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Particle deposition with thermal and electrical effects in turbulent flows
International Journal of Thermal Sciences 50 (2011) 1867e1877
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Particle deposition with thermal and electrical effects in turbulent flows M.C. Chiou a, *, C.H. Chiu a, H.S. Chen b a b
Department of Vehicle Engineering, National Formosa University, Yunlin 632, Taiwan Department of Material Science and Engineering, National Formosa University, Yunlin 632, Taiwan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 March 2010 Received in revised form 30 November 2010 Accepted 20 March 2011 Available online 17 June 2011
Development of relationships for the particle concentration and convection velocity profile has been obtained by the adaptation of the surface renewal model to the particle continuity and momentum equations of the turbulence boundary-layer flow in the presence of thermal field [1]. The predictions obtained on the basis of this model for nonisothermal deposition velocity of particles have been found to be in good agreement with the experimental measurements for fully-developed turbulence tube flow conditions. The aim of this work is to extend the previous model for an applied electric field, with the inclusion of the effect of Coulombic force in addition to the Brownian and turbulent diffusion, the eddy impaction, the turbophoresis, and the thermophoresis. The calculations show an interaction between thermophoresis and turbophoresis in the presence of an applied electric field. The effect of electric force in nonisothermal flows can have a dramatic effect on thermophoretic deposition for sþ p < 0:02, where turbophoretic effect has ceased. The effect of axial pressure gradient is also included. Ó 2011 Elsevier Masson SAS. All rights reserved.
Keywords: Random surface renewal model Eddy impaction Particle inertia Thermophoresis Analytical equation Deposition velocity
1. Introduction The free-flight model was one of the most used calculation methods for the observed large increase in deposition velocities [2e5]. The fundamental difference between different calculation methods of this model lies in prescribing the initial velocity that the particles possess at the distance where they effectively breaks away from the containing eddies and embarks on a free flight toward the wall. This model yielded reasonable agreement with deposition rate measurements for intermediate relaxation times, but poor agreement at high values. The measured deposition velocities, which are generally accepted as one of most dependable data set, have been found to be changed fairly to a slowly falling value with increasing the particle relaxation time sþ p [6]. The previous paper [7] gave an alternative approach to formulate the thermophoretic velocity and the particle concentration profiles in a nonisothermal turbulence flow with fully-developed boundary layer. The characteristic features of this approach model were based on the consideration that a net particle flux J arises mainly from the Brownian diffusion Db and thermophoretic force suspended in a flowing fluid,
J ¼ Db
vC þ yth C; vy
(1)
* Corresponding author. E-mail address: [email protected] (M.C. Chiou). 1290-0729/$ e see front matter Ó 2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2011.03.013
where y is the wall-normal distance and vC/vy is the wall-normal gradient of mean particle concentration C. The mean thermophoresis velocity yth depends on the wall-normal gradient in mean fluid temperature. Incorporating the Cunningham correction as shown by Hinds [8]
Cc ¼ 1 þ
1 15:6 þ 7:0exp 0:059Pdp ; Pdp
(2)
the Brownian diffusion Db for a rarefied gas effect can be calculated by
Db ¼ Cc
Kb T ; 3pmdp
(3)
where P is the absolute pressure in kPa, dp the particle diameter in mm, m the dynamical viscosity, T the absolute temperature, and Boltzmann’s constant Kb ¼ 1.38 1023 J/K. The proposed relationships for the particle concentration distribution and transport coefficient within the average sublayer growth period s was obtained by adaptation of the surface rejuvenation model [9,10] to the particle continuity equation,
vC v vC Db C yth ; ¼ vs vy vy
(4)
where s is the residence time between two successive eddies. The calculations of particle transport coefficient yd H=Db within the average sublayer growth period s were presented for various values
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Nomenclature c C Cc d Db dp Dp e E ep f
y0f yd ye yp y0p ypc yr yth
x y F H J Kb Ke Kth Ni np P
mean thermal speed mean particle concentration Cunningham slip correction factor tube diameter Brownian diffusion particle diameter particle diffusion coefficient electronic unit charge electric field intensity relative permittivity or dielectric constant friction factor fluctuation fluid velocity in radial direction particle deposition velocity electric drift velocity mean particle velocity fluctuating particle velocity particle convection velocity particle drift velocity thermophoretic velocity distance along wall distance from wall Coulombic force sublayer layer growth thickness particle mass flux Boltzmann constant proportional constant thermophoretic coefficient ion concentration maximum saturation charge number pressure
pffiffiffiffiffiffiffiffi of H= Db s on the basis of the previous analyses by Refs. [11,12]. The behavior of thermophoretic depositions within the average sublayer growth period obtained on the basis of this model is useful in stressing the importance of thermophoretic effect on the deposition processes. The small particles have been found to benefit most from this effect because with their low inertia they tend to follow the flow more closely. The predicted trend of average particle deposition velocities in an isothermal turbulence flow has been found to be in good agreement with both the Harriott technique [11] and the formulation proposed by Ref. [12]. However, because the order of Bessel function has to be a positive integer, the expression of analytical equations obtained by this formulation scheme is limited to the determination of average transport properties in accordance with specified transport parameters. Further, as compared to the measured deposition velocities [6], the validity of this calculation scheme seems to be restricted in an intermediate range of particle relaxation time. A simple stochastic theory was developed and used in the quantitative predictions for the deposition velocity of higher inertia particles [13]. The calculations of particle motion were based on the free-flight model [15] in which the fluid motion was determined by direct numerical simulation of the NaviereStokes equations. The Eulerian computational methods of deposition [14,16e19] have been developed by solving both the particle continuity and momentum equations. It was represented a considerable progress in the physical understanding of deposition processes. When the particles with certain range of inertia move against the wall-normal gradient in turbulent fluctuation intensity, they would get trapped into the low turbulence energy regions. The wall-normal component of particle Reynolds stresses in the regions was assumed to
Pr ps q r rp Re
Prandtl number statistic distribution for s total electric charge tube radius particle radius Reynolds number Lagrangian correlation coefficient time mean temperature friction velocity
Greek letters a thermal diffusivity rg fluid density rp particle density n kinematic viscosity m dynamical viscosity 3t turbulent eddy viscosity 3m turbulent eddy diffusivity 3p particle eddy diffusivity lp mean free path s sublayer growth period sg integral time scale sp particle relaxation time Superscripts þ dimensionless parameters _ average with respect to statistic distributions Subscripts N bulk stream conditions w wall conditions
play an important role in particle deposition processes. Therefore, the absence of the use of the particle momentum equation in the previous paper [7] is considered to be its major weakness, and so does the free-flight model. Recently, the work of Ref. [1] presented another alternative approach for calculating the deposition velocities in connection with the random surface renewal model [20e24]. Both the particle continuity and momentum equations in a simulate turbulence fluid field were written as
vC v v vC 3p ; ¼ C yp þ vs vy vy vy
(5)
vy0p 2 yp vyp D vC yth ¼ b þ ; sp sp C vy sp vs vy
(6)
where yp is the mean particle velocity, y0p the particle fluctuation velocity, 3p the particle turbulence diffusivity, and sp the particle relaxation time. The key to this analytical approach is that the mean particle velocity yp can be simplified as the sum of a diffusive and a convective part, in the manner suggested by Refs. [16,17]. With this simplification, the net particle flux can thus be separated into the diffusive and convective components by defining
vC J ¼ D b þ 3p C yth þ C ypc : vy
(7)
Consequently, the concentration dependent terms of the external forces imposed by the surrounding fluid are shifted from the momentum equation into the mass conservation equation. The unsteady equation for the mass balance of particles in an individual turbulence element approaching to the wall becomes
M.C. Chiou et al. / International Journal of Thermal Sciences 50 (2011) 1867e1877
vC vC v v Db þ 3p ¼ C yth C ypc : vs vy vy vy
(8)
The change in time-dependent convection velocity ypc of aerosol particles can be solved independently from the linear differential equation
dy0p 2 dypc ypc þ ¼ : sp ds dy
(9)
It is important to note that the convective acceleration of aerosol particles induced by the gradient in turbulent intensity is significantly dependent on the particle fluctuation velocity y0p, which may be different from the fluctuating velocity of surrounding fluid if the particles are large enough to have sufficient inertia. When the particles are small enough to effectively follow the fluid eddies, the two fluctuation velocities are essentially the same. The contribution from the gradient in turbulent fluctuation intensities is negligible. In this case, the net particle flux given by equation (1) is, therefore, an adequate description for the transport phenomena. Since the time mean values of the fluid and particle turbulence flux arise from the same mechanism of mean fluid convection, a linear differential equation for Reynolds normal stress of the particles proposed by Ref. [18] has been modified by substituting the residence time s of an individual eddy element for the instantaneous contact time and written as
dy0p ds
þ
1
sp
y0p ¼
1
sp
y0f :
(10)
The particle turbulence diffusivity 3p in the wall region has been protested to be correlated by the dimensionless rms fluctuation velocity y0f þ2 ¼ y0f 2 =u2* and defined as [5]
3p v
¼
3t þ y0f þ2 sþ p; n
(11)
2 where sþ p ¼ u* sp =n is the dimensionless particle relaxation time, n the kinematic viscosity, and u* the friction velocity. The dimensionless mean fluctuation velocity y0f þ of turbulent fluid has been defined as [13]
y0f þ ¼
0:005yþ2 for 0 < yþ < 200 1 þ 0:002923yþ2:128
(12)
The turbulent eddy diffusivity 3t has been considered as the turbulent eddy viscosity and calculated by
"
þ
3t þ0:08 2:5 ¼ yþð4y Þ n Re
107
# y 400 þ yþ
103 for all yþ ¼ yu* =n; (13)
where Reynolds number Re ¼ 2uNr/n, uN is the fluid velocity in central part of the pipe, and r is the radius of the pipe [3]. In order to keep the study focused, the electrical force was not retained in this proposed model, but it certainly provides a useful framework for coupling the Brownian and turbulent diffusion, thermophoresis, eddy impaction, and turbophoresis with an analytical solution of particle deposition velocity. It has been shown that the particles with very large inertia may acquire sufficient drift velocity imparted by the turbophoresis, which acts as an active transport mechanism right through the viscous sublayer. This convective drift velocity may not be sufficient for small particles to coast all the way to the wall, and thus the steep gradient in particle concentrations at the wall is necessary to maintain the particle flux by Brownian diffusion when the fluid turbulence level is very low.
