b). Considering the first $n$ ELPI+-stages, for example, the
density of the corresponding aerosol part is given by

$\displaystyle\rho_{\rm
aerosol}^{(n)}(t)\vcentcolon=\sum_{i=1}^{n}\rho_{\text{M},i}(t)\,.\qquad
n\in\\{1,\dots,14\\}\,,$ (6)

In the considered setup, each $\rho_{\rm aerosol}^{(n)}$ is mainly governed by
the particle densities $\rho_{\text{part},i}$. This dependence, however, is
only weak. The reason is that individual mass concentrations
$\rho_{\text{M},i}$ are determined indirectly from volumes and mass densities
of the Stokes equivalent particles, see Eq. (1b). Since an increase of the
particle density at the same time corresponds to smaller Stokes equivalent
diameters and therefore to smaller volumes, both effects compensate each other
and result in a weak dependence on the actual value of $\rho_{\text{part},i}$.
In other words, obtained values for $\rho_{\rm aerosol}^{(n)}$ are
approximately a direct output of an ELPI+-measurement as a consequence of
internal conversion processes.444 Overall, ELPI+conversion factors for mass
density are proportional to
$\rho_{M}\\!\propto\\!\rho_{\text{part},i}\\!\cdot\\!(d_{\rm S})^{3-a}$ with
$a\in\\{1.225,1.515,1.085\\}$, depending on the respective Stokes equivalent
diameter, see Eq. (1b) and Eqs. (24) in Appendix A1. Stokes equivalent
diameters in turn are proportional to $d_{\rm
S}\\!\propto\\!(\rho_{\text{part},i})^{-\frac{1}{2}}$ which can be seen in Eq.
(4). In total, conversion factors are therefore proportional to
$\rho_{M}\\!\propto\\!(\rho_{\text{part},i})^{\frac{a-1}{2}}$ with a maximum
dependence given by
$\rho_{M}\\!\propto\\!(\rho_{\text{part},i})^{\frac{1.515-1}{2}}\\!=\\!(\rho_{\text{part},i})^{0.2575}$.
Doubling $\rho_{\text{part},i}$, for example, then only leads to a mass
concentration increase of at most $20\,\%$. In practice, the increase is
typically much smaller in the order of a few percent, which is shown in Sec.
3.

#### Average distance between aerosol particles

A quantity closely related to the aerosol density is the average distance
$\overline{l}_{d}$ between the aerosol particles. It can be obtained via
dividing the unit volume $V_{\rm u}$ by the absolute particle number $N_{\rm
u}$ present in that volume,

$\displaystyle\overline{l}_{d}^{3}=\frac{V_{\rm u}}{N_{\rm
u}}=\frac{\qty{1}{\cubed}}{N_{\rm u}}\,.$ (7)

As it turns out, in the experiments conducted in this contribution, the
maximum particle number within the steady states is $N_{\rm u}\approx
5\\!\cdot\\!10^{6}$, resulting in a minimal distance of
$\overline{l}_{d}\\!\approx\\!\qty{60}{\text{\textmu}}$. Accordingly, the
average distance between the aerosol particles is much larger than the size of
the equivalent spheres which is (much) smaller than $d_{\rm
S}\\!=\\!\qty{10}{\text{\textmu}}$. Aerosol particles can therefore be assumed
to interact independently of each other in terms of single-particle
interactions [7].

###  2.2 Transition measurements with MIREX

As a second, independent measuring system for light interaction of aerosol
particles we use the well-established extinction measuring equipment MIREX
[14]. The device analyses smoke densities based on optical measurements in the
infrared domain of the spectrum. More precisely, the system uses pulsed light
from a diode as light source and sends it to a reflecting mirror in
approximately one meter distance from the emitter. The reflected light is then
measured by a receiver, resulting in an overall length of the optical path of
$l\\!=\\!\qty{2}{}$. Accordingly, the MIREX constantly measures light-
intensity reduction in an area of one meter extension. The specification of
the light spectrum is such that the main part of the power is emitted in the
wave length range between $\qty{0.8}{\text{\textmu}m}$ and
$\qty{0.95}{\text{\textmu}m}$. The peak wave length of the distribution is
$\lambda_{\rm in}\\!\approx\\!\qty{.88}{\text{\textmu}m}$.

As an output of an individual MIREX-measurement, one obtains time-dependent
values for the (volume) extinction coefficient $\alpha_{\rm V}$ that enters
the Beer–Lambert–Bouguer law [7]

$\displaystyle\frac{I(t)}{I_{0}}=e^{-\alpha_{\rm V}(t)\cdot l}\,.$ (8)

Here, $I(t)$ is the measured time-dependent light intensity and $I_{0}$ is the
intensity of the light source without aerosol. Sample output of the conducted
experiments is shown in Fig. 2.

Figure 2:  Time-dependent light extinction measured by the MIREX-system. Shown
are results for soot from flaming $n$-heptane fires obtained in four
independent experiments. Details of the experimental setup are provided in
Sec. 3.

###  2.3 Light extinction in classical electrodynamics

(a) $m\\!=\\!1.5\\!+\\!0.0\,i$

(b) $m\\!=\\!1.5\\!+\\!0.5\,i$

Figure 3:  Extinction efficiency depending on the equivalent diameter and
refractive index of the interacting particle. Shown are results for a non-
absorbing aerosol (left) and an absorbing one (right). The wave length of the
incident light is $\lambda_{\rm in}\\!=\\!\qty{0.88}{\text{\textmu}}$ in both
cases. The shown diameter range corresponds to the one covered by the
ELPI+-device.

One main goal of the present contribution is to describe light extinction in
the presence of different aerosols by using ELPI+-measurements (discussed in
Sec. 2.1) and MIREX measurements (discussed in Sec. 2.2). In the following, we
provide theoretical background that is necessary to evaluate ELPI+-data
accordingly. Our approach is the following: In a first step, we start with a
basic theory that is as simple as possible but covers, nevertheless, all
relevant aspects of the underlying physical processes. Then, we discuss
possible modifications of the theory that may (or may not) improve the
predictive power of the initial model.

In a first-order approach, the interaction of a light beam with a spatially
extended aerosol is an electrodynamic process that can be adequately described
by classical electrodynamics where relativistic and quantum effects are
neglected. Moreover, to obtain meaningful results in the context of the
present work, we make the following additional assumptions:

  * •

In Sec. 2.1 it is shown that the ELPI+-device assigns dedicated Stokes
equivalent diameters $d_{\rm S}$ to each of the $14$ measuring channels. In
the following, we therefore assume interacting particles to be homogeneous
spheres, each with an individual geometrical diameter $d_{\rm
geom}\\!\equiv\\!d_{\rm S}$ and refractive index $m$. All particles are
monodisperse and interact independently of each other in terms of single-
particle interactions.

  * •

In Sec. 2.2 it is mentioned that the intensity distribution of the MIREX light
source is comparatively narrow. In the following, we therefore assume the
incident light beam to consist of plane, unpolarized, and monochromatic
electromagnetic waves. As wave length $\lambda_{\rm in}$ we take the peak wave
length of the intensity distribution which is in the infrared and given by
$\lambda_{\rm in}\\!=\\!\qty{0.88}{\text{\textmu}}$.

