Numerical study to predict the particle deposition under the influence of operating forces on a tilted surface in the turbulent flow

Advanced Powder Technology 22 (2011) 405–415

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Original Research Paper

Numerical study to predict the particle deposition under the influence of operating forces on a tilted surface in the turbulent flow M. Abdolzadeh a,*, M.A. Mehrabian a, A. Soltani Goharrizi b a b

Department of Mechanical Engineering, Faculty of Engineering, Shahid Bahonar University of Kerman, Kerman, Iran Department of Chemical Engineering, Faculty of Engineering, Shahid Bahonar University of Kerman, Kerman, Iran

a r t i c l e

i n f o

Article history: Received 23 February 2010 Received in revised form 6 June 2010 Accepted 14 June 2010 Available online 25 June 2010 Keywords: Particle deposition V2-f turbulence model Tilt angle Eulerian approach Operating forces

a b s t r a c t This study uses a v2-f turbulence model with a two-phase Eulerian approach. The v2-f model can accurately calculate the near wall fluctuations in y-direction, which mainly represent the anisotropic nature of turbulent flow. The model performance is examined by comparing the rate of particle deposition on a vertical surface with the experimental and numerical data in a turbulent channel flow available in the literature. The effects of lift, turbophoretic, electrostatic and Brownian forces together with turbulent diffusion are examined on the particle deposition rate. The influence of the tilt angle and surface roughness on the particle deposition rate were investigated. The results show that, using the v2-f model predicts the rate of deposition with reasonable accuracy. It is observed that in high relaxation time the effect of lift force on the particle deposition is very important. It is also indicated that decreasing the tilt angle from 90° to 0° enhances the deposition rate especially for large size particles. Furthermore, the results show that increasing the Reynolds number at a specific tilt angle decreases the rate of particle deposition and the tilt angle has insignificant impact on the particle deposition rate in high shear velocity or high Reynolds number. Ó 2010 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved.

1. Introduction The deposition of suspended particles from turbulent gas flow on adjacent surfaces has been extensively investigated during the last 30 years. The understanding and prediction of deposition mass flux is of great interest in areas like pollution control, gas cleaning, design of industrial reactors or transport of particles in two-phase flow systems. Many publications explain calculations of the inertial deposition of particles in 2D flow fields while the effect of turbulent flow has not been considered. Most researchers used a Lagrangian approach in which the particle equations of motion were integrated along the particle pathlines. In such deterministic flows, a few pathline calculations give a good representation of the particle velocity field. Gosman and Ioannides [1] obtained the flow field using the random sampling of a crude turbulence model at each time-step. A similar method was reported by Kallio and Reeks [2] but problems remained in coping with very small particles when Brownian diffusion was important. Li and Ahmadi [3] developed a near-wall model using DNS analysis to capture the near wall fluctioatins. Ounis et al. [4] and, Brooke et al. [5] predicted the motion of particles where the fluid motion was predicted by di* Corresponding author. Tel.: +983413201086. E-mail address: [email protected] (M. Abdolzadeh).

rect numerical simulation (DNS). Using large eddy simulation [6] or direct numerical simulation methods [7] improved representations of the turbulence but consumed larger computational time. Healy and Young [8] showed that the particle concentration field can also be predicated accurately and efficiently if the so-called full-Lagrangian approach is used, but complications arise when the particles respond to the turbulence flow regime because Lagrangian approach includes a stochastic element in the governing equations. Tian and Ahmadi [9] successfully applied the nearwall model with a Reynolds stress model (RSM) to predict particle deposition in channel flows. Lai and Chen [10] adopted the RNG k-e turbulence model to predict indoor particle dispersion, and deposition rate was used to quantify the wall-normal turbulent fluctuation within the viscous layer near the wall. Marchioli et al. [11] reported detailed statistics for velocity and deposition rates of heavy particles dispersed in turbulent boundary layers using DNS. However, Lagrangian approach typically involves the determination of trajectories of a very large number of particles (to establish statistically meaningful average quantities) and may be too time consuming to be effective as a practical calculation method, especially for small particles. Therefore, the two-phase Eulerian approach which is computationally more efficient was adopted for the present work. The air flow properties which can be obtained analytically or experimentally are used as the input parameters

0921-8831/$ - see front matter Ó 2010 The Society of Powder Technology Japan. Published by Elsevier B.V. and The Society of Powder Technology Japan. All rights reserved. doi:10.1016/j.apt.2010.06.005

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M. Abdolzadeh et al. / Advanced Powder Technology 22 (2011) 405–415

