penetration of aerosol undergoing combined electrostatic dispersion and diffusion in a cylindrical...

Aerosol Science 38 (2007) 481–493www.elsevier.com/locate/jaerosci

Penetration of aerosol undergoing combined electrostatic dispersionand diffusion in a cylindrical tube

Manuel Alonso∗, Francisco José AlguacilNational Center for Metallurgical Research (CSIC), Gregorio del Amo, 8. 28040 Madrid, Spain

Received 20 April 2006; received in revised form 2 March 2007; accepted 2 March 2007

Abstract

The deposition of unipolarly charged aerosol particles by simultaneous Brownian diffusion and electrostatic dispersion (space-charge) in laminar flow tubes has been modelled in two different ways. In the first approach, the rigorous transport equation has beensolved numerically under the restriction of negligible axial components of the particle Brownian motion and of the space-chargeelectric field. The thus calculated theoretical penetrations agree satisfactorily with the experimental results obtained for air ions.Second, the system has also been modelled from a phenomenological point of view, based on the fact that deposition by diffusionand space-charge follows first- and second-order kinetics, respectively. The latter approach yields a simple and practical equationcorrelating particle penetration with tube geometry, aerosol flow rate, and particle mobility and concentration.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Aerosol deposition; Diffusion; Space-charge

1. Introduction

Knowledge of particle deposition in tubes is required in many applications of aerosol science and technology, suchas particle synthesis, sampling, sizing, counting, dosing, toxicology, etc. It thus seems logical that the literature on thesubject is very extensive.

Particle loss to the tube wall can occur as a result of a variety of mechanisms: diffusion, inertial deposition, gravita-tional settling and electrostatic forces. Electrostatic deposition, in turn, can take place because of the image (polarization)force between charged particles and a conductive wall, or because of electrostatic repulsion between particles chargedwith the same polarity (space-charge). In the present study we are interested in small particles, say those with diameterbelow 100 nm, for which the inertial and gravitational mechanisms are irrelevant.

Diffusion is possibly the deposition mechanism to which most of the research effort in the field has been de-voted (Davies, 1973; Gormley & Kennedy, 1949; Ingham, 1975; and many others). Most of these works were aimedat the solution of the convective–diffusive equation for the common case in which diffusion along the tube axiscan be neglected in comparison with the convective motion of the aerosol particles. The series solution reported byGormley and Kennedy (1949) has been confirmed by numerous experiments to be valid for the entire particle size

∗ Corresponding author.E-mail address: [email protected] (M. Alonso).

0021-8502/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.jaerosci.2007.03.001

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482 M. Alonso, F.J. Alguacil / Aerosol Science 38 (2007) 481–493

range, even for particle diameters as small as 2 nm (Alonso, Kousaka, Hashimoto, & Hashimoto, 1997; Otani, Emi,Cho, & Namiki, 1995; Ramamurthi, Strydom, & Hopke, 1990; Scheibel & Porstendörfer, 1984).

The effect of image forces on non-diffusive particle deposition has been addressed by Yu (1977), Yu and Chandra(1978), Becker, Anderson, Allen, Birkhoff, and Ferrell (1980), and Ljepojevic and Balachandran (1993). According toYu and Chandra (1978), the effect of image force on particle deposition is negligible for particles carrying less than 10elementary charges.

Particle deposition by electrostatic repulsion has been studied for over a century; a brief history can be found inKasper (1981). In most of those early works, additional particle loss mechanisms were not considered. Simultaneousdiffusion and space-charge were studied theoretically by Chen (1978), who solved the transport equation by an integralmethod, for which he had to assume a particle concentration profile consisting of a uniform core and a third-orderboundary layer polynomial concentration profile. Kasper (1981) presented the results of an experimental investigationaimed at the determination of the so-called coefficient of electrostatic dispersion, which represents the particle loss rateconstant by mutual electrostatic repulsion. For this, he had to account for diffusion and sedimentation losses to correctthe measured particle number concentrations.

