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Photophoresis of an aerosol sphere normal to a plane wall
Huan J. Keh
∗, Fu C. Hsu
Department of Chemical Engineering, National Taiwan University, Taipei 106-17, Taiwan, Republic of China
Received 22 November 2004; accepted 21 March 2005 Available online 27 April 2005
Abstract
A combined analytical–numerical study is presented for the quasisteady photophoretic motion of a spherical aerosol particle of arbitrary thermal conductivity and surface properties exposed to a radiative flux perpendicular to a large plane wall. The Knudsen number is assumed to be so small that the fluid flow is described by a continuum model with a temperature jump, a thermal slip, and a frictional slip at the surface of the radiation-absorbing particle. In the limit of small Peclet and Reynolds numbers, the appropriate equations of conservation of energy and momentum for the system are solved using a boundary collocation method and numerical results for the photophoretic velocity of the particle are obtained for various cases. The presence of the neighboring wall causes two basic effects on the particle velocity: first, the local temperature gradient on the particle surface is enhanced or reduced by the wall, thereby speeding up or slowing down the particle; second, the wall increases viscous retardation of the moving particle. The net effect of the wall can decrease or increase the particle velocity, depending upon the relative conductivity and surface properties of the particle as well as the relative particle–wall separation distance. In general, the boundary effect of a plane wall on the photophoresis of an aerosol particle can be quite significant in some situations. In most aerosol systems, the boundary effect on photophoresis is weaker than that on the motion driven by a gravitational field.
2005 Elsevier Inc. All rights reserved.
Keywords: Photophoresis; Aerosol sphere; Thin Knudsen layer; Boundary effect
1. Introduction
Small particles, when suspended in a gaseous medium and exposed to an intense light beam, will migrate paral-lel to the direction of the light. This phenomenon is a result of the uneven heating of the light-absorbing particle (and therefore of its adjacent gas molecules) and is known as pho-tophoresis[1]. The photophoretic (or thermophoretic) effect can be explained in part by appealing to the kinetic theory of gases[2,3]. The higher energy molecules in the hot region of the gas impinge on the particle with greater momenta than molecules coming from the cold region, thereby leading to the migration of the particle in a direction opposite to the surface temperature gradient. Thus, the photophoretic force on an aerosol particle can be directed either toward (neg-ative photophoresis) or away from (positive photophoresis)
*Corresponding author.
E-mail address:huan@ntu.edu.tw(H.J. Keh).
the light source, depending upon the optical characteristics of the particle. If the particle is opaque and the incident light energy is absorbed and dissipated directly at the front surface of the particle, positive photophoresis occurs. Conversely, if the light beam is partially transmitted and focused on the rear side of the particle, negative photophoresis may appear.
Photophoresis has been observed for many particulate materials in the diameter range between 10−8and 10−3m, and at pressures from above 1 atm down to below 1 Torr, under illumination intensities comparable with those of sun-light[4]. Therefore, the results of photophoretic investiga-tions are of interest to a wide variety of fields including cloud physics, aerosol science, and environmental engineering. For example, measurements of the photophoretic force or the re-versal point from positive to negative photophoresis with the elaboration of photophoretic spectroscopy can be used to determine the physical properties, such as the complex re-fractive index, and the chemical composition of aerosol par-ticles[5]. The photophoretic phenomena of aerosol particles 0021-9797/$ – see front matter 2005 Elsevier Inc. All rights reserved.
subjected to coherent light beams have been applied to the development of laser atmospheric monitoring methods[6]. It was found that, due to the effect of both positive and negative photophoresis, some stratospheric aerosol particles may be caused to rise against gravity, whereas others are induced to fall considerably more rapidly than they would under gravity alone [7]. Considering that radiative heat transfer can ac-count for around 95% of the total heat flux in pulverized-coal furnaces, the driving force for photophoresis of small parti-cles in combustion environments can be significantly greater than that for thermophoresis[8].
When the Knudsen number (l/a, where a is the radius of the particle and l is the mean free path of the surrounding gas molecules), Reynolds number, and Peclet number are small, the photophoretic velocity of an aerosol sphere illuminated by an intense light beam can be expressed as[8,9]
(1)
U0= −
2J CsηI
3(1+ 2Cml/a)(2k+ kp+ 2kpCtl/a)ρfT0
.
In the above equation, I is the intensity of the incident light (incoming illumination energy flux); ρf, η, and k are the density, viscosity, and thermal conductivity, respectively, of the gas; kp is the thermal conductivity of the particle;
T0is the absolute temperature of the bulk gas; J is the so-called photophoretic asymmetry factor[10], which can be either positive (negative photophoresis) or negative (positive photophoresis); Cs, Ct, and Cm are dimensionless coeffi-cients accounting for the thermal slip, temperature jump, and frictional slip phenomena, respectively, at the particle sur-face and must be determined experimentally for each gas– solid system. A set of reasonable kinetic-theory values for complete thermal and momentum accommodation appear to be Cs = 1.17, Ct = 2.18, and Cm= 1.14 [11]. Recently, kinetic-theory values of these slip coefficients have been ob-tained accurately under various conditions[12,13]. Note that the photophoretic velocity given by Eq.(1)is proportional to the fluid viscosity due to the existence of the thermal slip ve-locity at the particle surface.