Nomenclatures Cc CD dp DB FD FL FB FE g J k ks q Rep R Res Sc T TL tp us up

Cunningham coefficient drag coefficient particle diameter Brownian diffusivity drag force lift force Brownian force electrostatic force gravitational acceleration wall mass flux Boltzmann constant effective roughness height particle’s electrical charge Reynolds number based on velocity of particles relative to air radius of channel Reynolds number based on the shear velocity Schmitt number fluid absolute temperature Langragian time scale relaxation time shear velocity particle velocity in x-direction

of the Eulerian model. Cleaver and Yates [12] developed a theory for particle deposition based on the bursts and ejections of turbulent fluid into the laminar sublayer based on Eulerian approach. They did not account for the finite dispersion in the core flow, but even so they reproduced acceptable experimental data. Reeks [13] deduced the migration effect (turbophoresis) and showed that turbulent migration could have a strong impact on particle deposition. Guha [14] developed a unified model which considers all the transfer mechanisms of particle deposition. His results were validated when compared with the experimental data. Nazaroff and Lai [15] developed an Eulerian particle deposition model, in which some of transport mechanisms in the particle deposition were not considered. Their model prediction agreed reasonably well with the experimental data. Zhao and Wuo [16] considered the turbophoretic forces in the Nazorrof model and improved their results. However, they did not consider the lift and electostatic forces, but borrowed the turbulence features from the literature. Zhao and Chen [17] used the Nazaroff model with the k-e turbulence model for predicting the air flow turbulence features. However, they did not consider the turbophoretic force and their results underestimated the measured data in some cases. Even if the turbophoretic force was taken into account, the k–e model was not able to give accurate near wall fluctuations. The previous studies were mainly concentrated on the particle phase and borrowed the required information of air flow from literature, this makes Eulerain approach less flexible. This paper tends to adopt a numerical approach based on boundary layer analysis to obtain detailed information of turbulence features. The turbulence features of flow were found using v2-f turbulence model. This is the beauty of v2-f model which provides real quantities for the normal fluctuations near the wall. The numerical results were validated when compared with the measured data from literature, and then applied to study the particle deposition rate on a tilted surface under different conditions. The effects of Brownian, turbophoretic, lift, electrostatic and gravitational forces on the particle deposition rate were examined. The deposition of particles at different tilt angles and shear velocities was then studied.

uF

vF vp

VE Vdep

v 0p v 0F

air flow velocity in x-direction air flow velocity in y-direction particle velocity in y-direction electrostatic drift velocity particle deposition velocity particle fluctuation velocity in normal direction air flow fluctuation velocity in normal direction

Greek symbols qp particle concentration l fluid viscosity b tilt angle l fluid viscosity qp0 mean particle concentration qpmax maximum particle concentration f fraction of the maximum charge e0 electric permittivity of vacuum k mean free path of the air molecules qf fluid density vt air flow turbulent viscosity v pt particle turbulent viscosity  mean + non-dimensional

2. Modeling of air flow and turbulent features Accurate prediction of air flow and turbulence is very important to the success of modeling the particle deposition on the surfaces [9]. The v2-f model for predicting the air flow in this study was developed by Lien and Kalitzin [18] and Davidson et al. [19]. It has been shown that this model has a good accuracy in predicting the mean flow and the turbulence features compared to other turbulence models [20]. The model formulation has the general form of:

q

  @/ @/ @ @/ ¼ S/ þ quj  C/;eff @t @xj @xj @xj

ð1Þ

where / represents the independent flow variables, C/;eff the effective diffusion coefficient, Su the source term, q the flow density and the bars denote the Reynolds averaging. In Table 1 the mathematical form of each transport equation of the v2-f model are summarized. p is the air pressure, lt the turbulent viscosity, S the rate of strain, f a part of the v 02 source term and T the turbulent time scale. The appropriate boundary conditions of turbulence variable near the wall are as follows:

k ¼ v 02 ¼ f ¼ 0;

e ¼ 2t

k y2p

ð2Þ

yp is the distance from the cell center to the wall. 3. Particle phase modeling The particle phase as well as the fluid phase is described in the Eulerian frame of reference. The Ramshaw approach [21] described the motion of a particle cloud in a fluid flow. It is assumed that a dilute particle phase with no coupling between the particles and the fluid is under investigation. A criterion for having a dilute suspension is that the particle phase bulk density is negligible when compared with the gas phase density, i.e., qBulk,p << qBulk,gas.. In this study the particles are assumed to be heavy, qparticle >> qfluid. The motion of particle phase is governed by the continuity and momentum equations. Fig. 1 shows a tilted surface where the particle phase is moving above the surface.

