Self-consistent mathematical model and simulation of carbon nanoparticle deposition from nonisothermal gas flow

International Journal of Heat and Mass Transfer 101 (2016) 1086–1092

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Self-consistent mathematical model and simulation of carbon nanoparticle deposition from nonisothermal gas flow Sergey P. Fisenko ⇑, Dmitry A. Takopulo A.V. Luikov Heat and Mass Transfer Institute, National Academy of Sciences of Belarus, 15 P. Brovka Str, 220728 Minsk, Belarus

a r t i c l e

i n f o

Article history: Received 2 March 2016 Received in revised form 28 May 2016 Accepted 28 May 2016 Available online 11 June 2016 Keywords: Brownian diffusion Thermophoresis Nanostructured film Morphology Growth rate Nanoparticle beam

a b s t r a c t The new mathematical model of nanoparticle deposition and correlated nanostructured film growth has been developed. It is shown that the Brownian diffusion determines the final deposition rate but thermophoresis can drastically enhance it. Qualitative estimations and simulation results of nanostructured film growth are presented. For relatively small nanoparticles, some morphological parameters of the film are calculated. Ó 2016 Elsevier Ltd. All rights reserved.

1. Introduction The fundamentals of the Brownian diffusion of nanoparticles in a gas phase can be already found in many physics textbooks [1]. Technical applications of the Brownian diffusion have emerged only recently. In particular, considerable interest has focused on deposition of nanoparticles from gas flow on a reactor wall [2–6]. It is due to the importance of this process for wide variety of modern technologies concerned with coating and thin film production [7], nanoparticles transport [8], human health [9], membrane production [10], and others. Therefore, insight into the physics of deposition process of nanoparticles, wherein the nanostructured film is formed, is really of great importance [10]. The formation and growth of a nanostructured film, resulting from nanoparticle deposition, depends on a variety of parameters. The temperature gradient in gas flow is one of the most important one. This temperature gradient affects the deposition due to the thermophoresis of nanoparticles. The theoretical investigation of this complex process is based on the joint solution of the hydrodynamic problem for gas flow and the problem of heat transfer between the gas and the growing film, and subsequently, between the film and the reactor walls. Knowledge of the gas velocity profile and temperature profiles along the reactor is essential for solving the problem of convective ⇑ Corresponding author. Tel.: +375 17 284 2222; fax: +375 17 292 2513. E-mail address: [email protected] (S.P. Fisenko). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.05.123 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.

diffusion for nanoparticle distribution inside the reactor. After this we can determine the deposition rate of nanoparticles on the wall. It is obvious that a growing nanostructured film should change the conditions of the deposition process. This effect is especially important for a relatively thick deposited layer. Since the film grows relatively slowly, we consider the hydrodynamics and the heat transfer in the gas phase in quasi-steady state approximation. Also, it is worthy to note that experimental results depend on the shape of nanoparticles [11]. We consider only spherical nanoparticles in what follows. It is important to note that a vast majority of publications deal with relatively large nanoparticles with radii larger 100 nm [12]. Our consideration is limited to significantly smaller nanoparticles, less than 10 nm. Therefore we take into account the effects related with the Brownian diffusion. In particular, the Brownian diffusion of such nanoparticles determines the growth rate of a nanostructured film on the reactor wall. For some industrial applications related with power engineering and chemical engineering special interest has dust deposition on tube surfaces from high speed compressible flows. There are many publications devoted to this problem. It is worthy to mention several reviews which consider this part of deposition problem [13–15]. The paper is structured the follows. First, we give a mathematical model of heat and mass processes that govern the deposition of nanoparticles. Then a self-consistent algorithm of the solution of our model is proposed and discussed. It is worthy to emphasize that we use the free molecular approximation for describing the