Of course, these assumptions only represent a first-order description of the
optical effects actually taking place. We show, however, that even with these
strong restrictions it is possible to obtain reliable numerical results under
various different conditions. In order to show this, we explicitly determine
the following quantities:

#### Extinction efficiency $q_{\rm ext}$

This parameter quantifies the ability of a single particle to attenuate parts
of a light beam from its original path. Starting from Maxwell’s equations, it
can be computed from basic principles and contains the two major components
scattering and absorption [7]:

$\displaystyle q_{\rm ext}=q_{\rm sca}+q_{\rm abs}=q_{\rm ext}(\lambda_{\rm
in},d_{\rm S},m)\,.$ (9)

Absorption is not present for every aerosol and governed by the imaginary part
of the refractive index $m$. Moreover, extinction efficiency depends on the
wave length $\lambda_{\rm in}$ of the incident light as well as on the
equivalent diameter $d_{\rm S}$ of the interacting particle. It can be
computed e. g. by using the freely available software MiePlot [16] which
itself is based on the code published in the appendix of [7]. Sample output of
the software for two different refractive indices is shown in Fig. 3. As can
be seen, the measuring range of the ELPI+-device, which ranges from
$\qty{0.006}{\text{\textmu}}$ to $\qty{10}{\text{\textmu}}$ [13], covers the
global maximum of the efficiency distributions, respectively.

#### Extinction cross section $\sigma_{\rm ext}$

Figure 4:  Extinction cross section of one interacting particle depending on
the equivalent diameter of the particle. Shown are results obtained from Eq.
(10) for different refractive indices. Both diagrams have in fact the same
content. In the right one, however, the $y$-axis has a logarithmic scale. The
wave length of the incident light is again $\lambda_{\rm
in}\\!=\\!\qty{0.88}{\text{\textmu}}$.

This quantity is directly related to the extinction efficiency. In contrast to
the latter, however, it is a physical observable that can actually be measured
experimentally. Cross section and efficiency are related through the visible
particle area in the direction of the light beam which is a circle in case
spherical objects are considered. The extinction cross section and its
constituting parts are for an equivalent particle therefore given by

$\displaystyle\sigma_{j}(\lambda_{\rm in},d_{\rm
S},m)\vcentcolon=q_{j}(\lambda_{\rm in},d_{\rm
S},m)\cdot\pi\cdot\Big{(}\frac{d_{\rm S}}{2}\Big{)}^{2},\qquad j\in\\{\rm
ext,\,sca,\,abs\\}\,.$ (10)

Examples of extinction cross sections for different refractive indices are
shown in Fig. 4. In first approximation, the distributions are monotonously
increasing. Larger particles therefore have a larger extinction cross section
than smaller ones. Moreover, relevant dependence on the refractive index is
only present for equivalent diameters smaller than the wave length of the
incident light. Particles larger than that almost behave identically which
corresponds to the principles of geometrical optics [7].

#### Extinction coefficient555In some references, this quantity is also
denoted as ‘attenuation’ coefficient, see e. g. [7]. $\alpha_{\rm V}$

This quantity is the cumulative sum of all particle extinction cross sections
of an aerosol. Several different definitions can be found in the literature.
In the present contribution, we use the so-called volume extinction
coefficient which is normalized to unit volume $V_{\rm
u}\\!=\\!\qty{1}{\cubed}$. In this form, it is exactly the quantity that
enters the Beer–Lambert–Bouguer law in Eq. (8) [7]. Using particle number
densities of the $14$ ELPI+-channels defined in Eq. (1a) as well as extinction
cross sections given in Eq. (10), $\alpha_{\rm V}$ can be written as

$\displaystyle\alpha_{\rm V}(t)$
$\displaystyle=\sum_{i=1}^{14}\alpha_{\text{V},i}(t)=\sum_{i=1}^{14}\frac{N_{i}(t)}{V_{\rm
u}}\cdot\sigma_{\rm
ext}(d_{\text{S},i},m)=\sum_{i=1}^{14}\rho_{\text{N},i}(t)\cdot\sigma_{\rm
ext}(d_{\text{S},i},m)$ (11a)
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:elpi_densities_a},\eqref{eq:sigma}}}{{=}}\pi\sum_{i=1}^{14}q_{\rm
ext}(d_{\text{S},i},m)\cdot\left(\frac{d_{\text{S},i}}{2}\right)^{2}\cdot
X_{N,i}(d_{\text{S},i})\cdot I_{i}(t)\,.$ (11b)

Eq. (11b) is one of the most important formulas in the present contribution.
In this formula, extinction coefficients are expressed in terms of extinction
efficiencies $q_{\rm ext}$ (and therefore depend on the refractive index $m$),
equivalent diameters $d_{\text{S},i}$ (and therefore depend indirectly on
corresponding particle densities $\rho_{\text{part},i}$, see Eq. (4)),
conversion factors $X_{N,i}$, and electrical currents $I_{i}(t)$ measured by
the ELPI+-device.

In the following, we assume the same effective particle density $\rho_{\rm
part}$ for each ELPI+-channel,

$\displaystyle\rho_{\text{part},i}\equiv\rho_{\rm part}\,,\qquad
i\in\\{1,\dots,14\\}\,,$ (12)

which turns out to be sufficient to consistently combine model predictions and
experimental results from MIREX- and ELPI+-measurements.666 The authors of the
present contribution are aware of the fact that individual particle densities
$\rho_{\text{part},i}$ can actually be assumed to be different for individual
ELPI+-channels, see e. g. [17]. We further comment on limitations of
assumption (12) in the course of the contribution, in particular in Sec. 3.
Using Eq. (12), we therefore only have three independent aerosol properties in
Eq. (11b) that can be used to adjust the model, namely

  * •

the real part $\mathfrak{Re}(m)$ of the refractive index
which describes scattering properties of the aerosol particles,

  * •

the imaginary part $\mathfrak{Im}(m)$ of the refractive index
which describes absorbing properties of the aerosol particles, and

  * •

the effective particle density $\rho_{\rm part}$.

In Sec. 3, we consider different aerosols and show that, using known
literature values of $m$ and $\rho_{\rm part}$, it is possible to consistently
describe all obtained experimental results within a unified framework. For
some applications it turns out to be useful to also define the cumulative
extinction coefficient obtained from the first $n$ ELPI+-channels only,
similar to Eq. (6):

$\displaystyle\alpha_{\rm
V}^{(n)}(t)\vcentcolon=\sum_{i=1}^{n}\alpha_{\text{V},i}(t)\,,\qquad
n\in\\{1,\dots,14\\}\,.$ (13)

#### Mass-specific extinction coefficient $\alpha_{\rm M}$

This quantity is closely related to the extinction coefficient $\alpha_{\rm
V}$ and describes light extinction per mass density unit of the aerosol.
According to [7], it is defined as

$\displaystyle\alpha_{\rm M}$ $\displaystyle\vcentcolon=\frac{\alpha_{\rm
V}(t)}{{\rho}_{\rm aerosol}(t)}\,.$ (14)

To be able to estimate the size of $\alpha_{\rm M}$, we first consider a
single equivalent particle and determine its contribution to the total value
of $\alpha_{\rm M}$. Using Eqs. (11b), (6), and (1), mass-specific extinction
of a single particle associated with ELPI+-channel $i$ can be written as