In the present work, we examine the deposition of unipolarly charged ultrafine aerosol particles in laminar flow tubesby simultaneous diffusion and space-charge. Instead of assuming a concentration profile as Chen (1978), we will solvenumerically the transport equation (convection + diffusion + space-charge) and compare the numerical results withexperimental penetration data obtained for positively charged air ions. In addition, a simplified model derived from aphenomenological point of view (in a way similar, though not identical, to that followed by Kasper, 1981 to accountfor diffusion and sedimentation losses), will be presented and compared with the numerical solution of the transportequation as well as with the experimental results.

2. Theory

The development that follows is restricted to particles carrying very few elementary charges of the same sign. Inthese circumstances, the effect of image force in particle deposition is insignificant (Scheibel & Porstendörfer, 1984;Yu & Chandra, 1978). In general, particles with few charges are small in size and, thus, possess a relatively high electricmobility, so that the contribution of space-charge to the deposition process can be noticeable even at moderate aerosolconcentration.

2.1. Rigorous approach

The transport equation for the system under study is

u · ∇n = −∇ · (−D∇n + ZnE), (1)

where u is the air flow velocity, whose only non-vanishing component is that in the axial direction, ux ; n(r, x) is theparticle number concentration, r and x are the radial and axial coordinates, D and Z are the diffusion coefficient andthe electric mobility of the particles, and E is the electric field. The latter obeys Poisson’s equation

∇ · E = pe

�0n, (2)

where pe is the particle charge, and �0 the dielectric constant of a vacuum. In this study, we will take p=1 for simplicity,although the equations to be derived are also valid for not too large values of p, inasmuch as image force effects can beneglected. Assuming a constant diffusion coefficient, Eqs. (1) and (2) yield

u · ∇n = D∇2n − ZE · ∇n − e

�0Zn2. (3)

Particle diffusion in the axial direction is negligible for Peclet number Pe?1 (Ingham, 1984), where Pe = 2uxR/D

(ux is the mean air flow velocity, R the tube radius). Under these conditions, Eq. (3) in cylindrical coordinates becomes

(ux + ZEx)�n

�x= D

(�2n

�r2 + 1

r

�n

�r

)− ZEr

�n

�r− e

�0Zn2 (P e?1), (4)

M. Alonso, F.J. Alguacil / Aerosol Science 38 (2007) 481–493 483

where Er and Ex are, respectively, the radial and axial components of the electric field. Introducing the dimensionlessvariables

r∗ = r/R; x∗ = x/L; n∗ = n/n0;

u∗x = ux/ux; vs = eZn0R/ux�0; a = R/L; � = DL/uxR

2;

E∗r = �0Er/en0R; E∗

x = �0LEx/en0R2, (5)

where L is the tube length, and n0 the particle number concentration at the tube inlet, Eq. (4) can be written as

(u∗x + avsE

∗x )

�n∗

�x∗ = �

(�2n∗

�r∗2 + 1

r∗�n∗

�r∗

)− vsE

∗r

a

�n∗

�r∗ − vs

an∗2. (6)

In the set of definitions (5), vs represents the dimensionless particle velocity due to the space-charge field, a is the tube‘aspect ratio’, and � is the parameter which appears in the well-known Gormley–Kennedy equation (1949), shownbelow as Eq. (20), which gives the diffusional particle penetration through a laminar flow tube in the absence ofelectrical effects.

It will be shown next that the term containing the axial component of the electric field can be neglected in comparisonwith that containing the radial component. First, when written in terms of the same dimensionless variables introducedabove, Poisson’s equation becomes

1

r∗�

�r∗ (r∗E∗r ) + a2 �E∗

x

�x∗ = n∗. (7)

From ∇ ×E=0, it is found that �Er/�x =�Ex/�r , so that Ex/Er is of the order of R/L. Therefore, the dimensionlessfield components E∗

r and E∗x are of the same order of magnitude. Eq. (7) then shows that the axial field term is of

the order of a2 times the radial term. In the experiments to be discussed below, we have used tubes of radii between0.2 and 0.4 cm and lengths between 5 and 45 cm so that a2 ranged between 1.6 × 10−5 and 6.4 × 10−3. When a2>1the axial term in Poisson’s equation can be safely neglected in comparison with the radial term. For the same reason,avsE

