The growth kinetics and polydispersity of condensational aerosols

The Growth Kinetics and Polydispersity of Condensational Aerosols E. JAMES DAVIS AND S. C. LIAO Department of Chemical Engineering, Clarkson College of Technology, Potsdam, New York 13676 Received M a y 15, 1974; accepted September 16, 1974 This work is a theoretical analysis of aerosol growth in a recently developed laminar flow aerosol generator, a modification of the Sinclair-LaMer generator. T h e parameters which affect the aerosol size distribution are elucidated by rigorous analysis of the droplet growth kinetics and transport p h e n o m e n a associated with vapor condensation on nuclei in a nonisothermal flow field. T h e results of the analysis are compared with available experimental data. Theory and experiment are in excellent agreement with respect to the effects of using nitrogen rather t h a n helium as a carrier gas, and other experimentally observed characteristics are predicted. INTRODUCTION

The Sinclair-LaMer aerosol generator (1) has been widely used to prepare aerosols in the submicron range having a narrow size distribution of liquid droplets. Aerosols are produced by condensation of vapor on nuclei under flow conditions. Earlier versions of this generator involved problems of stability of the output over an extended period of time and lack of reproducibility from run to run. A considerably improved falling film/condensation generator has been developed recently (2-5), and it has been used to study aerosol growth kinetics. By providing better control over the variables affecting aerosol formation this generator produces stable, highly monodisperse aerosols that are reproducible from experiment to experiment. A schematic diagram of this generator is shown in Fig. 1. A carrier gas stream, usually helium for reasons that will become apparent, is passed through a combustion furnace where NaC1 or AgC1 nuclei are produced. The nucleicarrier gas stream then flows through a jacketed temperature controlled wetted-wall column in which evaporation of a heated organic liquid,

usually dibutylphthalate (DBP) in the studies mentioned above, occurs. The liquid which flows uniformly down the wall of the wettedwall evaporator is continuously recycled. The resulting nuclei-gas-vapor mixture is finally cooled in a tubular condenser that is either air-cooled or water-cooled. In the condenser the vapor condenses on the nuclei to produce an aerosol. A more monodisperse aerosol is obtained if the primary aerosol is then evaporated and recondensed. The aerosol is then passed to a light scattering cell to measure the aerosol size distribution. Shahriari and Goodrich (6, 7), who shared in the development of the falling film generator, eliminated the tubular condenser, permitting the aerosol to form in the jet emanating from an orifice at the end of the heated zone. The3" used their flow reactor to study the growth kinetics of DBP aerosols. They attempted a theoretical analysis (7) to interpret their growth data, but because they could not take into account vapor diffusion and vapor depletion in the ensemble of droplets their theoretical analysis cannot be used to predict polydispersity. Furthermore, they failed to

488 Journal of Colloid and Interface Science, Vol. 50. No. 3, March 1975

Copyright {~ 1975 by Academic Press, Inc. All rights of reproduction in any form reserved.

GROWTH KINETICS OF AEROSOLS

- -

EVAPORATOR HOT O L

parameters are specified, and their effects on the aerosol are elucidated. Furthermore, the theoretical constructs developed herein can, in principle, be extended to atmospheric phenomena such as fog and cloud formation in nonuniform temperature, humidity and velocity fields.

ORGANIC

LIQUID RECfOILRCULATED

"--~: REC~CULA' ORGANICi'~D

CONDE~S21

489

CO01JNG

WATER

THE GOVERNING EQUATIONS

AEROSOL FIG. 1. T h e aerosol g e n e r a t o r .

account for the significant temperature gradients that occur in the jet. In 1951 Reiss (8) pointed out that the depletion of material in an ensemble of growing spherical particles must be taken into account when determining the diffusion-controlled growth rate of single particles, and he developed a mathematical model for single-particle growth, which is a quasisteady state approximation. We shall have more to say of Reiss's result below. Nicolaon and Kerker (5) attempted to use their generator to study the condensation nuclei by measurements of the positions in the condenser where the aerosol first exhibits a visible Tyndall cone, but their results are ambiguous because of the complexity of the system. Because of the nonuniform temperature, concentration and flow fields it has been difficult to interpret many experimental results in an unambiguous manner. Furthermore, extensive measurements of the aerosol characteristics as a function of flow rates, nuclei size, condenser wall temperatures and other parameters of the system have led to surprising and confusing results. It is the purpose of this paper to show, by rigorous theoretical analysis, what factors affect the droplet size and size distribution of aerosols formed by vapor condensation on nuclei in a nonisothermal laminar flow system. Qualitative and, in most respects, quantitative information can be obtained from the general theory to predict the operating characteristics of the falling film aerosol generator. The system

The part of the aerosol generator of principal interest here is the condenser, for it is in the condenser that nuclei activation and growth occur. Davis and Nicolaon (4) have analyzed the operating characteristics of the wettedwall evaporator, which has the primary function of saturating the carrier gas with vapor. But if the evaporator is not sufficiently long to attain a uniformly saturated stream, the resulting aerosol can be adversely affected by the nonuniform concentration distribution at the inlet of the condenser. Davis and Nicolaon's theoretical criterion for sufficient length was given in terms of the nondimensional length, ~r =- L / R Pe~, that is, ~L > 0.7 for uniform saturation. The Peclet number, Pe2, for mass transfer is defined by Pe~ = Ruo/D, where R is the tube radius, u0 is the velocity at the centerline of the tube, D is the diffusion coefficient of the vapor in the carrier gas, and L is the tube length of the wetted-wall column. In what follows we shall assume that the vapor concentration at the inlet to the condenser corresponds to uniform saturation in the evaporator. Because of the well defined flow conditions, geometry, and thermal conditions associated with this system it is possible to model the system with considerable rigor. Since the volume fraction, ¢, of the aerosol is very small the physical properties of the aerosol-vaporgas mixture can be taken to be those of the vapor-gas mixture. For example, Einstein (9) showed that the viscosity of a dilute suspension of rigid spheres uniformly dispersed in an incompressible Newtonian fluid of viscosity #0 is given by

