Acta Polytechnica


doi:10.14311/AP.2019.59.0051
Acta Polytechnica 59(1):51–58, 2019 © Czech Technical University in Prague, 2019

available online at http://ojs.cvut.cz/ojs/index.php/ap

MODELLING, SIMULATION AND PARAMETER
IDENTIFICATION OF ACTIVE POLLUTION

REDUCTION WITH PHOTOCATALYTIC ASPHALT

Jens Kruschwitzb, Martin Linda, Adrian Munteana, ∗,
Omar Richardsona, Yosief Wondmagegnea

a Department of Mathematics and Computer Science, Karlstad University, Universitetsgatan 2, Karlstad, Sweden
b Capital City Kiel, Transportation Infrastructures Department, Fleethörn 9, Kiel, Germany
∗ corresponding author: adrian.muntean@kau.se

Abstract. We develop and implement a numerical model to simulate the effect of photocatalytic
asphalt on reducing the concentration of nitrogen monoxide (NO) due to the presence of heavy traffic
in an urban environment. The contributions in this paper are threefold: we model and simulate the
spread and breakdown of pollution in an urban environment, we provide a parameter estimation process
that can be used to find missing parameters, and finally, we train and compare this simulation with
different data sets. We analyse the results and provide an outlook on further research.

Keywords: pollution; environmental modelling; parameter identification; finite element simulation.

1. Introduction
Pollution in urban environments has been a major is-
sue for several decades and efforts of combating it have
spanned across many areas of research. Emerging tech-
nologies, such as those based on photo-catalysis target
the removal of vehicular nitrogen oxides (NOx) to mit-
igate the roadside air pollution problem. These tech-
niques have proven effective and research in this area
is still ongoing. Recently, a lot has been done from
the experimental perspective as well, see i.e. [1] for
experimental studies on visible–light activated photo-
catalytic asphalt, or [2–5] for photo-catalytic concrete
products as well as [6] for actual observational studies.
A comprehensive overview of the underlying princi-
ples in photo-catalysis processes in connection with
removal of air pollutants is presented in [7–9] and
the references therein. However, from the mathe-
matical modelling and simulation point of view, this
setting is less studied. A remotely related situation
is handled via a fluid dynamics-based study on the
pollutant propagation in the near ground atmospheric
layer as discussed in [10]. We strongly believe that
there is a need for a deeper insight via mathematical
modelling and simulations in the connections between
experiments, theory and practical applications of the
methodology. This is the place where we wish to
contribute. Similar techniques as the ones we apply
are detailed in [11], to which we refer the interested
reader for more details on the modelling structure.
In this paper, we report on the use of numerical

simulations to mimic the effect of the presence of a
street paved with photocatalytic asphalt has on the
NO reduction in the local ambient. Comparisons are
based on data collected at a neighbourhood of one
of the streets in Kiel, Germany. The findings in this
work contribute to enhance the understanding on the

interplay between the various factors involved in the
air pollution control. The contributions of this paper
are threefold:
• We model and simulate the spread and breakdown
of pollution in an urban environment.

• We provide a parameter estimation process that can
be used to determine relevant missing parameters.

• We train and compare this simulation with different
data sets.

The paper is structured as follows: first, we present
the model equations in a dimensionless formulation
and explain what they are describing. Then, we pro-
vide an insight in the reference set of parameters we
use. As a next step, we use the designed model to for-
mulate and solve a parameter identification problem
required to identify the effect of the local environment
on the NO evolution. Finally, we compare our numer-
ical results with the measured data from both before
and after the photocatalytic asphalt was placed.

2. NO pollution model
2.1. Model description
The urban environment we model is the cross-section
of a road. This is a three-lane highway within city
limits. In our two-dimensional model, we represent
the introduction of the pollution by the cars, which is
diffused through the air. We model sunlight as a reac-
tive factor [7, 8]. Finally, we model interaction with
the rest of the environment by using Robin boundary
conditions. These boundary conditions include envi-
ronmental effects from neighbouring locations (espe-
cially molecular diffusion and dispersion), represented
in a single parameter later referred to as σ. We do

51

http://dx.doi.org/10.14311/AP.2019.59.0051
http://ojs.cvut.cz/ojs/index.php/ap


J. Kruschwitz, M. Lind, A. Muntean et al. Acta Polytechnica

Figure 1. Sketch of the cross-section. Γ represents
the face of the asphalt, the box is a cross-section of
the street. The grey rectangular box inside of the
cross-section illustrates the location of NO emission.

not incorporate wind flux in our model. Our moti-
vation for this is two-fold: (1) we are not aware of
measurements of wind data in the region under obser-
vation; (2) our model can handle a slow-to-moderate
wind by a simple translation of the fluxes with an
averaged convective term. However, the size effects
that a strong drift introduces would be too significant
for our model to capture in the current setting and
with the current data available. Introducing genuine
wind effects in our model would necessarily force us
to handle a big area of the urban environment, where
more localized measurements would be needed to de-
termine a correct NO pollution level. Instead we opt
for introducing some level of dispersion correcting the
molecular diffusion of the main pollutant agent; see
Section 6 for a further discussion of this topic.