∗x�n∗/�x∗ can be disregarded in comparison with (vsE

∗r /a)�n∗/�r∗ in Eq. (6). Therefore, the transport equation

can be finally written as

u∗x

�n∗

�x∗ = �

(�2n∗

�r∗2 + 1

r∗�n∗

�r∗

)− VS

(E∗

r

�n∗

�r∗ + n∗2)

, (8)

which is valid for Pe?1 and (R/L)2>1. In Eq. (8) we have introduced a new definition of the dimensionless particlevelocity due to space-charge:

VS = vs

a= eZn0L

ux�0. (9)

It is further assumed that the Poiseuille flow velocity profile is already developed at the tube inlet (see Experimentalbelow), so that

u∗x = 2(1 − r∗2). (10)

Eq. (8) must be solved along with Poisson’s equation approximated for a2>1:

1

r∗�

�r∗ (r∗E∗r ) = n∗, (11)

and the following boundary conditions, expressing uniform particle concentration at the tube inlet, zero particle con-centration at the wall, and electric field symmetrical about the tube centreline:

n∗(r∗, 0) = 1; n∗(1, x∗) = 0; E∗r (0) = 0. (12)


Finally, particle penetration through the tube can be calculated as

P =∫ 1

0 r∗u∗xn

∗(r∗, 1) dr∗∫ 1

0 r∗u∗x dr∗ = 4

∫ 1

0r∗(1 − r∗2)n∗(r∗, 1) dr∗. (13)

2.2. Phenomenological approach

In the phenomenological approach, it is simply assumed that diffusional deposition is a first-order process withrespect to the particle number concentration, and that space-charge deposition is a second-order process. This is sobecause a diffusive particle undergoes Brownian motion even in the absence of other particles, but a charged particlecannot feel electrostatic repulsion unless other charged particle(s) is (are) present. Denoting by n the average particlenumber concentration at a distance x from the tube inlet, different from the local concentration n(r, x) discussed before,we can write

ux

dn

dx= dn

dt= −�Dn − �Sn2. (14)

In the above expression, �D and �S are, respectively, the particle loss rate constants for the diffusion and space-chargedeposition processes. The �S in Eq. (14) is just the coefficient of electrostatic dispersion discussed by Kasper (1981).Note that Eq. (14) can be also derived from (8) and (11) using an order of magnitude analysis.

The solution of Eq. (14) with initial condition n = n0 at t = 0 (or x = 0) is

P = n

n0= exp(−�Dt)

1 + (�S/�D)n0[1 − exp(−�Dt)] . (15)

In the absence of diffusion, Eq. (14) is reduced to dn/dt = −�Sn2, with solution

nS = n0

1 + �Sn0t. (16)

The subscript S attached to n is a reminder that (16) is only valid for non-diffusive particles. Comparing this equationwith the well-known electrostatic dispersion equation (see for instance Kasper, 1981; Yu, 1977)

nS = n0

1 + (e/�0)Zn0t, (17)

it follows that the space-charge deposition rate constant can be calculated theoretically as

�S = eZ

�0. (18)

As for the diffusional deposition rate constant �D, an analogous procedure does not yield a satisfactory result. Indeed,in the absence of space-charge effects, Eq. (15) is reduced to

nD = n0 exp(−�Dt). (19)

If one compares the last expression with that given by Gormley–Kennedy (1949),

PD = nD

n0=

{0.8191 exp(−3.657�) + 0.0975 exp(−22.3�) + 0.0325 exp(−57�) for ��0.0312,

1 − 2.56�2/3 + 1.2� + 0.177�4/3 for � < 0.0312,(20)

where � is the dimensionless parameter defined in Eq. (5), and which can be rewritten as

� = Dt

R2 , (21)

one sees the difficulty in establishing a theoretical expression for the determination of �D.