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Nomenclature f n

distribution function of nanoparticles in a reactor number of all nanoparticles per unit volume in the reactor (m3) radial coordinate (m) axial coordinate (m) gas velocity (m s1) average gas velocity (m s1) nanoparticle radius (m) Brownian diffusion coefficient (m2 s1) Boltzmann constant (J K1) temperature (K) pressure (Pa) mass of a gas molecule (kg) mobility of nanoparticle (m/kg) heat capacity at constant pressure of a gas (J kg1 K1) thickness of a deposited film (m) porosity of a deposited film deposition flow density (m2 s1) half-width of an inlet nanoparticle beam (m)

r z u
u RB DB k T P m b cp h

e

j r⁄

interaction of nanoparticles with gas flow and at the same time we use approximation of continuous medium for describing of gas flow. In the next chapter we present a qualitative analysis of these equations and obtain qualitative analytical estimations of the nanostructured film. Then we present our simulation results. We summarize our results in the final section. 2. Mathematical model of the Brownian deposition of nanoparticles

lt lB p v s

thermal characteristic length (m) characteristic length of the Brownian deposition (m) relative number of nanoparticles in a deposited film volumetric ratio of nanoparticles in a deposited film relative surface area of nanoparticle group in deposited film number density of nanoparticles on the reactor axes efficiency of deposition of nanoparticles dynamic viscosity of a gas (Pa s) density of a gas (kg m3) thermophoresis velocity (m s1) heat conductivity of a gas (W m1 K1)

n⁄ K

g q

v th

k

Subscripts in value at the inlet of a reactor w reactor wall c deposited carbon film i ith group of particles

The field of gas temperature T(r, z) along the reactor is governed by the advective heat diffusivity equation

qcp uðr; zÞ


 @Tðr; zÞ 1 @ @Tðr; zÞ ¼ rkðTÞ : @z r @r @r

We neglected a heat flow related to nanoparticles. Thus we cannot use Eq. (5) if the number density of nanoparticles is high. For the gas velocity profile we use the Poiseuille profile:


ðzÞ 1  uðr; zÞ ¼ 2u The steady-state continuity equation for the distribution of the number density of nanoparticles f (r, z, RB) via laminar flow in a cylindrical reactor is


 @fu 1 @r v th f 1 @ @f þ ¼ rDB : @z r @r r @r @r

ð1Þ

The number of all nanoparticles n per unit volume in the reactor is

Z

1

nðr; zÞ ¼ 0

f ðr; z; RB ÞdRB :

For a reactor with length L it is easy to show that the contribution of the longitudinal Brownian diffusion of nanoparticles could be 2

neglected if the inequality ðR=LÞ  1 is valid; this condition is true for most practical cases. The second term on the left side of Eq. (1) takes into account the thermophoresis of nanoparticles in a radial direction. For the free molecular regime of the interaction of nanoparticles with gas flow the thermophoretic velocity vth is determined by the expression [16]

v th ¼ 

3 g 3 gk rT ¼  rT: 4 qT 4 mP

ð2Þ

The Brownian diffusion coefficient is calculated by means of the Einstein relation [1]

DB ¼ kTb;

ð3Þ

the mobility of a spherical nanoparticle b is calculated by means of the expression below

b¼

rffiffiffiffiffiffiffiffiffiffiffiffi 2pkT : m 16pR2B P 3

ð4Þ

ð5Þ

!

r2 2

ðR  hðzÞÞ

:

ð6Þ


ðzÞ is recalculated using the integral The averaged gas velocity u form of the continuity equation for mass gas flow rate in the reactor. It was shown earlier that such approximation for a velocity profile is good enough even in nonisothermal gas flow in a cylindrical reactor [17]. The equation for evolution of the film thickness is obtained by means of the mass conservation law for nanoparticles. In particular, for a monodisperse beam of nanoparticles in gas flow we have the following equation:

dhðzÞ 4 pR3B @nðR  h; zÞ ¼ : DB dt 3 ð1  eÞ @r

ð7Þ

It is important to emphasize that only the Brownian diffusion affects the nanostructured film growth because the size of nanoparticles is much smaller than the Knudsen layer. It is well known that the temperature gradient is absent in the Knudsen layer. We recall that thickness of the Knudsen layer is about twothree times larger than the mean free path of gas molecules. For atmospheric pressure the mean free path of air molecules is about 100 nm. In a similar manner we obtain the flow of nanoparticles j on the reactor wall

j ¼ DB

@nðR  h; zÞ : @r

ð8Þ

For a polydisperse beam of nanoparticles we have a more complicated equation for change of the film thickness:

dhðzÞ 4 p ¼ dt 3 ð1  eÞ

Z

0

1

R3B DB

@f ðR  h; zÞ dRB : @r

ð9Þ

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Eqs. (7) and (9) contain the porosity e of a nanostructured film as a parameter. The porosity e is defined as the ratio of the volume of voids in a body to its total volume. In most cases this parameter is resolved by standard experimental measurements, see for example [18,2]. For Eq. (5) the initial condition is