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When the particles suspended in a nonisothermal turbulence fluid migrate against the thermal intensity gradient, the variations of kinetic energy intensity between colder and hotter regions of fluid cause the thermophoretic effect on suspended particle transports toward the colder region. In general, the effect of thermophoresis on the motion of spherical particles is distinguished by the thermophoretic velocity yth. Approximate simulations indicated clearly that the magnitude of thermophoretic effect on the submicron particle deposit in the cold surface can be greater than 30% and the calculated results for the hot surface are counter to that of the cold surface. The small particles benefit most from this effect because with their low inertia they tend to follow the flow more closely. For intermediate-size particles, there is a strong interaction between thermophoresis and turbophoresis, and the thermophoresis will cease to have any effect on the particle deposition beyond certain particle size. The combined effects of these coupling mechanisms were added in the momentum equation as the appropriate forces, which come out naturally from this analysis in a physically satisfying manner. Hence, there is scope to add other effects in analytical formulation through the equilibrium among the acceleration terms. Based on the fact that all natural and artificial aerosol particles are electrically charged to some extent, the main objective of the present work is to extend this framework for finding the general behavior of particle deposition in the presence of an applied electric field. Aerosol particles are considered to be exposed to unipolar ions, and become charged due to collision of the particles and the ions. As well as the number of charges acquired by a particle, the effect of Coulombic forces can be very significant depending on the charging mechanisms of diffusion charging, field charging, and combined diffusion and field charging. It is expected that the inclusion of Coulombic forces in addition to the combined effects of different transport mechanisms would play an important role. 2. Physical description of transport mechanisms The particle deposition under the influence of electrostatic force has become a very important issue in many engineering applications as well as in many natural phenomena. Generally, the gaseous ions are produced by ionizing radiation, high voltage discharge or high temperature. Many aerosol particles carried electrical charges are essentially the same as the electrical charge of natural atmospheric particles, which may be acquired when gaseous ions are captured by the particle due to the frequent collision between the particles and ions. In the presence an external electric field, it has been found that the deposition velocity for particles with saturation charge distribution is much higher than that of particles with Boltzmann charge distribution. Hence, a charged aerosol near a solid wall generally experiences the Coulombic force due to an applied electric field, the image force due to the induced charges on the wall, the dielectrophoretic force exerted by the gradient of the field from the image charge, and the dipoleedipole force due to the interaction of the induced dipole and its image [25e28]. The image force has been found to be restricted in quite short range near the wall and its effect on particle deposition is not significant as compared to the effects of Coulombic force. The dielectrophoretic force and dipole-dipolar interactions are usually small [29]. In this study, it is main attempt to understand the combined effects of electric force and turbulent transport mechanisms on deposition of particles. When an applied electric field is present, each particle is assumed to carry a single unit of charge, and the force acting on a charged particle suspended in a gas near the solid surfaces is considered to be dominated mainly by the Coulombic force. Hence, with the inclusion of Coulombic force in addition to the turbulent transport mechanisms, the momentum equation of
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aerosol particles in an eddy approaching to the wall may be expressed as
vy0p 2 yp vyp D vC yth ye ¼ þ ; b þ sp sp C vy sp sp vs vy
(14)
where ye is the mean migration velocity of the charged particles. If the net flux of particles in a turbulent boundary layer can be defined as
J ¼
Db þ 3p
vC C yth þ C ypc þ C ye ; vy
(15)
then the momentum equation (14) is reduced to equation (9). This procedure has the effect of shifting the concentration dependent terms into the mass conservation equation and leads to the relation
vC vC v v ypc þ ye C ; D b þ 3p ¼ yth C vs vy vy vy
(16)
It is important to note that the key to the surface renewal model involves the transformation of the mean transport properties into the average domain prior to the solution of the particle mass balance equation. As mentioned in Ref. [1], equation (9) relates the convective velocity of aerosol particles with the gradient in turbulent intensity (turbophoresis) and the viscous drag resistance. The drag resistance always balances the gradient of turbulent intensity for not too large particles. The convective acceleration of aerosol particles may only be importance for larger particles and should no be neglected. The larger particles with their high inertia respond poorly to the
surrounding fluid turbulence and begin to accelerate toward the wall at very early stage of their travel through the viscous wall region, where the turbulent field is relatively homogeneous. Hence, the convective velocity gained by the gradient of turbulent fluctuation intensity may be negligible small at the time just after the arrival of a turbulent eddy, ypc ¼ 0 at s ¼ 0. Same as the transformation of carrier fluid properties into the spatial variation of average time domain, the exponentially distributed density function
ps ðsÞ ¼
1
s
s exp ;
(17)
s
proposed by Refs. [30e32], is taken into account for predicting the transport properties during the average residence time s of all eddy lifetimes, which also establishes a measure of the average sublayer growth period. The mean convection velocity of aerosol particles can be transformed by multiplying each term of equation (9) with ps(s), and then by integrating with respect to s. ConseRN þ quently, the convective velocity yþ pc ¼ ypc =u* ¼ 0 ypc ps ðsÞds of particles within the average sublayer growth period can be calculated by
yþ pc
¼
dy0p þ2 sþ p þ dy
!
sþ p 1þ þ : sp þ sþ2
(18)
From the Lagrangian correlation coefficient
y0f ð0Þy0f ðsÞ=y0f 2 of the fluid motion in the stationary homogeneous uniform flow [33], the solution of equation (10), which satisfies the initial condition yp0 ¼ 0 at s ¼ 0, can be derived and expressed as
Fig. 1. Effects of mean axial pressure gradient on isothermal deposition of particles with imposed electric fields of E ¼ 10 kv/m and 20 kv/m at Re ¼ 5000.
M.C. Chiou et al. / International Journal of Thermal Sciences 50 (2011) 1867e1877
y0p 2 ðsÞ ¼ y0f 2 1 2exp
sg þ sp s sg sp
2 þ exp s ;
(19)
sp
yþ th ¼
y0p þ2 ¼ y0f þ2 1
þ 2sþ g sp þ2 þ sþ2 sþ g þ s sp
þ þ sþ g sp
þ
sþ p þ2
2s
þ sþ p
! ;
(20)
RN where y0p þ2 ¼ 0 y0p þ2 ps ðsÞds. In the wall-bounded turbulence shear flows with thermal intensity gradients, the heat transports of a nonisothermal fluid element intermittently moving to near the surface are analogous to its momentum. When temperature gradient is present, the aerosol particles experience a thermophoretic force in addition to the drag resistance and Brownian diffusion. The effects of thermophoresis would give rise to the containing particle migrations toward the cooler fluid because the molecular bombardment of particles is more energetic on the hot side than on the cold side. The thermophoretic force per unit particle mass is usually transformed into the corresponding thermophoretic velocity yth ¼ (Kthn/T)vT/vy, where Kth is thermophoretic coefficient [34]. Based on the analytical solution of temperature distributions T ¼ RN þ 0 Tps ðsÞds obtained by Ref. [1], the thermophoretic velocity yth ¼ yth =u* within the average sublayer growth period can be calculated by
2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 1 3t 6 sþ þ 6expB @ Pr n 4
TN Tw TN 1
3;
(21)
C yþ TN Tw 7 7 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC A TN 5 1 3t þ s þ Pr n
where sg ¼ 3t =y0f 2 is integral time scale of the Lagrangian correlation. Simultaneously, within the average sublayer growth period a constitutive relation for the wall-normal fluctuation velocities of immersed particles and carrier fluid can be calculated by
Kth 0
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where Prandtl number Pr ¼ n/a. Since the temperature fluctuation in the region very close to the wall is higher than elsewhere as well as the temperature gradient, a location of about on particle radius from the wall is taken into account for the diffusive flux by thermophoretic effect. An applied electrical field is concerned with dilute particle concentration in which the space charge effect is ignored. The migration of particles caused by Coulombic force can be assumed to be constant within the extremely thin boundary layer and treated as another independent mechanism like the thermophoretic and turbophoretic mechanisms. Hence, for an aerosol particle carrying np elementary units of electrical charge in an applied electrical field of intensity E, the total electrical charge by a particle is q ¼ npe, then the Coulombic force due to the imposed field is F ¼ qe ¼ npeE. Consequently, this electrical force causes the particle to move through the viscous wall region. When the viscous drag force acting on the particle becomes equal to the applied electrical force, the final migration velocity attained by the particle can be obtained from Stokes law for spherical particles
ye ¼
np eECc ; 3pmdp
Fig. 2. Effects of fluid friction velocity on isothermal deposition of particles with different charged mechanisms at E ¼ 20 kv/m and Re ¼ 50,000.
