Vol.3, No.10, 884-888 (2011) Natural Science
http://dx.doi.org/10.4236/ns.2011.310113
Deposition of charged nano-particles in the human
airways including effects from cartilaginous rings
Hans O. Åkerstedt
Department of Applied Physics and Mechanical Engineering, Luleå University of Technology, Luleå, Sweden;
Received 17 August 2011; revised 20 September 2011; accepted 28 September 2011.
ABSTRACT
This paper presents a numerical study of the
deposition of spherical charged nano-particles
caused by convection, Brownian diffusion and
electrostatics in a pipe with a cartilaginous ring
structure. The model describes the deposition
of charged particles in the different generations
of the tracheobronchial tree of the human lung.
The upper airways are characterized by a certain
wall structure called cartilaginous rings which
modify the particle deposition when compared
to an airway with a smooth wall. The problem is
defined by solving Naver-Stokes equations in
combination with a convective-diffusion equation
and Gauss law for electrostatics. Three non-
dimensional parameters describe the problem,
the Peclet number 2PeUa D, the Reynolds
number 2
R
eUa

 and an electrostatic pa-
rameter

22
00
4acq T

 . Here U is the mean
velocity, a the pipe radius and D the diffusion
coefficient due to Brownian motion given by
3π
D
TCu d  , where Cu is the Cunningham-
factor
 
12.341.05exp0.39Cu dd

   Here
d is the particle diameter and

the mean free
path of the air molecules. Results are provided
for generations G4-G16 of the human airways.
The electrostatic parameter is varied to model
different concentrations and charge numbers.
Keywords: Charged Particles; Nanoparticles;
Convection; Brownian Motion; Deposition;
Respiratory Airways; Cartilaginous Rings
1. INTRODUCTION
Use of Carbon-nanotubes in material design enables
the development of new materials with superior proper-
ties. A drawback of this development is that these parti-
cles when inhaled may be toxic and can cause substan-
tial health risks of the human lung [1]. In experiments
these particles are known to be electrically charged
which probably leads to an increase in particle deposition
in the lung. Another important application of charged
nano-particles is in the design of methods for optimal
drug-aerosol targeting to predetermined lung sites [2].
The morphology of the human respiratory airways of
the human lung is represented by a system of branching
tubes where each tube belongs to a certain generation [3].
The first tube labeled generation zero (G0) is a single
tube called trachea. After the trachea the tube bifurcates
into two tubes of generation 1 (G1). Beyond generation
G16 we have the alveolar region (generation G17-G23),
which consists of tubes with multiple sacs (the alveoli)
expanding and contracting periodically during the breath-
ing cycle. This is the region where the gas exchange
takes place with the blood-vessel system.
In the present paper we analyze how nano-particles
are deposited in the human lung airways and especially
we consider the effects caused by charged nanoparticles.
We also include the wall structure of the upper airways
called cartilaginous rings.
For low space-charge densities the dominant transport
mechanism is the electric field that occurs from the im-
age charge caused by the interaction of a particle and a
grounded wall. This case has been studied by [4-6].
For a large number of particles, however, the transport
to the wall is dominated by the mutual electrical repul-
sion of the charged particles. This case has been ana-
lyzed by [5,7].
To find the transport and particle deposition in general
flow geometry there are essentially two methods. One
method is to solve the equations of motion of the parti-
cles in the flow field which is generated by solutions of
Navier-Stokes equation [8]. To get statistical measures of
the deposition a large number of particles (N) need to be
simulated with an error of the order

12
ON
 This
approach is difficult if we wish to find the electric field
and transport from a high concentration of particles. An
alternative method is to more directly consider the equa-

H. O. Åkerstedt / Natural Science 3 (2011) 884-888
885
885
tion that describe the probability density of the particles,
i e. the Fokker-Planck equation (Risken (1976) [9], Åker-
stedt et al. (2010) [10]) or a convective diffusion equa-
tion for the concentration (Åkerstedt et al. (2010) [11]).
In the present paper we consider this latter approach
of solving the convective-diffusion equation combined
with Navier-Stokes equations for the fluid flow and
Poisson’s equation for the electrostatic field. We con-
sider the situation where the space-charge density is
large so that the transport mechanism is dominated by
the electric field due to charge repulsion. In previous
studies [5,7,12] of this case fully developed fluid flow is
assumed. This is a good approximation for the higher
generations (>G10) of the airways, but for the upper
lower generation airways, the effect of a developing
fluid flow is important. In this paper where we consider
generations G4 - G16 we therefore assume the fluid to
develop from a uniform velocity profile at entry. We also
consider the additional effects of the cartilaginous ring
wall structure.
2. GOVERNING EQUATIONS
Each generation of the human airways is supposed to
be described by a tube with a smooth surface or with a
cartilaginous ring structure (Figure 1). For the individ-
ual airways of each generation we assume axial symme-
try. The following set of equations describes the physics
of the problem.