Journal of Colloid and Interface Science, Vol. 50, No. 3. March 1975

490

DAVIS AND LIAO

Similarly, for the diffusion coefficient, D, and the thermal diffusivity, a, we may neglect the effect of nuclei or aerosol droplets. Since the flow is Poiseuillian at the relatively low flow rates used in aerosol generation, we may write the velocity distribution as

particles entering near the tube wall will collide with the wall due to Brownian motion, and many of these particles will stick to the wall in an actual experiment. We have neglected both this wall loss and the reduction in the number density of particles due to coagulation in these calculations. For the rather u = u(r) -- uo(l -- rS), [2~ low number densities used here and in experiwhere r is the nondimensional tube radius ments these approximations are probably defined by r*/R, r* is the radial position, and valid, but for systems involving a high number R is the tube radius as shown in Fig. 2. If density of nuclei and droplets, wall losses and we assume that nuclei and droplets move coagulation should be taken into account. along steamlines of the flow, then Eq. I-2-] Since the condenser is cooled by heat indicates that nuclei near the wall move at a transfer through the tube wall, the stream much lower velocity than particles at the temperature varies radially and axially. In the centerline. The residence time of a particle at experiments of Shahriari and Goodrich (6), radius r*, therefore, is given by however, the jet cooled by direct contact with the surrounding air. In the case of the jacketed X* X* water-cooled condenser it is possible to specify t= t(r) , [33 the thermal boundary conditions, so it is this u(r) u0(1 -- r 2) type of condenser that we shall analyze. The where t is the time required to travel axial analysis proposed here can be adapted to distance x* from the inlet of the condenser. other systems provided that the boundary Although the nuclei and smallest droplets conditions can be specified a priori. As the (as small as 0.01 vm in this study) can be air-cooled tube is approximated by a constant expected to be affected by Brownian motion, heat flux boundary condition, it is readily it can be shown that even the smallest particles treated in a manner similar to the constant do not deviate appreciably from the stream- wall temperature problem analyzed below. lines along which they enter the condenser. The nondimensional temperature field is For the residence times encountered here, described by the steady-state energy equation, except very near the tube wall, the displacement due to Brownian motion in our system is calculated to be less 10-2 cm. I t is likely that (1 r 2) Oxl r Or Pel 2 Oxl 2 =

-

-

-1[- ff#l(Xl, r ) , r~

E4-1

where the nondimensional variables are defined as follows : T-

t

Tw

r*

x*

To -- T~

R

R Pel

0 =

FIo. 2. T h e s y s t e m coordinates.

and the Peclet number for heat transfer is defined by Pel = Ruo/a. The nondimensional heat source term, ¢1(xl, r), arises because the condensation of vapor involves the release of the heat of condensation. If S(Xl, r) is the condensation rate per unit volume of fluid, then

Journal of Colloid and Interface Science, Vol. 50, No. 3, March 1975

491

G R O W T H K I N E T I C S OF AEROSOLS

the nondimensional source term is given by

The nondimensional vapor concentration, X, is described by the convective diffusion equation

R2AH~p ~I(Xl, r) --

k ( T o - T~)

S(Xl, r),

[5]

(I

where To is the inlet temperature of the vaporgas mixture, Tw is the constant wall temperature, k is the thermal conductivity of the fluid mixture, and AHv~ is the latent heat of vaporization (considered to be positive here). We have assumed that the condensation rate per unit volume, S(xl, r), which is a function of the ensemble of droplets, can be represented by a continuous function of the coordinates xl and r. This term requires further elaboration below. In addition, we shall neglect axial conduction compared with convective energy transport. It is clear, from Eq. r4-], that if Pel is sufficiently large we may neglect the axial conduction term. Hsu (10), Davis (11), and Jerri and Davis (12) have shown that for Pe~ > 100 axial conduction can be neglected, and for Pel < 10 axial conduction has a large effect on the temperature field. We shall neglect axial conduction in this analysis, but we must note here that in the experiments of Nicolaon heat transfer Peclet numbers as low as 11 were encountered. The Peclet number, Pe~, may be written in terms of the Reynolds number, Re = Ruo/~, and the Prandtl number, Pr = v/a, that is, P e ~ - - R e P r . As the Prandtl number is of order unity for the gases of interest here, it is clear that for Re < 100, the neglect of axial conduction can lead to inaccuracies in the calculation of the temperature field. For these conditions, then, our analysis cannot be expected to yield precise quantitative results, but good qualitative results can be obtained. The boundary conditions associated with Eq. [-4] are (i) 0(0, r) = 1 (constant, uniform inlet temperature), (ii) O(xl, 1) = 0 (constant, uniform wall temperature), (iii) (O0/Or)(x~, 0) = 0 (symmetry about the centerline).