(22)
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where e ¼ 1.602 1019C is the electronic unit charge and dp is the particle diameter. When aerosol particles are exposed to unipolar ions, most aerosol particles carry some electric charges. The charge levels depend on the charging mechanisms. The effects of the Coulombic force are examined by two frequently encountered charging mechanisms, such as the diffusion and the field charging mechanism. In the absence of an applied electric field, the aerosol particles become charged due to collision of the particles and the ions. This collision is primarily caused by the random thermal motion of the ions. Hence, it has been considered as a diffusion process that the gaseous ions diffuse toward the particle surface and impart their electric charges to the particle. According to Ref. [8], an approximate diffusion charging expression for the number of charges np acquired by a particle of diameter dp during time duration t is given by
! dp Kb T Ke dp cpe2 Ni t ; np ¼ ln 1 þ 2Kb T 2Ke e2
(23)
where Ni is the concentration of ions and, for Nit > 1012 s/m3 and Nit > 1013 s/m3, equation (23) is accurate to within a factor of two for particles from 0.07 to 1.5 mm and from 0.05 to 40 mm, respectively. It has been mentioned that, however, the diffusion charging is the predominant mechanism for charging particles less than 0.2 mm in diameter, even in the presence of an electrostatic field [8]. Therefore, a value of Ni ty1012 s=m3 is assumed for the subsequent analysis. Ke ¼ 9.0 109 N m2/C2 is a constant of proportionality. Because aerosol particles are exchanging energy with the surrounding gas molecules, both the particles and the gas
molecules have the same average kinetic energy. Therefore, the mean thermal speed of the gaseous ions is calculated by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ 48Kb T=p2 rp d3p . With an imposed electric field of intensity E, the ordered motion of the ions in the direction of the applied electric field increases the frequency of collision. The cause of this motion has been considered as the predominant charging mechanism that rapidly drives the unipolar ions onto the particle. Under these conditions the random thermal motion of the ions may be ignored. Assuming the motion of the ions to be confined along the electric line of force, the field charging equation for the maximum saturation number of charges np acquired by particles is derived and written as [8]
np ¼
3ep ep þ 2
Ed2p
; 4Ke e
(24)
where the relative permittivity or dielectric constant ep reflects the strength of electrostatic field produced in different materials by a fixed potential relative to that produced in a vacuum under the same conditions. ep is 1.00059 for air, 4.3 for quartz, 80.0 for pure water, and is infinite for conducting particles. Same as the transformations of carrier fluid properties, the mean concentration of aerosol particles can be transformed into its average time domain by multiplying each term of equation (16) with ps(s)ds, and then by integrating. The solution of timeRN averaged particle concentration, CðyÞ ¼ 0 Cðy; sÞps ðsÞds, which satisfies the initial condition of C ¼ cN at s ¼ 0 and the boundary conditions of C ¼ cw at y ¼ rp and C ¼ cN as y / N, can be derived and expressed as the form
Fig. 3. Effects of imposed electric field intensity on isothermal deposition of particles with different charging mechanisms.
M.C. Chiou et al. / International Journal of Thermal Sciences 50 (2011) 1867e1877
2 C
þ
6 ¼ exp6 4
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dp þ þ 4 þ sþ2 yþ2 r s yr
n
2
Dp
n
sþ
3 7 yþ rpþ 7 5:
(25)
where Dp/n ¼ DB/n þ 3p/n indicates a comparable effect between molecular diffusion and turbulent diffusivity of the particles. The external properties caused by the turbulent, thermal and electrical effects are manifested in the form of the particle drift velocity þ þ þ yþ r ¼ ypc þ yth þ ye . Once each concentration dependent term of the external forces imposed by the surrounding fluid has been evaluated prior to the concentration boundary development, the concentration developments within the average sublayer growth period can be obtained. Subsequently, the particle deposition y velocity yþ d ¼ d =u* can be predicted quantitatively by means of the form
yþ ¼ d
J u* ðcN cw Þ
yþ ¼rpþ
¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dp þ2 þ 4 þ yþ2 þ yþ r s r s
n
2sþ
:
(26)
3. Results and discussions In accordance with the exponential-distributed density function ps ðsÞ ¼ ð1=sÞexpðs=sÞ, the proposed relationship for average sublayer growth period s has been obtained by adaptation of the surface renewal model to the fluid momentum transfer, with the inclusion of the effect of mean axial pressure gradient [1]. pffiffiffiffiffiffiffi þ s s ¼ u =n with the Reynolds number The variances in * pffiffiffiffiffiffiffiffi Re ¼ 2rþ 2=f obtained on the basis of this approach have been
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found to be in basic agreement with experimental data obtained by Refs. [21,35] for fully-developed tube flow conditions. It is important to note that, when the Reynolds number is increased, the viscous turbulence transition at the point on the wall is delayed or equivalently the period of fluid sublayer growth is increased. Calculations are also shown for the case in which the effect of axial pressure gradient is neglected. A quite dramatic influence of mean axial pressure gradient on the calculated sþ is observed for Re < 104, and as a result the apparent effect of deeper molecular penetration on the viscous turbulence transition is associated with low Reynolds number flow conditions. The development of relationships for the turbophoretic and thermophoretic velocity profile obtained on the basis of equations (18) and (21) has also been well documented by Ref. [1], respectively. When the solid wall is heated or cooled, the near-wall temperature gradients generate thermophoretic forces that act as an additional effect on the particle transport against the gradient of thermal intensities. The temperature fluctuation in the region very close to the wall is found to be higher than elsewhere and results in distinct heat flux against the temperature gradient. The dependence of thermophoretic velocity yþ th distribution upon particle relaxation time sþ p is used in directly characterizing the interaction of particle inertia and thermal drift mechanisms. It has been found sþ that the variances of yþ th with p caused by the thermal drift mechanisms appear to be much more pronounced on the small particles. The predicted trend of yþ pc profiles has been found to rise slowly from the main stream boundary to a maximum at a short distance close to the wall before dropping rapidly to zero at the wall. The peak position of yþ pc profiles is shown to be located at yþ zrpþ þ 12, and becomes closer to the wall with increasing
Fig. 4. Effects of density ratio rp/rg on isothermal deposition of particles with imposed electric fields of E ¼ 10 kv/m and 20 kv/m.
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particle diameter as well as increasing the peak value. This demonstrates that the wall-normal gradient of turbulent fluctuation intensities induces an additional impaction momentum, which accelerates the particles toward the wall in the regions within the peak position, but in the regions beyond the peak position pushes the particles toward the main stream boundary. Usually, the varisþ ations of dimensionless deposition velocity yþ d with p are divided into three distinct regimes, such as the turbulent diffusion deposition regime, the turbulent diffusion-eddy impaction regime and the particle inertia-moderated regime. The predictions obtained by Ref. [1] for neutral particle deposition velocity have been found in reasonable agreement with existing numerical and experimental data. Therefore, when the particles are merged into an imposed electric field, this deposition behavior can be used in clarifying the relative importance of turbulent transport mechanisms on the deposition process of charged particles. The charged particle transport to the wall obtained on the basis of the surface renewal model is based on the assumption that the aerosol particles are produced by olive oil with a dielectric constant of ep ¼ 4.0 and carry a single unit of charge. The Coulombic force is considered to be directed toward the wall, and once the particles reach the wall surface they will stick to it. Fig. 1 shows the deposition velocity of charged aerosol particles calculated for various values of particle relaxation time sþ p with (solid curves) and without including the effects of mean axial pressure gradient. A saturation charging condition of aerosol particles reaching the maximum charge number at combined diffusion charging (DC) and field charging (FC) mechanisms is used and denoted by DC þ FC. The resulting deposition velocity of neutral particles is also shown in this figure for comparison [2–4, 6, 13, 36]. Interestingly, the
deposition curves of neutral particle are somewhat lower for nonzero axial pressure gradient but, as the electric forces are in effect, the mean axial pressure gradient on the characteristic of deposition process has ceased to have any effect. In order to explain this deposition behaviour, Fig. 2 represents the dependence of deposion the friction velocity u* of containing fluid adjation velocity yþ d cent to the wall in the absence and the presence of an imposed electric field. For neutral particles, it is clearly shown that, although the mean axial pressure gradient is operating, its influence on the particle depositions will be significantly attenuated at high Reynolds number flow conditions. Also this is evident from the due to the increase of fluid friction enhancement in predicted yþ d velocity u*. Based on kinetic theory, the momentum carried across the streamlines of the average flow by turbulence becomes more effective with the increase of Reynolds number, which in turn results in an enhanced fluid friction velocity, so that the effect of increasing friction velocity actually causes steeper velocity gradient with the wall in a very short distance. The outer edge of growing sublayer under this flow conditions is always farther from the wall than the depth of molecular penetration. Consequently, the particles moving against the deeper turbulence penetrations can reach much more quickly the region very near the wall, where the effective resistance against the particle transport to the wall is relatively small in comparison with the Brownian diffusion. It is to be noted that the small particles benefit most from this effect because with their low inertia they tend to follow the flow more closely. In the presence of an imposed electric field, this characteristic dependence of predicted yþ d on the fluid friction velocity u* is quite similar to that for neutral particles, except that the predicted values of yþ d always becomes larger in the presence of than
Fig. 5. Effects of imposed electric intensity on isothermal deposition of particles with different dielectric constants of particle material.
M.C. Chiou et al. / International Journal of Thermal Sciences 50 (2011) 1867e1877
that of neutral particles. This provides a clearer picture that the near-wall turbulent flow can also affect the deposition of charged particles and becomes more prominent in the presence of an imposed electric field. sþ Fig. 2 also displays the variations of yþ d with p for the particles that are charged under different charging mechanisms. From the case that the aerosol particle merges into the diffusion charging mechanism, it is shown especially in the turbulent diffusion deposition regime that the variations in predicted yþ d between the charged and neutral particle with sþ p are not too much deviated from each other but, in the range sþ p < 0:02, the deposition curves of charged particle rise much more steeply with the decrease of þ sþ p . This signifies that the steep rise in yd curves against the þ decrease of sp is caused not by turbulent transport mechanisms but rather by the effects of diffusion charging, which serves to enhance these mechanisms restrictively in the range sþ p < 0:02. As a logical argument, because the diffusion charging effect primarily arises from the random collisions between the particles and ions due to the Brownian motion, the most significant enhancement in the unassisted deposition velocity occurs due to this charging mechanism only when the diffusive mass flux toward the wall is entirely predominated by the Brownian diffusion. In the large sþ p range, it is observed that the steep rise in yþ is relatively immune to the diffusion charging effect and, as d mentioned by Ref. [1], that the transport of aerosol particles to the wall then depends upon their inertia. As a result, the inertial effect allows discrete particles to acquire an impacting momentum against the sharp decay in turbulent fluctuation intensities toward the wall, which will consequently make the sþ dramatically by several orders of increase of yþ d with p
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magnitude in the turbulent diffusion-eddy impaction regime. In beyond the maximum addition, the subsequent decrease in yþ d with increased particle inertia in the particle inertia-moderated regime is related to the peak position of yþ pc profiles. In the outer region of yþ > rpþ þ 12, an additional turbophoresis induced by a strong gradient in particle fluctuation velocity tends to push the particles away from the wall. Fig. 3 reconstructs the resulting deposition velocities for various electric field intensities of 10, 20 and 100 kv/m. There certainly exists a significant quantitative difference between the deposition curves if one includes the effect of saturation charging distribution. Apparently, a definite range 0:1 sþ p 1:0 is eliminated from the turbulent diffusion-eddy impaction regime, while the turbulent eddy impaction has ceased to have any effect on particle deposition by including this charging effect in the turbulent transport mechanisms. It is shown that, on the other hand, there is an overlap between two yþ d profiles such as that the saturation charge overlaps the functions of the field charging already in existence. This draws an important conclusion that one of most striking effects of aerosol particle interaction with an imposed electric field is an existence of field charging region. As can be seen in Figs. 2 and 3, this region has a wide range of time scales while the near-wall resistance is decreased or the electric field intensity is increased, respectively. It is shown that, when the particle sizes involved fall in this region, the deposition velocities are much higher than the values predicted under the diffusion charging effect and the discrepancy becomes smaller with decreased sþ p . This implies that the field charging region is then succeeded by an overlapping region where the combination of diffusion and field charging mechanisms is in effect. Consequently,
Fig. 6. Effects of imposed electric field intensity on thermophoretic deposition of particles for different thermal intensity gradients.