2
1
uu upRe   (1a)
0u  (1b)

2
2
11
4uc cc c
PePe Pe
   (1c)
24c

  (1d)
The dimensionless parameters of the problem are the
Reynolds number, the Peclet-number and an electrostatic
parameter.
22
0
0
2
Re
2
1
4
Ua
Ua
Pe D
cqa
T







(2)
Here U is the mean uniform velocity at inlet and a
is the pipe radius, v is the kinematic viscosity of air and
D is the Brownian diffusion constant defined as
3π
12.34 1.05exp0.39
TCu
Dd
d
Cu d







 




(3)
where Cu is the Cunningham factor which is a correction
factor needed to bridge the gap between the continuum
limit and the free molecular limit for the flow past a
sphere. Here
 is the collision mean free path and d is
the particle diameter.  is Boltzmanns constant, T is the
absolute temperature and  is the dynamic viscosity of
the air. The charge of the particles is q and the concen-
tration of particles at inlet is c0.
Eq.1c is an equation describing the evolution of the
concentration c, which is an equation of convective-
diffusion type with an extra source-term including the
effects from the electric field. Eqs.1a and 1b are the
Navier-Stokes equations describing the evolution of the
laminar fluid flow and Eq.1d is Poisson’s equation
which gives the link between the charged particles and
the electric field.
The set of Eqs.1a-1d needs to be solved together nu-
merically as a system and for this purpose we use the
commercial software Comsol Multiphysics 4.2. The ap-
plication modes in Comsol that have been applied are
stationary Navier-Stokes, stationary transport of diluted
species (convective-diffusion equation) and electrostat-
ics. The boundary conditions for the fluid flow are no
slip on the pipe wall ra

. At the inlet 0x

a uni-
form velocity is chosen and at
x
L outlet conditions
are applied. For the application of the diffusion-conve-
ction mode, the concentration at the inlet is uniform
0
cc

.
The boundary condition of the absorbing wall is c

0 and at the outlet convective flux is chosen. For the
electrostatics mode, zero charge/symmetry is chosen at
the inlet and outlet. Since the wall at ra of the res-
piratory airways consists of a so called mucus-layer in-
cluding mainly saline water, the wall is treated as a good
conductor and therefore the electric potential is taken as
zero. For the meshing we apply the extremely fine phys-
ics-controlled mesh supplied by the software.
3. NUMERICAL RESULTS
We have chosen to provide numerical results from
generation G4-G16 for which the airway has a ring
structure with an amplitude of 0.1 diameters [13,14]
(Figure 1). In generations G4 - G16 the flow is usually
laminar in contrast with the upper generations G0 - G3
which is usually characterized by turbulent flow.
The deposition of particles is calculated using the total
flux of particles out of the tube 2
 and the total flux of
particles into the tube 2

which are defined as
11
ˆˆˆ
dd du
jjj
j
SSS
cnS cnSnS
Pe Pe
   

(4)
Here ˆ n is the unit normal out of the tube surfaces
,1,2
j
Sj

. The first terms in (4) correspond to convec-

H. O. Åkerstedt / Natural Science 3 (2011) 884-888 886
tive flux, the second to Brownian diffusion flux and the
third to electrical mobility flux. Since the Pe-number is
usually large the convective flux dominates.
The deposition is then calculated from
2
1
1P



(5)
We first discuss some results for the tube of airway
generation G4 with uncharged particles. For light breath-
ing conditions we have a mean velocity of 1.47 m/s. The
tube radius is 2.25 mm and the length of the pipe is 12.4
mm. The Reynolds number is then 348Re =. We chose
a particle with a diameter of 10 nm, which gives a Pe-
clet-number of 5 1.41 10Pe .
In Figures 1 and 2 the result for generation G4 is pre-
sented. The distribution of concentration and the stream-
lines are shown. In the figures fluid enters from the top
with uniform velocity and uniform concentration 0 c. In
the figures high concentration corresponds to red color
and low concentration to blue color. From the behavior
of the streamlines it is noted that the flow separates in
the regions between the rings. In the separated regions
the concentration is about half the concentration at inlet.
Figure 1. Tube with a cartilaginous ring wall structure. Concen-
tration and streamlines. Red color denotes high concentration
and blue color low concentration. Fluid and particles enter from
the left at x = 0.
Figure 2. Concentration and streamlines close to entry of G4.
Red color denotes high concentration and blue color low con-
centration. Fluid and particles enter from the left x = 0.
In general the local Brownian diffusion flux or parti-
cle deposition is smaller in the separated regions with a
maximum deposition at the point of reattachment of the
fluid flow. More details of this case for uncharged parti-
cles have been presented by Åkerstedt et al. (2010) [11].
Next we consider the effect of charged particles. For a
large number of particles the transport to the wall is
dominated by the mutual electrical repulsion effect. The
dimensionless parameter that describes this space-charge
electrostatic effect is
22
0
0
1
4
cqa
T