2.3. The connection between the two approaches

We have just seen the difficulty in finding a direct relation between the constant �D and the Gormley–Kennedyparameter �. According to Eq. (19), the phenomenological diffusional penetration PD (no space-charge) is given byexp(−�Dt), but we know that the correct expression for PD is Eq. (20). Therefore, taking

�D = − ln PD

t= −ux

Lln PD (22)

ensures that Eq. (15) approaches the correct equation (20) when space-charge deposition becomes insignificant.Inserting Eqs. (18) and (22) into (15), and using the dimensionless space-charge particle velocity VS given by Eq.

(9), we have

P = PD

1 − VS(1 − PD)/ ln PD. (23)

We have thus arrived at a very simple expression giving particle penetration in terms of the diffusional penetrationPD (Eq. (20)) and the space-charge parameter VS (Eq. (9)). Eq. (23) differs from that Kasper (1981) used to accountfor sedimentation and diffusion losses. For negligible sedimentation, the Kasper equation analogous to (23) wouldread P = PD/(1 + VS) which, as (23), approaches the correct one, P → PD, as VS → 0, but does not include the−(1 − PD)/ ln PD factor required by (15).

Of course, it remains to be seen whether the simple equation (23) is in agreement with (i) the theoretical penetrationcalculated by numerical integration of the transport equation (8), and (ii) experimental measurements. These aspectswill be dealt with in the following sections.

3. Numerical solution of the transport equation

Eq. (8) has been solved by an explicit method (Farlow, 1993). The algorithm starts by solving Poisson’s equation(11) at the tube inlet, x∗ = 0, using the particle number concentration profile n∗(r∗, 0) given by the boundary condition(12). Poisson’s equation is solved following a simple forward integration scheme from r∗ = 0 up to r∗ = 1. With thethus calculated electric field profile E∗

r (r∗, 0) at the tube inlet, the transport equation (8) is integrated from r∗ =1 downto r∗ = 0 using the central difference for the second derivative, and the backward difference for the first derivative�n∗/�r∗. This yields the concentration profile n∗(r∗, �x∗) at the next tube section. Next, the electric field is calculatedat x∗ = �x∗ and the concentration profile at x∗ = 2�x∗. This process is repeated marching on along the axial directionup to x∗ = 1. It is well known that this simple explicit method only works when the intervals �r∗ and �x∗ are chosenso as to satisfy the condition �x∗/�r∗2 �0.5 (Smith, 1965). We have taken �r∗ = 0.02 and �x∗ = 10−4, so that�x∗/�r∗2 = 0.25.

The transport equation has been solved numerically for nearly 2000 cases, varying the tube radius R, the tubelength L, the particle electric mobility Z, the initial particle number concentration n0, and the aerosol flow rate q.For each particular case, the values of the aforementioned variables were selected randomly within the followingintervals: R between 1 and 5 mm; L between 100R and 1000R (so that a2 = (R/L)2 was always >1); Z between10−4 (∼ 180 nm) and 1.5 (air ions) cm2 V−1 s−1; n0 between 104 and 108 cm−3; and q between 1 and 20 l min−1.Some of these combinations led to Reynolds numbers exceeding 2000 and were discarded. The results reported beloware all for laminar flow conditions. The dimensionless parameters �, VS, Re and Pe took values within the followingintervals: � between 7×10−5 and 2600; VS between 2×10−5 and 1.6×104; Re between 0.1 and 2000; and Pe between100 and 107.

The numerically calculated penetrations are plotted in Fig. 1 against the right-hand side of Eq. (23). The agreementis very poor and we conclude that Eq. (23) is wrong. However, if VS is replaced by kV S, where k is a numerical factor,it results that the 2000 data collapse remarkably well onto the diagonal straight line of Fig. 1 for k = 0.49. We thusrewrite Eq. (23) as

P = PD

1 − 0.49VS(1 − PD)/ ln PD. (24)


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

num

erical penetr

ation, P

[-

]

PD / {1 - VS[(1 - PD) / lnPD]}

Fig. 1. Comparison between the penetrations obtained from numerical solution of the transport equation (8) and those calculated with Eq. (23).

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

num

erical penetr

ation, P

[-

]

PD / {1 - 0.49 VS [(1 - PD) / lnPD]}

Fig. 2. Comparison between the penetrations obtained from numerical solution of the transport equation (8) and those calculated with Eq. (24).