In many applications of photophoresis, aerosol particles are not isolated and will move in the presence of neighbor-ing particles and/or boundaries. For example, the mechanism and rate of deposition of photophoretic particles on various surfaces are of practical interest. It is of some importance, therefore, to examine the behavior of a particle under pho-tophoretic forces as it is in the proximity of rigid boundaries. Recently, the quasisteady photophoresis of an aerosol sphere located at the center of a spherical cavity has been inves-tigated and an analytical expression for the wall-corrected particle velocity was derived in a closed form[14]. In this work we present an analysis of the photophoretic motion of an aerosol sphere perpendicular to an infinite plane wall. The quasisteady equations of conservation applicable to the sys-tem are solved by a combined analytical–numerical method using the boundary collocation technique. Results for the wall-corrected particle velocity are obtained with good con-vergence for various cases.
Fig. 1. Geometrical sketch for the photophoretic motion of a spherical par-ticle perpendicular to a plane wall.
2. Analysis
We consider the quasisteady photophoretic motion of a spherical particle of radius a with arbitrary thermal conduc-tivity and surface properties in a gaseous medium transpar-ent to radiation in the direction normal to an infinite plane wall located at a distance d from the particle center, as illus-trated inFig. 1. Here, (ρ, φ, z) and (r, θ, φ) denote the cir-cular cylindrical and spherical coordinate systems, respec-tively, and both origins of coordinates are set at the center of the particle. The incident light is imposed to the particle in the z direction (perpendicular to the wall) with intensity I and the photophoretic velocity of the particle is U ez, where ez is the unit vector in the positive z direction. Both of the bulk gas and the plane wall are kept at a constant temperatu-re T0. The Knudsen numbers l/a and l/(d− a) are assumed to be so small that the fluid flow is in the continuum regime and the Knudsen layer adjacent to the surface of the particle is relatively thin. The fluid is allowed to slip, both thermally and frictionally, and the temperature jump may occur at the particle surface. Our objective is to determine the correction to Eq.(1)for the photophoretic particle due to the presence of the plane wall.
To determine the photophoretic velocity of the particle, it is necessary to ascertain the temperature and fluid velocity distributions.
2.1. Temperature distribution
The Peclet number of this axisymmetric problem is as-sumed to be small. Hence, the equation of energy governing the temperature distribution T (r, θ ) for the fluid of constant thermal conductivity k is the Laplace equation,
(2)
∇2T = 0.
The temperature distribution Tp(r, θ ) inside the radiation-absorbing particle is described by
(3) ∇2T p= − 1 kp Q(r, θ ),
where kp is the thermal conductivity of the particle and
re-sulting from local radiation absorption. For a plane mono-chromatic wave, the source function Q(r, θ ) is related to the electric field E(r, θ ) inside the particle according to the Lorenz–Mie theory[7,15], (4) Q(r, θ )=4π υκI λ |E(r, θ)|2 |E0|2 = 4π υκI λ B(ζ, θ ).
Here, υ and κ are the real and imaginary parts of the com-plex refractive index N (N= υ + iκ) of the particle, E0is the incident electric field strength, λ is the wavelength of the incident radiation, B(ζ, θ ) is the dimensionless electric field distribution function, and ζ = r/a is the dimensionless ra-dial spherical coordinate.
The boundary conditions at the particle surface (r= a) require that the normal heat fluxes be continuous and a tem-perature jump that is proportional to the normal temtem-perature gradient[3] occur. Also, the fluid temperature at the plane wall or far removed from the particle approaches the bulk-gas temperature T0, which is a constant. Thus,
(5a) r= a: k∂T ∂r = kp ∂Tp ∂r , (5b) T − Tp= Ctl ∂T ∂r, (6) z= d: T = T0, (7) r→ ∞ and z d: T = T0,
where Ctis the temperature jump coefficient on the particle surface. In Eq.(5a), the flux due to radiative heat transfer, which was taken into account by Akhtaruzzaman and Lin
[16]in studying the photophoresis of a spherical particle in the free molecule regime (in a rarefied gas with large Knud-sen number), is neglected on the assumption that the surface temperature of the particle is not very high. The Knudsen layer adjacent to the isothermal plane wall is neglected for the case of small Knudsen number.
Since the governing equations and boundary conditions are linear, one can write the temperature distribution T as the superposition
(8)
T = T0+ Tw+ Ts.
Here, Tw is a Fourier–Bessel integral solution of Laplace’s equation in cylindrical coordinates that represents the distur-bance produced by the plane wall and is given by
(9) Tw= ∞ 0 ωR(ω)J0(ωρ)eωzdω,
where J0is the Bessel function of the first kind of order zero and R(ω) is an unknown function of ω. The last term on the right-hand side of Eq.(8), Ts, is the solution of Laplace’s equation in spherical coordinates, representing the distur-bance generated by the particle, and is given by an infinite series in harmonics, (10) Ts= aI k ∞ n=0 Tnζ−(n+1)Pn(cos θ ),
where Pnis the Legendre polynomial of order n and Tnare the unknown coefficients to be determined. Note that a so-lution for T in the form given by Eqs.(8)–(10)immediately satisfies boundary condition(7).
Since the temperature is finite for any position in the in-terior of the particle, the solution to Eq.(3)can be written as[8,9] (11) Tp= T0+ aI kp ∞ n=0 Hnζn+ Sn(ζ ) Pn(cos θ ), where Sn(ζ )= 2π υκa λ × ζn 1 ζ t−n+1 π 0 B(t, θ )Pn(cos θ ) sin θ dθ dt + ζ−n−1 ζ 0 tn+2 π 0 B(t, θ )Pn(cos θ ) sin θ dθ dt , (12) and Hnare unknown coefficients.