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M. Abdolzadeh et al. / Advanced Powder Technology 22 (2011) 405–415 Table 1 Coefficients and source terms for Eq. (1). Independent flow variable /

Effective diffusion coefficient

C/;eff

Source term S/

Continuity x-Component of momentum

1 u

0

0 @ @u @ @v  @p @x þ @x ðlt @x Þ þ @y ðlt @x Þ

y-Component of momentum

v

lt þ l lt þ l

Turbulent kinetic energy (k) Turbulent kinetic energy dissipation rate (e)

k

Gk  qe

e

Wall normal turbulence fluctuations (v 0 2)

v02

lt =rk;t þ l lt =re;t þ l lt =rk;t þ l

Name of conservation equation

@ @v  @p @y þ @y ðlt @y Þ þ

@

@x

ðlt @u @yÞ

ðC e1 Gk e  C e2 qe2 Þ=k h i minðkf ;  T1 ðC 1  6Þv 0 2  23 kðC 1  1Þ þ C 2 P k Þ

pffiffi 0 @2 f C2 Pk C 1 0v 0 2 2  1T ð6 vk2  23Þ; P k ¼ 2C l v 0 2TS2ij ; T ¼ maxðke ; 6 meÞ. L2 @x 2  f ¼ T ð k  3Þ  k qffiffiffiffiffi 3=2 2 3=4 L ¼ C L maxðk e ; C g te1=4 Þ; lt ¼ minfq C l;k ke ; q C l;v 2f v 02 Tg; C e1 ¼ 1:4ð1 þ 0:05 vk02 Þ. C e2 ¼ 1:9; C 1 ¼ 1:4; C 2 ¼ 0:3; C l;v 2f ¼ 0:22; C L ¼ 0:23; C g ¼ 70;

rk;t ¼ 1; re;t ¼ 1:3.

F Ly

rffiffiffiffiffiffiffiffi u  0:171 @ F ¼ qp dp ðuF  uP Þ tp @y t

ð11Þ

The electrostatic force due to mirror charging is given by Boothroyd [24] as:

g F Ey ¼ qp

x

y

In the steady state condition, the equations of motion in a two dimensional flow field are:

@ðqp up Þ @ðqp v p Þ þ ¼0 @x @y @ðqp u2p Þ @ðqp up v p Þ ¼ ðF Dx  qp g sin bÞ þ @y @x @ðqp up v p Þ @ðqp v 2p Þ þ ¼ ðF Dy  qp g cos b þ F Ly þ F Ey þ F By Þ @x @y

ð3Þ ð4Þ ð5Þ

where up and vp are the particle velocities in x- and y-directions, respectively, qp the particle concentration, FD the drag force, FL the lift force, FB the Brownian force and FE the electrostatic force. The Drag force is defined as:

F D ¼ qp ðVF  VP ÞC D Rep =ð24t p Þ

ð6Þ

VF ¼ uF i þ v F j;

ð7Þ

V p ¼ up i þ v p j

ð8Þ ð9Þ

VF is the fluid velocity, tp the relaxation time, qf the fluid density, l the fluid viscosity, Cc the Cunningham coefficient [22] dp the particle diameter, k the mean free path of the air molecules, CD the drag coefficient, Rep the Reynolds number based on velocity of particles relative to air, g the gravitational acceleration and b the tilt angle. The drag coefficient is either derived by Stokes equation (Rep < 1) or by an empirical relation (1 < Rep < 1000) [22]:

CD ¼

The lift force is given by Saffman [23] as:

ð13Þ ð14Þ

DB is the Brownian diffusivity, T the fluid absolute temperature and k the Boltzmann constant. Eqs. (3)–(14) are now simplified assuming negligible axial gradient of the particle field compared with normal gradient, (@/@x = 0). The up, vp, qp, uF and vF are now decomposed into a mean and a fluctuating component:

uF ¼ U F þ u0F ; up ¼ U p þ u0p ;

v F ¼ V F þ v 0F ; qp ¼ qp þ q0p ; v p ¼ V P þ v 0p

ð15Þ

The average y-component of fluid velocity is assumed to be zero, i.e., V F ¼ 0. Inserting Eq. (15) into Eqs. (3)–(14) and time averaging of the resulting equations give the following [14]1:

@ qp @y

ð16Þ !

 @U p @ @U q C Re  qp tpt p þ p D p U F  U p  qp g sin b ¼ @y @y @y 24tp



 @ qp @V p qp V p C D Rep @ @q D @q tpt p  B p J  tpt ¼ þ Vp @y @y @y tp 24 @y t p @y J

l

for Rep < 1   0:678 for 1 < Rep < 1000 24=Rep 1 þ 0:15Rep

DB @ðln qp Þ @y tp kT DB ¼ 3pdp l

F By ¼ 

J ¼ qp V p  tpt

2

 dp qop C c 2k  tp ¼ ; Cc ¼ 1 þ 1:257 þ 0:4eð1:1dp =2kÞ dp 18l jV P  V F jdp qf Rep ¼

24=Rep

ð12Þ

VE is the electrostatic drift velocity, q the particle’s electrical charge, e0 the electric permittivity of vacuum and y the distance between the wall and the center of the particle. It should be noted that the effects due to mirror charging exist if either the walls or the particles are charged. The Brownian force is:

Tilt Angle, β

Fig. 1. Schematic diagram of the tilted surface.