2.2. Setting of the model equations
The equations are defined in the aforementioned urban
environment, denoted by Ω. The geometric represen-
tation of the environment is illustrated in Figure 1.
To keep the presentation concise, we introduce the
model directly into a dimensionless form. We refer
the reader to the Appendix for a concise explanation
of the non-dimensionalization procedure.

Let x be the variable denoting the position in space
and t ∈ [0, 1) be the variable denoting the time of
day. The unknown concentration profile is denoted
by u(x, t) and represents the NO concentration at the
position x and time t. We refer here to NO as air
pollutant. We choose L to represent a reference length
scale: the width of the two times three-lane highway
plus the two corresponding banks (see Section 2.3).
tr is a reference time scale.
Concerning the concentration of the air pollutant,

let u0 denote the initial concentration value, uT a
preset (threshold) value and ur a reference concentra-
tion value. The preset concentration of the pollutant

might correspond, for example, to the concentration
of NO naturally present in the environment.

The transport (dispersion) coefficient for NO is de-
noted by D, while T = 0 and T = 1 denote the
dimensionless initial and final times of the process
observation; the start and the end of one day, re-
spectively. Let f denote the traffic intensity on the
street, measuring the average number of vehicles pass-
ing through the observation point during a specified
period, with fr as a reference value. Let s denote the
effect of the solar radiation, with sr as a reference
value. Introducing non-dimensional variables and
rewriting gives the main equation of this study:

∂u

∂t
−∇·

(
∇u
)

= Aff(x, t) −κAss(t)u, (1)

where Af f represents the contribution from the emis-
sion of NO by motor vehicles. The dependence of
the coefficient Af on the other parameters can be ex-
pressed as Af = fr L

2

ur D
= 5.5, a Damköhler-like number

that expresses the relation between the NO emitted
and the density of the traffic. The baseline value for
the molecular diffusion coefficient D is 0.146 cm2/s;
this reference value is taken from [12]. To account
for the effects of dispersion and slow winds, we use
values for this coefficient that are of order 102 higher.
This brings the numerical output in the range of emis-
sion measurements for large ranges of all the other
model parameters. To quantify the evolution of the
air pollutant, one has to first get a grip on the typical
sizes of σ, κ and γ. The term As s(t) represents the
contribution from the reaction where NO is converted
to NO2 with a reaction rate κ. The dependence of As
on the other parameters is given by As = sr L

2

D
.

The following initial and boundary conditions are
imposed on (1).

u(x, 0) =
u0
ur

for x ∈ Ω,

−∇u · n = 0 at ΓN × (0, 1),

∇u · n =
σ L

D

(
u−

uT
ur

)+
at ΓR × (0, 1),

∇u · n =
γ κL

D
u at Γ × (0, 1), (2)

where g+ denotes the positive part of g, namely
g+(x) = max(g(x), 0) for x ∈ ΓR.
We refer to σ as the environmental parameter, to

κ as the reaction rate and to γ as the asphalt reac-
tivity. All parameters can be seen as mass-transfer
coefficients; σ represents the exchange of NO with
the ambient atmosphere, κ expresses the speed of the
reaction from NO to NO2 while γ expresses the ca-
pacity of the photo-voltaic asphalt. In our model, we
choose a value of σ = 300 m3/µg, consistent with the
base level of NO concentration in the environment
according to the measurements. When simulating the
scenario prior to the photo-voltaic asphalt, γ = 0.

It is worth mentioning that κ and γ are influenced
by a multitude of effects, including but not limited

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vol. 59 no. 1/2019 Modelling and Simulation of Active Pollution Reduction

0 0.5 1

0

0.5

1

1.5

2

2.5

Figure 2. Traffic density m(t) as a function of t.

to the local atmospheric conditions, the effect of UV,
temperature and humidity and the porosity and the
chemical composition of the asphalt. For this reason,
these parameters are situation specific and require
tuning for each scenario that one wants to model.
We will do so by applying a parameter identification
technique, described in Section 4.