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the dependence of saturation charge upon the diffusion charging mechanism becomes more and more important with decrease of sþ p , and tries to bring out the dependence on the field charging mechanism. Clearly, the total number of charges acquired by the particle is crucial to explain this shift in dominant changing mechanism. sþ Fig. 4 demonstrates that the variations of yþ d with p in the presence of an applied electric field can also be compared with the ratio of immersed particle density rp to carrier fluid density rg. increase The predictions obtained by the present analysis for yþ d with the increase of particle density. This can be attributed to the greater particle concentrations near the wall, resulting in an appropriate development of particle concentration profiles close to the deposition surfaces. In the presence of an imposed electric field, the particles acquire electric force against the sharp gradient in particle concentration toward the wall through interactions with the Brownian and turbulent diffusion, and as a result the of charged particles becomes more increase in the predicted yþ d prominent with the increase of particle density. However, in large sþ p range, it is observed also that the inertia of larger particles imparted from the strong gradient in turbulent fluctuation intensity near the wall causes a suppression of this effect. This deposition behavior is useful in stressing the importance of the particle-fluid density ratio on deposition and confirming the relative independence of deposition upon electric force effect at very large inertia particles. Fig. 5 displays the deposition velocity of particles having combined charging mechanism for various dielectric constants, proposed by Refs. [8]. It should be noted that an accurate representation of the field charging effect on the particle deposition for
these values is difficult because particle turbulence properties are different from those of carrier flow due to the particle drift relative to the surrounding fluid. The result obtained on the basis of present analysis is one of the simplest means for testing the dependence of particle deposition velocity on the field charging mechanism for various dielectric constants. The tendency of predicted deposition velocities is shown to be higher for a large dielectric constant ep, and will eventually close to each other in accordance with the particle diameter dp and the electric field intensity E. It is of interesting to note that the field charging equation (24) predicts a particle charge that is proportional in the square of particle diameter. For fixed electric field intensity, the effect of electric force on the particle deposition caused by the field charging mechanism spontaneously decays with decreases of the particle relation time sþ p , but is at the same time generated by the diffusion charging mechanism. As a result, for extremely small sþ p , the deposition velocity becomes independent of ep and increases with decreases of particle diameter dp. Figs. 6 and 7 characterize the dependence of thermophoretic deposition velocities in the presence of electric force effect as a function of the parameters TN Tw and Pr ¼ n/a, respectively. The results for neutral particle are also shown in these figures. According to the discussions given in the previous study [1], the neutral particle deposition due to thermophoretic effect are summarized as follows: (a) When the variations in thermal intensity between colder and hotter region of containing fluid are maintained, the dependence of thermophoretic deposition on these characteristic parameters will be most marked for the low inertia particles with sþ p < 0:1. (b) The temperature gradients generate thermophoretic forces on the diffusive mass flux which
Fig. 7. Effects of imposed electric field intensity on thermophoretic deposition of particles for different Prandtl numbers.
M.C. Chiou et al. / International Journal of Thermal Sciences 50 (2011) 1867e1877
can be very significant in modifying the deposition velocities, and as a result the turbulent particle transport to the cold and hot surface becomes significantly enhanced and collapsed, respectively. (c) The dependence of thermophoretic deposition velocity on Pr can be characterized by considering that temperature fluctuations imparted from the near-wall turbulent fluctuations render the thermophoretic force a fluctuating quantity; consequently, the effects of thermophoretic force will become more prominent in a nonisothermal flow field with the increase of Prandtl number Pr. In the presence of an imposed electric field, the variations of sþ predicted yþ d with p are shown to be disordered due to the effects of electric force, particularly for small particles with sþ p < 0:1. However, as compared with the results for neutral particles suspended in the nonisothermal flow conditions, it is observed that the dependence of charged particle deposition on both the thermal intensity gradient and the Prandtl number Pr becomes more prominent only in a definite range 0:001 < sþ p < 0:1. Indeed, this characteristic dependence is essentially the same as the neutral particles except that the values of predicted yþ d always become larger in the presence of an imposed electric field. Therefore, there can be a strong interaction between the electric effect and thermophoresis by which the electric force generates a significant drift toward the wall, resulting in a significant effect on the enhancement of diffusive mass flux against the thermal intensity gradient toward the wall. On the other hand, as compared with the results for charged particles suspended in the isothermal flow conditions, it is clearly shown that the dependence of predicted yþ d on the thermal intensity gradient and Prandtl number Pr has ceased approximately around sþ p z0:001, sþ where the predicted yþ d profile rises steeply with decreasing p without being influenced by these additional mechanisms. This is because the electrical force due to diffusion charging has significant effect on making the immersed particles to move faster than the containing fluid. In addition, it can be seen that the effect of electric forces on deposition processes in the large sþ p range has become comparable to that of thermophoresis around sþ p z0:2. The turbulent particle transport against the gradient of thermal intensities then depends on the particle inertia, which would influence the enhancement of deposition due to thermophoresis. For particles of the intermediate-size range ð1 < sþ p < 10Þ, there is an interaction between thermophoresis and turbophoresis. Through which the thermophoresis loses importance, and the particle transport to the wall then depends on the interaction between the particle inertia and turbophoresis. 4. Conclusions A useful framework for estimating the combined effects of turbulent transport mechanisms has been established in the previous work [1] by modifying stochastic formulation of the surface renewal model. The quantitative predictions obtained by this calculation scheme for the neutral particles are used in this work in terms of the average sublayer growth period sþ , the , and the turbophoretic velocity yþ thermophoretic velocity yþ pc. th The proposed relationship for deposition velocity yþ d has been obtained by adaptation of the previous work to include the effects of Coulombic force in addition to the combined effects of turbulent transport mechanisms. The predicted trend of charged particle deposition velocity obtained on the basis of this analysis has been found in reasonable agreement with available data. This confirms that the stochastic formulation is to be preferred as the method adapted for engineering calculations.
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References [1] M.C. Chiou, C.H. Chiu, H.S. Chen, Formulation for predicting deposition velocity of particles in turbulence, Int. J. Therm. Sci. 49 (2010) 290e301. [2] S.K. Friedlander, H.F. Johnstone, Deposition of suspended particles from turbulent gas stream, Ind. Eng. Chem. 49 (1957) 1150e1156. [3] C.N. Davies, Deposition of aerosol from turbulent flow through pipes, Proc. Roy. Soc. Lond. Math. Phys. Sci. 289 (1966) 235e246. [4] S.K. Beal, Deposition of particles in turbulent flow on channel or pipe walls, Nucl. Sci. Eng. 40 (1970) 1e11. [5] B.Y.H. Liu, T.A. Illori, Inertial deposition of aerosol particles in turbulent pipe flow, in: ASME Symposium on Flow Studies in Air and Water Pollution, Atlanta, Georgia, 1973, pp. 103e113. [6] B.Y.H. Liu, J.K. Agarwal, Experimental observation of aerosol deposition in turbulent flow, J. Aerosol Sci. 5 (1974) 145e155. [7] C.H. Chiu, C.M. Wang, M.C. Chiou, Turbulent thermophoresis effect on particle transport processes, Int. J. Therm. Sci. 45 (2006) 1238e1250. [8] W.C. Hinds, Aerosol Technology e Properties, Behavior, and Measurement of Airborne Particles, vol. 2, John Wiley & Sons. Inc., 1999. [9] L.C. Thomas, The surface rejuvenation model of wall turbulence: inner laws for uþ and Tþ, Int. J. Heat Mass Transf. 23 (1980) 1099e1104. [10] R. Rajagopal, L.C. Thomas, Adaptation of the stochastic formulation of the surface rejuvenation model to turbulent convection heat transfer, Chem. Eng. Sci. 29 (1974) 1639e1644. [11] P. Harriott, A random eddy modification of the penetration theory, Chem. Eng. Sci. 17 (1962) 149e154. [12] J.A. Bullin, A.E. Dukler, Random eddy models for surface renewal: formulation as a stochastic process, Chem. Eng. Sci. 27 (1972) 439e442. [13] G.A. KAllio, M.W. Reeks, A numerical simulation of particle deposition in turbulent boundary layers, Int. J. Multiphase Flow 15 (1989) 433e446. [14] S.T. Johansen, The deposition of particles of vertical walls, Int. J. Multiphase Flow 17 (1991) 355e376. [15] J.W. Brooke, T.J. Hanratty, J.B. McLaughlin, Free-flight mixing and deposition of aerosols, Phys. Fluids 6 (1994) 3404e3415. [16] A. Guha, A unified Eulerian theory of turbulent deposition to smooth and rough surfaces, J. Aerosol Sci. 28 (8) (1997) 1517e1537. [17] S.A. Slater, A.D. Leeming, J.B. Young, Particle deposition from two-dimensional turbulent gas flow, Int. J. Multiphase Flow 29 (2003) 721e750. [18] M. Shin, D.S. Kim, J.W. Lee, Deposition of inertia-dominated particles inside a turbulent boundary layer, Int. J. Multiphase Flow 26 (2003) 892e926. [19] S. Cerbelli, A. Giusti, A. Soldati, ADE approach to predicting dispersion of heavy particles in wall-bounded turbulence, Int. J. Multiphase Flow 27 (2001) 1861e1879. [20] R. Nijsing, Predictions of momentum, heat and mass transfer in turbulent channel flow with aid of boundary layer growth-breakdown model, Warmeund Stoffubertragung 2 (1969) 65e86. [21] R.L. Meek, A.D. Baer, The periodic viscous sublayer in turbulent flow, AIChE J. 16 (1970) 841e848. [22] W.V. Pinczewski, S.A. Sideman, Model for mass (heat) transfer in turbulent flow, Moderate and high Schmidt (Prandtl) numbers, Chem. Eng. Sci. 29 (1974) 1969e1976. [23] J.M.H. Fortuin, E.E. Musschenga, P.J. Hamersma, Transfer processes in turbulent pipe flow described by the ERSR model, AIChE J. 38 (1992) 343e362. [24] E.E. Musschenga, P.J. Hamersma, J.M.H. Fortuin, Momentum, heat and mass transfer in turbulent pipe flow: the extended random surface renewal model, Chem. Eng. Sci. 47 (1992) 343e362. [25] G.C. Hartmann, L.M. Marks, C.C. Yang, Physical models for photoactive pigment electrophotography, J. Appl. Phys. 47 (1976) 5409e5420. [26] G.M. Hidy, Aerosls, An Industrial and Environmental Science. Academic Press, New York, 1984. [27] D.W. Cooper, M.H. Peters, R.J. Miller, Predicted deposition of submicron particles due to diffusion and electrostatics in viscous axisymmetric stagnation-point flow, Aerosol Sci. Technol. 