(

q2 qp VE ¼ tp 12sp2 e0 ldp y2

ð17Þ

@ < v 02 q2 p > þ     qp 2 12sp e0 tp ldp y2 @y v ffiffiffiffiffiffiffiffiffi 0 0u 11 U   @ u 0:171 F þ dp qp U F  U P @ @t AA t tp @y

 qp

)

 qp g cos b

ð18Þ

ð10Þ 1

The details regarding the derivation of Eqs. (16)–(18) are given in Guha [14].

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M. Abdolzadeh et al. / Advanced Powder Technology 22 (2011) 405–415

J is the wall mass flux, v 02 p the particle fluctuation velocity in normal direction and the bars denote the Reynolds averaging. 4. Dimensionless formulation of governing equations Eqs. (16)–(18) can be solved exactly if an accurate expression for the particle rms velocity as a function of y is available. The fluid rms velocity has been found as a function of y by the proposed CFD model in this paper. The ratio of particle fluctuation mean–square velocity to the mean fluid square velocity is defined as:

R¼

< v 02 p >

ð19Þ

< v 02 F >

1 1 þ 0:7ðtp =T L Þ

ð20Þ

TL is the Langragian time scale and is defined as:

TL ¼

tt

Eqs. (16)–(18) are converted into dimensionless form: J

þ

Jþ

¼ V þdep

¼q

þ

@U @ ¼ @yþ @yþ

q

þ pþ @ p þ þ p Vp  t @yþ

t

ð23Þ



þ @ qþp @V þp qþp V þp C D Rep pþ @ qp þ @ J þ  tpþ ¼ V t þ t t p þ @yþ @yþ @yþ tp 24 @yþ 

@ < v 0þ2 > 1 @ qþp p  qþp þ þ Sc t p @y @yþ |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} 

q2 u3s

0:171 þ þ þ dp qþp þ tp 12tþp p2 e0 ldp yþ2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!   @  þ   U þF  U þp UF @yþ

 g  cos bt  qþp u3s þ p

q

ð24Þ

It should be noted that the fourth term in the right hand side (*) of Eq. (24) is the turbophoretic force. The non-dimensional parameters are defined as:

V þdep

¼ J=ðqp0 us Þ; 2

u t þp ¼ t p s ;

t

< m0þ2 >¼ F

þ

y ¼ yus =m;

þ

dp ¼ dp us =m; < m02 F > ; u2s

< m0þ2 p

þ p

q ¼ qp =qp0 ;

Up ; us < m02 p > >¼ ; u2s

U þp ¼

Sc1 ¼

DB

t

ð25Þ

us is the shear velocity, Vdep the particle deposition velocity and qp0 the mean concentration. The particle turbulent viscosity is given by Hinz [26] as:

tpþ t

tþt

¼

1 tp 1þ TL

dp 2

ð27Þ

ks is the effective roughness height as shown in Fig. 2. The boundary conditions are now:

at yþ ¼

dþ p 2

þ

þ 0:45ks  gttþ cos b  þ þ p 24 þ V P ¼  u3 at y ¼ R C D Rep s gttþ sin b    p 24 U þP ¼ U þF Rþ  at yþ ¼ Rþ C D Rep u3

ð28Þ

s

ð22Þ

!

 @U þp qþp CD Rep  þ þ  þ g  sinbt qþp tpþ U  U q þ  t F P p @yþ 24t þp u3s

y0 ¼ 0:45ks þ

qþp ¼ 0 ð21Þ

< v 02 F >

particle is deposited when its center of mass is at a distance dp/2 away from the effective roughness height, ks [14]. The origin of velocity profile is taken as:

yþ ¼ yþ0 ¼ y0  us =t;

where R is recommended by Binder and Hanaraty [25] as:

R¼

Fig. 2. Schematic diagram for roughness parameter.

@U þ P @yþ

at yþ ¼

¼0

dþ p 2

where R+ is the non-dimensional radius of channel, Rþ ¼ R us =t. 5. Numerical method The air phase equations were discretized and converted into algebraic equations using the finite volume method. The coupled velocity and pressure equations were solved using SIMPLE algorithm. The resulting set of discretized equations for each variable were solved by a line-by-line procedure, using the tri-diagonal matrix algorithm (TDMA) and the linear under-relaxation iteration [27]. The governing equations describing qp and V p were discretized using a standard explicit technique, while the U p equation which is an elliptic differential equation was discretized by a standard implicit technique. The TDMA method was applied to solve the U p equation. A non-uniform grid system was employed in the present study with finer grids near the surface to resolve the boundary layer properly. 6. Validation of the air flow numerical model In order to execute the model, a good representation of the turbulent kinematic viscosity, the fluid rms velocity and the mean fluid velocity throughout the boundary layer are required. As mentioned before, the fully developed turbulent channel flow is solved and the required information was taken for the particle phase. Fig. 3 shows the geometry of channel with the dimensions of 0.0127  0.8 m2. Fig. 4a shows the computational channel grid. The computational domain and the flow conditions were selected according to Liu and Agrawal [28] experiment. It should be mentioned that the computational grid of the flat plate is identical to one half of the computational grid of the channel, this is indicated