Fig. 2 shows the numerically calculated penetrations against the phenomenological penetration given by Eq. (24).As seen, except for a few data points, the vast majority of the numerical penetrations, calculated with the somehowinvolved rigorous equation (8), are very well predicted by the simple and practical equation (24). In a certain sense,Eq. (24) can be regarded as a fairly accurate solution to the differential equation (8).

It is not easy to find an interpretation for the 0.49 factor that we have been forced to introduce in Eq. (24) so as toreproduce the penetration values calculated by numerical solution of Eq. (8). First, it must be noted that in the absenceof diffusion, i.e. when PD =1, Eq. (24) simplifies to P =1/(1+0.49VS), which differs from the electrostatic dispersionequation (17) by just the 0.49 factor. Thus, whereas (24) is valid in the limit of negligible space-charge (i.e., P → PD asVS → 0), it is not valid in the limit of negligible diffusion. The reason, we think, resides in the following fact: accordingto the electrostatic dispersion equation (17), the particle number concentration is independent of the radial coordinate;it only depends on the axial coordinate x = uxt . This is so because (17) was obtained for non-diffusive particles.However, when particles diffuse along the radial direction, it cannot be maintained that the number concentration beindependent of r; on the contrary, there is a definite concentration gradient along r, and the very existence of this


gradient is in conflict with Eq. (17). It seems, thus, that the 0.49 factor represents a sort of correction factor to accountfor the appearance of a radial concentration gradient in the system. Summarizing, Eq. (24) is valid for simultaneousdiffusion and space-charge, and also for diffusion alone; but it is not valid for space-charge alone. For the latter case,non-diffusive particles, one should use the correct electrostatic dispersion equation (17).

We have thus arrived at a fairly simple expression which allows accurate estimation of particle penetration whendiffusion and electrostatic dispersion act simultaneously, by using a constant value, 0.49, for the correction factor k.However, it will be shown next that k is actually a function of the tube length.

4. The correction factor k

Since the k factor has been introduced to account for the existence of a radial concentration gradient which isgradually developed along the tube, it seems reasonable to accept that k might actually depend on the tube length.To examine this possibility we have first carried out a set of numerical experiments to estimate the tube length requiredfor full development of the concentration profile.

Fig. 3 shows an example of particle concentration profiles along the tube. The profiles obtained by numericalsolution of Eq. (8) with VS = 0 are shown as full lines; these are typical concentration profiles for the case in whichdiffusion is the only mechanism for particle deposition. These profiles should be compared with those obtained whenelectrostatic dispersion acts simultaneously along with diffusion (dashed lines). The same data are replotted in Fig. 4using n∗(r∗, x∗)/n∗

max(x∗) in the x-axis, where n∗

max(x∗) = n∗(0, x∗) is the maximum value of the particle number

concentration in the cross section; the maximum concentration value is always attained at the tube centerline. In thecase of diffusion alone (VS very small), the shape of the ‘reduced’ profiles n∗(r∗, x∗)/n∗

max(x∗) varies along the tube

up to about x∗ ≈ 0.4 and remains unchanged thereafter: for the specific example shown in this figure, the tube lengthrequired for fully developed diffusion boundary layer would then be LD ≈ 0.4L= 1000 cm (the subscript D stands fordiffusion). For the case of non-negligible electrostatic dispersion (shown as symbols in Fig. 4), the full developmentof the concentration profile takes place further downstream, for a value of x∗ somewhere between 0.4 and 0.7. Thuswe see that, for this example, LDS > LD, where the subscript DS means diffusion + space-charge. Actually, LDS > LDis a result with general validity: when diffusive particles also spread out by mutual electrostatic repulsion, the lengthrequired for a fully developed concentration profile becomes larger.