$\displaystyle\alpha_{\text{M},i}\vcentcolon=\frac{\alpha_{\text{V},i}}{\rho_{\text{aerosol},i}}=\frac{3}{2}\frac{q_{\rm
ext}(d_{\text{S},i},m)}{d_{\text{S},i}\cdot\rho_{\rm part}}\,.$ (15)

Accordingly, $\alpha_{\text{M},i}$ is a constant in time that depends on the
extinction efficiency, on the corresponding Stokes equivalent diameter
(directly and indirectly), as well as on the particle density. Exemplary
results for two different refractive indices are shown in Fig. 5. Since
$q_{\rm ext}\\!\to\\!2$ in the limit of large (equivalent) particles, see Fig.
(3), mass-specific extinction vanishes for large $d_{\rm S}$:

$\displaystyle\lim_{d_{\text{S},i}\,\to\,\infty}\alpha_{\text{M},i}=0\,.$ (16)

The reason for this effect is that $\alpha_{\rm V}$ is related to the
circular-shaped surface of the equivalent particle in the direction of the
light beam (and therefore $\propto\\!d_{\rm S}^{2}$, see Eq. (10)), whereas
$\rho_{\text{aerosol}}$ is related to the particle volume (and therefore
$\propto\\!d_{\rm S}^{3}$, see Eq. (1b)). For increasing values of $d_{\rm
S}$, the overall dependence is therefore $\propto\\!d_{\rm S}^{-1}$. In other
words, contributions to mass-specific extinction from the largest particles
measured by the ELPI+-device are smaller than the ones from intermediate-size
particles since larger particles proportionally contribute more to the aerosol
density than to light extinction. Accordingly, $\alpha_{\rm M}$ in Eq. (14)
describing the whole aerosol does not depend on numerical values of
$\alpha_{\rm V}$ and $\rho_{\rm aerosol}$ alone, but also on the size-
distribution of the individual equivalent diameters. This fact is a general
prediction of classical electrodynamics and already evident within the
measuring range of the ELPI+-device, as can be seen in Fig. 5.

(a) $m\\!=\\!1.5+\\!0.0\,i$

(b) $m\\!=\\!1.5\\!+\\!0.5\,i$

Figure 5:  Mass-specific extinction of a single aerosol particle depending on
the equivalent diameter of the particle. The curves are obtained from Eq. (15)
using $m\\!=\\!1.5\\!+\\!0.0\,i$ (left) and $m\\!=\\!1.5\\!+\\!0.5\,i$
(right). The particle density and the wave length of the incident light are in
both cases fixed as $\rho_{\rm part}\\!=\\!\qty{1}{\per\cubed}$ and
$\lambda_{\rm in}\\!=\\!\qty{0.88}{\text{\textmu}}$, respectively.

##  3 Application of the model

In the previous section, we deduce Eq. (11b) in which light-intensity
reduction is expressed in terms of the parameters $m$ and $\rho_{\rm part}$.
For a given aerosol, it is in principle possible to use known literature
values of the input parameters to obtain corresponding results for extinction
coefficients. This, however, is complicated by the fact that both quantities
are comparatively difficult to determine experimentally and, more importantly,
through distinct independent measurements. Typically, the literature does
therefore not provide combinations of $m$ and $\rho_{\rm part}$ but only
values or value ranges for one parameter at a time. Then, it is often not
clear if considered measurements have actually been conducted under overall
comparable experimental conditions.

In this contribution, we therefore follow a different approach. Since light
extinction is determined independently through MIREX-measurements in each of
the conducted experiments, we consider $\alpha_{\rm V}(t)$ and its time
average $\overline{\alpha}_{\rm V}$ as known and try to reproduce obtained
MIREX-results by finding appropriate combinations of $m$ and $\rho_{\rm part}$
in Eq. (11b). We do this by scanning the refractive index over the known range
of literature values. For each fixed $m$, we then find the unique numerical
value of $\rho_{\rm part}$ for which the time average of $\alpha_{\rm V}(t)$
is the same for both measurements, i. e.
$\overline{\alpha}_{V,\text{ELPI+}}\\!\equiv\\!\overline{\alpha}_{V,\text{MIREX}}$.
Although in principle all ELPI+-channels can be used in the evaluation of the
measurement data, in the following we only consider the first $13$ stages.
This is done since contributions of channel $14$ to total light extinction are
only small (less than $\qty{1}{}$ in case of paraffin and less than
$\qty{5}{}$ in case of TF5). In this way, however, the total amount and
therefore the quality of valid data that can be evaluated increases
significantly. In this setup, the measuring range of the ELPI+is approximately
given by $\qty{0.006}{\text{\textmu}}\\!<\\!d_{\rm
S}\\!<\\!\qty{4}{\text{\textmu}}$.

###  3.1 Extinction coefficient $\alpha_{\rm V}$

#### Paraffin aerosol

All paraffin experiments discussed in this contribution have been conducted at
the department of Communication Systems at the University of Duisburg-Essen
using the test duct described in [20]. In each of the experiments, a ‘ramp’
setup is considered in which the aerosol concentration is successively
increased by external supply before the supply is stopped and a steady state
is forming. As value ranges of the model parameters we take into account

$\displaystyle\mathfrak{Re}(m)$
$\displaystyle=(1.3-1.5)\phantom{()\,\unit{\per\cubed}}\text{\cite[cite]{[\@@bibref{}{Cooper:82,Li2015,kontges:2016}{}{}]}\,,}$
(17a) $\displaystyle\mathfrak{Im}(m)$
$\displaystyle=0\phantom{,(-1.5)\quad\unit{\per\cubed}()}\text{\cite[cite]{[\@@bibref{}{Cooper:82,Li2015,kontges:2016}{}{}]}}\,,$
(17b) $\displaystyle\rho_{\rm part}$
$\displaystyle=(0.8-0.9)\,\unit{\per\cubed}\text{\ (at $20\,^{\circ}C$)}\,.$
(17c)

Note, that the imaginary part of the refractive index vanishes. Accordingly,
the aerosol does not absorb light and extinction is determined exclusively by
scattering processes, see also Eq. (9).

Figure 6:  Time-dependent extinction coefficient obtained from a paraffin ramp
experiment. The black curve is obtained from the MIREX-system, the blue curve
from Eq. (11b) using $m\\!=\\!1.42$ and $\rho_{\rm
part}\\!=\\!\qty{0.85}{\per\cubed}$. The white area indicates the time span in
which the aerosol concentration is increased by external supply. In the grey
area, the supply is stopped and the system passes into a steady state.

The comparison of MIREX- and ELPI+-measurements for one exemplary paraffin
ramp experiment is shown in Fig. 6. As can be seen, almost perfect agreement
between the two measuring systems can be obtained using appropriate model
parameters, in this case $m\\!=\\!1.42$ and $\rho_{\rm
part}\\!=\\!\qty{0.85}{\per\cubed}$. It is important to emphasize that these
numerical values are simultaneously compatible with the value ranges in Eqs.
(17) and, therefore, with known results from the literature. This fact serves
as a first indication that the developed model is able to correctly describe
MIREX- and ELPI+-measurements without the need of introducing additional
correction or conversion factors. Among other things, this comprises the
correct description of different experimental phases (‘ramp’ phase, steady
state) as well as subtleties in the time development of $\alpha_{\rm V}$ (in
Fig. 6 e. g. for times $\qty{450}{}\\!<\\!t\\!<\\!\qty{500}{}$).