A brief conceptual description of the solution procedure to determine R(ω) and Tnis given below to help the read-ers follow the mathematical development. At first, boundary condition(6) is exactly satisfied on the plane wall using a Hankel transform. This permits the unknown function R(ω) to be determined in terms of the coefficients Tn. Then, the boundary conditions on the particle surface given by Eqs.(5)
can be satisfied by making use of the collocation method and the solution of the collocation matrix provides numeri-cal values for the coefficients Tn.
Substitution of the solution T given by Eqs.(8)–(10)into the boundary condition (6) and application of the Hankel transform on the variable ρ lead to a solution for R(ω) in terms of the unknown coefficients Tn. After the substitution of this solution into Eq.(9), the resulting temperature field
T is given by (13) T = T0+ aI k ∞ n=0 Tnan+1 Bn(ρ, z)− Bn(ρ, 2d− z),
where the function Bn(ρ, z) is defined by Eq. (A.5b) in
Appendix A. Equation (13)provides an exact solution for the temperature distribution in the fluid phase and the un-known coefficients Tnmust be determined from the remain-ing boundary conditions on the particle surface.
Utilizing the relation
(14) ∂ ∂r = sin θ ∂ ∂ρ + cos θ ∂ ∂z,
one can apply Eqs.(5)to Eqs.(11) and (13)to yield
(15a) ∞ n=0 Tnan+1αn(ρ, z)− nHn+ Sn(ζ ) Pn(cos θ ) = 0,
∞ n=0 Tnk∗an+1 Bn(ρ, z)− Bn(ρ, 2d− z) − C∗tαn(ρ, z) (15b) −Hn+ Sn(ζ ) Pn(cos θ ) = 0
at ζ= 1. Here, the function αn(ρ, z) is defined by Eq.(A.1a),
k∗= kp/k, Ct∗= Ctl/a, and the prime on Sn(ζ ) means dif-ferentiation with respect to ζ . Note that the dimensionless parameter k∗C∗t denotes the relative resistance caused by the temperature jump at the particle surface with respect to heat conduction inside the particle.
To satisfy the conditions in Eqs.(15)exactly along the entire surface of the particle would require the solution of the infinite series of unknown coefficients Tn. However, the collocation technique[17–19]enforces the boundary condi-tions at a finite number of discrete points on the semicircular generating arc of the particle and truncates the infinite series in Eqs.(11) and (13)into finite ones. If the spherical bound-ary is approximated by satisfying conditions in Eqs.(5) at
N discrete points (values of θ ) on its generating arc, then
the infinite series in Eqs.(11) and (13)are truncated after N terms, resulting in a system of 2N simultaneous linear al-gebraic equations in the truncated form of Eqs.(15). This matrix equation can be solved to yield the 2N unknown coefficients Tn and Hn required in the truncated form of Eqs.(11) and (13)for the temperature distribution. The ac-curacy of the truncation technique can be improved to any degree by taking a sufficiently large value of N . Naturally, as N→ ∞ the truncation error vanishes.
2.2. Fluid velocity distribution
Having obtained the solution for the temperature distrib-ution on the particle surface which drives the migration, we can now proceed to find the flow field. The fluid surrounding the particle is assumed to be incompressible and Newtonian. Due to the low Reynolds number, the fluid motion caused by the photophoretic migration of the particle is governed by the quasisteady fourth-order differential equation for ax-isymmetric creeping flows,
(16)
E4Ψ = E2 E2Ψ= 0,
where the Stokes stream function Ψ is related to the velocity components in cylindrical coordinates by
(17a) vρ= 1 ρ ∂Ψ ∂z, (17b) vz= − 1 ρ ∂Ψ ∂ρ,
and the operator E2has the form
(18) E2= ρ ∂ ∂ρ 1 ρ ∂ ∂ρ + ∂2 ∂z2.
Owing to the thermal and frictional slip velocities along the particle surface, the boundary conditions for the fluid veloc-ity at the particle surface are[20]
(19a) r= a: vr= U cos θ, (19b) vθ= −U sin θ + Cml η τrθ + Csη ρfT0r ∂T ∂θ.
Here, vr and vθ are the velocity components in spherical coordinates, Cm and Cs are the frictional and thermal slip coefficients, respectively, on the surface of the particle, τrθis the shear stress for the fluid flow,
(20) τrθ= η r ∂ ∂r vθ r +1 r ∂vr ∂θ ,
and U is the photophoretic velocity of the particle, to be determined. The derivative ∂T /∂θ at the particle surface can be evaluated from the temperature distribution given by Eq.(13)with coefficients Tndetermined from Eqs.(15). The thermal slip velocity expressed by the last term in Eq.(19b)
is proportional to the fluid viscosity due to the fact that the tangential stress of the fluid caused by the temperature gra-dient along the particle surface is proportional to the product of the temperature gradient (or momentum gradient) and the kinematic viscosity of the fluid, and the validity of this ex-pression is based on the assumption that the fluid tempera-ture is only slightly nonuniform on the length scale of the particle radius. Generally speaking, the slip condition be-comes increasingly more important for small particles. At the isothermal plane wall and far away from the particle, the boundary conditions for the fluid velocity are
(21)
z= d: vρ= vz= 0,
(22)
r→ ∞ and z d: vρ= vz= 0.