11 (1989) 133e143. [28] F.G. Fan, G. Ahmadi, On the sublayer model for turbulent deposition of aerosol particles in the presence of gravity and electric field, J. Aerosol Sci. 21 (1994) 49e71. [29] A. Li, G. Ahmadi, Aerosol particle deposition with electrostatic attraction in turbulent channel flow, J. Colloid Interface Sci. 158 (1993) 476e482. [30] P.V. Danckwerts, Significance of liquid-film coefficients in gas absorption, Ind. Eng. Chem. 43 (1951) 1460e1467. [31] T.J. Hanratty, Turbulent exchange of heat and momentum with a boundary, AIChE J. 2 (1956) 359e362. [32] J.M.H. Fortuin, P.J. Klijn, Drag reduction and random surface renewal in turbulent pipe flow, Chem. Eng. Sci. 37 (1982) 611e623. [33] J.O. Hinze, Turbulence. McGraw-Hill Book Company, New York, 1975. [34] L. Talbot, R.K. Cheng, R.W. Schefer, D.R. Willis, Thermophoresis of particles in a heated boundary layer, J. Fluid Mech. 101 (1980) 737e758. [35] E.R. Corino, R.S. Brodkey, A visual investigation of the wall region on turbulent flow, J. Fluid Mech. 37 (1969) 1e30. [36] T.L. Montgomery, M. Corn, Aerosol deposition in a pipe with turbulent airflow, Aerosol Sci. 1 (1970) 185e213.
Contents lists available at ScienceDirect
International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts
Particle deposition with thermal and electrical effects in turbulent flows M.C. Chiou a, *, C.H. Chiu a, H.S. Chen b a b
Department of Vehicle Engineering, National Formosa University, Yunlin 632, Taiwan Department of Material Science and Engineering, National Formosa University, Yunlin 632, Taiwan
a r t i c l e i n f o
a b s t r a c t
Article history: Received 10 March 2010 Received in revised form 30 November 2010 Accepted 20 March 2011 Available online 17 June 2011
Development of relationships for the particle concentration and convection velocity profile has been obtained by the adaptation of the surface renewal model to the particle continuity and momentum equations of the turbulence boundary-layer flow in the presence of thermal field [1]. The predictions obtained on the basis of this model for nonisothermal deposition velocity of particles have been found to be in good agreement with the experimental measurements for fully-developed turbulence tube flow conditions. The aim of this work is to extend the previous model for an applied electric field, with the inclusion of the effect of Coulombic force in addition to the Brownian and turbulent diffusion, the eddy impaction, the turbophoresis, and the thermophoresis. The calculations show an interaction between thermophoresis and turbophoresis in the presence of an applied electric field. The effect of electric force in nonisothermal flows can have a dramatic effect on thermophoretic deposition for sþ p < 0:02, where turbophoretic effect has ceased. The effect of axial pressure gradient is also included. Ó 2011 Elsevier Masson SAS. All rights reserved.
Keywords: Random surface renewal model Eddy impaction Particle inertia Thermophoresis Analytical equation Deposition velocity
1. Introduction The free-flight model was one of the most used calculation methods for the observed large increase in deposition velocities [2e5]. The fundamental difference between different calculation methods of this model lies in prescribing the initial velocity that the particles possess at the distance where they effectively breaks away from the containing eddies and embarks on a free flight toward the wall. This model yielded reasonable agreement with deposition rate measurements for intermediate relaxation times, but poor agreement at high values. The measured deposition velocities, which are generally accepted as one of most dependable data set, have been found to be changed fairly to a slowly falling value with increasing the particle relaxation time sþ p [6]. The previous paper [7] gave an alternative approach to formulate the thermophoretic velocity and the particle concentration profiles in a nonisothermal turbulence flow with fully-developed boundary layer. The characteristic features of this approach model were based on the consideration that a net particle flux J arises mainly from the Brownian diffusion Db and thermophoretic force suspended in a flowing fluid,
J ¼ Db
vC þ yth C; vy
(1)
* Corresponding author. E-mail address: [email protected] (M.C. Chiou). 1290-0729/$ e see front matter Ó 2011 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.ijthermalsci.2011.03.013
where y is the wall-normal distance and vC/vy is the wall-normal gradient of mean particle concentration C. The mean thermophoresis velocity yth depends on the wall-normal gradient in mean fluid temperature. Incorporating the Cunningham correction as shown by Hinds [8]
Cc ¼ 1 þ
1 15:6 þ 7:0exp 0:059Pdp ; Pdp
(2)
the Brownian diffusion Db for a rarefied gas effect can be calculated by
Db ¼ Cc
Kb T ; 3pmdp
(3)
where P is the absolute pressure in kPa, dp the particle diameter in mm, m the dynamical viscosity, T the absolute temperature, and Boltzmann’s constant Kb ¼ 1.38 1023 J/K. The proposed relationships for the particle concentration distribution and transport coefficient within the average sublayer growth period s was obtained by adaptation of the surface rejuvenation model [9,10] to the particle continuity equation,
vC v vC Db C yth ; ¼ vs vy vy
(4)
where s is the residence time between two successive eddies. The calculations of particle transport coefficient yd H=Db within the average sublayer growth period s were presented for various values
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Nomenclature c C Cc d Db dp Dp e E ep f
y0f yd ye yp y0p ypc yr yth
x y F H J Kb Ke Kth Ni np P
mean thermal speed mean particle concentration Cunningham slip correction factor tube diameter Brownian diffusion particle diameter particle diffusion coefficient electronic unit charge electric field intensity relative permittivity or dielectric constant friction factor fluctuation fluid velocity in radial direction particle deposition velocity electric drift velocity mean particle velocity fluctuating particle velocity particle convection velocity particle drift velocity thermophoretic velocity distance along wall distance from wall Coulombic force sublayer layer growth thickness particle mass flux Boltzmann constant proportional constant thermophoretic coefficient ion concentration maximum saturation charge number pressure
pffiffiffiffiffiffiffiffi of H= Db s on the basis of the previous analyses by Refs. [11,12]. The behavior of thermophoretic depositions within the average sublayer growth period obtained on the basis of this model is useful in stressing the importance of thermophoretic effect on the deposition processes. The small particles have been found to benefit most from this effect because with their low inertia they tend to follow the flow more closely. The predicted trend of average particle deposition velocities in an isothermal turbulence flow has been found to be in good agreement with both the Harriott technique [11] and the formulation proposed by Ref. [12]. However, because the order of Bessel function has to be a positive integer, the expression of analytical equations obtained by this formulation scheme is limited to the determination of average transport properties in accordance with specified transport parameters. Further, as compared to the measured deposition velocities [6], the validity of this calculation scheme seems to be restricted in an intermediate range of particle relaxation time. A simple stochastic theory was developed and used in the quantitative predictions for the deposition velocity of higher inertia particles [13]. The calculations of particle motion were based on the free-flight model [15] in which the fluid motion was determined by direct numerical simulation of the NaviereStokes equations. The Eulerian computational methods of deposition [14,16e19] have been developed by solving both the particle continuity and momentum equations. It was represented a considerable progress in the physical understanding of deposition processes. When the particles with certain range of inertia move against the wall-normal gradient in turbulent fluctuation intensity, they would get trapped into the low turbulence energy regions. The wall-normal component of particle Reynolds stresses in the regions was assumed to
Pr ps q r rp Re
Prandtl number statistic distribution for s total electric charge tube radius particle radius Reynolds number Lagrangian correlation coefficient time mean temperature friction velocity
Greek letters a thermal diffusivity rg fluid density rp particle density n kinematic viscosity m dynamical viscosity 3t turbulent eddy viscosity 3m turbulent eddy diffusivity 3p particle eddy diffusivity lp mean free path s sublayer growth period sg integral time scale sp particle relaxation time Superscripts þ dimensionless parameters _ average with respect to statistic distributions Subscripts N bulk stream conditions w wall conditions
play an important role in particle deposition processes. Therefore, the absence of the use of the particle momentum equation in the previous paper [7] is considered to be its major weakness, and so does the free-flight model. Recently, the work of Ref. [1] presented another alternative approach for calculating the deposition velocities in connection with the random surface renewal model [20e24]. Both the particle continuity and momentum equations in a simulate turbulence fluid field were written as
vC v v vC 3p ; ¼ C yp þ vs vy vy vy
(5)
vy0p 2 yp vyp D vC yth ¼ b þ ; sp sp C vy sp vs vy
(6)