2.3. Other parameters
2.3.1. Initial and threshold concentrations

of NO
u0 represents the initial mass concentration of NO
present in the environment. In this simulation, we
choose a concentration of u0 = 37 µg/m3. This value
corresponds to the lowest available NO concentra-
tion level from the measurements. uT represents the
threshold concentration level from which NO disperses
out of the environment we consider. In this model,
we choose uT = 0 µg/m3, which means we assume low
ambient levels of NO, ensuring natural dispersion even
at low concentrations. In the case of high ambient
levels of NO due to, e.g., nearby factories or other
highways, uT can be higher.

2.3.2. Patch-wise vehicle distribution
The emission of NO is proportional to the amount
of motor vehicles passing through this cross-section.
The distribution of motor vehicles is derived from
measurements of [13], a German municipal service that
performs automated traffic counts for a large number
of cities. The traffic count we collect our data from
is located at 3.5 kilometers from the NO measuring
point. Because the roads in question are similar in
size and geographically close, we expect this data to
provide a reliable estimate. Daily, 72278 cars passed

are counted in both directions of the road. The traffic
count is aggregated on an hourly level, so in order to
obtain a vehicle distribution, we interpolate the hourly
data with cubic splines and normalize the resulting
function. We end up with a nominal density m(t) of
cars for each time t ∈ [0, 1) such that

∫ 1
0 m(t) dt = 1.

A plot of m(t) is presented in Figure 2.

2.3.3. Patch-wise NO emission (per vehicle)
We model the emission of NO by choosing a specific
shape for the source term f(x, t) from (1). This source
term is defined as follows:

f(x, t) =
{
m(t), if x ∈ A,
0, if x /∈ A,

(3)

for all x ∈ Ω and t ∈ [0, 1), where A is a rectangle
of 15 × 0.4 square meters, located 0.1 meter over
the asphalt. This box represents the location of the
emission of the vehicles. It is illustrated by the grey
box in Figure 1.

2.3.4. Cross-section geometry
The dimensions of the simulated cross-section (dis-
played in Figure 1) are 40 meters by 8 meters. In the
simulation, the road has a width of 15 meters (2 × 3
lanes of each 2.5 meters wide) and is located in the
middle of the cross section. This is a simplified rep-
resentation of the road under consideration, but can
be generalized to model roads of any dimension. The
measuring point we use to evaluate the simulated NO
concentration profile is located 1.75 meters above the
centre of the road, in accordance to the real measuring
point that provided us the NO concentration data.

53



J. Kruschwitz, M. Lind, A. Muntean et al. Acta Polytechnica

0 0.5 1

0

0.5

1

1.5

2

2.5

Figure 3. Nondimensional UV strength s(t) as a function of t.

2.3.5. Solar effects on NO
The natural conversion from NO to NO2 due to sun-
light is represented by term κAss(t). In this term,
the intensity of the UV radiation is expressed by the
dimensionless factor s(t). We compute this factor
by interpolating the sunrise, sunset and solar noon
data from [14] and normalizing to obtain a function
s(t) such that

∫ 1
0 s(t) dt = 1. This is done under the

assumption that there is no UV radiation between
sunset and sunrise and that the maximum of UV radi-
ation takes place at solar noon. For our simulation, we
choose a reference value sr = 1 UVI (see Appendix),
where by UVI we mean the UV Index of the sunlight,
a dimensionless quantity. A plot of the shape of s(t)
is presented in Figure 3.

2.3.6. Pollution-reducing effects
As mentioned above, the values of κ and γ strongly de-
pend on the environment. By using the measurement
data detailed in the next section, we have enough
information to derive the values of these parameters
for our model.

3. Measurements
The NO measurements used in the simulation cover a
period from 2012 to 2017, where NO was measured
in 30-minute intervals. We clean the data by limiting
ourselves to a specific period: from September 1st
to December 10th: 101 measurements outside of the
holiday period, with an average intensity of the sun
(between summer and winter). We interpolate this
data to obtain the emission profile of an average day.
Data in this period from 2017 corresponds to the
newly installed photocatalytic asphalt. To compare

this scenario with the initial situation, we use the
measured data from 2016.
To assert that the traffic intensity remained rela-

tively constant, we compare the measurements from
2016 to the same period in 2015. Figure 4 shows
comparable concentration profiles. This indicates, as
a reasonable assumption, that the only major change
between 2016 and 2017 was the new asphalt, given
the local weather conditions incorporated in the pa-
rameter κ. Figure 4 reveals a large reduction in NO
pollution after 2016.