-

Ox~

r

Pe~20x2 2 -

[6]

q)2(x2, r),

-

where C

X=--, Co

x*

x2-

Ruo

,

Pe2-

R Pe~

, D

and the nondimensional mass depletion term, av2(x2, r), is de~ned by R2 av2(x2, r) -- - - S ( x 2 ,

DCo

r).

[7]

The mass transfer Peclet number, Pe2, can be written Pe2 = Re Sc, where the Schmidt number is defined by Sc - v/D. As Sc > 10 for the systems of interest here, we can neglect the axial diffusion term without hesitation for Re as low as 10. Again we assume that the vapor depletion rate per unit volume is a continuous function of the axial and radial coordinates. The depletion term can be expected to be particularly significant in the lower temperature regions of the flow where the largest rate of condensation occurs. The appropriate boundary conditions are (iv) X(0, r) = 1 (constant, uniform inlet concentration), (v) (OX/Or)(x2, 1) = 0 (no diffusion through the wall), (vi) (OX/Or)(x2, 0) -- 0(symmetry about the centerline). Note that the nondimensional mass sink term, ¢2, is related to the nondimensional heat source term, ¢1, by ¢'~ = ~®_,,

1-83

where ~ = DCo&H~/k (To -- Tw). If ¢/<< 1, it is clear that the heat source will have a much smaller effect on the temperature field than the mass depletion will have on the concentration field since Eqs. [-4-] and [6-] have the

Journal of Colloid and Interface Science, Vol. 50, No. 3, March 1975

492

DAVIS AND LIAO

same form, and the dependent variables have been normalized, that is 0 < 0 < 1 and 0 _< X <_ 1. For the DBP-helium system studied by Nicolaon et al., 13 = 0 (10-3) because of the low vapor concentration, Co, at the inlet to the condenser. Furthermore, it can be shown that the spatial average of qh is not too large, so we may neglect the heat source. This approximation is not essential to the analysis, but it greatly simplifies the solution procedure by decoupling the temperature field from the concentration field, as we shall see. Under no conditions, however, is it possible to neglect q52 in Eq. [67, for is it the depletion of vapor together with vapor diffusion that determines the aerosol size distribution. MASS DEPLETION, S(x*, r*) In the foregoing we have tacitly assumed that the laws of continuum mechanics apply and that the rate of vapor depletion per unit volume, S(x*, r*), is a continuously distributed function. The latter assumption requires further scrutiny because, on a microscopic scale, condensation occurs on individual nuclei distributed throughout the flow field. If the individual nuclei or droplets can be considered to have no relative motion with respect to the fluid immediately surrounding them and if droplet growth is controlled by Fickian diffusion from the surrounding vaporgas mixture to tile surface of the spherical droplet, then the vapor concentration in the immediate vicinity of the droplet is described by the unsteady state diffusion equation, OC* Or

1

- D---

0 [

p 20p

k 02

OC* \

)

~p / '

[-9]

where r = t -- to, tc is the residence time at the start of condensation, C* refers to the vapor concentration in the neighborhood of the droplet, and p is the m i c r o s c o p i c coordinate measured from the center of the droplet as shown in Fig. 3. Now this microscopic view of droplet growth must be made consistent with the continuum theory of the previous section, for the concentration field around the

f

/

\

/

f \

\

/

FIG. 3. The microscopiccoordinates. droplets must also satisfy the convective diffusion equation, Eq. [-6]. To this purpose it should be noted that the sizes of the nuclei and the ultimate sizes of the fully grown aerosol droplets are very much smaller than the dimensions of the condenser. Typical aerosols produced by the falling film/condensation generator are of the order 0.1 gm in radius, a, while the tube radius, R, is of order 1 cm (104 am). Furthermore, the volume fraction, ¢, of grown aerosol droplets is typically of order 10.8 or 10.7 since the number density, N, of nuclei is of order 107 nuclei/cm 3 (that is, ¢ = }Tra3N .-~ 10-7). Thus, we can consider that a small annular volume, AV = 2zrrArAx, of the vapor-gas mixture contains a statistically large number of nuclei and/or droplets. From the continuum point of view this small volume can be considered to be located at some point (x*, r*) in the condenser tube, and hence, the nuclei and droplets in this volume can be considered to be identified by the coordinate (x*, r*) of the tube. Let us further assume that the temperature of the particles and fluid within volume AV is uniform at the value predicted by solution of Eq. [-4] and that the vapor concentration far from the droplet surface, p = a, (but within the volume, AV) is described by Eq. [-6]. The condensation rate in volume AV is thus obtained by sumrning the rates over the individual droplets. For any single droplet of radius a~ the rate of condensation or evaporation is given by Eq. [-10].