It is important to notice that agreement between MIREX- and ELPI+-measurements
is not only obtained for one particular combination of $m$ and $\rho_{\rm
part}$ but for different ones. Considering the average of eight independent
paraffin ramp experiments and scanning over the entire value range in Eq.
(17a), we obtain further parameter combinations for which agreement between
MIREX- and ELPI+-measurements is achieved. The results are summarized in Fig.
7.

$\mathfrak{Re}(m)$ |  $\rho_{\rm part}\,[\unit{\per\cubed}]$
---|---
$1.30$ |  $0.56\pm 0.06$
$1.35$ |  $0.68\pm 0.07$
$1.40$ |  $0.77\pm 0.08$
$1.45$ |  $0.88\pm 0.12$
$1.50$ |  $0.99\pm 0.13$
Figure 7:  Values of $\rho_{\rm part}$ for which MIREX-and ELPI+-measurements
of the paraffin aerosol agree best. Shown is the dependence on the value of
$\mathfrak{Re}(m)$ (blue solid line and corresponding values in the table),
including the $3\sigma$-range (blue band and uncertainties in the table). The
grey band indicates the value range in Eq. (17c); the dashed vertical line
refers to the refractive index obtained from an analytical formula provided in
[21].

Since obtained fit values of $\rho_{\rm part}$ differ slightly from one
experiment to another, particle densities are subject to small variations for
which we provide the $3\sigma$-range. In general, larger values of
$\mathfrak{Re}(m)$ correspond to larger particle densities and therefore to
smaller (Stokes) equivalent particles, see for instance Eq. (4). Since smaller
particles correspond in first approximation to less light extinction of the
aerosol (see Fig. 4), overall scattering in the presence of a paraffin aerosol
therefore increases with increasing values of $\mathfrak{Re}(m)$. In this
regard, it should be noted that the blue band in Fig. 7 is simultaneously
compatible with both value ranges in Eqs. (17).

Finally, we use Eq. (13) to evaluate individual contributions of the different
ELPI+-channels in the left diagram of Fig. 8.

(a) paraffin

(b) soot from flaming $n$-heptane fire

Figure 8:  Relative cumulative contribution of the different ELPI+-channels to
extinction. Shown are results obtained from Eq. (13). In the left diagram, the
first $10$ channels combined contribute $\qty{92}{\%}$ to entire light
extinction; the main part ($\qty{91}{\%}$) stems from channels $8$, $9$, and
$10$ only. In the right diagram, the first $9$ channels contribute
$\qty{77}{\%}$ to light extinction; the main part ($\qty{76}{\%}$) stems from
channels $6$, $7$, $8$, and $9$.

As can be seen, the main part of extinction can be assigned to only a small
number of neighboring ELPI+-channels.

#### Soot from flaming $n$-heptane fire

A second application case of the model is a room-scale setup investigating
soot from flaming $n$-heptane fires (test fire TF5 according to standard EN54
[24]). Details of the experimental setup are provided in [25]. In general, the
aerosol is characterized by two major differences compared to paraffin: First,
the particle density $\rho_{\rm part}$ is larger than unit density
$\qty{1}{\per\cubed}$. Stokes equivalent spheres are therefore smaller than
corresponding aerodynamic ones. Second, soot from an $n$-heptane fire absorbs
parts of a light beam. Light extinction is therefore a combination of
scattering and absorption processes and not of scattering alone, see e. g. Eq.
(9). Compared to paraffin, we therefore have three independent variables
instead of two:

$\displaystyle\mathfrak{Re}(m)$
$\displaystyle=(1.5-1.9)\phantom{\,\unit{\per\cubed}}\text{\cite[cite]{[\@@bibref{}{Sorensen_2001,
Smyth_Shaddix_1996, Wu1997, Chang_1990, Dalzell:1969 }{}{}]}}\,,$ (18a)
$\displaystyle\mathfrak{Im}(m)$
$\displaystyle=(0.4-0.7)\phantom{\,\unit{\per\cubed}}\text{\cite[cite]{[\@@bibref{}{Sorensen_2001,
Smyth_Shaddix_1996, Wu1997, Chang_1990, Dalzell:1969 }{}{}]}}\,,$ (18b)
$\displaystyle\rho_{\rm part}$
$\displaystyle=(1.8-2.1)\,\unit{\per\cubed}\phantom{}\text{\cite[cite]{[\@@bibref{}{Botero_2018,
COUDRAY2009, Slowik2007, Schneider2006, Park2004, Widmann2003MeasurementOT,
Hueglin1997, Dobbins1994ComparisonOA}{}{}]}}\,.$ (18c)

Choosing the most commonly used refractive index for carbonaceous soot,
$m\\!=\\!1.57\\!+\\!0.56\,i$ (see [27] and references therein), the comparison
between MIREX- and ELPI+-measurements for one exemplary experiment is shown in
Fig. 9.

Figure 9:  Extinction coefficient obtained from an $n$-heptane fire
experiment. The black curve is the result of the MIREX-system; the blue curve
is the result of Eq. (11b) using $m\\!=\\!1.57\\!+\\!0.56\,i$ and $\rho_{\rm
part}\\!=\\!\qty{2.21}{\per\cubed}$. The differently shaded areas indicate
different phases of the soot generation, namely the flame-on, the flame-off,
and the exhaust-on phase, respectively.

Compared to a paraffin aerosol, the distribution is characterized by larger
dynamics from one time step to another which is due to the fact that we are
evaluating a room-scale experiment instead of a bench-scale one. As before,
however, subtleties in the time development of $\alpha_{\rm V}$ like different
experimental phases (flame-on, flame-off, exhaust-on phase) and local extrema
(in particular within the exhaust-on phase) are correctly recovered. Taking
the average of the four independent $n$-heptane fire experiments in Fig. 2 and
choosing exemplary values for $m$ compatible with Eqs. (18), we again obtain
distinct parameter combinations for which the result of Eq. (11b) coincides
with the respective MIREX-measurement, see Fig. 10.

$\mathfrak{Re}(m)$ | $\mathfrak{Im}(m)$ |  $\rho_{\rm part}\,[\unit{\per\cubed}]$
---|---|---
$1.50$ | $0.40$ |  $1.80\pm 0.28$
$1.90$ | $0.40$ |  $2.13\pm 0.29$
$1.50$ | $0.50$ |  $2.02\pm 0.32$
$1.90$ | $0.50$ |  $2.27\pm 0.31$
$1.57$ | $0.56$ |  $2.18\pm 0.35$
$1.50$ | $0.60$ |  $2.23\pm 0.36$
$1.90$ | $0.60$ |  $2.40\pm 0.34$
$1.50$ | $0.70$ |  $2.45\pm 0.41$
$1.90$ | $0.70$ |  $2.53\pm 0.37$
Figure 10:  Values of $\rho_{\rm part}$ for which MIREX-and ELPI+-results
agree best. Shown is the dependence on the values of $\mathfrak{Re}(m)$ and
$\mathfrak{Im}(m)$ (blue solid line and values in the table), including the
$3\sigma$-range (blue band and uncertainties in the table). In the diagram,
the imaginary part of $m$ is fixed as $\mathfrak{Im}(m)\\!=\\!0.5$, whereas
the dashed vertical line refers to the commonly used real part
$\mathfrak{Re}(m)\\!=\\!1.57$ [27]. The dotted horizontal lines denote the
density of crystalline graphite ($\rho_{\rm
part}\\!=\\!\qty{2.26}{\per\cubed}$) and the commonly used density for
carbonaceous soot ($\rho_{\rm part}\\!=\\!\qty{1.86}{\per\cubed}$ [8]),
respectively.