4. Parameter identification
As stated in Section 2, we require situation-specific
values for the parameters κ and γ. To obtain these, we
propose and solve a parameter identification problem
based on the datasets described in Section 3. In a
two-step procedure, we first use the measurements
prior to the installation to obtain κ, knowing that in
this case γ = 0, and then use the measurements after
the installation to obtain γ.
Mathematically speaking, we wish to solve the fol-

lowing problem: Let u( · ; κ,γ) be the solution to (1)
for given parameters κ and γ and ur be the measure-
ment data. Find κ and γ that solve the following
optimization problem:

min
κ∈Pκ

min
γ∈Pγ

∥∥ur(x, t) −u(x, t; κ,γ)∥∥L2(Ω×(0,1)). (4)
Here Pκ and Pγ are compact sets in R where parame-
ters are searched.

5. Simulation
This section describes the setup of determining the
effectiveness of the photocatalytic asphalt.

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vol. 59 no. 1/2019 Modelling and Simulation of Active Pollution Reduction

0 2 4 6 8 10 12 14 16 18 20 22 24

0

50

100

150

200

250

300

350

2015

2016

2017

Figure 4. NO measurements at the Theodorus-Heuss-Ring in Kiel, Germany. The green curve (2017) has the same
shape as the other two curves, but presents a significant reduction of NO levels.

0 0.5 1 1.5 2 2.5 3 3.5

κ
×10

4

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

R
e
la
t
iv
e
d
e
v
ia
t
io
n
fr
o
m

m
e
a
s
u
r
e
m
e
n
t
s
in

L
2
−
n
o
r
m

Parameter identification for κ

Figure 5. The relative discrepancy between the simulated NO concentrations and the measurements for a range of κ
values and γ = 0. The optimum is reached for κ = 1.85 × 104 1/[day UVI], displayed as the minimum of this curve.

5.1. Simulation framework
We numerically compute the solution to (1) with (2)
with a finite element simulation using the FEniCS
library [15]. We investigate the process within the
specified cross-section, leading to a two-dimensional
reaction-diffusion process. The finite element mesh
has 30 × 30 elements on a rectangular grid. The
solution to (1) is approximated with quadratic basis
functions.

We simulate two scenarios: (A) the NO concentra-
tion profile prior to the photocatalytic asphalt (cor-

responding to measurements from 2015 and 2016);
and (B) the NO concentration profile after the instal-
lation of the photocatalytic asphalt (corresponding
to measurements from 2017). In these simulations,
we use the parameters as described in Section 2 and
choose a diffusion coefficient of D = 43.8, which shows
an agreement with the current setting. In case (A)
specifically, we fix γ = 0.
Our goal is to use the data set from (A) to train

our simulation on the value of parameter κ. Then
we use the data set from (B) to determine the effect

55



J. Kruschwitz, M. Lind, A. Muntean et al. Acta Polytechnica

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01

γ

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

R
e
la
t
iv
e
d
e
v
ia
t
io
n
fr
o
m

m
e
a
s
u
r
e
m
e
n
t
s
in

L
2
−
n
o
r
m

Parameter identification for γ

Figure 6. The relative discrepancy between the simulated NO concentrations and the measurements for a range of
γ values and κ = 1.85 × 104 1/[day UVI]. The optimum is reached for γ = 3.0 × 10−3, displayed as the minimum of
this curve.

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (of day)

0

50

100

150

200

250

300

350

N
O

(

µ
g
/
m

3
)

Simulated concentration of NO

Passive/Before

Active/After

Figure 7. Simulated NO concentration profiles. The dash-dotted line corresponds to the period in 2016, the
continuous line has corresponds to the period in 2017.

the photocatalytic asphalt has on the reduction of the
local NO concentration, i.e., find γ.
Figure 5 displays this process, where for a range

of κ, we plotted the relative discrepancy with the
measurements. More precisely, let ur denote the in-
terpolation of the measurements and uκ the result
of our simulation for a specific κ, then the relative
discrepancy eκ(u) is defined as

eκ(u) =
‖ur −uκ‖L2(Ω×(0,1))
‖ur‖L2(Ω×(0,1))

. (5)

Having captured the nature of the process, we can
estimate the effect of the photocatalytic asphalt in
terms of γ by starting a new parameter identification
process using the data set from the second scenario.
Figure 6 shows the relative discrepancy (similarly

defined as in (5)) for uγ (with an optimal value of
κ). Figure 5 and Figure 6 suggest that we are dealing
with a convex minimization problem.