Journal of Colloid and Interface Science, Vol. 50, No. 3, March 1975

d

oc*

re: = - - (OL]zca, 3) = 4zra,2D - , dt Op O=a~

[,10]

GROWTH KINETICS OF AEROSOLS where C*(p, r) is the solution of Eq. (9). If the number density of the droplets of size a~ is N~, then the total condensation rate in volume AV is (per unit volume) OC* S(x*, r*) = Y'~ 4~ra~2NiD i=I Op p=~

493

a unit volume to consist of N unit cells each of volume 1 / N and each containing one nucleus. If the unit cell described in this way is spherical, each has an effective radius, b, given by b = (~X) ~

Reiss has shown, largely from intuitive arguments, that if b >> a, Eq. [-9-] can be replaced by the quasisteady state equation

I

= Z 4~raflNiJe.~,

[13]

[11]

i=1

0 /

where

0

__

je.~ = D OC* Op p=~

4rra2n(a)D ao

da, Op

kV) = 0

[-14]

with boundary conditions

is the mass flux to a droplet of radius a~. To obtain Eq. [-11-] we have assumed that the particle size distribution is not continuous, that is, the nuclei entering and the resulting droplets consist of I groups of droplets. Each group is assumed to contain particles of a single size. At each radial position (in our computations, the region between r and r 4- At) the particles in a particular group all have the same growth rate, but the growth rate varies from group to group. If the size distribution is represented by the distribution function n(a), where n(a)da is the number of droplets having sizes in the range a to a 4- da, then Eq. 1-11] can be replaced by Eq. [-12-]. S(x*, r*) =

OC*\ p2

c*(<

C*(b, t) = Cb(t) = C(x*, r*),

where a0 is the size of the smallest nuclei. For purposes of numerical computation Eq. [11] will be used here. It remains to determine the mass flux, J,, from which the condensation rate is calculated. Although Eq. [9"] applies in the vicinity of a drop, we must introduce appropriate auxiliary conditions to solve the equation. This type of problem was examined by Reiss (8) in his analysis of the growth of colloidal dispersions in a nonflow system, and our analysis at this point parallels his treatment of growth of a competitive assembly of spheres. If N is the number of nuclei per unit volume of the vapor-gas stream, then we can consider

[15]

[-16-]

where Ce(x*,r*) is the equilibrium concentration at the droplet surface at the temperature, T(x*, r*), obtained by solving Eq. [-4-], and Cb(t) is the concentration C(x*, r*) determined by solution of Eq. [-6-]. Again we note that the residence time t is related to tube coordinates x* and r* by Eq. [-3-]. Following Reiss, the flux of vapor to the droplet becomes OC*

D

b

Je = D Op o=a

a (b--a)

X EC(x*, r*) -- Ca(x*, r*)],

[-12]

p=.

t) = ce(t) = co(~*, ~*)

and

El7-]

which leads to the droplet growth rate, from Eq. [-10], da

D

b

dt

pL a(b -

a)

x [-C(x*, ~*) - ce(**, ~*)].

E18]

It is more convenient to consider the growth rate with respect to the axial coordinate xe than with respect to time, so we may write Oa

Ol da

Ox2

Ox2 dl

R2

da

D(1 -- r 2) dl

R2b

[ c ( . 2 , ~) - Ce(*2, ~)]

pLa(b -- a)

(1 -- ~2)

-

[19]

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494

DAVIS AND LIAO

In writing boundary condition [-15-] we have ignored the fact that for very small droplets the Kelvin effect is important. If p0° is the vapor pressure over a flat surface, then the vapor pressure over a spherical surface of radius a is p0 = p0o exp [-27M/pLRGTa-].

[-203

The combined effects of noncontinuum vapor transport and the Kelvin effect substantially reduce condensation rates compared with the uncorrected Eq. [17~. Taking these corrections into account the total condensation rate becomes

f 4rrDN,:Fiaib S(x*, r*) = E

Using Eq. [-20~, boundary condition [-15~ can be corrected to read

C*(a, t) = C,(a, x*, r*) = C,°(x *, r*) X exp E2"yM/pLRGT(X*, r*)a],

[-213

where C,°(x *, r*) is the equilibrium concentration at temperature T(x*, r*) over a flat surface. One further correction should also be made to the mass flux expression, Eq. [17-]. When the droplets are so small that the mean free path of the vapor molecules is of the same order as the droplet radius, the continuum theory upon which Eq. [-17-] is based is no longer valid. This so-called Knudsen aerosol regime has been discussed in considerable detail by Hidy and Brock (13), and recently Fuchs and Sutugin (14, 15) derived an interpolation formula which fitted the exact solution of Sahni (16) for the noncontinuum regime. If J~ is the flux based on continuum theory, then the flux in the noncontinuum regime is given by

J

= JcFi

J~ I

{ 1+EKn~(1.333 Km+0.71)/(Kn~+l)~}

i~1

(b - - ai)

X EC(x*, r*)

C,(a~, x*, r*)-]. 1-23~]

-

Finally, using Eq. E23-], the nondimensional heat source term is I ]~~F i iai • 1(xl, r) = 4rrR2bfl E - - - ( X -- Xe), [-24-]

~=1 (b -

a~)

and the mass sink term is

I NiFiai qh(x2, r) = 4rrR2b E - ( X -- Xe). i=1

(b

[-25-]

ai)

--

Neglecting the axial diffusion of heat and mass, the governing equations reduce to

O0 lO(rO0 ~ ( 1 - - r 2) Oxl - r Or\ O r / + q h

[-26-]

and

2) Ox2

(1--r

--

-

r Or

r

--q~2.

[-27-]

Since the equilibrium concentration, X,, is a function of temperature, 0, and the heat source, ¢1, involves concentration, X, as well as X,, Eqs. F26-] and [27-] together with the

[-22-] where Kni is the Knudsen number defined by Kni = l/a~, where l is the mean free path of the vapor molecules. The correction factor, F, = {1 + rKm(1.333 K m + 0 . 7 l ) / ( K m + 1)-]}-1, then, should be made to Eqs. [-17-][19-]. For the system studied extensively by Nicolaon et al., dibutylphthalate with helium carrier gas, the correction factor is very significant for the whole range of aerosol droplets encountered as indicated by Fig. 4, a plot of F~ as a function of a~.