ð26Þ

Since the majority of data on particle deposition is available for tube flow, it is natural to verify the model against reference experimental data in tube flow. The region under investigation in this study is a small segment close to the surface and the curvature effects in the governing equations are negligible. It is assumed that a

þ

þ 0:45ks

h=0.0127m

L=0.8m Fig. 3. Geometrical information of the channel.

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M. Abdolzadeh et al. / Advanced Powder Technology 22 (2011) 405–415

a

b

0.012

0.006

0.01

0.005

Y(m)

Y(m)

0.008 0.006

0.004 0.003

0.004

0.002

0.002

0.001

0.005

0

0.01

0

0.002

0.004

0.006

X(m)

X(m)

Fig. 4. (a) Non-uniform grid used to calculate the flow field, (b) the computational domain of tilted plate with higher resolution.

a

6.5

b

6 v+2 DNS Kim et al. [29] v+2 CFD v2-f k+ DNS Kim et al.[29] k+ CFD v2-f

5.5

6 5.5 5

4.5

4.5

4

4

k+,v+2

k+,v+2

5

6.5

3.5 3

3.5 3

2

2

1.5

1.5

1

1

0.5

0.5

0

v+2 DNS Kim et al. [29] v+2 CFD v2-f k+ DNS Kim et al. [29] k+ CFD v2-f

2.5

2.5

0

100

200

0

300

0

5

10

15

20

y+

y+

Fig. 5. Comparison of predicated turbulent flow features with DNS results [29]: (a) normal fluctuation velocity and turbulent kinetic energy near the wall; (b) a higher resolution of normal fluctuation velocity and turbulent kinetic energy near the wall.

a

b 20

20

15

15

DNS-Kim et. al [29] CFD v2f

U+

U+

DNS-Kim et. al [29] CDF v2-f 10

5

5

0

10

0

100

200

y+

300

0

0

5

10

15

y+

Fig. 6. Comparison of predicated mean flow with DNS results [28]: (a) mean flow velocity; (b) a higher resolution of mean flow velocity.

20

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M. Abdolzadeh et al. / Advanced Powder Technology 22 (2011) 405–415

in Fig. 4b. The CFD Model has been validated against the fully developed turbulent channel flow at Res = 395 [29]. Res is the Reynolds number based on the shear velocity, us. As shown in Figs. 5 0 and 6, the non-dimensional predicted profile of k, v 2 and UF are in reasonable agreement with DNS data of Kim et al. [29].

7. Results and discussion 7.1. Particle model validation when electrostatic force and roughness are not present In this part the most important factors influencing the particle deposition, i.e., the Brownian, drag, lift and gravitational forces

0

log(Vdep+)

1

Present Work Liu & Agarwal [28] Sehmel [31] Wells & Chamberlain [32]

-1

are considered. The effects of these factors on particle deposition are studied. The rate of deposition under the effects of the above factors were then compared with reference experimental and theoretical data. Figs. 7 and 8 are plots of non-dimensional deposition þ velocity, V þ dep , versus non-dimensional relaxation time t p . The Reynolds number and the density ratio q0p =qf are taken as 10,000 and 770, respectively, as in Liu and Agarwal [28] experiments. The particle size ranges between 0.1 and 100 lm. Fig. 7 shows the variation of deposition velocity with relaxation time. To compare the present work with the reference experimental data [31,32], three sets of data were chosen. Fig. 7 also shows, the present calculation is compatible with the measured deposition velocity both qualitatively and quantitatively. In Fig. 8 the present work is compared with the large eddy simulation (LES) results of Wang [6] and

ρp/ρpmax

0.8

0.6

-2

0.4

-3

0.2

-4

|Up-Uf|/|(Up-Uf)max|

|Vp|/|Vpmax|

0 -5

-2

-1

0

1

2

0

3

100

200

300

y+

log(tp+) Fig. 7. Comparison of predicted deposition velocity with experimental data, þ Re = 10,000, us = 0.745 m/s, f = 0, ks = 0, b = 90.

Fig. 9. Distribution of qp/qpmax, |Up  UF|/|(Up  UF)max| and |Vp|/|Vpmax| for t þ p = 0.1, þ Re = 10,000, f = 0, ks = 0, b = 90.