The simulations with optimally fitted parameters,
before and after the construction of the photocatalytic
asphalt, are displayed in Figure 7.

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vol. 59 no. 1/2019 Modelling and Simulation of Active Pollution Reduction

0 2 4 6 8 10 12 14 16 18 20 22 24

Time (of day)

0

50

100

150

200

250

300

350

N
O

(

µ
g
/
m

3
)

Simulated and measured concentration of NO

Before (Simulation)

Before (Measurements, 2016)

After (Simulation)

After (Measurements, 2017)

Figure 8. NO measurements compared to simulations in the two cases. The simulations show good quantitative
agreement.

We use the measurements of NO to compare the
simulation results with. Figure 8 shows the measured
and simulated data combined. The simulation agrees
qualitatively and quantitatively, with a mass error
of 58.64 µg/m3 in the pre-photocatalytic case and
22.68 µg/m3 in the post-photocatalytic one. This mass
error is defined as∫ T

0

∫
Ω

∣∣u(x, t) −ur(x, t)∣∣ dx dt, (6)
where u represents the simulation solution and ur the
measurements.

6. Discussion and outlook
This report shows that it is possible to use a math-
ematical model like the one presented in Section 2
to describe the evolution of NO concentration as a
function of time in an urban environment where both
intense motorized vehicle traffic and photocatalytic
asphalts are present.
A number of things can be done to continue this

investigation and improve the quantitative predictions
obtained in this study.
• Perform a 3D simulation along the whole length of
the photocatalytic asphalt.

• Account for the presence of uncertainties in the
weather condition (especially the variation in the
UV radiation and the effect of precipitation).

• Numerically identify scenarios leading to an extreme
NO pollution.

• By neglecting wind flux, we emphasize that our
interest lies in quantifying the reactive part of this
reaction-diffusion process. This approach allows

us to identify two extreme scenarios: (1) excellent
performance of the asphalt, and (2) low-response
on NO pollution induced by the presence of the
asphalt. Including more detailed wind effects is
certainly of an interest (both from a theoretical
and a practical perspective), since it is known that
the location under investigation (Kiel, Germany) is
prone to varied weather conditions.

7. Acknowledgements
We gratefully thank the State Agency for Agricul-
ture, Environment and Rural Areas Schleswig-Holstein
(LLUR) for providing the data measured at the
Theodor-Heuss-Ring in Kiel, Germany. The authors
acknowledge the very valuable input from the side of
the referees regarding the shaping of the final form of
the manuscript.

8. Appendix
For x ∈ Ω and t ∈ [0,T), we consider the following
equation

∂u

∂t
−∇·

(
D∇u

)
= f(x, t) −κs(t) u(x, t). (7)

Let us also introduce the following rescalings: x̄ = x
L
,

t̄ = t
tr
, ū = u

ur
, f̄ = f

fr
and s̄ = s

sr
, where

fr = max
(x,t)∈Ω×[0,T]

∣∣f(x, t)∣∣.
Rewriting (7) using these new rescaled functions and
variables yields

∂ū

∂t̄
−
trD

L2
∇̄ ·
(
∇̄ū
)

=
tr fr
ur

f̄ −κtr sr s̄ ū. (8)

57



J. Kruschwitz, M. Lind, A. Muntean et al. Acta Polytechnica

This equation is posed in a rescaled space domain
Ω̄ = Ω

L
. Choosing tr = L

2

D
reduces (8) to the form

∂ū

∂t̄
−∇̄ ·

(
∇̄ū
)

=
L2 fr
Dur

f̄ −κ
L2 sr
D

s̄ū. (9)

Setting Af = L
2 fr
Dur

, As = L
2 sr
D

and abusing the no-
tation, namely rewriting Ω instead of Ω̄, u instead
of ū, f instead of f̄, and s for s̄ in (9) posed in Ω̄
gives (1) posed in Ω. A similar discussion yields the
structure of the initial and boundary conditions in a
non-dimensional form.

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	Acta Polytechnica 59(1):51–58, 2019
	1 Introduction
	2 NO pollution model
	2.1 Model description
	2.2 Setting of the model equations
	2.3 Other parameters
	2.3.1 Initial and threshold concentrations of NO
	2.3.2 Patch-wise vehicle distribution
	2.3.3 Patch-wise NO emission (per vehicle)
	2.3.4 Cross-section geometry
	2.3.5 Solar effects on NO
	2.3.6 Pollution-reducing effects


	3 Measurements
	4 Parameter identification
	5 Simulation
	5.1 Simulation framework

	6 Discussion and outlook
	7 Acknowledgements
	8 Appendix
	References