Here c0 is the concentration of particles at the inlet
and q is the charge of the particles.
As an example for generation G4 with a tube radius of
2.25 mm and particles that carry 10 elementary charges
and where the particle concentration at inlet is 1011 par-
ticles/m3 the electrostatic parameter is 8.5

.
In Figure 3 the upper part of the tube of generation
G4 is shown for 100


. Here the arrows correspond
to the electric field and the color to the norm of the elec-
tric field. It is noted that the electric field is weaker in
the regions between the rings, i.e. in the regions of sepa-
rated flow. This can be explained by the lower values of
concentration in the separated regions (see Figure 2).
This also has some consequences also for the amount
of particles deposited to the wall. In Figure 4 the parti-
cle deposition is plotted for generation G4 for different
values of the parameter

. For comparison the deposi-
tion for a smooth tube is also presented.
We note that the deposition is lower for a tube with a
cartilaginous ring structure, which can be explained by
the lower concentration of particles and the lower values
of the electric field in the separated flow regions.
Figure 3. Red color indicates high electric field and blue low
electric field. Arrows of the electric field and the streamlines
close to entry of G4.

H. O. Åkerstedt / Natural Science 3 (2011) 884-888
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The amount of deposition can be controlled to a great
extent with the value of the paramete α, which in turn
depends on the concentration at entry c0 and the square
of the charge of the particles q.
This result may be of importance in the application of
designing charged nano-particles for optimal drug-aerosol
targeting to predetermined lung sites. For generation G4
although the deposition rates are small, we note that in-
creasing the parameter α from zero to 350 the deposition
increases by 80%. It is expected that this effect is larger
for higher generations. Therefore it is of interest to con-
sider particle deposition also for generations G6 to G16.
In Figure 5 the particle deposition as a function of the
parameter α is presented for generations G6 - G10 and in
Figure 6 the corresponding deposition is presented for
generations G12 - G16.
Figure 4. Deposition for different values of α for a smooth
walled tube and a tube with cartilaginous rings. Generation G4.
Figure 5. Deposition for different values of α for a smooth
walled tube and a tube with cartilaginous rings. Generations
G12 - 16.
Figure 6. Deposition for different values of α for a smooth
walled tube and a tube with cartilaginous rings. Generations
G12 - 16.
We note that there is a considerable increase in the
deposition for large values of α especially for the higher
generations for which even a rather small value of α
leads to considerable deposition. We also note that the
presence of the cartilaginous rings only leads to a slight
decrease in the deposition. The effect of these rings is
probably larger for particles of micron size and for lower
generations which has been observed in model experi-
ments of the upper airways [13].
4. CONCLUSIONS AND DISCUSSION
The deposition of nano-sized charged spherical parti-
cles on the walls of cylindrical tube with a periodically
spaced cartilaginous ring structure is investigated. The
model includes convective and Brownian diffusion tran-
sport as well as effects from the electric field created by
the charged particles.
The deposition of charged particles depends to a large
extent upon the single parameter α, which includes the
concentration at inlet, the particle charge and the particle
radius. For the higher generations G14 - 16 and large α
the deposition is as large as 80% - 90%, so almost all
particles are deposited. The effect of the cartilaginous is
in general to decrease the deposition, which can be ex-
plained by the separation of the flow in the regions be-
tween the rings. In the separated area the concentration
is about half the concentration at inlet which gives a
smaller electric field in these areas and therefore also
leads to less deposition. The reduction of deposition due
to the cartilaginous rings is in general not large with the
largest reduction of about 30% for G4 and almost no
reduction for G16.

H. O. Åkerstedt / Natural Science 3 (2011) 884-888 888
Due to decisive role of the parameter α for deposition,
these results should be useful for manufactures of in-
haler-devices in their strive for optimal design of thera-
peutic aerosols.
To validate the results experimentally is extremely
difficult especially in vivo measurements of nano-parti-
cles in the higher generations. As mentioned earlier a
physical model of the first two generations including a
cartilaginous wall structure has been conducted for mi-
cron particles [13]. The behavior of micron particles is
however quite different than for nano-particles. The lat-
ter are also much more difficult to detect. So up to the
knowledge of the author there are no known results
which can be used for validation of the present numeri-
cal results.
Future work involves studying the lower generations
G0 - G3 where the flow may become turbulent. This ana-
lysis requires the use of a low-Reynolds-number turbu-
lence model for the flow and a modification of the con-
vective-diffusion equation to include turbulent diffusion.
Of interest is also the study of larger particles of micron-
size, for which the alternative technique of Lagrangian
tracking is a more suitable approach. (Högberg et al. [8]).
For validation of deposition of nano-particles in sim-
ple geometries, we will also consider a recent technique
to track the individual transport of particles using digital
holography [14].
5. ACKNOWLEDGEMENTS
This work is sponsored by the Swedish research council, Swedish
Agency for Economic and Regional Growth and Centre for biomedical
engineering and physics.
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