1 0

Present work(v2-f) LES [6] k-epsilon model [9] RSM model [9] DNS[11]

-0.5 -1

0.8

-1.5

log(Vdep+)

-2

0.6

-2.5 -3

0.4

-3.5

ρp /ρ pmax

-4 -4.5

0.2 |Up-Uf|/|(Up-Uf)max|

-5 -5.5 -6

0 -3

-2

-1

0

1

2

3

log(tp+) Fig. 8. Comparison of predicted deposition velocity with numerical data, þ Re = 10,000, us = 0.745 m/s, f = 0, ks = 0, b = 90.

|Vp|/|Vpmax|

100

200

300

y+ Fig. 10. Distribution of qp/qpmax |Up  UF|/|(Up  UF)max| and |Vp|/|Vpmax| for t þ p = 1, þ Re = 10,000, f = 0, ks = 0, b = 90.

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M. Abdolzadeh et al. / Advanced Powder Technology 22 (2011) 405–415

DNS results of Marchioli [11] as well as the k–e and RSM results of Tian and Ahamdi [9]. It is clear that the particle deposition velocity is over-predicated when calculated by the k–e model. This is because of the inability of k–e model to predict the normal fluctuations near the surfaces accurately. The v2-f model, the RSM model with the quadratic correction and the LES and DNS methods predict the near wall turbulence correctly. This comparison confirmed that it is important to predict the near wall turbulence accurately in order to model the particle deposition correctly. It should be mentioned that only the data of Liu and Agarwall [28] were plotted as the experimental reference, since these data are generally accepted as one of the most reliable data sets. Figs. 9–15 show how V p ; U p  U F and qp =qpmax vary in the velocity boundary layer for tþ p = 0.1, 1, 10, 100 and 1000. As shown in these figures, the value of U p  U F at the wall increases with tþ p , be-

cause the particles do not obey the no-slip condition. The peak of normal particle velocity at yþ ffi 15 corresponds to the maximum turbophoretic force, for t þ p = 0.1, 1 and 10. This is clearly shown in Fig. 11, where the effect of lift force on the particle deposition is negligible. For t þ p = 100 and 1000, the effect of lift force is clearly shown as the particles continue to accelerate towards the wall, despite the reduction of the turbophoretic force for y+ < 5. This fact is apparently shown in Fig. 15. However, this figure suggests for y+ > 5, the turbophoretic force is still the dominant force. The reason of increasing the lift force for high t þ p is increasing the value of U p  U F at the wall. The distribution of non-dimensional particle concentration for tþ p = 0.1 as shown in Fig. 9, indicates that turbulent and Brownian diffusion dominate. The very steep gradient of concentration distribution is necessary to maintain the particle

1 0

0.8 Turbophoretic and Lift Forces Included Turbophoretic Force Excluded Lift Force Excluded

-0.01

ρ/ρmax

Vp+

0.6 -0.02

0.4 -0.03

|Vp|/|Vpmax|

0.2 |Up-Uf|/|(Up-Uf)max|

-0.04 100

200

100

300

200

300

y+

y+ Fig. 11. The effect of lift and turbophoretic forces on wall-normal particle velocity, þ tþ p = 1, Re = 10,000, f = 0, ks = 0, b = 90.

Fig. 13. Distribution of qp/qpmax, |Up  UF|/|(Up  UF)max| and |Vp|/|Vpmax| for þ tþ p = 100, Re = 10,000, f = 0, ks = 0, b = 90.

1

1

0.8

0.8 ρp /ρpmax

0.6

0.6

0.4

ρp /ρpmax

0.4

|Vp|/|Vpmax|

|Vp|/|Vpmax|

0.2

0.2 |Up-Uf|/|(Up-Uf)max|

|Up-Uf|/|(Up-Uf)max|

0 100

200

300

y+ Fig. 12. Distribution of qp/qpmax, |Up  UF|/|(Up  UF)max| and |Vp|/|Vpmax| for t þ p = 10, þ Re = 10,000, f = 0, ks = 0, b = 90.

100

200

300

y+ Fig. 14. Distribution of qp/qpmax, |Up  UF|/|(Up  UF)max| and |Vp|/|Vpmax| for þ tþ p = 1000, Re = 10,000, f = 0, ks = 0, b = 90.

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flux by the Brownian diffusion when the fluid turbulence intensity is very low. The major difference in concentration distribution is in þ þ tþ p < 10 and t p > 10, because in t p < 10 the value of V p at wall is zero and U p  UF is small and therefore, the particle build-up in the þ boundary layer increases with increasing t þ p . For t p P 10 the near-wall particle concentration assumes lower values, because V p and U p  U F at the wall take much larger values. A comparison of Figs. 9–15 with results extracted from [33] to [34] shows that there is good qualitative agreement between the present results and the reference results that were obtained using DNS data as the input of the Eulerian Particle model. Fig. 16 shows the impact of lift force, Brownian diffusion, turbulent dispersion and turbophoretic force on the particle deposition rate. For tþ p 6 0.1, the deposition is completely controlled by the Brownian diffusion and turbulent dispersion. In the range of 0.1 6 tþ p 6 10 the predicted

deposition rate is controlled by the turbophoretic force. For tþ p P 10, the lift and turbophoretic forces dominate. When the lift and turbophoretic forces are excluded, for tþ p P 4, particles are too large to respond to the rapid fluctuations of near wall eddies and the transport to the wall due to turbulent diffusion is very slow. 7.2. Validation of the model when electrostatic force and roughness are present In this part the effects of electrostatic force, roughness as well as lift, drag, Brownian and gravity forces are considered on the particle deposition rate. The electrostatic force due to mirror charging is