~IO-I

~'÷ -" ~ -~

16 3

I0-3

,

I() 2 AEROSOL

;

I0" RADIUS,

I

f CI ( ~ m )

1o

FIG. 4. T h e K n u d s e n aerosol m a s s flux correction factor.

Journal of Colloid and Interface Science, Vol. 50, No. 3, March 1975

GROWTH KINETICS OF AEROSOLS TABLE I

appropriate boundary conditions constitute a formidable system of coupled equations. The principal reason for the complexity in both the mathematical analysis and in the interpretation of experimental results is the coupling between the temperature and concentration fields arising from the source and sink terms, (I)1 and ¢2, respectively.

THE EIGENVALUESCALCULATEDFROM EQS. [32] AND [36] J

O(Xl, r)

=

r'(1 -- r'2)Gl(xl, r; O, r')dr',

[-283

Gffxl, r; Xl', r')

$&)$&')

s:o

~-Xi2(xl-xI') ,

2.704364 6.679031 10.673379 14.671078 18.669871 22.669143 26.668661 30.668323 34.668073 38.667883

problems in terms of the confluent hypergeometric function, M ( A , B , Z ) , which has well-known mathematical properties and has been tabulated (18). The eigenfunctions are ~s(r) = e-Xir~/2MF(2 - Xi)/4, 1, Xir2-], [31-]

[-32-]

which have been tabulated by Liao (17). A partial list of these eigenvalues is given in Table I. The concentration field is determined in a similar manner to give

X(x2, r) = 1 -- j l x2 f01 r'qoe(x2', r')

[29]

jl$sl! ~

X G2(x2, r; x2', r')dr'dx(,

r'(1 -- r'2)~b~2(rt)dr ',

[l¢jll 2 =

5.067505 9.157606 13.197224 17.220229 21.235517 25.246531 29.254905 33.261523 37.266908 41.271389

M[-(2 -- hi)/4, 1, Xy3 = 0,

where the Green's function is

= ~

A~

and the eigenvalues are the zeros of the transcendental equation,

//fo 1rt~l(Xl ', rt)G1(xl, r; Xl', r')dx(dr' +

Xi

1 2 3 4 5 6 7 8 9 10

SOLUTION PROCEDURE Fortunately, in the systems of practical interest the vapor concentration is so small and the aerosol is so dilute that the heat source can be neglected, as discussed above. Although we shall neglect qs1 in our computations, it is enlightening to include it in the formulation of the solution. We shall develop the solutions of Eqs. [-26-] and [27-] in terms of the Green's function. The energy equation, Eq. [-26-], has a solution of the form

495

[-33-]

where the Green's function in this case is and the eigenfunctions, ~j, and eigenvalues, Xj, satisfy the following Sturm-Liouville problem

G2(x2, r; x2', r') =4+Y]

--

r

+ Xj2r(1 -- r2)~ks = 0,

[-30a-]

dr

and dC,j

(0) = $i(i) = O. [-30b,c-]

dr

Davis (11) has solved this problem and related

e-Ai2(~2-x~')

[-34-]

~=~ II~A~

lleJP = f [

r'(1 -- r'~)'~ fl(r')dr '.

The eigenfunctions, ~'s, and eigenvalues, Aj,

Journal of Colloid and Interface Science,

Vol. 50, No. 3, March 1975

496

DAVIS AND LIAO

satisfy the problem dr r - ~ r ! + &2r(1 -- r2)'I'~ = 0,

C35a]

d'I9 d,I,j - - (0) = (1) = 0,

[35b,c~

dr

dr

The eigenfunctions, 'I~i, have the same form as the ~b~.,replacing ),i by Aj, but in this case the eigenvalues are the zeros of (2 -- A j)

2

Ai

6 --

M ( T '

) 2'Aj

/

--M - - , 1 ,

hi =0.

4

[36]

A partial set of these eigenvalues is given in Table I, and more are tabulated by Liao (17). Now, if the source term in Eq. [28-] can be neglected, the temperature field is directly obtained from Eq. [37-], which is the classical Graetz solution for laminar flow heat transfer in a tube. 1

if0

r'(1 -- r'2)~bj(r')dr '

0(xl, r) =

j=l

jl~jl~

N ~bj(r)e-xi~xl.

[37]

Once the temperature is calculated from Eq. [37] the equilibrium concentration, Ce°, can be calculated from vapor pressure information on the chemical species in question, and Eq. [33-] can be solved by an iterative procedure. An iterative method is necessary because the sink term, dP2,involves the unknown concentration, X(x2, r) as well as the temperature-dependent equilibrium concentration, X~ (a~-,x2, r). Hence, the solution procedure reduces to a method of successive approximations involving double quadrature. If ~1 cannot be neglected, then Eq. r28] must be included in the iterative calculations. Starting at the condenser inlet (where the initial size distribution of nuclei, inlet temperature, and inlet concentration are all known), calculations are performed to determine the

temperature distribution at an axial position Ax2 downstream, where &x2 is a suitably sized increment. The radial concentration distribution is assumed at various radial increments at axial position, x~ = Ax2, and the droplet sizes, at, are approximated by applying the truncated Taylor series,

~i(,ax~) = adO) + (oa~/ox2)(o)z~x~,

[3s]

where Oai/Ox2 is obtained from Eq. [-19]. Equation [33] is then solved by successive approximation until satisfactory convergence is attained. The procedure is then repeated at 2Ax2 and again downstream until calculations indicate that the vapor concentration has reached its equilibrium concentration corresponding to the condenser wall temperature, at which point no further aerosol growth occurs.