Tðr; 0Þ ¼ T 0 ðrÞ at z ¼ 0:

ð10Þ

For a steady-state case the temperature on the interface between the deposited film and the gas flow is determined by the condition of the equality of heat flows on the interface. At r = R–h(z) we equalize the heat fluxes

krT ¼ kc rT c

ð11Þ

where kc is the heat conductivity of the deposited carbon nanostructured film. We assume that the reactor wall temperature is equal to a constant value, Tw. At the axis of the cylindrical reactor we use the symmetry condition for the temperature field:

at r ¼ 0 :

@T ¼ 0: @r

ð12Þ

For the Brownian diffusion equation (1) the initial conditions are

at z ¼ 0 and 0 < r 6 r : f ðr; RB Þ ¼ f 0 ðRB Þ; at z ¼ 0 and r < r 6 R : f ðr; RB Þ ¼ 0;

ð13Þ

where r⁄ is the half-width of the beam of nanoparticles. On the reactor wall and on the deposited layer surface we have:

The initial condition for Eq. (20) is

at t ¼ 0 : hðzÞ ¼ 0:

ð21Þ

The growth of deposited layer affects on fields of gas velocity and temperature. In order to take into account the effect of growth of deposited layer on thermodynamic fields and, correspondingly, vise versa we have to solve nonlinear problem of the deposition. Therefore we have developed the self-consistent algorithm in order to solve our nonlinear boundary problem using different temporal scales of transfer processes in the gas phase and in the film. This algorithm is as follows: 1. The gas temperature field is determined as a result of the solution of convective heat transfer problem (5), (10)–(12). 2. Based on the known gas velocity and temperature distributions, the convective diffusion problem (16)–(19) is solved, and then, deposition flows are determined according Eq. (9). 3. We calculate the deposited film thickness for some selected period. 4. The gas velocity distribution in the reactor is recalculated in accordance with the updated deposited layer thickness. Following this procedure, steps 1–4 are repeated. Thus, the proposed procedure permits us to obtain the approximate selfconsistent solution of the problem under investigation.

ð14Þ

In other words, the Exp. (14) is assumption of the full adhesion of nanoparticles after collisions with reactor wall or the deposited layer surface. Physical background of this assumption is small velocities of nanoparticles. Typical velocity of nanoparticles is much smaller than the mean gas velocity due to much larger mass. At the reactor axis we use the symmetry condition for the number density of nanoparticles f:

@f ¼ 0: @r

ð15Þ

We seek the numerical solution of Eq. (1) in the so-called group approximation. For group approximation, f(r, z, RB) is approximated by the histogram with a finite number of groups of nanoparticles. We assume that all nanoparticles of the ith-group have the same radii RBi . By means of the group approximation Eq. (1) is converted into a system of differential equations for the number of nanoparticles of each group ni per unit volume. Thus, we have the system of equations for ni


 @ni u 1 @rv th ni 1 @ @ni þ rDBi ¼ ; @z r @r r @r @r

ð20Þ

2.1. Qualitative estimations

at r ¼ R  hðzÞ f ðr; z; RB Þ ¼ 0:

at r ¼ 0

dhðzÞ 4 p X 3 @ni ðR  hÞ ¼ : R DBi dt 3 ð1  eÞ i Bi @r

i ¼ 1; 2; . . . N:

ð16Þ

Before carrying out numerical simulation, let us make some qualitative estimation using our mathematical model. We start with the temperature field. It was shown in [5] that for a cylindrical reactor the thermal characteristic length lt is

lt ¼ 0:26

uR2 qcp ; kðTÞ

ð22Þ

and on the axis of the reactor the temperature difference T–Tw decreases according to the expression