0 0

Zeta=0.003 Zeta=0.03 Zeta=0.3 Liu & Agarwal [28]

-1

log(Vdep+)

-0.1

Vp+

-0.2

-2

-3

-0.3 Turbophoretic Force Exlcuded Lift Force Exlcuded Lift & Turbophoretic Forces Included

-0.4

-4

-5 -0.5

100

200

-2

-1

0

1

2

3

log(tp+)

300

y+ Fig. 15. The effect of lift and turbophoretic forces on the wall-normal particle þ velocity, tþ p = 1000, Re = 10,000, f = 0, ks = 0, b = 90.

Fig. 17. The influences of electrostatic charge on the deposition velocity, þ Re = 10,000, ks = 0, b = 90.

0 All Forces Included Lift Force Excluded Turbophoretic Force Excluded Turbophoretic and Lift Forces Excluded Liu & Agarwall [28]

0

ks+=0 ks+=0.35 ks+=1 Liu & Agarwall [28]

-1

log(Vdep+)

log(Vdep+)

-1

-2

-3

-2

-3

-4

-4 -5 -2

-1

0

1

2

3

log(tp+) Fig. 16. Effect of lift force, turbophoretic force and Brownian and turbulent diffusion on the particle deposition velocity, Re = 10,000, f = 0, b = 90.

-5

-2

-1

0

1

2

3

log(tp+) Fig. 18. The influence of surface roughness on the deposition velocity, Re = 10,000, f = 0, b = 90.

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now investigated. The maximum charging of a particle, according to Hesketh [30] is:

qmax ¼ 500eðdp =106 mÞ2 ;

dp > 107 m

ð29Þ

where e is the elementary charge, e = 1.602  1019 C. The particle charge is therefore assumed to be a fraction of the maximum charge qmax given as:

q ¼ qmax  f;

f61

ð30Þ

Calculated deposition rates for f = 0.003, 0.03 and 0.3 are shown in Fig. 17. The deposition velocity is strongly increased by electrical charges for intermediate tþ p . This strong effect is caused by the turbulent migration of particles into the wall boundary layer which is

enhanced as a result of the electrostatic force. The electrostatic force is dominant in a short range of t þ p , it is even more important than the Brownian diffusion and turbulent dispersion near the wall. Hence, near-wall resistance is lowered when the particle charge is increased and the deposition rate will increase accordingly. The theoretical deposition rates are now compared with experimental results of Liu and Agarwall [28]. The electrical charge reported in Liu and Agarwall correspond to f = 0.003. However, the charge distributions were measured at exit of their aerosol generator and it is reasonable to expect that the non-charged particles might pick up a few more charges on their way from the aerosol generator to the test section. The results are shown in Fig. 17. The value of f = 0.003, given in the experiment of Liu and Agarwall leads to under-prediction of the deposition rate for tþ p 6 6. However, as shown in this figure the value of 0.03 gives good agreement

0 Tilt Angle=0 Tilt Angle=40 Tilt Angle=60 Tilt Angle=90 Liu & Agarwall [28]

0.8

Tilt Angle=0 Tilt Angle=40 Tilt Angle=90

-2 0.6

ρ/ρmax

log(Vdep+)

-1

-3

0.4

-4 0.2

-5

-2

-1

0

1

2

3 0

log(tp+) Fig. 19. Comparison of predicted deposition velocity at different tilt angles, þ Re = 10,000, f = 0, ks = 0.

100

200

300

y+ Fig. 21. Comparison of particle concentration at different tilt angles, tþ p = 1000, þ Re = 10,000, f = 0, ks = 0.

0 Tilt Angle=0 Tilt Angle=40 Tilt Angle=90

-0.1

20

-0.2 19 Tilt Angle=0 Tilt Angle=40 Tilt Angle=90

Up+

Vp+

-0.3

-0.4

-0.5

18

17

-0.6 16

100

200

300

y+ Fig. 20. Comparison of the wall-normal particle velocity at different tilt angles, þ tþ p = 1000, Re = 10,000, f = 0, ks = 0.

100

200

300

y+ Fig. 22. Comparison of particle streamwise velocity at different tilt angles, þ tþ p = 1000, Re = 10,000, f = 0, ks = 0.