It should be pointed out that there is a singularity at r = 1 associated with Eq. [19]. This is a result of the velocity vanishing at the wall, thereby leading to an infinite residence time for nuclei immediately adjacent to the wall. The singularity is avoided by assuming that all droplets in the radial increment nearest the wall reach their equilibrium sizes instantaneously and are maintained at the equilibrium size corresponding to the wall temperature. RESULTS The nondimensional temperature distribution calculated using Eq. [35] is shown in Fig. 5 as a function of x2 and r. At axial positions very near the inlet the temperature remains at the inlet value in the central core of the tube, dropping sharply to the prescribed boundary condition near the wall. From this information alone it is possible to draw numerous qualitative conclusions without resort to the tedious solution for the concentration field. It is clear that condensation on nuclei will first occur in the vicinity of the wall. Although it is likely that some condensation will occur on the wall itself, we shall assume that all of the vapor that condenses does so on nuclei. Because

Journal of Colloid and Interface Science, Vol. 50, No. 3, M a r c h 1975

G R O W T H K I N E T I C S OF AEROSOLS

497

FIG. 5. The nondimensional temperature distribution for the reference case, DBP-He.

the nuclei near the wall have a large residence time in the tube they can be expected to reach their equilibrium sizes within a short distance from the inlet. The resulting depletion of vapor in the wall region leads to radial concentration gradients that cause diffusion of vapor to the wall region. As a result of this diffusion, nuclei in the center of the tube either cannot become active or cannot grow to the same size as droplets near the wall because of vapor depletion. Thus, there is a tendency to form polydisperse aerosols because of the nonuniform temperature and concentration fields. Davis and Nicolaon (4) confirmed this conclusion experimentally by taking samples of a dibutylphthalate-helium aerosol at various radial positions in a condenser of the modified Sinclair-LaMer type. Samples taken near the wall consisted of larger droplets than those taken near the centerline. The degree of polydispersity depends on the numerous parameters involved, and quantitative estimates require solution of Eq. [-33] together with Eq. [-38] to determine the aerosol characteristics. The aerosol size distribution is usually described in terms of the average radius, a, and the relative standard deviation, a,, defined by = f l a l + f2a2 + " " + f~a~

[-39]

and o" ~r =

-

It =

[E

(~i -

a)~/I5~/¢

[40]

where f i is the fraction of droplets of size ai, It is the total number of droplets, and o- is the normal standard deviation. Nicolaon et al. (19) reported size distributions in terms of aM, the modal value of a, defined by aM = a [ ( r / ; ) 2 + i ] - ~

[41]

and the parameter ~0 defined by ¢0 = {In [-(¢/4) 2 q- 1]}3.

[-42]

The latter parameter, ~0, is a measure of the width of the zero-order logarithmic distribution. We have calculated ar in our computations, but Nicolaon and his associates reported experimental results in terms of aM and ¢0. For a narrow size distribution ~0 = at, and we shall arbitrarily refer to an aerosol as polydisperse if ar > 0.3 and monodisperse when ar < 0.3. To examine the solution of Eq. [-33] and to calculate aerosol size characteristics we must carry out numerical computations using specific values of the parameters. Since Nicolaon and his co-workers extensively studied the system DBP-helium-NaC1 nuclei, we shall choose parameters corresponding to that system. As a reference case, we shall first consider the parameters listed in Table II. Except for our specification of perfectly monodisperse nuclei of size, a = 0.01 /xm, and a uniform wall temperature of 60°C and a tube radius, R = 1 cm (Nicolaon's tube radius was 1.27 cm), the other parameters closely match

Journal of Colloid and Interface Science, Vol. 50, No. 3, March 1975

498

DAVIS AND LIAO TABLE II

THE PARAMETERS USED FOR THE ]~EFERENCE

CASE CALCULATIONS

Thermal parameters To = 110°C, T~ = 60°C, aue = 1.80 cm2/sec Geometrical and flow parameters R = lcm,

Pc1 = 11.4, Pc2= 141

Mass transfer parameters Co* = 1.12 X 10-6 g/cma, DnsP-~e = 0.16 cm2/sec Nuclei parameters N = 1.2 X 10Tparticles/cm 3, a0 = 0.01 ~m

some typical runs of Nicolaon. Note that Pel = 11.4 for this reference case. For this low value of Pel, axial heat conduction, which we have neglected, can be significant. Because of the assumption of monodisperse nuclei and the neglect of axial conduction precise quantitative agreement between theory and experiment should not be anticipated. The nondimensional concentration distributions computed from Eq. F333 for the parameters of Table I I are plotted in Fig. 6. As expected, vapor depletion occurs only in the wall region when x2 is small. At x2 ~ 5 )< 10.3 aerosol begins forming at the centerline and for x2 > 2 X 10-2 the asymptotic equilibrium vapor concentration is reached, and no further growth occurs. Comparison of Figs. 5 and 6 indicates that the rate at which radial temperature profiles