The effect of gas slip at the plane wall is negligible since the plane is isothermal and the Knudsen number is small.
The stream function for the fluid flow is linearly com-posed of two parts:
(23)
Ψ = Ψw+ Ψs.
Here Ψw is a solution of Eq.(16)in cylindrical coordinates that represents the disturbance produced by the plane wall and is given by a Fourier–Bessel integral,
(24) Ψw= ∞ 0 ρJ1(ωρ) X(ω)+ zY (ω)eωzdω,
where J1is the Bessel function of the first kind of order one, and X(ω) and Y (ω) are unknown functions of ω. The sec-ond part of Ψ , denoted by Ψs, is a solution of Eq.(16)in spherical coordinates representing the disturbance generated by the particle and is given by
(25) Ψs= Ua2 ∞ n=2 Bnζ−n+1+ Dnζ−n+3 G−1/2n (cos θ ),
where G−1/2n is the Gegenbauer polynomial of the first kind of order n and degree −1/2, and Bn and Dn are unknown constants. Note that boundary condition(22)is immediately satisfied by a solution of the form given by Eqs.(23)–(25).
As was the case with the solution for the temperature distribution, the determination of X(ω), Y (ω), Bn, and Dn
will be undertaken by a two-step procedure. First, bound-ary condition(21)is exactly satisfied on the plane wall by a Hankel transform; then, the boundary conditions given by Eq. (19) are satisfied numerically at collocation points on the particle surface. Application of boundary condition(21)
to Eqs.(23)–(25)using Eq.(17)leads to a solution for X(ω) and Y (ω) in terms of the unknown coefficients Bn and Dn. This solution is substituted back into Eq.(24), leading to a new expression for the stream function. The result for the velocity components is (26a) vρ= U ∞ n=2 Bnan+1βn(ρ, z)+ Dnan−1δn(ρ, z) , (26b) vz= U ∞ n=2 Bnan+1βn(ρ, z)+ Dnan−1δn(ρ, z) ,
where the functions βn, δn, βn, and δn are defined by Eqs.(A.2) and (A.3).
The only boundary conditions that remain to be satisfied are those on the particle surface. Substituting Eq.(13)into Eq.(19), one obtains
vρ= Cm∗a η τrθcos θ+ CsηI ρfT0k cos θ ∞ n=0 Tnan+1αn(ρ, z), (27a) vz= U − Cm∗a η τrθsin θ− CsηI ρfT0k sin θ ∞ n=0 Tnan+1αn(ρ, z) (27b) at r= a, where Cm∗ = Cml/a and the definition of αn(ρ, z) is given by Eq.(A.1b). Here the first N coefficients Tnhave been determined through the procedure given in the pre-vious section. Application of these boundary conditions to Eqs.(26)can be accomplished by utilizing the collocation technique presented for the solution of the temperature field. At the particle surface, boundary conditions in Eq.(27)are applied at M discrete points (values of θ ) and the infinite se-ries in Eqs.(26)are truncated after M terms. This generates a set of 2M linear algebraic equations for the 2M unknown coefficients Bnand Dn. The fluid velocity field is completely solved once these coefficients are determined.
2.3. Velocity of the particle
The drag force exerted by the fluid on the spherical boundary r= a can be determined from[21]
(28) F = ηπ π 0 r3sin3θ ∂ ∂r E2Ψ r2sin2θ r dθ.
Substitution of Eqs.(23)–(25)into the above integral and ap-plication of the orthogonality properties of the Gegenbauer polynomials result in the simple relation
(29)
F = 4πηD2.
This shows that only the coefficient D2 contributes to the drag force on the particle.
Since the photophoretic particle is freely suspended in the fluid, the net force exerted by the fluid on the surface of the particle must vanish. From Eq.(29), we have D2= 0. The photophoretic velocity U of the particle can be obtained by satisfying this constraint.
The normalized particle velocity U/U0will be a function of the parameters k∗, Ct∗, Cm∗, and a/d, where U0is the pho-tophoretic velocity of the particle in the absence of the plane wall and is given by Eq.(1), or
(30)
U0= −
2J CsηI
3(1+ 2Cm∗)(2+ k∗+ 2k∗Ct∗)kρfT0
.
Here J is the so-called photophoretic asymmetry factor,
(31) J=6π υκa λ 1 0 π 0 B(ζ, θ )ζ3cos θ sin θ dθ dζ,
which depends on the complex refractive index (N= υ +iκ) and the normalized size (2π a/λ) of the particle. The asym-metry factor represents a weighted integration of the heat source function over the particle volume and defines the sign and magnitude of the photophoretic force. If J < 0, the particle moves in the direction of the light beam (positive photophoresis). If J > 0, the particle moves in the opposite direction (negative photophoresis). Equation(30)shows that the magnitude of U0decreases with an increase in k∗, Ct∗, or
Cm∗ for a given value of J CsηI /kρfT0.
For a completely opaque spherical particle the heat sources are concentrated on the illuminated part of the parti-cle surface[9,14]; namely,
(32) B(ζ, θ )= −λ
2π υκacos θ δ(ζ − 1), for π
2 θ π,
0, for 0 θ π2,
where δ(ζ−1) is a Dirac delta function which equals infinity if ζ= 1 and vanishes otherwise. The substitution of Eq.(32)
into Eq.(31)results in J= −1/2 knowing that 1
0
g(ζ )δ(ζ− 1) dζ = (1/2)g(1).