T  T w  expðz=lt Þ: Thus, we can make an important conclusion that a radial temperature gradient exists in the region (2–3) lt. Thermal characteristic length lt depends on the flow mass rate, reactor geometry, and gas properties. Other details of calculation of lt are given in [5]. Similar, for the isothermal nanoparticle deposition we have obtained the characteristic length of the Brownian deposition lBi. For the ith group [4,5] the expression is

lBi  0:26

R2 u : DBi

ð23Þ

Accuracy of the group approximation increases with increase of the parameter N. It is worthy to note that we neglect the coagulation of nanoparticles. For the group approximation, boundary conditions (13)–(15) are transformed as:

For a more detailed description of a nanostructured film, let us introduce several additional parameters. We define the relative number of the ith group nanoparticles in a nanostructured film pi(z, t) as follows:

for z ¼ 0 and 0 6 r 6 r : ni ¼ n0i ;

ð17Þ

j ðz; tÞ pi ðz; tÞ ¼ P i ; k jk ðz; tÞ

for r ¼ R  hðzÞ : ni ¼ 0;

ð18Þ

for r ¼ 0 :

@ni ¼ 0: @r

In the group approximation, Eq. (9) is written as follows

ð19Þ

ð24Þ

where ji (z, t) is the ith group deposition flow defined by Eq. (8). Also, we determine the second important morphological characteristics of the nanostructured film: the volumetric ratio of nanoparticles of the ith group vi(z, t), which is calculated accordingly by the formula

S.P. Fisenko, D.A. Takopulo / International Journal of Heat and Mass Transfer 101 (2016) 1086–1092

j ðzÞR3Bk

v i ðz; tÞ ¼ P i

3 k jk ðzÞRBk

:

ð25Þ

It is worthy to note that we neglect possible deformations of nanoparticles in the film. Also, the relative surface area of the ith group si(z, t) in the nanostructured film near the axial coordinate z at the moment t is an important morphological characteristic. We calculate it from the formula

j ðzÞR2Bi si ðzÞ ¼ P i : 2 k jk ðzÞRBk

ð26Þ

Note, that in the Eq. (26) we neglect the contact area between nanoparticles. It is worthy to emphasize that the nominator and the denominator of the right hand side of the Eq. (26) does not depend on the nanoparticle radii. All these parameters satisfy the obvious normalization condition:

X

X

i

i

vi ¼

pi ¼

X

si ¼ 1:

i

Thus, if we consider N nanoparticle groups, there are only N  1 independent components for each of the relative parameters. For a monodisperse beam of nanoparticles we can transform the Eq. (7) and qualitatively integrate it. We have

dh R3B n ðzÞ RB n ðzÞ   DB ; dt ð1  eÞ R  h ð1  eÞðR  hÞ

ð27Þ

where n⁄(z) is the number density of nanoparticles on the reactor axes with z coordinate. An interesting feature follows from the Eq. (27) that the rate of the film growth increases with the film thickness. After the integration Eq. (27), we have a useful expression for thickness of the nanostructured film at the moment t

hðz; tÞ 

RB n ðzÞt : 1e

Thus the film thickness is directly proportional to the product RB t/ (1  e). The reactor radius R affects the film thickness only through the value of n⁄(z). 3. Simulations results We simulated our problem using the method of lines for partial differential equations [19], an explicit scheme for spatial derivatives and our the self-consistent approach. Simulation results are obtained for the initial nanoparticle size distribution function, which is shown in Fig. 1. The distribution function (smooth curve in Fig. 1.) is the lognormal type function with the mean value l = 1 and the standard deviation r = 0.5. For the group approximation we use only three groups. Let us denote the total number of nanoparticles per unit volume as nin. Then for the first group we have n01 ¼ 0:6nin

Fig. 1. Nanoparticle size distribution function and their three group approximation.

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(RB1 = 2 nm). Corresponding, n02 ¼ 0:3nin (RB2 = 4 nm) and n03 ¼ 0:1nin (RB3 = 6 nm). For all calculations we used the same initial temperature distribution of gas flow:

Tðr; 0Þ ¼


 r 2  ðT in  T w Þ þ T w ; 1 R

ð28Þ

where Tin is the temperature at the axis. This profile satisfies boundary conditions (11) and (12). Also for simulations we used the following parameters: reactor radius R = 5 mm, inlet gas velocity u0 = 0.1 m/s, the initial numerical density of all group nanoparticles in flow nin = 1018 v3, temperature Tin = 500 K, wall temperature Tw = 300 K. Pressure is equal to 1 bar and carrier gas is argon. For such parameters lt  0.0183 m = 4R and for nanoparticles with RB = 2 nm the characteristic diffusion length lB  1.2 m. Some calculated profiles of gas temperature are shown in Fig. 2. It is seen that the temperature decreases exponentially and the region of nonisothermal flow is up to 4 lt. These profiles are valid if we neglect the film growth. The simulations showed that the inlet gas temperature affect the parameters of the nanostructured carbon film. Such inlet parameters as the spatial initial nanoparticle distribution, nanoparticle radii, and some other parameters are important too. For two slightly different spatial initial distributions of nanoparticles our simulation results are shown in Fig. 3. The combined effect of the Brownian diffusion and thermophoresis of nanoparticles is obvious in Fig. 3. It is worthy to emphasize that the numerical density of carbon nanoparticles in the central radial region exceeds its inlet value. This effect occurs due to the cooling of the gas flow and its deceleration. The value of r⁄ affects the intensity of Brownian diffusion as the number density gradients increased in the case depicted in Fig. 3b. The influence of thermophoresis on the thickness of a carbon nanostructured film is shown in Fig. 4. It is obvious that the thermophoresis enhances the deposition rate of nanoparticles on cold wall. We recall that for these conditions characteristic length lt is about 4R. The influence of the inlet nanoparticle distributions on the film thickness along the reactor is shown in Fig. 5. Attention is drawn to the large-scale variations the film thickness. These plots are obtained for the monodisperse beam of nanoparticles. Such variations depend on the inlet distribution only at the beginning of the reactor. The position of the maximum thickness of deposited film can be estimated in accord with the expression given below [20]

Fig. 2. Temperature distribution along the reactor. Curve 1 is for r = 0, curve 2 is for r = 0.5R, curve 3 is for r = 0.7R, curve 4 is for r = 0.9R, h(z) = 0.

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a)

b)

Fig. 3. Radial distribution of monodisperse nanoparticles. RB = 2 nm; curve 1 is for z = R; curve 2 is for z = lt; curve 3 is for z = 5 lt. Inlet nanoparticle distribution is given by Eq. (15). (a) Is for r⁄ = 0.7R (b) is for r⁄ = 0.85R.

Fig. 4. Thickness of the deposited film. Deposition time t = 106 c, RB = 2 nm, inlet nanoparticle distribution is given by Eq. (15) with r⁄ = 0.85R, curve 1 is for Tin = 300 K (isothermal deposition); curve 2 is for 400 K; curve 3 is for 500 K.

Fig. 5. Morphology of the deposited film. Deposition time t = 106 s, RB = 2 nm, Tin = 500R; the inlet nanoparticle distribution is given by Eq. (15). Curve 1 is for r⁄ = 0.7R, curve 2 is for r⁄ = 0.85R.

z  u 0

RðR  R  hÞP ; jT w  T in j

where parameter R⁄ depends on initial distribution of nanoparticles as:

RR 2 r uðrÞnðrÞdr R ¼ R0R : ruðrÞnðrÞdr 0 The temporal scale of relatively large changes in the thickness of nanostuctured film along the reactor is shown in Fig. 6. The growth

Fig. 6. Evolution of the deposited film morphology during polydisperse deposition. Tin = 500 K, r⁄ = 0.85R, curve 1 is for deposition time t = 1.3  105 s, curve 2 is for t = 2.5  105 s, curve 3 is for t = 3.8  105 s.

rate of the maximum of the film thickness increases slightly in time. Simulation results shown that during growth of the film position of the maximum of film thickness practically was practically the same. The morphological parameters of nanostructured film are shown in Figs. 7–9. The relative contributions of each group to the total number of deposited nanoparticles are shown in Fig. 7. It is interesting that for z > 0.5lt the relationship between the groups does not change significantly. The first group is about 65% of all nanoparticles, the second one is 20%, and the third group is 15%. The relative volumes of deposited nanoparticles in the nanostructured film are shown in Fig. 8. At the beginning of the reactor the first group of nanoparticles (the smallest ones) is the main contributor to the film volume. The reason is that the significantly larger value of the Brownian diffusion coefficient for the first group of nanoparticles. The contributions of other groups nanoparticles increase gradually along the reactor. The relative contributions of each deposited nanoparticle group in such important parameter as the total surface of nanostructured carbon film are shown in Fig. 9. It is interesting to note that the main contribution is made by the second group of nanoparticles except those from the inlet region. The total surface area of all deposited nanoparticles versus the time is plotted in Fig. 10. It is obvious that the surface area is directly proportional to the deposition time. Also, the initial spatial distribution of nanoparticles affects the growth rate of the total surface area of the nanostructured film.