From Fig. 4 we have concluded that LDS is somewhere between 0.4 and 0.7 times the tube length; this is tooambiguous. Clearly, a criterion is needed to estimate LDS in a systematic manner. We have adopted the following one,

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

10.7

0.4

0.2

0.075

0.01

0.001 x∗=0

r∗ [-

]

n∗(r∗,x∗) [-]

Fig. 3. Concentration profile for a case in which space-charge is negligible (full lines, initial aerosol concentration n0 < 104 cm−3), and for a situationin which diffusion and electrostatic dispersion are both non-negligible (dashed lines, n0=5×105 cm−3). R=0.3 cm; L=2500 cm; Z=0.4 cm2/Vs;q = 6 l min−1; Re = 1400; Pe = 21000; � = 0.794. VS < 0.05 for the first case (full lines), and VS = 2.56 for the second case (dashed lines).


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

x∗

0.001

0.01

0.075

0.2

0.4

0.7

1

r∗ [-

]

n∗(r∗,x∗) / n∗max(x

∗)

Fig. 4. Concentration profile for the same two cases of Fig. 3, but expressing the concentration as n∗(r∗, x∗)/n∗max(x∗). Lines: case of negligible

electrostatic dispersion (VS < 0.05); symbols: VS = 2.56. In the first case (only diffusion), the profiles for x∗ = 0.4, 0.7 and 1 are practicallycoincident. For the diffusion + space-charge case, the coincident profiles are those for x∗ = 0.7 and 1, while that for x∗ = 0.4 is clearly different.

0.0 0.2 0.4 0.6 0.8 1.00.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

x∗DS = LDS / L

1.01 A∗min

A∗min

are

a u

nder

the p

rofile

curv

e, A

∗ [-

]

x∗ [-]

Fig. 5. Illustration of the method used to estimate LDS, the length required for fully developed diffusion boundary layer. The area un-der the curve is calculated as A∗ = ∫ 1

0 (n∗/n∗max) dr∗. R = 0.2 cm; L = 500 cm; Z = 1 cm2/Vs; q = 1 l min−1; n0 = 3.5 × 105 cm−3;

Re = 350; Pe = 2100; � = 2.38; VS = 2.39.

illustrated in Fig. 5. In this plot we have represented the area under the concentration profile,

A∗(x∗) =∫ 1

0

n∗(r∗, x∗)n∗

max(x∗)

dr∗, (25)

against the axial coordinate x∗. The area attains asymptotically a minimum value, A∗min. We take for LDS the value

of x∗L at which the area is 1% larger than A∗min. Of course, the selection of 1% is arbitrary as we could have chosen

equally well any other figure. However, for the specific purpose of the present investigation, which is none other thanto examine the possible dependency of the k factor on the degree of development of the concentration profile, the justdescribed technique for the estimation of LDS is sufficiently accurate.


0.01 0.1 1 10 1000.0

0.1

0.2

0.3

0.4

0.5

0.41-0.31 / [1+0.26 (VS/β)1.47]

LD

S / R

Pe [-]

VS / β [-]

Fig. 6. Length LDS required for fully developed diffusion boundary layer in the case of simultaneous diffusion and electrostatic repulsion.

The transport equation (8), along with Poisson’s equation (11), was solved for 100 cases with values of R, Z, q andn0 within the same intervals as mentioned above in Section 3. In all the cases, the tube length L was chosen sufficientlylarge so as to let the area under the profile, Eq. (25), to fully attain a clear minimum value. The estimated values ofLDS are shown in Fig. 6 as LDS/RPe = f (VS/�). The reason for choosing this representation is based on the fact thatEq. (8) can be rewritten as

u∗x

�n∗

�x∗ = 2L

RPe

[�2n∗

�r∗2 + 1

r∗�n∗

�r∗ −(

VS

�

) (E∗

r

�n∗

�r∗ + n∗2)]

. (26)

The obtained data are fairly well correlated by the expression

LDS

R= Pe

[0.41 − 0.31

1 + 0.26(VS/�)1.47

], (27)

shown in Fig. 6 as a full line.In the limiting case of negligible space-charge, VS → 0, the above expression reduces to LD = 0.1RPe. In the

other extreme, for very large VS, expression (27) approaches a maximum value LDS = 0.41RPe, but this asymptoticalexpression should be taken with care because we have not estimated LDS for values of VS/� above about 60 and wehave gathered very few data for this region.