where yp is the mean particle velocity, y0p the particle fluctuation velocity, 3p the particle turbulence diffusivity, and sp the particle relaxation time. The key to this analytical approach is that the mean particle velocity yp can be simplified as the sum of a diffusive and a convective part, in the manner suggested by Refs. [16,17]. With this simplification, the net particle flux can thus be separated into the diffusive and convective components by defining
vC J ¼ D b þ 3p C yth þ C ypc : vy
(7)
Consequently, the concentration dependent terms of the external forces imposed by the surrounding fluid are shifted from the momentum equation into the mass conservation equation. The unsteady equation for the mass balance of particles in an individual turbulence element approaching to the wall becomes
M.C. Chiou et al. / International Journal of Thermal Sciences 50 (2011) 1867e1877
vC vC v v Db þ 3p ¼ C yth C ypc : vs vy vy vy
(8)
The change in time-dependent convection velocity ypc of aerosol particles can be solved independently from the linear differential equation
dy0p 2 dypc ypc þ ¼ : sp ds dy
(9)
It is important to note that the convective acceleration of aerosol particles induced by the gradient in turbulent intensity is significantly dependent on the particle fluctuation velocity y0p, which may be different from the fluctuating velocity of surrounding fluid if the particles are large enough to have sufficient inertia. When the particles are small enough to effectively follow the fluid eddies, the two fluctuation velocities are essentially the same. The contribution from the gradient in turbulent fluctuation intensities is negligible. In this case, the net particle flux given by equation (1) is, therefore, an adequate description for the transport phenomena. Since the time mean values of the fluid and particle turbulence flux arise from the same mechanism of mean fluid convection, a linear differential equation for Reynolds normal stress of the particles proposed by Ref. [18] has been modified by substituting the residence time s of an individual eddy element for the instantaneous contact time and written as
dy0p ds
þ
1
sp
y0p ¼
1
sp
y0f :
(10)
The particle turbulence diffusivity 3p in the wall region has been protested to be correlated by the dimensionless rms fluctuation velocity y0f þ2 ¼ y0f 2 =u2* and defined as [5]
3p v
¼
3t þ y0f þ2 sþ p; n
(11)
2 where sþ p ¼ u* sp =n is the dimensionless particle relaxation time, n the kinematic viscosity, and u* the friction velocity. The dimensionless mean fluctuation velocity y0f þ of turbulent fluid has been defined as [13]
y0f þ ¼
0:005yþ2 for 0 < yþ < 200 1 þ 0:002923yþ2:128
(12)
The turbulent eddy diffusivity 3t has been considered as the turbulent eddy viscosity and calculated by
"
þ
3t þ0:08 2:5 ¼ yþð4y Þ n Re
107
# y 400 þ yþ
103 for all yþ ¼ yu* =n; (13)
where Reynolds number Re ¼ 2uNr/n, uN is the fluid velocity in central part of the pipe, and r is the radius of the pipe [3]. In order to keep the study focused, the electrical force was not retained in this proposed model, but it certainly provides a useful framework for coupling the Brownian and turbulent diffusion, thermophoresis, eddy impaction, and turbophoresis with an analytical solution of particle deposition velocity. It has been shown that the particles with very large inertia may acquire sufficient drift velocity imparted by the turbophoresis, which acts as an active transport mechanism right through the viscous sublayer. This convective drift velocity may not be sufficient for small particles to coast all the way to the wall, and thus the steep gradient in particle concentrations at the wall is necessary to maintain the particle flux by Brownian diffusion when the fluid turbulence level is very low.
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When the particles suspended in a nonisothermal turbulence fluid migrate against the thermal intensity gradient, the variations of kinetic energy intensity between colder and hotter regions of fluid cause the thermophoretic effect on suspended particle transports toward the colder region. In general, the effect of thermophoresis on the motion of spherical particles is distinguished by the thermophoretic velocity yth. Approximate simulations indicated clearly that the magnitude of thermophoretic effect on the submicron particle deposit in the cold surface can be greater than 30% and the calculated results for the hot surface are counter to that of the cold surface. The small particles benefit most from this effect because with their low inertia they tend to follow the flow more closely. For intermediate-size particles, there is a strong interaction between thermophoresis and turbophoresis, and the thermophoresis will cease to have any effect on the particle deposition beyond certain particle size. The combined effects of these coupling mechanisms were added in the momentum equation as the appropriate forces, which come out naturally from this analysis in a physically satisfying manner. Hence, there is scope to add other effects in analytical formulation through the equilibrium among the acceleration terms. Based on the fact that all natural and artificial aerosol particles are electrically charged to some extent, the main objective of the present work is to extend this framework for finding the general behavior of particle deposition in the presence of an applied electric field. Aerosol particles are considered to be exposed to unipolar ions, and become charged due to collision of the particles and the ions. As well as the number of charges acquired by a particle, the effect of Coulombic forces can be very significant depending on the charging mechanisms of diffusion charging, field charging, and combined diffusion and field charging. It is expected that the inclusion of Coulombic forces in addition to the combined effects of different transport mechanisms would play an important role. 2. Physical description of transport mechanisms The particle deposition under the influence of electrostatic force has become a very important issue in many engineering applications as well as in many natural phenomena. Generally, the gaseous ions are produced by ionizing radiation, high voltage discharge or high temperature. Many aerosol particles carried electrical charges are essentially the same as the electrical charge of natural atmospheric particles, which may be acquired when gaseous ions are captured by the particle due to the frequent collision between the particles and ions. In the presence an external electric field, it has been found that the deposition velocity for particles with saturation charge distribution is much higher than that of particles with Boltzmann charge distribution. Hence, a charged aerosol near a solid wall generally experiences the Coulombic force due to an applied electric field, the image force due to the induced charges on the wall, the dielectrophoretic force exerted by the gradient of the field from the image charge, and the dipoleedipole force due to the interaction of the induced dipole and its image [25e28]. The image force has been found to be restricted in quite short range near the wall and its effect on particle deposition is not significant as compared to the effects of Coulombic force. The dielectrophoretic force and dipole-dipolar interactions are usually small [29]. In this study, it is main attempt to understand the combined effects of electric force and turbulent transport mechanisms on deposition of particles. When an applied electric field is present, each particle is assumed to carry a single unit of charge, and the force acting on a charged particle suspended in a gas near the solid surfaces is considered to be dominated mainly by the Coulombic force. Hence, with the inclusion of Coulombic force in addition to the turbulent transport mechanisms, the momentum equation of
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aerosol particles in an eddy approaching to the wall may be expressed as
vy0p 2 yp vyp D vC yth ye ¼ þ ; b þ sp sp C vy sp sp vs vy
(14)
where ye is the mean migration velocity of the charged particles. If the net flux of particles in a turbulent boundary layer can be defined as
J ¼
Db þ 3p
vC C yth þ C ypc þ C ye ; vy
(15)
then the momentum equation (14) is reduced to equation (9). This procedure has the effect of shifting the concentration dependent terms into the mass conservation equation and leads to the relation
vC vC v v ypc þ ye C ; D b þ 3p ¼ yth C vs vy vy vy
(16)
It is important to note that the key to the surface renewal model involves the transformation of the mean transport properties into the average domain prior to the solution of the particle mass balance equation. As mentioned in Ref. [1], equation (9) relates the convective velocity of aerosol particles with the gradient in turbulent intensity (turbophoresis) and the viscous drag resistance. The drag resistance always balances the gradient of turbulent intensity for not too large particles. The convective acceleration of aerosol particles may only be importance for larger particles and should no be neglected. The larger particles with their high inertia respond poorly to the
surrounding fluid turbulence and begin to accelerate toward the wall at very early stage of their travel through the viscous wall region, where the turbulent field is relatively homogeneous. Hence, the convective velocity gained by the gradient of turbulent fluctuation intensity may be negligible small at the time just after the arrival of a turbulent eddy, ypc ¼ 0 at s ¼ 0. Same as the transformation of carrier fluid properties into the spatial variation of average time domain, the exponentially distributed density function
ps ðsÞ ¼
1
s
s exp ;
(17)
s
proposed by Refs. [30e32], is taken into account for predicting the transport properties during the average residence time s of all eddy lifetimes, which also establishes a measure of the average sublayer growth period. The mean convection velocity of aerosol particles can be transformed by multiplying each term of equation (9) with ps(s), and then by integrating with respect to s. ConseRN þ quently, the convective velocity yþ pc ¼ ypc =u* ¼ 0 ypc ps ðsÞds of particles within the average sublayer growth period can be calculated by
yþ pc
¼
dy0p þ2 sþ p þ dy
!
sþ p 1þ þ : sp þ sþ2
(18)
From the Lagrangian correlation coefficient
y0f ð0Þy0f ðsÞ=y0f 2 of the fluid motion in the stationary homogeneous uniform flow [33], the solution of equation (10), which satisfies the initial condition yp0 ¼ 0 at s ¼ 0, can be derived and expressed as
Fig. 1. Effects of mean axial pressure gradient on isothermal deposition of particles with imposed electric fields of E ¼ 10 kv/m and 20 kv/m at Re ¼ 5000.