If the incident light is transmitted and focused on the rear part of the particle surface, Eq.(32)should be modified in reverse with the angle θ and its substitution into Eq.(31)
yields J= 1/2. Thus, the range of the asymmetry factor is
−1/2 J 1/2. According to Eq.(30), the photophoretic velocity at on illumination of an intensity comparable to the solar constant (1353 W m−2) is on the order of 10−5m s−1.
By the linearity of the problem, the same value of the normalized photophoretic velocity is predicted for a given combination of k∗, C∗t, C∗m, and a/d whether the particle is approaching the plane wall or receding from it.
Table 1
Normalized photophoretic mobility of a spherical particle moving perpendicular to a plane wall
a/d U/U0 C∗t = 2Cm∗= 0.02 C∗t = 2Cm∗= 0.2 k∗= 0 k∗= 1 k∗= 10 k∗= 100 k∗= 0 k∗= 1 k∗= 10 k∗= 100 0 1 1 1 1 1 1 1 1 0.1 0.99615 0.99708 0.99875 1.00576 0.99733 0.99917 1.00681 1.07453 0.2 0.98225 0.98630 0.99356 1.02194 0.98677 0.99438 1.02538 1.29750 0.3 0.95589 0.96563 0.98316 1.04860 0.96541 0.98292 1.05400 1.67337 0.4 0.91522 0.93354 0.96709 1.08841 0.93091 0.96258 1.09221 2.21707 0.5 0.85791 0.88816 0.94508 1.14764 0.88062 0.93077 1.14048 2.95811 0.6 0.78017 0.82634 0.91688 1.23912 0.81066 0.88358 1.19956 3.94480 0.7 0.67557 0.74227 0.88209 1.39044 0.71479 0.81404 1.26856 5.24557 0.8 0.53302 0.62437 0.84021 1.67222 0.58226 0.70800 1.33672 6.91926 0.9 0.33199 0.44385 0.78874 2.33781 0.39035 0.52828 1.33283 8.69112 0.95 0.19723 0.30288 0.74658 3.15679 0.25188 0.37173 1.17040 8.67640 0.99 0.05885 0.11546 0.56164 4.21222 0.08011 0.13257 0.55408 4.67247 0.995 0.03570 0.07447 0.43936 3.79905 0.04577 0.07751 0.34294 2.95778 0.999 0.01071 0.02401 0.18049 1.85088 0.01086 0.01884 0.08861 0.78223
3. Results and discussion
The solution for the photophoretic motion of a spherical particle normal to a plane wall, obtained by using the bound-ary collocation method described in the previous section, is presented in this section. The system of linear algebraic equations to be solved for the coefficients Tnand Hnis con-structed from Eq.(15), while that for Bnand Dnis composed of Eqs.(26) and (27).
In specifying the points along the semicircular generat-ing arc of the sphere (with a constant value of φ) where the boundary conditions are to be exactly satisfied, the first point that should be chosen is θ= π/2, since this point defines the projected area of the particle normal to the direction of motion. In addition, the points θ= 0 and π are also impor-tant because they control the gap between the sphere and the plane. However, an examination of the systems of lin-ear algebraic equations given by Eqs.(15)and(27) shows that the matrix equations become singular if these points are used. To overcome this difficulty, these points are replaced by closely adjacent points, i.e., θ= δ, π/2 − δ, π/2 + δ, and
π− δ. Additional points along the boundary are selected as
mirror-image pairs about the plane θ = π/2 to divide the two quarter-circular arcs of the particle into equal segments. The optimum value of δ in this work is found to be 0.01◦, to which the numerical results of the particle velocity converge satisfactorily.
For the continuum-with-slippage approach employed in this work, the Knudsen number (l/a) of the system should be smaller than about 0.1. As mentioned in the first section, a set of well-adapted values for the temperature jump and frictional slip coefficients under the condition of complete thermal and momentum accommodations are 2.18 and 1.14, respectively. Consequently, the normalized coefficients Ct∗ and Cm∗ must be restricted to be less than unity. For con-venience we will use the ratio Ct∗/Cm∗ = 2 (rounded from
2.18/1.14= 1.91) throughout this section, without loss of
reality or generality. On the other hand, although the silica aerogel can have a low thermal conductivity comparable to that of nonmetallic gases [22], the thermal conductivity of an aerosol particle is typically much higher than that of the surrounding gas. Thus, the value of the relative conductivity
k∗will exceed unity under most practical circumstances. The collocation solutions for the wall-corrected reduced photophoretic mobility of an opaque spherical particle with the heat source distribution function given by Eq.(32) mov-ing perpendicular to a plane wall for various values of the parameters Cm∗ (= Ct∗/2), k∗, and a/d are presented in Ta-ble 1and depicted inFigs. 2–4. The cases of k∗< 1, which
are not likely to exist in practice, are considered here for the sake of numerical comparison. All of the results obtained
un-Fig. 2. Plots of the normalized velocity of a spherical particle undergoing photophoresis perpendicular to a plane wall versus the separation parame-ter a/d for various values of k∗. The solid curves represent the case of C∗t = 2Cm∗= 0, and the dashed curves denote the case of Ct∗= 2C∗m= 0.2.