As before, particle densities increase with increasing values of
$\mathfrak{Re}(m)$. However, the increase is less pronounced in general and
smaller for larger values of $\mathfrak{Im}(m)$. Overall, it turns out that
Eq. (11b) yields reliable results also for flaming $n$-heptane fires in a
room-scale setup. Thereby, compatible combinations of $m$ and $\rho_{\rm
part}$ are in accordance with value ranges provided by the literature.

Finally, we again use Eq. (13) to evaluate individual contributions of the
ELPI+-channels, see the r. h. s. of Fig. 8. Compared to paraffin, the slope of
the distribution is less steep. However, the main part of extinction again
stems from only a small number of neighboring channels.

###  3.2 Mass-specific extinction coefficient $\alpha_{\rm M}$

In the literature, mass-specific extinction has so far has been studied
empirically and mainly in small-scale experiments under well-controlled and
well-ventilated conditions. A review of such results is provided e. g. in
[11], finding an overall fit of the collected data as $\alpha_{\rm
M}\\!=\\!\qty{4.8081}{\squared\per}(\lambda_{\rm
in}/\unit{\text{\textmu}})^{-1.0088}$, depending on the wave length of the
incident light. Transferred to the present MIREX-experiments ($\lambda_{\rm
in}\\!=\\!\qty{0.88}{\text{\textmu}}$), the fit yields

$\displaystyle\alpha_{\rm M}=(5.5\pm
0.6)\,\unit{\squared\per}\qquad(95\%\text{
c.l.})\text{\cite[cite]{[\@@bibref{}{Widmann_2003}{}{}]}}\,,$ (19)

which is quoted to be valid (at least) for all kinds of carbonaceous soot. In
the following, we test the hypothesis of a (universal) mass-specific
extinction coefficient. More precisely, we first check if constant mass-
specific extinction does occur at all in the conducted experiments. If so, we
determine numerical values of the corresponding extinction coefficients and
compare them with Eq. (19). Before starting, it should be noted that the
quantities entering Eq. (14) are approximately fixed in our evaluation: On the
one hand, values of $\alpha_{\rm V}$ are obtained from independent MIREX-
measurements and correctly described by the developed model in terms of $m$
and $\rho_{\rm part}$. Aerosol densities $\rho_{\rm aerosol}$, on the other
hand, do not depend on $m$ and are almost independent of the particle density
$\rho_{\rm part}$ which is shown in Sec. 2.1. Accordingly, obtained values of
$\alpha_{\rm M}$ are approximately a direct output of the applied experimental
and theoretical setup.

#### Paraffin aerosol

Figure 11:  Mass-specific extinction obtained from a paraffin ramp experiment
using Eq. (14). A time average of $\overline{\alpha}_{\rm
M}\\!=\\!\qty{4.32}{\squared\per}$ is obtained within the supply-off phase.
For the model parameters, we use the same values as in Fig. 6, namely
$m\\!=\\!1.42$ and $\rho_{\rm part}\\!=\\!\qty{0.85}{\per\cubed}$.

To start, we consider a paraffin ramp experiment and show results obtained
from Eq. (14) in Fig. 11. First of all, it is obvious that the value of
$\alpha_{\rm M}(t)$ increases in the supply-on phase before it stabilizes for
times $t\\!>\qty{300}{}$. Accordingly, a steady state regarding mass-specific
extinction is indeed forming with an average value in the supply-off phase of
$\overline{\alpha}_{\rm M}\\!=\\!\qty{4.32}{\squared\per}$.

Second, in order to check the impact of the particle density $\rho_{\rm
part}$, we consider the average of in total eight experiments as before and
summarize best fit values in Tab. 1.

Table 1:  Average $\alpha_{\rm M}$-values obtained from paraffin experiments. Shown are results obtained from Eqs. (14), (11b), and (6). Given uncertainties of $\overline{\alpha}_{\rm M}$ correspond to the $3\sigma$-range and include variations from one experiment to another as well as uncertainties of $\rho_{\rm part}$.  $\mathfrak{Re}(m)$ | $\rho_{\rm part}\,[\unit{\per\cubed}]$ |  $\overline{\alpha}_{\rm M}\,[\unit{\squared\per}]$
---|---|---
$1.30$ | $0.56\pm 0.07$ |  $4.54\pm 0.92$
$1.35$ | $0.68\pm 0.07$ |  $4.42\pm 0.80$
$1.40$ | $0.77\pm 0.08$ |  $4.38\pm 0.93$
$1.45$ | $0.88\pm 0.12$ |  $4.32\pm 1.08$
$1.50$ | $0.99\pm 0.12$ |  $4.26\pm 0.90$

As expected, the $\rho_{\rm part}$-dependence of $\overline{\alpha}_{\rm M}$
is only weak. For instance, a $\qty{77}{\%}$ increase of $\rho_{\rm part}$ in
Tab. 1 corresponds to an $\overline{\alpha}_{\rm M}$-reduction of only
$\qty{6}{\%}$. Combining the results, we therefore estimate mass-specific
extinction for a paraffin aerosol at a wave length of $\lambda_{\rm
in}\\!=\\!\qty{0.88}{\text{\textmu}}$ as

$\displaystyle\alpha_{\rm M}=(4.4\pm
1.1)\,\unit{\squared\per}\qquad(99.7\,\%\,\text{c.l.})\,.$ (20)

Figure 12:  Mass-specific extinction depending on the diameter of the
equivalent particle. The curves are obtained from Eq. (15) using
$m\\!=\\!1.40$, $\rho_{\rm part}\\!=\\!\qty{0.77}{\per\cubed}$ for paraffin
and $m\\!=\\!1.57\\!+\\!0.56\,i$, $\rho_{\rm
part}\\!=\\!\qty{2.18}{\per\cubed}$ for soot from a flaming $n$-heptane fire,
respectively. These values are compatible with obtained results summarized in
Figs. 7 and 10. The bands indicate a variation of $\rho_{\rm part}$ by
$\pm\,\qty{15}{\%}$, respectively. The wave length of the light is
$\lambda_{\rm in}\\!=\\!\qty{0.88}{\text{\textmu}}$.

Comparing the result with the literature, it is noticeable that Eq. (20)
agrees with Eq. (19) within the specified uncertainties, but is in general
slightly smaller. This fact, however, is compatible with fundamental
predictions of classical electrodynamics, which can be seen for instance in
Fig. 12. Only within a small range of $d_{\rm S}$ is it at all possible to
obtain values equal to or larger than $\alpha_{\rm
M}\\!=\\!\qty{5.5}{\squared\per}$, e. g. for $d_{\rm
S}\\!\approx\\!\qty{1}{\text{\textmu}}$ (blue curve). Since the considered
aerosol is made of particles that differ in their individual aerodynamic and
gravimetric properties (and therefore in corresponding equivalent diameters
$d_{\rm S}$), average values of $\alpha_{\rm M}$ can indeed be expected to be
smaller than the maximum possible value of the distribution. This is true
despite the fact that the $d_{\rm S}$-distribution of a paraffin aerosol is
comparatively narrow, see for instance the left diagram in Fig. 8.