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Fig. 7. Relative deposition flow of nanoparticles. Tin = 500 K, r⁄ = 0.85R, curve 1 is for RB = 2 nm, curve 2 is for 4 nm.

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Fig. 10. The growth of total nanoparticle surface area for z = 4lt. Curve 1 is for r⁄ = 0.7R, curve 2 is for r⁄ = 0.85R.

Fig. 11. Relative composition of deposited film at z = 4lt.

Fig. 8. Volumetric ratio of each nanoparticle group. Tin = 500 K, curve 1 is for RB = 2 nm, curve 2 is for RB = 4 nm; inlet nanoparticle distribution is given by Eq. (15) where r⁄ = 0.85R.

Fig. 12. Deposition coefficient vs. the size of nanoparticles at z = 4lt. Curve 1 is for r⁄ = 0.7 R, curve 2 is for r⁄ = 0.85R.

4. Conclusions Fig. 9. Relative surface area of nanoparticle groups vs. distance. Tin = 500 K, curve 1 is for RB = 2 nm, curve 2 is for RB = 4 nm; inlet nanoparticle distribution is given by Eq. (15) where r⁄ = 0.85R.

For the position along the reactor axis z = 4lt the composition of the nanostructured film is shown in Fig. 11. Due to the more intensive Brownian diffusion of the smallest nanoparticles, the contribution of the first group is higher than their fraction in the initial composition of nanoparticles beam. It is useful to remind that the thermophoretic velocity does not depend on the nanoparticles radii.

The mathematical model of the deposition of a polydispersed beam of carbon nanoparticles and the self-consistent growth of nanostructured film on wall of cylindrical reactor has been developed. For gas flow we used the description of a continuous medium and the free molecular approximation for describing of interaction of nanoparticles with gas flow. We consider that the temperature of the gas flow differs from the wall temperature, therefore we calculated fields of temperature in the gas flow and in the film. For film porosity we use our experimental data [18]. The simulation results are presented for the porosity equal to 0.7.

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It was shown that the Brownian diffusion of nanoparticles is the key factor of the growth nanostructured film and nanoparticle deposition. Thermophoresis can drastically enhance this process if the temperature difference between cold wall and flow is several hundred degrees. In order to take into account the feedback between the film growth and deposition process, for simulation we use a selfconsistent approach. In our simulation the film thickness reached several percentages of the reactor radius. A qualitative analysis of the equations of our model showed that for a monodisperse beam of nanoparticles the thickness of the deposited film h(t) is

h

RB n t : Pð1  eÞ

For a polydisperse beam we have generalization of this expression

h

X t RBi ni : Pð1  eÞ

The actual calculation of porosity of a nanostructured film has to be based on more sophisticated mathematical models. The qualitative analysis of the mathematical model predicts that nanoparticle deposition is a self-accelerating process. The simulation results confirm this conclusion. Finally, it was shown that the thickness and the morphology of nanostructured film can be controlled by several initial parameters, such as the initial gas temperature and the radius of nanoparticles beam. It is worthy to note, that the efficiency of deposition of nanoparticles can be characterized by the parameter K. To calculate K we use the formula from [20]

RR K¼

0

½n0 ðrÞu0 ðrÞ  nout ðrÞuout ðrÞrdr ; RR n ðrÞu0 ðrÞrdr 0 0

where nout and uout are the numerical density of nanoparticles and outlet gas velocity respectively. The K = 0 corresponds to the absence of nanoparticle deposition on the reactor wall. If R = 1, all of the nanoparticles are deposited. The lower estimate of the deposition efficiency can be obtained as