Fig. 3. Plots of the normalized velocity of a spherical particle undergo-ing photophoresis perpendicular to a plane wall versus the relative thermal conductivity of the particle for various values of the separation parameter a/d. The solid curves represent the case of Ct∗= 2Cm∗= 0, and the dashed curves denote the case of Ct∗= 2C∗m= 0.2.
der the collocation scheme converge satisfactorily to at least the significant figures shown in the table. The accuracy and convergence behavior of the truncation technique is princi-pally a function of the relative spacing (d− a)/a. For the difficult case with a/d= 0.999, the numbers of collocation points N= 250 and 250 are sufficiently large to achieve this convergence. As expected, the particle will move with the velocity that would exist in the absence of the wall, given by Eq. (30), as a/d= 0. In general, the wall effect on the photophoresis of a particle can be quite significant in prac-tical situations. It should be pointed out that the continuum assumption for the fluid flow in the gap between the parti-cle and the wall might have failed for the table and figures as a/d > 0.99 in the case of Cm∗ = 0.01 and as a/d > 0.9 in the case of Cm∗ = 0.1.
It is found that the normalized photophoretic mobility
U/U0 of the particle increases with an increase in the rel-ative conductivity k∗, keeping the other factors (Ct∗, Cm∗, and a/d) unchanged. The increase in the particle mobility becomes more pronounced as a/d increases. This behavior is expected knowing that the temperature gradients on the particle surface near an isothermal plane wall increase as
k∗increases[23]. The value of U/U0 becomes a sensitive function of k∗if the value of Cm∗ or a/d is relatively large. On the other hand, the normalized photophoretic mobility of the particle in general increases with an increase in Cm∗ for specified values of k∗and a/d, and U/U0is not a sensitive function of Cm∗ if the value of Cm∗ is small.
Examination of the results shown inTable 1 andFig. 2
reveals an interesting feature. For the case of a sphere with a relatively small value of k∗, the photophoretic mobility of
Fig. 4. Plots of the normalized velocity of a spherical particle with Ct∗= 2C∗mundergoing photophoresis perpendicular to a plane wall versus the normalized frictional slip coefficient for various values of the separa-tion parameter a/d. The solid curves represent the case of k∗= 0, and the dashed curves denote the case of k∗= 10.
the particle near the plane wall is a monotonically decreas-ing function of a/d. For the case of a sphere with a relatively large value of k∗, however, the photophoretic mobility of the particle increases with an increase in a/d when a/d is small, but decreases steadily from a maximum with increasing a/d when a/d is sufficiently large, going to zero in the limit. The locations of the maximum shift to larger values of a/d as the parameter k∗increases or C∗mdecreases. At the maxi-mum point, the particle can move much faster than it would at a/d= 0. For example, as Ct∗= 2C∗m= 0.2, k∗= 100, and a/d= 0.9, the photophoretic velocity can be more than seven times higher than the value with the wall far away from the particle. This interesting feature that U/U0may not be a monotonic function of a/d and can even be greater than unity is understandable because the wall effect of hydrody-namic (viscous) resistance on the particle is in competition with the wall effect of thermal enhancement when the par-ticle with a relatively large value of k∗ is undergoing pho-tophoretic motion perpendicular to an isothermal plate. It is understood that the value of l/(d− a) may be greater than unity for the case of a/d→ 1 (as Ct∗= 2Cm∗ = 0.2 or 0.02), where the continuum model for the fluid is not likely to be valid in practice. This limiting case is considered here for the sake of numerical comparison. Since the magnitude of
U0 can be quite small at large values of k∗ (to which the particle surface temperature is almost uniform), the strong thermal enhancement on the particle velocity can make the value of U/U0much greater than unity. This unexpected be-havior of wall enhancement on the particle migration also occurs in the case of thermophoresis of an aerosol sphere
Fig. 5. Plots of the normalized photophoretic mobility (solid curves, with Ct∗= 2C∗mand k∗= 1) and sedimenting mobility (dashed curves) of a
spherical particle migrating perpendicular to a plane wall versus the sep-aration parameter a/d for different values of C∗m.
with a relatively large value of k∗normal to an isothermal plane wall[24].
For the creeping motion of a spherical aerosol particle with a frictional (but isothermal) slip surface on which a constant body force F ez (e.g., a gravitational field) is ex-erted perpendicular to an infinite plane wall, exact results of the particle velocity have been obtained by using spherical bipolar coordinates[25]. A comparison of the boundary ef-fects on the motion of an aerosol sphere under gravity (in which U0= (F/6πηa)(1 + 3Cm∗)/(1+ 2Cm∗)) and on the photophoresis of the particle (with k∗= 1, so that the effect of heat conduction will not be important) is given inFig. 5. Similar to the case of photophoresis, the wall effect on the particle motion in a gravitational field is stronger when the value of Cm∗ becomes smaller. Evidently, the wall effect on a photophoretic particle in general is much weaker than that on a sedimenting particle.