#### Soot from flaming $n$-heptane fire

To study mass-specific extinction for soot from a flaming $n$-heptane fire, we
apply the same setup as for the paraffin aerosol. Corresponding results are
summarized in Tab. 2 and Fig. 13.

Figure 13:  Mass-specific extinction from an $n$-heptane fire experiment using Eq. (14). A time average of $\overline{\alpha}_{\rm M}\\!=\\!\qty{2.15}{\squared\per}$ is obtained within the flame-off phase. For the model parameters, we use the same values as in Fig. 9, namely $m\\!=\\!1.57\\!+\\!0.56\,i$ and $\rho_{\rm part}\\!=\\!\qty{2.21}{\per\cubed}$.  Table 2:  Average $\alpha_{\rm M}$-values obtained from $n$-heptane fire experiments. Shown are results obtained from Eqs. (14), (11b), and (6). Given uncertainties of $\overline{\alpha}_{\rm M}$ correspond to the $3\sigma$-range and include variations from one experiment to another as well as uncertainties of $\rho_{\rm part}$.  $\mathfrak{Re}(m)$ | $\mathfrak{Im}(m)$ | $\rho_{\rm part}\,[\unit{\per\cubed}]$ |  $\overline{\alpha}_{\rm M}\,[\unit{\squared\per}]$
---|---|---|---
$1.50$ | $0.40$ | $1.80\pm 0.28$ |  $2.38\pm 0.96$
$1.90$ | $0.40$ | $2.13\pm 0.29$ |  $2.38\pm 1.02$
$1.50$ | $0.50$ | $2.02\pm 0.32$ |  $2.37\pm 0.96$
$1.90$ | $0.50$ | $2.27\pm 0.31$ |  $2.38\pm 0.99$
$1.57$ | $0.56$ | $2.18\pm 0.35$ |  $2.38\pm 1.00$
$1.50$ | $0.60$ | $2.23\pm 0.36$ |  $2.38\pm 0.98$
$1.90$ | $0.60$ | $2.40\pm 0.34$ |  $2.38\pm 1.01$
$1.50$ | $0.70$ | $2.45\pm 0.41$ |  $2.38\pm 1.01$
$1.90$ | $0.70$ | $2.53\pm 0.37$ |  $2.38\pm 1.02$

As for paraffin, a steady state with constant average values of $\alpha_{\rm
M}$ is forming in Fig. 13. Moreover, almost all $\overline{\alpha}_{\rm
M}$-values in Tab. 2 are identical, irrespective of the actual values of $m$
and $\rho_{\rm part}$. We therefore estimate mass-specific extinction for soot
from flaming $n$-heptane fires as

$\displaystyle\alpha_{\rm M}$ $\displaystyle=(2.4\pm
1.1)\,\unit{\squared\per}\qquad(99.7\,\%\,\text{c.l.})\,,$ (21)

considering again particle sizes between $d_{\rm
S}\\!=\\!\qty{0.006}{\text{\textmu}}$ and $d_{\rm
S}\\!\approx\\!\qty{4}{\text{\textmu}}$ as well as $\lambda_{\rm
in}\\!=\\!\qty{0.88}{\text{\textmu}}$.

In the assessment of the result, the following points should be noted: First,
the obtained value is significantly smaller than the value in Eq. (19) and
even outside the provided uncertainties. Similar discrepancies towards the
referenced literature value, however, are reproduced within the scope of a
study that has been conducted recently [25]. Here, the setup of the experiment
was mapped by a numerical CFD model assuming a mass-specific extinction
coefficient as defined in Eq. (14). Local optical measurements with the MIREX-
device revealed an overestimation of the computed extinction coefficient by a
factor of approximately four. This result is compatible with Eq. (21) if one
compares it with the default value of $\alpha_{\rm
M}\\!=\\!\qty{8.7}{\squared\per}$ used in the simulations [6, 12]. On a
spatial and temporal scale, the deviation showed an almost linear correlation
over the entire duration of the fire. Finally, Eq. (21) is also compatible
with [10] when using it to compute the corresponding dimensionless extinction
coefficient $K_{\rm e}$, see Tab. 4 in Appendix A2.

From the model point of view, the fact that the result in Eq. (21) is smaller
than the one in Eq. (20) is actually compatible with predictions of classical
electrodynamics, see Fig. 12. At first glance, it is obvious that the maximum
possible value of $\alpha_{\rm M}$ is considerably smaller in case of heptane.
Moreover, for larger $d_{\rm S}$, $\alpha_{\rm M}$ approaches zero more
quickly. Since convolution with the actual size-distribution of the aerosol
particles does not change this fact, a smaller average of $\alpha_{\rm M}$ is
the consequence. At this point, it should be noted explicitly that the curve
progressions in Fig. 12 are not a particular feature of the developed model,
but a general prediction of classical electrodynamics. The observations made
are underpinned by the fact that the average of $\alpha_{\rm M}$ increases if
one only considers ELPI+-channels associated with small and intermediate-size
particles. For instance, considering only the first ten ELPI+-channels (which
still provide $\approx\\!\qty{80}{\%}$ of total light extinction, see the r.
h. s. of Fig. 8) and using the most commonly used particle density $\rho_{\rm
part}\\!=\\!\qty{1.86}{\per\cubed}$ [8], mass-specific extinction increases to
$\alpha_{\rm M}\\!=\\!(3.8\\!\pm\\!0.2)\unit{\squared\per}$. This value
corresponds to a smaller measuring range of the ELPI+-device as it is the
average for equivalent particles up to a size of $d_{\rm
S}\\!\approx\\!\qty{1}{\text{\textmu}}$. In this case, $\alpha_{\rm M}$ is
still slightly smaller than the value in Eq. (19), but approaches the order of
magnitude of the fit result.

###  3.3 Light extinction from ELPI+-measurements

Now that values of $\alpha_{\rm M}$ are obtained for the considered aerosols,
the results can be used to determine visibility conditions in future
applications of the ELPI+-system or measuring devices that work in a similar
way. This can be done by reversing the logic that has been used to determine
$\alpha_{\rm M}$-values in the first place. For this, we rewrite Eq. (14) as

$\displaystyle\alpha_{\rm V}(t)$ $\displaystyle=\alpha_{\rm M}\cdot\rho_{\rm
aerosol}(t)\,.$ (22)

Accordingly, the only quantity that has to be extracted from ELPI+-data in
future applications is the overall mass density of the aerosol under
investigation, e. g. via Eq. (6). Multiplication with $\alpha_{\rm M}$ then
yields results of $\alpha_{\rm V}$ which can ultimately be used in the
Beer–Lambert–Bouguer law in Eq. (8). Results obtained using this approach are
shown in Fig. 14.

Figure 14:  Extinction coefficient $\alpha_{\rm V}$ obtained from aerosol
density measurements. Shown are results for a paraffin aerosol (upper diagram)
and soot from a flaming $n$-heptane fire (lower diagram). In both diagrams,
the wave length of the light source is $\qty{0.88}{\text{\textmu}}$.