We have now the tools required to examine the possible dependency of the k factor on the tube length. For eachof the 2000 cases tested before (as explained in Section 3), the value of LDS was estimated using Eq. (27). The 2000cases were ordered according to increasing values of L/LDS and subdivided into 20 intervals, each containing 100cases. For each interval, we calculated the k factor which best fitted the numerical penetrations. Fig. 7 shows the thusobtained values of k against the average value of L/LDS in each of the 20 intervals. There is a clear dependency of kon the degree of development of the concentration profile, and this fact may be taken as a confirmation that k indeedrepresents a correction factor accounting for the existence of a developing radial concentration gradient in the tube.If instead of LDS we use the length required to attain a fully developed parabolic flow velocity profile, LM = 0.1RRe

(Fuchs, 1964), the results shown in Fig. 8 are obtained. This plot shows no systematic dependency of k on L/LM . FromFigs. 7 and 8 we conclude that k depends, not on the tube length itself, but on the degree of development of the particleconcentration profile.

Therefore, the use of a unique, constant value k = 0.49 in Eq. (24) is an approximation, certainly a very good one asthe plot in Fig. 2 showed. Furthermore, the values of k in Fig. 7 which most notably depart from 0.49 are those for eithervery small or very large values of L/LDS, which correspond, respectively, to very large or very small penetrations, i.e.very close to the extremes of the straight line drawn in Fig. 2. In these regions, Eq. (24) is not very sensitive to changes


10-2 10-1 100

0.2

0.4

0.6

0.8

1.0

k [-]

L / LDS [-]

Fig. 7. The k-factor as a function of L/LDS (L = tube length; LDS = length required for fully developed diffusion boundary layer). The dashedhorizontal line, k = 0.49, is the value of the k-factor which best fits the whole set of numerically calculated penetrations.

0 5 10 15 20 250.40

0.45

0.50

0.55

0.60

k [-]

L / LM [-]

Fig. 8. The k-factor as a function of the L/LM ratio (L = tube length; LM = 0.1RRe = length required for fully developed laminar flow). Thedashed horizontal line, k = 0.49, is the value of the k-factor which best fits the whole set of numerically calculated penetrations.

in the value of k, which means that, for purely practical purposes, the use of a single, constant value 0.49 for the entirerange of penetrations is fairly accurate.

Finally we also examined the possible dependency of the k factor on Reynolds and Peclet numbers. The correspondingplots, not shown, resulted to be very similar to that of Fig. 8, in the sense that k varied about 0.49 with no systematictendency.

5. Comparison with experimental results

5.1. Experimental method

Tube penetration experiments of positive air ions generated by corona discharge were carried out. Clean dry air at acertain flow rate q was passed through a corona ionizer (RAMEM model IONER CC08010) operated at a fixed positive


0 10 20 30 40 500.5

0.6

0.7

0.8

0.9

1.0

diffusion +

space-charge

only diffusion

penetr

ation, P

[-]

tube length, L [cm]

Fig. 9. Experimental data (symbols) for 4 mm ID tube and 3 l min−1 flow rate. n0 = 2.08 × 106 cm−3. Full line curve: penetration obtained fromnumerical solution of the full equation (8). Dashed line curve: penetration obtained from numerical solution of Eq. (8) with VS = 0 (this curvecoincides exactly with Eq. (20)).

voltage of 3.4 kV. Circular grounded copper tubes of varying length and diameter were connected to the outlet of theionizer. Ion number concentration at the tube outlet was measured with an electrometer (TSI model 3068A).

For a given flow rate, the ion concentration measured at the outlet of the shortest tube employed, L0 = 50 cm, wastaken as the initial concentration n0; from the measured concentration n(L + L0) at the outlet of a tube of a largerlength L + L0, the penetration through the length L was then calculated as P(L) = n(L + L0)/n0.