4. Concluding remarks
In this work, the quasisteady photophoresis of a spheri-cal particle in a gaseous medium normal to a plane wall has been analyzed by using the boundary collocation method. On the basis of the assumption of small Knudsen, Peclet, and Reynolds numbers, the temperature and fluid flow fields for this axisymmetric motion are solved and the wall-corrected particle velocity is obtained. This photophoretic velocity rel-ative to its undisturbed value increases monotonically with an increase in the relative thermal conductivity k∗ of the particle. When the value of k∗is sufficiently large, the pho-tophoretic mobility of a particle near a wall may not be a monotonic decreasing function of the separation parameter
a/d, and the wall effect can speed up or slow down the
par-ticle velocity with respect to its isolated value. This behavior reflects the competition between the weak hydrodynamic retardation exerted by the neighboring wall on the particle migration and the possible relatively strong thermophoretic enhancement due to the thermal interaction between the par-ticle and the wall. The results show that the boundary effect of the plane wall on the photophoretic motion of an aerosol particle can be significant in some situations. In practical aerosol systems, the boundary effect on photophoresis of a particle in general is weaker than that on the particle motion driven by gravity.
In the previous section, we have presented the results of wall-corrected photophoretic mobility calculated from a simplified model for a completely opaque sphere whose heat sources are concentrated on the illuminated part of the par-ticle surface. A calculation according to the Lorenz–Mie theory showed bad applicability of this model even for large strongly absorbing spherical particles[26]. Evidently, if we perform a corresponding calculation for a model transpar-ent sphere with heat sources conctranspar-entrated completely on the rear half of the particle surface, the same results but with op-posite direction (negative photophoresis) will be obtained. If one uses a more consecutive energy-absorbing model than that defined by Eq. (32)for the asymmetry factor J given by Eq.(31)based on the Lorenz–Mie theory, it is expectable that the resulted photophoretic mobility of the sphere will be qualitatively similar to but quantitatively smaller than that obtained from using Eq.(32).
It is worth repeating that our results for the photophoretic velocity are obtained on the basis of a continuum model for the gas phase with slip-flow boundary conditions at the par-ticle surface. For a perfect gas, the kinetic theory predicts that the mean free path of gas molecules is inversely propor-tional to the pressure[27]. As examples, the mean free path of air molecules at 25◦C is about 67 nm at 1 atm and is about 51 µm at 1 Torr. Therefore, our results obtained with the as-sumption of small Knudsen number can be used for a broad range of particle sizes around atmospheric pressure but are only applicable to relatively large particles at low pressures.
Acknowledgment
Part of this research was supported by the National Sci-ence Council of the Republic of China.
Appendix A. Definitions of some functions in Section2 For conciseness the definitions of some functions in Sec-tion2are listed here:
αn(ρ, z)= ρAn(ρ, z)− An(ρ, 2d− z)
− (n + 1)zBn+1(ρ, z)+ Bn+1(ρ, 2d− z),
αn(ρ, z)= zAn(ρ, z)− An(ρ, 2d− z) + (n + 1)ρBn+1(ρ, z)+ Bn+1(ρ, 2d− z), (A.1b) βn(ρ, z)= Bn(ρ, z)− Bn(ρ, 2d− z) (A.2a) + 2(n + 1)(d − z)Bn+1(ρ, 2d− z), βn(ρ, z)= Bn(ρ, z)− Bn(ρ, 2d− z) (A.2b) − 2(n + 1)(d − z)Bn+1(ρ, 2d− z), δn(ρ, z)= Dn(ρ, z)− Dn(ρ, 2d− z) − (2/n)(n − 1)(n − 3)(d − z)Bn−1(ρ, 2d− z) (A.3a) + 2(2n − 3)d(d − z)Bn(ρ, 2d− z), δn(ρ, z)= Dn(ρ, z)− Dn(ρ, 2d− z) + 2(n − 2)(d − z)Bn−1(ρ, 2d− z) (A.3b) − 2(2n − 3)d(d − z)Bn(ρ, 2d− z), where An(ρ, z)= nz2− (n + 1)ρ2 ρ(ρ2+ z2)(n+3)/2Pn z (ρ2+ z2)1/2 − nz ρ(ρ2+ z2)(n+2)/2Pn−1 z (ρ2+ z2)1/2 , (A.4) (A.5a) Bn(ρ, z)= n+ 1 ρ(ρ2+ z2)n/2G −1/2 n+1 z (ρ2+ z2)1/2 , (A.5b) Bn(ρ, z)= 1 (ρ2+ z2)(n+1)/2Pn z (ρ2+ z2)1/2 , Dn(ρ, z)= n+ 1 ρ(ρ2+ z2)(n−2)/2G −1/2 n+1 z (ρ2+ z2)1/2 − 2z ρ(ρ2+ z2)(n−1)/2G −1/2 n z (ρ2+ z2)1/2 , (A.6a) Dn(ρ, z)= 2 (ρ2+ z2)(n−1)/2G −1/2 n z (ρ2+ z2)1/2 (A.6b) + 1 (ρ2+ z2)(n−1)/2Pn z (ρ2+ z2)1/2 .
In the above equations, Pn is the Legendre polynomial of order n and G−1/2n is the Gegenbauer polynomial of the first kind of order n and degree−1/2.