As can be seen, the prediction of Eq. (22) agrees much better with MIREX-
results when using $\alpha_{\rm M}$-values obtained in Eqs. (20) and (21)
instead of the one in Eq. (19) or the FDS default value $\alpha_{\rm
M}\\!=\\!\qty{8.7}{\squared\per}$ [6, 12]. In particular the latter leads to
an overestimation of light extinction of in part more than a factor of three.

In general, the determination of $\rho_{\rm aerosol}$ from ELPI+-data is
comparatively simple since it is approximately a direct output of the
measuring device, see e. g. Eq. (6) and the discussion below that equation.
Moreover, due to the fact that $\rho_{\rm aerosol}$ is almost independent of
the particle density $\rho_{\rm part}$, it is even possible to assume
$\rho_{\rm part}\\!=\\!\qty{1}{\per\cubed}$ in the determination of the
aerosol density.888 In fact, this corresponds to using aerodynamic equivalent
particles in the determination of $\rho_{\rm aerosol}$. Therefore, the actual
value of $\rho_{\rm part}$ does not necessarily have to be known when using
Eq. (22) to determine visibility conditions in future applications of the
ELPI+-device.

##  4 Conclusions and outlook

In this contribution, we develop and apply a model describing light extinction
in the presence of arbitrary aerosols. We do this by combining aerosol
measurements performed with the cascade impactor ELPI+ and light-intensity
reduction measurements performed with the extinction measuring equipment
MIREX. A main motivation in developing the model was that it should yield
reliable predictions of visibility conditions, but should at the same time be
as simple as possible. Accordingly, several assumptions were made in order to
focus on most relevant characteristics of the aerosol. This results in the
fact that model predictions essentially depend on only three independent
parameters: the universal particle density $\rho_{\rm part}$ as well as real
and imaginary parts of the refractive index $m$. Despite the strong
restrictions, Eq. (11b) is able to correctly describe time-dependent light
extinction in the presence of an arbitrary aerosol. This comprises, among
other things, different phases of the aerosol generation (flame-on, flame-off,
exhaust-on phase) as well as other experimental conditions (bench scale and
room scale, ventilation conditions, etc.). As it turns out, valid values of
the input parameters $\rho_{\rm part}$, $\mathfrak{Re}(m)$, and
$\mathfrak{Im}(m)$ are simultaneously compatible with well-established value
ranges found in the literature.

The reason why the model works so well is that ELPI+-measurements
simultaneously combine, through the working principle of the device,
aerodynamic and gravimetric properties of the collected aerosol particles.
This fact can be capitalized on especially when Stokes equivalent spheres are
used in the evaluation of the measurement data instead of aerodynamic ones.
Only in the former case, particle properties are adequately considered and
introduced equivalent spheres have a size compatible with the size of the
actual aerosol particles. In this way, the combination of ELPI+-data and
developed model is able to correctly take into account large hierarchies of
magnitudes which range for individual cross sections from
$\mathcal{O}(\sigma_{\rm ext})\\!=\\!10^{-22}\,\unit{\squared}$ for the
smallest particles up to $\mathcal{O}(\sigma_{\rm
ext})\\!=\\!10^{-10}\,\unit{\squared}$ for the largest particles. In doing so,
the model gets by without the need of introducing additional ‘correction’ or
‘conversion’ factors and correctly predicts extinction coefficients that are
typically in the order $\mathcal{O}(\alpha_{\rm V})\\!=\\!\qty{1}{\per}$.

As a first application of the developed model, it is found that main
contributions to extinction stem from only a small number of neighboring
ELPI+-channels that correspond to intermediate-size particles. More precisely,
for a paraffin aerosol, more than $\qty{90}{\%}$ of extinction originate from
only three ELPI+-channels, whereas for soot from a flaming $n$-heptane fire,
$\approx\qty{80}{\%}$ of extinction stem from only four channels. This is one
of the main reasons, why the concept of a universal particle density (which is
considered to be the same for each ELPI+-channel) works at all and why valid
numerical values are close to known literature values. In a second
application, we verify the existence of constant mass-specific extinction, at
least for the two considered aerosols. For them, we get the following results
which are valid for the wave length $\lambda_{\rm
in}\\!=\\!\qty{0.88}{\text{\textmu}}$ using the first $13$ channels of the
ELPI+-device (particle sizes between $d_{\rm
S}\\!=\\!\qty{0.006}{\text{\textmu}}$ and $d_{\rm
S}\\!\approx\\!\qty{4}{\text{\textmu}}$):

$\displaystyle\text{{paraffin aerosol}}:\qquad\alpha_{\rm M}$
$\displaystyle=(4.4\pm
1.1)\,\unit{\squared\per}\qquad(99.7\,\%\,\text{c.l.})\,,$ (23a)
$\displaystyle\text{{soot from flaming $n$-heptane fire}}:\qquad\alpha_{\rm
M}$ $\displaystyle=(2.4\pm
1.1)\,\unit{\squared\per}\qquad(99.7\,\%\,\text{c.l.})\,.$ (23b)

The results are not universal in the sense that they have the same numerical
value irrespective of the present aerosol. It is important to note, however,
that this is not a drawback. On the contrary, each of the values rather
considers specific aerosol characteristics that are relevant regarding light
extinction, in particular size-distributions and value ranges of the
equivalent particles. Therefore, Eqs. (23) can ultimately be used to reliably
predict visibility conditions in future applications of the ELPI+-device. For
this, it is sufficient to determine aerosol densities $\rho_{\rm
aerosol}^{(13)}$ via Eq. (6) and to multiply them with the respective value of
$\alpha_{\rm M}$. The determination of $\alpha_{\rm M}$-values describing
further aerosols as well as the investigation of their wave-length dependence
are beyond the scope of the present contribution. This work is left for future
investigations.

## Acknowledgements

We would like to thank Alica Kandler from the Institute for Advanced
Simulation, Forschungszentrum Jülich, for active support in the implementation
of the aerosol measurements. Parts of the present contribution are funded by
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under the
project number 465392452.

##  Appendix A Appendix

### A1 ELPI+ conversion factors

According to [13], ELPI+ conversion factors are given by

$\displaystyle X_{\rm N}$ $\displaystyle=\frac{1}{1.8300}\cdot\frac{J_{\rm
cal}}{J_{\rm imp}}\cdot X_{\rm dil}\cdot\left(\frac{1}{d_{\rm
S}}\right)^{1.225}\,,\qquad d_{\rm S}<\qty{1.035}{}\,,$ (24a) $\displaystyle
X_{\rm N}$ $\displaystyle=\frac{1}{1.8114}\cdot\frac{J_{\rm cal}}{J_{\rm
imp}}\cdot X_{\rm dil}\cdot\left(\frac{1}{d_{\rm
S}}\right)^{1.515}\,,\qquad\qty{1.035}{}<d_{\rm S}<\qty{4.282}{}\,,$ (24b)
$\displaystyle X_{\rm N}$ $\displaystyle=\frac{1}{3.3868}\cdot\frac{J_{\rm
cal}}{J_{\rm imp}}\cdot X_{\rm dil}\cdot\left(\frac{1}{d_{\rm
S}}\right)^{1.085}\,,\qquad d_{\rm S}>\qty{4.282}{}\,,$ (24c)

where $J_{\rm cal}$, $J_{\rm imp}$, and $X_{\rm dil}$ denote the calibration
flow rate, the impactor flow rate, and the dilution factor that can be
adjusted for each ELPI+-measurement, respectively.