change in the x2 direction is greater than that rate of change of the concentration profiles. This suggests that vapor begins condensing in the core of the tube, due to the temperature decrease, before significant radial diffusion occurs. In this case the resulting aerosol can be expected to be relatively monodisperse. Figure 7 shows the aerosol sizes calculated for various positions in the condenser. Near the inlet only nuclei near the wall become active and grow, but as x2 increases the zone in which droplets grow sweeps inward from the wall as the temperature decreases. At x2 ~ 3 X 10.2 the droplets are fully grown. For this reference case, calculations give g = 0.26/~m and ~r = 0.23, which correspond to aM = 0.24 ~m and ~0 = 0.23, respectively. The data of Nicolaon et al. (2, 4), taken under rather similar conditions, give aM = 0.244 ~m and ~0 = 0.16. The agreement between theory and experiment is probably better than should be expected, considering the neglect of axial conduction and the heat source. This agreement is not an adequate test of the theoretical analysis, for there are numerous other experimental observations that must be predicted and explained by the analysis. Among these are the observations of Shahriari and Goodrich (17) as well as those of Nicolaon et al., that under m a n y conditions condensation begins along an isotherm of the flow field, and the temperature of that isotherm depends on the evaporator temperature (the inlet concentration of vapor to the condenser). Once Eq.

5

4

FIG. 6. The nondimensional concentration distribution for the reference case, DBP-He. Journal of Colloid and Interface Science, Vol. 50, No. 3, March 1975

GROWTH KINETICS OF AEROSOLS 5xlO -I '

'

'

''"'1

'

'

r=l ¢0

¢D o .¢

16. 3

10.2 4x10-z DIMENSIONLESS AXIAL DISTANCE, Xa

FIG. 7. The aerosol radii as a function of position for the reference case, DBP-He.

[-337 has been solved the onset of condensation can be determined theoretically, and Fig. 8 shows the calculated positions of the onset of condensation compared with the calculated 100°C isotherm. In our calculations the inlet concentration was based on a saturated DBPhelium stream, so the condensation isotherm ishigher than the 83 to 97°C isotherms observed experimentally because the vapor-gas stream was not usually saturated in the experiments. We should note here that our calculations are based on perfectly monodisperse nuclei whereas experiments always involve some distribution of nuclei sizes. As we shall show below, the introduction of nuclei of different sizes leads to further complications. An important experimental problem that has not been adequately explained is that the replacement of helium by nitrogen as the carrier gas very adversely affects the quality of the aerosol. With nitrogen, the aerosol is always found to be significantly more polydisperse than with helium under otherwise identical conditions. The present analysis makes it possible to clearly elucidate the reasons for the polydispersity. The principal cause of the polydispersity is the lower rate of heat transfer with nitrogen compared with helium. At 100°C the thermal conductivities are kN2= 7 . 0 X 10.5 and k H , = 4 . 0 X 10-*

499

cal/cm-sec-°k. For the flow velocity and other parameters of our reference case the Peclet numbers for heat transfer are Pei,~{e = 11.4 and Pel,N~ = 91. The diffusion coefficients also are substantially different, that is, DDBp-~e = 0.16 cm2/sec and DDBP--N2 = 0.056 cm2/sec, SO the Peclet number for mass transfer changes from Pe2,He = 133 to Pe~,N~ = 379. Using these new Peclet numbers the temperature and concentration fields have been calculated, and the results are plotted in Figs. 9 and 10. The aerosol size distributionpredicted is plotted in Fig. 11. As in the previous case the first aerosol formed is in the wall region, and farther downstream aerosol forms farther from the wall. But because of the relatively low rate of heat transfer compared with the rate of diffusion, aerosol which forms in the core of the tube reevaporates as vapor diffusion proceeds. At any axial plane, for x2 < 0.04, the aerosol is highly polydisperse, and in the region beyond x2 = 0.04 fully grown aerosol is to be found only in the region near the wall. I t is likely that significant condensation on the wall would occur under the Conditions specified for the calculations, and only nuclei would be found in the core of the tube. In this case the lower rate of heat transfer causes the temperature distribution to change rather gradually

0.02

~

/ t

0.01

I

I

I

I

o CALCULATEDONSET OF CONDENSATION 100°C ISOTHERH

--

< 0.005

~, z

o 5

0,001

I 0.2 0.z, 0.6 0.8 r, OIMENSIONLESS RADIUS

1.0

FIG. 8. A c o m p a r i s o n b e t w e e n t h e c a l c u l a t e d o n s e t of c o n d e n s a t i o n a n d t h e 100°C i s o t h e r m for t h e r e f e r e n c e case.

Journal of Colloid and Interface Science, Vol. 50, No. 3, March 1975

500

DAVIS AND LIAO

FIG. 9. The nondimensional temperature distribution ;[or DBP-N~.

with respect to x2, and, therefore, the central region of the tube remains at a higher temperature at any axial plane than in the case of helium. Although the decrease in the diffusion coefficient would tend to reduce the radial diffusion and lead to a more monodisperse aerosol, the effect of the temperature field predominates here. This is an excellent example of the complexity associated with the coupling between the concentration and temperature fields. To produce aerosols of a similar quality using nitrogen or helium the flow conditions should be selected such that the Peclet numbers, Pc1, for the systems are identical. Since Pel and Pc2 cannot be varied independently by changing the carrier gas it is more important to control ]?el. Shahriari and Goodrich (6, 7) used the DBP-nitrogen system to obtain

relatively monodisperse aerosols at very low carrier gas flow rates, a result that is consistent with our theoretical considerations. It should be pointed out that at very low Peclet numbers the effects of axial conduction of heat should be taken into account to make quantitative predictions. Axial conduction tends to smooth out temperature gradients, an effect that should lead to more uniform size droplets. Furthermore , at low flow rates and with large temperature differences, T o - T~, natural convection caused by density variations in the flow field appreciably distorts the velocity, temperature and concentration fields from those calculated here. The most ambiguous effect on aerosol polydispersity is that of the nuclei characteristics. Nicolaon and Kerker (5) were unable to explain the complex relation observed be-