Appendix B. Nomenclature
An functions of ρ and z defined by Eq.(A.4)(m−n−2)
a particle radius (m)
B(ζ, θ ) dimensionless electric field distribution function
defined by Eq.(4)
Bn, Dn coefficients in Eqs.(25) and (26)for the flow field
Bn, Bn functions of ρ and z defined by Eqs.(A.5)(m−n−1)
Cm coefficient accounting for the frictional slip at the particle surface
Cm∗ = Cml/a
Cs coefficient accounting for the thermal slip at the particle surface
Ct coefficient accounting for the temperature jump at the particle surface
Ct∗ = Ctl/a
Dn, Dn functions of ρ and z defined by Eqs.(A.6)(m−n+1)
d distance between the particle center and the plane wall (m)
E2 the Stokes operator defined by Eq.(18)(m−2)
G−1/2n the Gegenbauer polynomial of the first kind of or-der n and degree−1/2
Hn coefficients in Eq.(11)for the internal temperature field
I intensity of the incident light beam (W m−2)
J the photophoretic asymmetry factor of the particle defined by Eq.(31)
Jn the Bessel function of the first kind of order n
k thermal conductivity of the fluid (W m−1K−1)
kp thermal conductivity of the particle (W m−1K−1)
k∗ = kp/k
l mean free path of the gas molecules (m)
Pn the Legendre polynomial of order n
r radial spherical coordinate (m)
Sn(ζ ) function of ζ defined by Eq.(12)
T temperature distribution in the fluid phase (K)
Tn coefficients in Eqs. (10) and (13) for the external temperature field
Tp temperature distribution inside the particle (K)
T0 temperature of the bulk gas and the plane wall (K)
U photophoretic velocity of a particle (m s−1)
U0 photophoretic velocity of an isolated particle (m s−1)
vr, vθ components of the fluid velocity in spherical coor-dinates (m s−1)
vρ, vz components of the fluid velocity in cylindrical co-ordinates (m s−1)
z axial cylindrical coordinate (m)
Greek letters
αn, αn functions of ρ and z defined by Eqs.(A.1)(m−n−1)
βn, βn functions of ρ and z defined by Eqs.(A.2)(m−n−1)
δn, δn functions of ρ and z defined by Eqs.(A.3)(m−n+1)
ζ = r/a
η viscosity of the fluid (kg m−1s−1)
θ, φ angular spherical coordinates
λ the wavelength of the incident light beam (m)
υ, κ real and imaginary parts of the complex refractive index N of the particle
ρ radial cylindrical coordinate (m)
ρf density of the fluid (kg m−3)
References
[1] C. Orr, E.Y.H. Keng, J. Atmos. Sci. 21 (1964) 475–478. [2] J.C. Maxwell, Philos. Trans. R. Soc. 170 (1879) 231–256.
[3] E.H. Kennard, Kinetic Theory of Gases, McGraw–Hill, New York, 1938, pp. 291–337.
[4] O. Preining, in: C.N. Davies (Ed.), Aerosol Science, Academic Press, New York, 1966, pp. 111–135.
[5] S. Arnold, M.J. Lewittes, J. Appl. Phys. 53 (1982) 5314–5319. [6] V. Chernyak, S. Beresnev, J. Aerosol Sci. 24 (1993) 857–866. [7] M. Kerker, D.D. Cooke, J. Opt. Soc. Am. 72 (1982) 1267–1272. [8] D.W. Mackowski, Int. J. Heat Mass Transfer 32 (1989) 843–854. [9] L.D. Reed, J. Aerosol Sci. 8 (1977) 123–131.
[10] Yu.I. Yalamov, V.B. Kutukov, E.R. Shchukin, J. Colloid Interface Sci. 57 (1976) 564–571.
[11] L. Talbot, R.K. Cheng, R.W. Schefer, D.R. Willis, J. Fluid Mech. 101 (1980) 737–758.
[12] C. Cercignani, Rarefied Gas Dynamics: From Basic Concepts to Ac-tual Calculations, Cambridge Univ. Press, Cambridge, UK, 2000. [13] Y. Sone, Kinetic Theory and Fluid Dynamics, Birkhauser, Boston,
2002.
[14] H.J. Keh, Aerosol Air Quality Res. 1 (2001) 21–30.
[15] W.M. Greene, R.E. Spjut, E. Bar-Ziv, A.F. Sarofim, J.P. Longwell, J. Opt. Soc. Am. B 2 (1985) 998–1004.
[16] A.F.M. Akhtaruzzaman, S.P. Lin, J. Colloid Interface Sci. 61 (1977) 170–182.
[17] V. O’Brien, AIChE J. 14 (1968) 870–875.
[18] M.J. Gluckman, R. Pfeffer, S. Weinbaum, J. Fluid Mech. 50 (1971) 705–740.
[19] P. Ganatos, R. Pfeffer, S. Weinbaum, J. Fluid Mech. 99 (1980) 739– 753.
[20] J.R. Brock, J. Colloid Sci. 17 (1962) 768–780.
[21] J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics, Mart-inus Nijhoff, The Netherlands, 1983.
[22] J.H. Lienhard, A Heat Transfer Textbook, second ed., Prentice-Hall, Englewood Cliffs, NJ, 1987.
[23] H.J. Keh, L.C. Lien, J. Fluid Mech. 224 (1991) 305–333. [24] S.H. Chen, H.J. Keh, J. Aerosol Sci. 26 (1995) 429–444. [25] S.H. Chen, H.J. Keh, J. Colloid Interface Sci. 171 (1995) 63–72. [26] S.A. Beresnev, L.B. Kochneva, Atmos. Oceanic Optics 16 (2003) 119–
126.
[27] D.P. Shoemaker, C.W. Garland, J.I. Steinfeld, J.W. Nibler, Experi-ments in Physical Chemistry, fourth ed., McGraw–Hill, New York, 1981.