M.C. Chiou et al. / International Journal of Thermal Sciences 50 (2011) 1867e1877
y0p 2 ðsÞ ¼ y0f 2 1 2exp
sg þ sp s sg sp
2 þ exp s ;
(19)
sp
yþ th ¼
y0p þ2 ¼ y0f þ2 1
þ 2sþ g sp þ2 þ sþ2 sþ g þ s sp
þ þ sþ g sp
þ
sþ p þ2
2s
þ sþ p
! ;
(20)
RN where y0p þ2 ¼ 0 y0p þ2 ps ðsÞds. In the wall-bounded turbulence shear flows with thermal intensity gradients, the heat transports of a nonisothermal fluid element intermittently moving to near the surface are analogous to its momentum. When temperature gradient is present, the aerosol particles experience a thermophoretic force in addition to the drag resistance and Brownian diffusion. The effects of thermophoresis would give rise to the containing particle migrations toward the cooler fluid because the molecular bombardment of particles is more energetic on the hot side than on the cold side. The thermophoretic force per unit particle mass is usually transformed into the corresponding thermophoretic velocity yth ¼ (Kthn/T)vT/vy, where Kth is thermophoretic coefficient [34]. Based on the analytical solution of temperature distributions T ¼ RN þ 0 Tps ðsÞds obtained by Ref. [1], the thermophoretic velocity yth ¼ yth =u* within the average sublayer growth period can be calculated by
2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 1 3t 6 sþ þ 6expB @ Pr n 4
TN Tw TN 1
3;
(21)
C yþ TN Tw 7 7 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC A TN 5 1 3t þ s þ Pr n
where sg ¼ 3t =y0f 2 is integral time scale of the Lagrangian correlation. Simultaneously, within the average sublayer growth period a constitutive relation for the wall-normal fluctuation velocities of immersed particles and carrier fluid can be calculated by
Kth 0
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where Prandtl number Pr ¼ n/a. Since the temperature fluctuation in the region very close to the wall is higher than elsewhere as well as the temperature gradient, a location of about on particle radius from the wall is taken into account for the diffusive flux by thermophoretic effect. An applied electrical field is concerned with dilute particle concentration in which the space charge effect is ignored. The migration of particles caused by Coulombic force can be assumed to be constant within the extremely thin boundary layer and treated as another independent mechanism like the thermophoretic and turbophoretic mechanisms. Hence, for an aerosol particle carrying np elementary units of electrical charge in an applied electrical field of intensity E, the total electrical charge by a particle is q ¼ npe, then the Coulombic force due to the imposed field is F ¼ qe ¼ npeE. Consequently, this electrical force causes the particle to move through the viscous wall region. When the viscous drag force acting on the particle becomes equal to the applied electrical force, the final migration velocity attained by the particle can be obtained from Stokes law for spherical particles
ye ¼
np eECc ; 3pmdp
Fig. 2. Effects of fluid friction velocity on isothermal deposition of particles with different charged mechanisms at E ¼ 20 kv/m and Re ¼ 50,000.
(22)
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where e ¼ 1.602 1019C is the electronic unit charge and dp is the particle diameter. When aerosol particles are exposed to unipolar ions, most aerosol particles carry some electric charges. The charge levels depend on the charging mechanisms. The effects of the Coulombic force are examined by two frequently encountered charging mechanisms, such as the diffusion and the field charging mechanism. In the absence of an applied electric field, the aerosol particles become charged due to collision of the particles and the ions. This collision is primarily caused by the random thermal motion of the ions. Hence, it has been considered as a diffusion process that the gaseous ions diffuse toward the particle surface and impart their electric charges to the particle. According to Ref. [8], an approximate diffusion charging expression for the number of charges np acquired by a particle of diameter dp during time duration t is given by
! dp Kb T Ke dp cpe2 Ni t ; np ¼ ln 1 þ 2Kb T 2Ke e2
(23)
where Ni is the concentration of ions and, for Nit > 1012 s/m3 and Nit > 1013 s/m3, equation (23) is accurate to within a factor of two for particles from 0.07 to 1.5 mm and from 0.05 to 40 mm, respectively. It has been mentioned that, however, the diffusion charging is the predominant mechanism for charging particles less than 0.2 mm in diameter, even in the presence of an electrostatic field [8]. Therefore, a value of Ni ty1012 s=m3 is assumed for the subsequent analysis. Ke ¼ 9.0 109 N m2/C2 is a constant of proportionality. Because aerosol particles are exchanging energy with the surrounding gas molecules, both the particles and the gas
molecules have the same average kinetic energy. Therefore, the mean thermal speed of the gaseous ions is calculated by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c ¼ 48Kb T=p2 rp d3p . With an imposed electric field of intensity E, the ordered motion of the ions in the direction of the applied electric field increases the frequency of collision. The cause of this motion has been considered as the predominant charging mechanism that rapidly drives the unipolar ions onto the particle. Under these conditions the random thermal motion of the ions may be ignored. Assuming the motion of the ions to be confined along the electric line of force, the field charging equation for the maximum saturation number of charges np acquired by particles is derived and written as [8]
np ¼
3ep ep þ 2
Ed2p
; 4Ke e
(24)
where the relative permittivity or dielectric constant ep reflects the strength of electrostatic field produced in different materials by a fixed potential relative to that produced in a vacuum under the same conditions. ep is 1.00059 for air, 4.3 for quartz, 80.0 for pure water, and is infinite for conducting particles. Same as the transformations of carrier fluid properties, the mean concentration of aerosol particles can be transformed into its average time domain by multiplying each term of equation (16) with ps(s)ds, and then by integrating. The solution of timeRN averaged particle concentration, CðyÞ ¼ 0 Cðy; sÞps ðsÞds, which satisfies the initial condition of C ¼ cN at s ¼ 0 and the boundary conditions of C ¼ cw at y ¼ rp and C ¼ cN as y / N, can be derived and expressed as the form
Fig. 3. Effects of imposed electric field intensity on isothermal deposition of particles with different charging mechanisms.
M.C. Chiou et al. / International Journal of Thermal Sciences 50 (2011) 1867e1877
2 C
þ
6 ¼ exp6 4
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dp þ þ 4 þ sþ2 yþ2 r s yr
n
2
Dp
n
sþ
3 7 yþ rpþ 7 5:
(25)
where Dp/n ¼ DB/n þ 3p/n indicates a comparable effect between molecular diffusion and turbulent diffusivity of the particles. The external properties caused by the turbulent, thermal and electrical effects are manifested in the form of the particle drift velocity þ þ þ yþ r ¼ ypc þ yth þ ye . Once each concentration dependent term of the external forces imposed by the surrounding fluid has been evaluated prior to the concentration boundary development, the concentration developments within the average sublayer growth period can be obtained. Subsequently, the particle deposition y velocity yþ d ¼ d =u* can be predicted quantitatively by means of the form
yþ ¼ d
J u* ðcN cw Þ
yþ ¼rpþ
¼
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Dp þ2 þ 4 þ yþ2 þ yþ r s r s
n
2sþ
:
(26)
3. Results and discussions In accordance with the exponential-distributed density function ps ðsÞ ¼ ð1=sÞexpðs=sÞ, the proposed relationship for average sublayer growth period s has been obtained by adaptation of the surface renewal model to the fluid momentum transfer, with the inclusion of the effect of mean axial pressure gradient [1]. pffiffiffiffiffiffiffi þ s s ¼ u =n with the Reynolds number The variances in * pffiffiffiffiffiffiffiffi Re ¼ 2rþ 2=f obtained on the basis of this approach have been
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found to be in basic agreement with experimental data obtained by Refs. [21,35] for fully-developed tube flow conditions. It is important to note that, when the Reynolds number is increased, the viscous turbulence transition at the point on the wall is delayed or equivalently the period of fluid sublayer growth is increased. Calculations are also shown for the case in which the effect of axial pressure gradient is neglected. A quite dramatic influence of mean axial pressure gradient on the calculated sþ is observed for Re < 104, and as a result the apparent effect of deeper molecular penetration on the viscous turbulence transition is associated with low Reynolds number flow conditions. The development of relationships for the turbophoretic and thermophoretic velocity profile obtained on the basis of equations (18) and (21) has also been well documented by Ref. [1], respectively. When the solid wall is heated or cooled, the near-wall temperature gradients generate thermophoretic forces that act as an additional effect on the particle transport against the gradient of thermal intensities. The temperature fluctuation in the region very close to the wall is found to be higher than elsewhere and results in distinct heat flux against the temperature gradient. The dependence of thermophoretic velocity yþ th distribution upon particle relaxation time sþ p is used in directly characterizing the interaction of particle inertia and thermal drift mechanisms. It has been found sþ that the variances of yþ th with p caused by the thermal drift mechanisms appear to be much more pronounced on the small particles. The predicted trend of yþ pc profiles has been found to rise slowly from the main stream boundary to a maximum at a short distance close to the wall before dropping rapidly to zero at the wall. The peak position of yþ pc profiles is shown to be located at yþ zrpþ þ 12, and becomes closer to the wall with increasing
Fig. 4. Effects of density ratio rp/rg on isothermal deposition of particles with imposed electric fields of E ¼ 10 kv/m and 20 kv/m.
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particle diameter as well as increasing the peak value. This demonstrates that the wall-normal gradient of turbulent fluctuation intensities induces an additional impaction momentum, which accelerates the particles toward the wall in the regions within the peak position, but in the regions beyond the peak position pushes the particles toward the main stream boundary. Usually, the varisþ ations of dimensionless deposition velocity yþ d with p are divided into three distinct regimes, such as the turbulent diffusion deposition regime, the turbulent diffusion-eddy impaction regime and the particle inertia-moderated regime. The predictions obtained by Ref. [1] for neutral particle deposition velocity have been found in reasonable agreement with existing numerical and experimental data. Therefore, when the particles are merged into an imposed electric field, this deposition behavior can be used in clarifying the relative importance of turbulent transport mechanisms on the deposition process of charged particles. The charged particle transport to the wall obtained on the basis of the surface renewal model is based on the assumption that the aerosol particles are produced by olive oil with a dielectric constant of ep ¼ 4.0 and carry a single unit of charge. The Coulombic force is considered to be directed toward the wall, and once the particles reach the wall surface they will stick to it. Fig. 1 shows the deposition velocity of charged aerosol particles calculated for various values of particle relaxation time sþ p with (solid curves) and without including the effects of mean axial pressure gradient. A saturation charging condition of aerosol particles reaching the maximum charge number at combined diffusion charging (DC) and field charging (FC) mechanisms is used and denoted by DC þ FC. The resulting deposition velocity of neutral particles is also shown in this figure for comparison [2–4, 6, 13, 36]. Interestingly, the
deposition curves of neutral particle are somewhat lower for nonzero axial pressure gradient but, as the electric forces are in effect, the mean axial pressure gradient on the characteristic of deposition process has ceased to have any effect. In order to explain this deposition behaviour, Fig. 2 represents the dependence of deposion the friction velocity u* of containing fluid adjation velocity yþ d cent to the wall in the absence and the presence of an imposed electric field. For neutral particles, it is clearly shown that, although the mean axial pressure gradient is operating, its influence on the particle depositions will be significantly attenuated at high Reynolds number flow conditions. Also this is evident from the due to the increase of fluid friction enhancement in predicted yþ d velocity u*. Based on kinetic theory, the momentum carried across the streamlines of the average flow by turbulence becomes more effective with the increase of Reynolds number, which in turn results in an enhanced fluid friction velocity, so that the effect of increasing friction velocity actually causes steeper velocity gradient with the wall in a very short distance. The outer edge of growing sublayer under this flow conditions is always farther from the wall than the depth of molecular penetration. Consequently, the particles moving against the deeper turbulence penetrations can reach much more quickly the region very near the wall, where the effective resistance against the particle transport to the wall is relatively small in comparison with the Brownian diffusion. It is to be noted that the small particles benefit most from this effect because with their low inertia they tend to follow the flow more closely. In the presence of an imposed electric field, this characteristic dependence of predicted yþ d on the fluid friction velocity u* is quite similar to that for neutral particles, except that the predicted values of yþ d always becomes larger in the presence of than
Fig. 5. Effects of imposed electric intensity on isothermal deposition of particles with different dielectric constants of particle material.