K min ¼ 1  expðL=lB Þ  L=R2B R2 u0 : For different initial nanoparticle distributions, (Eq. (15)) the deposition efficiency K versus the size of nanoparticles is shown in Fig. 12 at z = 4lt. Our mathematical model can be used for simulating the deposition of any small nanoparticles from nonisothermal gas flow. But to

simulate of porosity of nanostructured film experimental data or a more sophisticated mathematical and physical model are be used. References [1] R. Balesky, Equilibrium and Nonequilibrium Statistical Mechanics, John Wiley and Sons, New York, 1975. [2] M. Kubo, Y. Ishihara, Y. Mantani, M. Shimada, Evaluation of the factors that influence the fabrication of porous thin films by deposition of aerosol nanoparticles, Chem. Eng. J. 232 (2013) 221–227. [3] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media, 2nd ed., Kluwer Academic, Dordrecht, 1991. [4] A.A. Brin, S.P. Fisenko, A.I. Shnip, Brownian deposition of nanoparticles from a laminar gas flow through a channel, Tech. Phys. 53 (2008) 1141–1145. [5] S.P. Fisenko, A.A. Brin, Heat and mass transfer and condensation interference in a laminar flow diffusion chamber, Int. J. Heat Mass Transfer 49 (2006) 1004– 1014. [6] D.A. Takopulo, S.P. Fisenko, Brownian diffusion and transfer of nanoparticle beam by a gas flow, J. Eng. Phys. Thermophys. 86 (2013) 1041–1045. [7] M. Shimada, T. Seto, K. Okuyama, Wall deposition of ultrafine aerosol particles by thermophoresis in nonisothermal laminar pipe of different carrier gas, Jpn. J. Appl. Phys. 33 (1994) 1174–1181. [8] X. Wang, F.E. Kruis, P.H. McMurry, Aerodynamic focusing of nanoparticles: I. Guidelines for designing aerodynamic lenses for nanoparticles, Aerosol Sci. Technol. 39 (2005) 611–623. [9] J. Storey-Bishoff, M. Noga, W.H. Finlay, Deposition of micrometer-sized aerosol particles in infant nasal airway replicas, J. Aerosol Sci. 39 (2008) 1055–1065. [10] Sh.Y. Lu, Ch.-M. Tsai, Membrane microstructure resulting from deposition of polydisperse particles, J. Membr. Sci. 177 (2000) 55–71. [11] D.J. Kraft, R. Wittkowski, B. Hagen, K.V. Edmond, D.J. Pine, H. Löwen, Brownian motion and the hydrodynamic friction tensor for colloidal particles of complex shape, Phys. Rev. E 88 (2013) 050301(R). [12] V.R. Gutti, S.K. Loyalka, Thermophoretic deposition in a cylindrical tube: computations and comparison with experiments, Nucl. Technol. 166 (2009) 121–133. [13] F.E. Spokoinyi, Z.R. Gorbis, Precipitation peculiarities of fine particles from gas flow on the perpendicular heat transfer surface, High-Temp. Thermophys. 19 (1981) 182–194 (in Russian). [14] Y.M. Tsirkunov, Modelling of the admixture flows in the problems of twophase aerodynamics. Boundary layer effects, Modell. Mech. 7 (2) (1993) (in Russian). [15] A.N. Osiptsov, Mathematical modeling of dusty-gas boundary layer flows, Appl. Mech. Rev. 50 (1997) 357–370. [16] L. Talbot, R.K. Cheng, R.W. Schefer, D.R. Willis, Thermophoresis of particles in a heated boundary layer, J. Fluid Mech. 101 (1980) 737–758. [17] Y.A. Stankevich, S.P. Fisenko, Reorganization of the Poiseuille profile in nonisothermal flows in the reactor, J. Eng. Phys. Thermophys. 84 (2011) 1083–1086. [18] D.A. Takopulo, S.P. Fisenko, Heat and mass transfer in the system hydrocarbon gas – porous carbon layer – metal and formation of supersaturated solid carbon solutions, J. Eng. Phys. Thermophys. 85 (2012) 539–548. [19] W.E. Schiesser, The Numerical Method of Lines: Integration of Partial Differential Equations, Academic Press, San Diego, 1991. [20] S.P. Fisenko, D.A. Takopulo, Brownian diffusion and large-scale morphology of nanostructured film on the wall of a flow reactor, Doklady Nat. Acad. Sci. Belarus 59 (2015) 108–114 (in Russian).