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between the theoretical prediction and the experiments. It is also observed that further increase in f increases the deposition velocþ ity in lower t þ p (t p 6 0.1) where the deposition is mainly governed by Brownian diffusion. In Fig. 18 the variation of deposition velocþ þ ity for three different roughness parameters ks = 0, ks = 0.35 and þ ks = 1 are shown when electrostatic effect is excluded. As shown in this figure a value of 0.35 can reduce the discrepancy.

7.3. The effect of tilt angle on the particle deposition In Fig. 19 the effect of the tilt angle, b, on the particle deposition is shown. According to this figure, for t þ p > 0.05 decreasing the tilt angle increases the deposition velocity, because as the tilt angle decreases the gravitational force in the normal direction takes a larger value. For t þ p > 200, the gravitational force dominates and much larger deposition rate (than t þ p < 200) happens. This is shown in Figs. 20–22. These figures suggest that, for tþ p = 1000 increasing the tilt angle from 0° to 90°, decreases the normal particle velocity

1

β= 90

Sh ea

rV

el oc ity =3

Sh ea

m /s

rV

β= 0 el oc ity =1 β= 90 .5 m /s β= 0

ar V β= elo 0 β=9 city= 0 .5 0 m/s

8. Summary

ρ/ρmax

0.6

She

0.8

0.4

0.2

0

500

1000

y+ Fig. 23. Comparison of the particle concentration under different shear velocities at þ 0° and 90° tilt angles, dp = 100 lm, f = 0, ks = 0.

Deposition Velocity(Vdep+)

0.7 0.6

This paper applies a numerical model for accurate prediction of particle deposition on a smooth and rough tilted surface. In this study a v2-f turbulence model compatible with the Eulerian approach is used to predict the deposition rate of particle on a tilted surface. The v2-f model can accurately calculate the normal fluctuations, which mainly represents the anisotropic nature of turbulence regime near the wall. The model performance is examined by comparing the predicted particle deposition rate on a vertical wall with the reference experimental and numerical data in a turbulent channel flow. The results show that, prediction of turbulence features using the v2-f model, gives reasonable accuracy when predicting the deposition rate. The effects of lift force, turbophoretic force, Brownaian force and turbulent diffusion are examined on the particle deposition rate. The results show that for low relaxation time the Brownian and turbulent diffusion are dominant. In intermediate range of relaxation time the turbophoretic force dominates. In the high relaxation time the effect of lift force on the particle deposition is very important. However, it is shown that the turbophoretic force still dominates. The present calculations show that the presence of small surface roughness enhances the deposition of small particles. It is indicated that changing the tilt angle from 90° to 0° enhances the deposition rate especially the large particles. Furthermore, the results show that the tilt angle has insignificant impact on the particle deposition rate in high shear velocity or high Reynolds number.

Tilt Angle=0

0.5

Tilt Angle=90 0.4 0.3 0.2 0.1 0

V p in the entire range of y+. This is also clearly shown in Fig. 21 (particle concentration, qp ) and Fig. 22 (streamwise velocity, U p ). Distribution of particle concentration under three different shear velocities for tilt angles of 0° and 90° is illustrated in Fig. 23. This figure suggests that increasing the shear velocity with changing the tilt angle from 0° to 90° decreases the particle concentration. To see this effect on the particle deposition rate, the particle deposition velocity is shown in Fig. 24. As shown in this figure, increasing the shear velocity when the particle diameter is 100 lm, decreases the particle velocity deposition. In fact when the tilt angle is 90°, as the velocity goes up, the inertia forces are increased and the deposition velocity, Vdep, is decreased. This is also true when the tilt angle is zero. Comparing the two tilt angles for each shear velocity reveals that as the shear velocity increases, the effect of tilt angle on the deposition velocity decreases. The reason for this is that the shear velocity takes higher values, the inertia forces are more effective than gravitational forces so that the particle deposition velocity gets lower values as compared to cases with lower shear velocity. The results indicate that the effect of tilt angle in high shear velocity or high Reynolds number on the particle deposition velocity is low.

0.5

1.5

3

Shear Velocity(m/s) Fig. 24. Comparison of the particle deposition velocity under different shear þ velocities at 0° and 90° tilt angles, dp = 100 lm, f = 0, ks = 0.

9. Conclusions The superiority of the v2-f model is predicting the normal fluctuations near the wall, where they can be inserted into the particle model. The previous investigations in which air flow properties obtained analytically were incapable of such prediction and therefore were less accurate. The other advantage of combining the v2-f model with particle model is reducing the computational time. To the best of authors’ knowledge, this paper probably for the first time takes into account the effect of tilt angel under different conditions on the particle deposition rate. The results show that the tilt angle has insignificant impact on the particle deposition rate in high shear velocity or high Reynolds numbers. This study has numerous applications, especially in the field of dust deposition on solar collectors.

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