Measurements were done at flow rates of 1, 2, 3, and 4 l min−1, with tube inside diameters of 4, 6, and 8 mm. Toeach of the twelve combinations of flow rate and tube diameter corresponded a certain initial concentration n0: theirvalues ranged between 1.9 × 105 and 2.6 × 106 cm−3. For each of these combinations, penetrations were measured fortube lengths of 5, 10, 15, 20, 25, 35 and 45 cm as described above, i.e. by comparing the concentration at the outlet oftubes of lengths 55, 60, 65, 70, 75, 85 and 95 cm with that measured at the outlet of a 50 cm tube of the same diameter.The penetrations reported below are the averages of three measurements.

The parabolic flow field in a tube is established beyond a distance of 0.1RRe from the entrance (Fuchs, 1964), whereRe is the Reynolds number. This critical length equals 7.03q cm if the flow rate q is expressed in l min−1. For the largestflow rate, 4 l min−1, the critical length is about 28 cm. Thus, our selection of 50 cm for the reference tube length ensuresfully the development of the parabolic flow velocity profile in all the cases. The use of Eq. (10) in the rigorous modelis, therefore, correct.

The positive air ion mobility was measured using a DMA (TSI 11.11 cm-column) with an aerosol-to-sheath flow rateratio of 6/40. The centroid mobility resulted to be 1.1 cm2 V−1 s−1. With this value, the air ion diffusion coefficient isD = kT Z/e = 0.028 cm2 s−1. The Peclet number is Pe = 2uxR/D = 3790q/R, with q expressed in l min−1 and Rin mm. In our experiments, the Peclet number ranged between about 950 and 7600: the assumption of negligible axialdiffusion in the theoretical model is more than justified.

5.2. Experimental results

Figs. 9 and 10 show the comparison between theoretical and experimental results for two specific cases. In the plotsshown, the experimental data (symbols) follow fairly well the theoretical curve calculated by numerical solution ofEq. (8) (full line, labelled as ‘diffusion + space-charge’). The dashed curve, labelled as ‘only diffusion’, represents thenumerical solution of Eq. (8) in the absence of space-charge effects, and coincides exactly with the Gormley–Kennedyequation (20).

The whole collection of experimental data is plotted in Fig. 11 against the phenomenological equation (24).The dashed lines represent the P ± 0.10 error interval. It can be noticed that, in general, Eq. (24) underpredictsthe experimental penetrations. The deviation between experiment and theory is minimized if one uses 0.46 in placeof the 0.49 factor in (24).


0 10 20 30 40 50

0.5

0.4

0.6

0.7

0.8

0.9

1.0

diffusion +

space-charge

only diffusion

penetr

ation, P

[-]

tube length, L [cm]

Fig. 10. Experimental data (symbols) for 8 mm ID tube and 4 l min−1 flow rate. n0 = 1.74 × 106 cm−3. Full line curve: penetration obtained fromnumerical solution of the full equation (8). Dashed line curve: penetration obtained from numerical solution of Eq. (8) with VS = 0 (this curvecoincides exactly with Eq. (20)).

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

experim

enta

l penetr

ation, P

[-]

PD / {1-0.49 VS [(1-PD) / lnPD]}

Fig. 11. Comparison between the experimentally measured penetrations of positive air ions and those calculated with Eq. (24).

From the results shown in Figs. 2 and 11 we are tempted to conclude that Eq. (24), with VS and PD given by Eqs. (9)and (20), respectively, represents a simple, practical, and sufficiently accurate correlation to estimate particle penetrationin laminar flow tubes under the restricting conditions imposed in the present development, namely, negligible axialdiffusion (large Peclet number), negligible axial space-charge (small tube aspect-ratio R/L), and negligible imageforce (few number of charges per particle).

6. Conclusions

Two different approaches have been used to estimate aerosol penetration through circular tubes by the simultane-ous mechanisms of diffusion and space-charge, namely, numerical solution of the rigorous transport equation, and aphenomenological model in which particle loss is assumed to follow first- (diffusion) and second-order (space-charge)


kinetics. The latter treatment has led to a simple equation, Eq. (24), relating particle penetration to tube geometry,aerosol flow rate, and particle mobility and number concentration. Penetrations calculated with this simple and prac-tical equation agree fairly well with those obtained by numerical integration of the transport equation, and also withexperimental data for positively charged air ions.

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