M.C. Chiou et al. / International Journal of Thermal Sciences 50 (2011) 1867e1877
that of neutral particles. This provides a clearer picture that the near-wall turbulent flow can also affect the deposition of charged particles and becomes more prominent in the presence of an imposed electric field. sþ Fig. 2 also displays the variations of yþ d with p for the particles that are charged under different charging mechanisms. From the case that the aerosol particle merges into the diffusion charging mechanism, it is shown especially in the turbulent diffusion deposition regime that the variations in predicted yþ d between the charged and neutral particle with sþ p are not too much deviated from each other but, in the range sþ p < 0:02, the deposition curves of charged particle rise much more steeply with the decrease of þ sþ p . This signifies that the steep rise in yd curves against the þ decrease of sp is caused not by turbulent transport mechanisms but rather by the effects of diffusion charging, which serves to enhance these mechanisms restrictively in the range sþ p < 0:02. As a logical argument, because the diffusion charging effect primarily arises from the random collisions between the particles and ions due to the Brownian motion, the most significant enhancement in the unassisted deposition velocity occurs due to this charging mechanism only when the diffusive mass flux toward the wall is entirely predominated by the Brownian diffusion. In the large sþ p range, it is observed that the steep rise in yþ is relatively immune to the diffusion charging effect and, as d mentioned by Ref. [1], that the transport of aerosol particles to the wall then depends upon their inertia. As a result, the inertial effect allows discrete particles to acquire an impacting momentum against the sharp decay in turbulent fluctuation intensities toward the wall, which will consequently make the sþ dramatically by several orders of increase of yþ d with p
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magnitude in the turbulent diffusion-eddy impaction regime. In beyond the maximum addition, the subsequent decrease in yþ d with increased particle inertia in the particle inertia-moderated regime is related to the peak position of yþ pc profiles. In the outer region of yþ > rpþ þ 12, an additional turbophoresis induced by a strong gradient in particle fluctuation velocity tends to push the particles away from the wall. Fig. 3 reconstructs the resulting deposition velocities for various electric field intensities of 10, 20 and 100 kv/m. There certainly exists a significant quantitative difference between the deposition curves if one includes the effect of saturation charging distribution. Apparently, a definite range 0:1 sþ p 1:0 is eliminated from the turbulent diffusion-eddy impaction regime, while the turbulent eddy impaction has ceased to have any effect on particle deposition by including this charging effect in the turbulent transport mechanisms. It is shown that, on the other hand, there is an overlap between two yþ d profiles such as that the saturation charge overlaps the functions of the field charging already in existence. This draws an important conclusion that one of most striking effects of aerosol particle interaction with an imposed electric field is an existence of field charging region. As can be seen in Figs. 2 and 3, this region has a wide range of time scales while the near-wall resistance is decreased or the electric field intensity is increased, respectively. It is shown that, when the particle sizes involved fall in this region, the deposition velocities are much higher than the values predicted under the diffusion charging effect and the discrepancy becomes smaller with decreased sþ p . This implies that the field charging region is then succeeded by an overlapping region where the combination of diffusion and field charging mechanisms is in effect. Consequently,
Fig. 6. Effects of imposed electric field intensity on thermophoretic deposition of particles for different thermal intensity gradients.
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the dependence of saturation charge upon the diffusion charging mechanism becomes more and more important with decrease of sþ p , and tries to bring out the dependence on the field charging mechanism. Clearly, the total number of charges acquired by the particle is crucial to explain this shift in dominant changing mechanism. sþ Fig. 4 demonstrates that the variations of yþ d with p in the presence of an applied electric field can also be compared with the ratio of immersed particle density rp to carrier fluid density rg. increase The predictions obtained by the present analysis for yþ d with the increase of particle density. This can be attributed to the greater particle concentrations near the wall, resulting in an appropriate development of particle concentration profiles close to the deposition surfaces. In the presence of an imposed electric field, the particles acquire electric force against the sharp gradient in particle concentration toward the wall through interactions with the Brownian and turbulent diffusion, and as a result the of charged particles becomes more increase in the predicted yþ d prominent with the increase of particle density. However, in large sþ p range, it is observed also that the inertia of larger particles imparted from the strong gradient in turbulent fluctuation intensity near the wall causes a suppression of this effect. This deposition behavior is useful in stressing the importance of the particle-fluid density ratio on deposition and confirming the relative independence of deposition upon electric force effect at very large inertia particles. Fig. 5 displays the deposition velocity of particles having combined charging mechanism for various dielectric constants, proposed by Refs. [8]. It should be noted that an accurate representation of the field charging effect on the particle deposition for
these values is difficult because particle turbulence properties are different from those of carrier flow due to the particle drift relative to the surrounding fluid. The result obtained on the basis of present analysis is one of the simplest means for testing the dependence of particle deposition velocity on the field charging mechanism for various dielectric constants. The tendency of predicted deposition velocities is shown to be higher for a large dielectric constant ep, and will eventually close to each other in accordance with the particle diameter dp and the electric field intensity E. It is of interesting to note that the field charging equation (24) predicts a particle charge that is proportional in the square of particle diameter. For fixed electric field intensity, the effect of electric force on the particle deposition caused by the field charging mechanism spontaneously decays with decreases of the particle relation time sþ p , but is at the same time generated by the diffusion charging mechanism. As a result, for extremely small sþ p , the deposition velocity becomes independent of ep and increases with decreases of particle diameter dp. Figs. 6 and 7 characterize the dependence of thermophoretic deposition velocities in the presence of electric force effect as a function of the parameters TN Tw and Pr ¼ n/a, respectively. The results for neutral particle are also shown in these figures. According to the discussions given in the previous study [1], the neutral particle deposition due to thermophoretic effect are summarized as follows: (a) When the variations in thermal intensity between colder and hotter region of containing fluid are maintained, the dependence of thermophoretic deposition on these characteristic parameters will be most marked for the low inertia particles with sþ p < 0:1. (b) The temperature gradients generate thermophoretic forces on the diffusive mass flux which
Fig. 7. Effects of imposed electric field intensity on thermophoretic deposition of particles for different Prandtl numbers.
M.C. Chiou et al. / International Journal of Thermal Sciences 50 (2011) 1867e1877
can be very significant in modifying the deposition velocities, and as a result the turbulent particle transport to the cold and hot surface becomes significantly enhanced and collapsed, respectively. (c) The dependence of thermophoretic deposition velocity on Pr can be characterized by considering that temperature fluctuations imparted from the near-wall turbulent fluctuations render the thermophoretic force a fluctuating quantity; consequently, the effects of thermophoretic force will become more prominent in a nonisothermal flow field with the increase of Prandtl number Pr. In the presence of an imposed electric field, the variations of sþ predicted yþ d with p are shown to be disordered due to the effects of electric force, particularly for small particles with sþ p < 0:1. However, as compared with the results for neutral particles suspended in the nonisothermal flow conditions, it is observed that the dependence of charged particle deposition on both the thermal intensity gradient and the Prandtl number Pr becomes more prominent only in a definite range 0:001 < sþ p < 0:1. Indeed, this characteristic dependence is essentially the same as the neutral particles except that the values of predicted yþ d always become larger in the presence of an imposed electric field. Therefore, there can be a strong interaction between the electric effect and thermophoresis by which the electric force generates a significant drift toward the wall, resulting in a significant effect on the enhancement of diffusive mass flux against the thermal intensity gradient toward the wall. On the other hand, as compared with the results for charged particles suspended in the isothermal flow conditions, it is clearly shown that the dependence of predicted yþ d on the thermal intensity gradient and Prandtl number Pr has ceased approximately around sþ p z0:001, sþ where the predicted yþ d profile rises steeply with decreasing p without being influenced by these additional mechanisms. This is because the electrical force due to diffusion charging has significant effect on making the immersed particles to move faster than the containing fluid. In addition, it can be seen that the effect of electric forces on deposition processes in the large sþ p range has become comparable to that of thermophoresis around sþ p z0:2. The turbulent particle transport against the gradient of thermal intensities then depends on the particle inertia, which would influence the enhancement of deposition due to thermophoresis. For particles of the intermediate-size range ð1 < sþ p < 10Þ, there is an interaction between thermophoresis and turbophoresis. Through which the thermophoresis loses importance, and the particle transport to the wall then depends on the interaction between the particle inertia and turbophoresis. 4. Conclusions A useful framework for estimating the combined effects of turbulent transport mechanisms has been established in the previous work [1] by modifying stochastic formulation of the surface renewal model. The quantitative predictions obtained by this calculation scheme for the neutral particles are used in this work in terms of the average sublayer growth period sþ , the , and the turbophoretic velocity yþ thermophoretic velocity yþ pc. th The proposed relationship for deposition velocity yþ d has been obtained by adaptation of the previous work to include the effects of Coulombic force in addition to the combined effects of turbulent transport mechanisms. The predicted trend of charged particle deposition velocity obtained on the basis of this analysis has been found in reasonable agreement with available data. This confirms that the stochastic formulation is to be preferred as the method adapted for engineering calculations.
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