### A2 Dimensionless extinction coefficient $K_{\rm e}$

In the already mentioned review [11] summarizing mass-specific extinction
coefficients obtained in various small-scale-experiments, the wave-length
dependence of $\alpha_{\rm M}$ is approximately fitted as $\alpha_{\rm
M}\\!\sim\\!\lambda_{\rm in}^{-1}$. Assuming such a functional dependence, it
is possible to define a dimensionless extinction coefficient $K_{\rm e}$ in
the following way:

$\displaystyle K_{\rm e}=\alpha_{\rm M}\cdot\rho_{\rm part}\cdot\lambda_{\rm
in}\,.$ (25)

Written in terms of $K_{\rm e}$, the Beer-Lambert-Bouguer law in Eq. (8) is
then given by

$\displaystyle\frac{I(t)}{I_{0}}=\text{exp}\big{[}-\alpha_{\rm V}\cdot
l\big{]}=\text{exp}\big{[}-\alpha_{\rm M}\cdot\rho_{\rm aerosol}\cdot
l\big{]}=\text{exp}\left[-K_{\rm e}\cdot\frac{\rho_{\rm aerosol}}{\rho_{\rm
part}}\cdot\frac{l}{\lambda_{\rm in}}\right].$ (26)

Using results obtained in Secs. 3.1 and 3.2, values of $K_{\rm e}$ for a
paraffin aerosol and soot from a flaming $n$-heptane fire are summarized in
Tabs. 3 and 4, respectively.

Table 3:  $K_{\rm e}$-values obtained for a paraffin aerosol. $\mathfrak{Re}(m)$ | $\rho_{\rm part}\,[\unit{\per\cubed}]$ | $\alpha_{\rm M}\,[\unit{\squared\per}]$ |  $K_{\rm e}\,[1]$
---|---|---|---
$1.35$ | $0.68$ | $4.42$ |  $2.6$
$1.40$ | $0.77$ | $4.38$ |  $3.0$
$1.45$ | $0.88$ | $4.32$ |  $3.3$
$1.50$ | $0.99$ | $4.26$ |  $3.7$
Table 4:  $K_{\rm e}$-values obtained for soot from a flaming $n$-heptane fire. $\mathfrak{Re}(m)$ | $\mathfrak{Im}(m)$ | $\rho_{\rm part}\,[\unit{\per\cubed}]$ | $\alpha_{\rm M}\,[\unit{\squared\per}]$ |  $K_{\rm e}\,[1]$
---|---|---|---|---
$1.50$ | $0.40$ | $1.80$ | $2.38$ |  $3.8$
$1.90$ | $0.40$ | $2.13$ | $2.39$ |  $4.5$
$1.50$ | $0.50$ | $2.02$ | $2.37$ |  $4.6$
$1.90$ | $0.50$ | $2.27$ | $2.38$ |  $4.8$
$1.57$ | $0.56$ | $2.18$ | $2.38$ |  $4.6$
$1.50$ | $0.60$ | $2.23$ | $2.38$ |  $4.7$
$1.90$ | $0.60$ | $2.40$ | $2.38$ |  $5.0$
$1.50$ | $0.70$ | $2.45$ | $2.38$ |  $5.1$
$1.90$ | $0.70$ | $2.53$ | $2.38$ |  $5.3$

### A3 Aerodynamic equivalent diameters

In Sec. 2.1 it is shown that Stokes equivalent diameters are better suited for
describing light interaction of aerosol particles than aerodynamic ones. In
the following, we explicitly verify and quantify this fact by introducing a
corresponding conversion factor

$\displaystyle C(\rho_{\rm part})=\frac{{\alpha}_{\rm V,\,Stokes}(\rho_{\rm
part})}{{\alpha}_{\rm V,\,aerodynamic}}\equiv\frac{{\alpha}_{\rm
V,\,Stokes}(\rho_{\rm part})}{{\alpha}_{\rm V,\,Stokes}(\rho_{\rm
part}\\!=\\!\qty{1}{\per\cubed})}\,.$ (27)

It is important to notice that, since Stokes diameters depend on the particle
density $\rho_{\rm part}$, this is of course also the case for values of $C$.

#### Paraffin aerosol

Since $\rho_{\rm part}$ is smaller than $\qty{1}{\per\cubed}$, Stokes
equivalent diameters are larger than corresponding aerodynamic ones. For given
values of $m$ and $\rho_{\rm part}$, extinction is therefore smaller when
using aerodynamic equivalent spheres which can be seen in Fig. 15.

Figure 15:  Extinction coefficient obtained when using aerodynamic equivalent
spheres. The value of $m$ is the same as in Fig. 6. The red curve is obtained
without correction factor, whereas the blue curve is the result of multiplying
$C\\!=\\!1.15$.

Agreement with MIREX-results can be obtained introducing correction factors
$C$ as defined in Eq. (27). $C$-factors for different values of $m$ (and
therefore $\rho_{\rm part}$) are summarized in Tab. 5.

Table 5:  $C$-factors for a paraffin aerosol. $\mathfrak{Re}(m)$ | $1.30$ | $1.35$ | $1.40$ | $1.42$ | $1.45$ | $1.50$
---|---|---|---|---|---|---
$\rho_{\rm part}\,[\unit{\per\cubed}]$ | $0.58$ | $0.70$ | $0.80$ | $0.85$ | $0.92$ | $1.03$
$C$ | $1.70$ | $1.41$ | $1.21$ | $1.15$ | $1.07$ | $0.98$

#### Soot from flaming $n$-heptane fire

Since $\rho_{\rm part}$ is larger than $\qty{1}{\per\cubed}$, Stokes
equivalent diameters are smaller than corresponding aerodynamic ones. For
given values of $\rho_{\rm part}$ and $m$, extinction is therefore larger when
using aerodynamic equivalent spheres which can be seen in Fig. 16.

Figure 16:  Extinction coefficient obtained when using aerodynamic equivalent
spheres. The value of $m$ is the same as in Fig. 9. The red curve is obtained
without correction factor, the blue curve is the result of multiplying
$C\\!=\\!0.48$.

Agreement with the MIREX-measurement can be obtained when using the overall
correction factors summarized in Tab. 6.

Table 6:  $C$-factors for soot from a flaming $n$-heptane fire. $\mathfrak{Re}(m)$ | $\mathfrak{Im}(m)$ | $\rho_{\rm part}\,[\unit{\per\cubed}]$ |  $C$
---|---|---|---
$1.50$ | $0.40$ | $1.80$ |  $0.57$
$1.90$ | $0.40$ | $2.13$ |  $0.46$
$1.50$ | $0.50$ | $2.02$ |  $0.52$
$1.90$ | $0.50$ | $2.27$ |  $0.44$
$1.57$ | $0.56$ | $2.18$ |  $0.48$
$1.50$ | $0.60$ | $2.23$ |  $0.48$
$1.90$ | $0.60$ | $2.40$ |  $0.42$
$1.50$ | $0.70$ | $2.45$ |  $0.45$
$1.90$ | $0.70$ | $2.53$ |  $0.41$

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