FIG. 10. The nondimensional concentration distribution for DBP-N2. Journal of Colloid and Interface Science, VoL 50, No. 3, March 1975

GROWTH KINETICS OF AEROSOLS 5xlo"

I

I

I

I I I I I]

I

:--r-I

\

d .I

o o

/oj

O

to

iO"t 4xlO"z DIMENSIONLESS AXIAL DISTANCE, X=

FIG. 11. The aerosol size distribution for DBP-N~. tween the size of nuclei and the supersaturation in the condensation zone of their condenser. To illustrate the theoretical complexity of this problem we have performed calculations for the simple case in which half the nuclei which enter have radii of 0.01 ~m and half have radii of 0.05 ~m. Although there is a greater range of nuclei encountered in experiments, these Sizes are useful for purposes of illustration. The total number density and all other parameters are as listed in Table II. The nondimensional temperature and concentration distributions calculated do not differ greatly from those of the reference case shown in Figs. 5 and 6, so we shall not present these results here. What is substantially changed is the aerosol size distributions, plotted in Fig. 12. The larger nuclei, as expected, begin to grow first, and as x2 increases to about 10.2 condensation occurs on the smaller nuclei. As x2 increases further the droplets which form on the small nuclei begin to decrease in size because of the depletion of vapor. The small droplets essentially evaporate and the vapor condenses on the larger droplets. The eventual result is a polydisperse aerosol with a~ = 0.35. Most of the aerosol mass is in the wall region, 0.5 < ~" < 1.0, and the aerosol in the central region is very polydisperse, for near the centerline both fully grown droplets and inactive nuclei exist. The calculations

501

indicate that the onset of condensation on the 0.05 /~m nuclei coincides with the 105°C isotherm, but condensation on the 0.01 ~m nuclei depends on the complex coupling of the temperature and concentration fields since the 0.05 ~m nuclei deplete vapor before the 0.01 ~m nuclei become active. A more complete study of the effects of the various parameters is provided elsewhere (17). I t suffices to point out that anything that tends to increase radial diffusion of vapor or decreases the rate of heat transfer will tend to degrade the quality of the resulting aerosol. Furthermore, the detailed kinetics related to the growth of single droplets are no more important than the vapor depletion rate, for the properties and behavior of the ensemble of droplets control the single particle growth rate. Our calculations as well as the published experimental results show that aerosols of the type considered here are fully grown within a few centimeters of length of the condenser. For example, for our reference case the aerosol was found to be fully grown within a nondimensional length x2 = x*/R Pc2 = 3 X 10-2. This corresponds to an actual length of 4 cm, but most of the growth occurs within 0.5 cm from the onset of growth. In fact the growth rates are so large that the droplet size distribution is determined primarily by the characteristics of the temperature and concentration 5xl°" L-

i

i

i

i i ,i,j

J

,

,/i/.} i f:.L_ //ooo,o II~I DIMEIqSIONLESS AXIAL DISTANCE,

4xlO"~' Xz

Fro. 12. The aerosol size distribution for bidisperse nuclei.

Journal of Colloid and Interface Science, Vol. 50, No. 3, M a r c h 1975

502

DAVIS AND LIA0

fields and not by the single particle growth kinetics. SUMMARY

The condensation of vapor on nuclei in a modified Sinclair-LaMer aerosol generator has been rigorously analyzed to take into account the flow characteristics, temperature and concentration fields and other system parameters. The most important theoretical conclusions are (1) The coupling between the temperature and vapor concentration fields, which arises in describing the vapor depletion rate of an ensemble of droplets, is the predominant factor in determining aerosol characteristics. (2) Single particle models of droplet growth cannot be applied to predict the growth kinetics of aerosols unless the vapor depletion caused by competing droplets is taken into account. (3) Isothermal models of aerosol condensation in systems of the type considered here are not appropriate because of the temperature gradients produced by cooling the system. (4) The parameters having the greatest effect on the polydispersity of the aerosol generated are the Peclet numbers for heat and mass transfer, Ruo/o~and Ruo/D, respectively. (5) Polydispersity is largely a result of radial diffusion of vapor to the cooler regions of the flow field where vapor depletion occurs. (6) The size distribution of available nuclei does not necessarily have a large effect on the final aerosol size distribution, particularly when radial diffusion of vapor is not a predominant factor. (7) The effects of using nitrogen rather than helium as the carrier gas are shown to be

very significant, and the predicted reduction of aerosol quality is in good agreement with experimental observations. REFERENCES 1. SINCLAIR,D., AND LAMER, V. K., Chem. Rev. 44, 245 (1949). 2. NICOLAON, G., COOKE,D. D., KERKER, M., AND MATIJEVIC,E., Y. Colloid Interface Sci. 34, 534 (1970). 3. NICOLAON, G., COOKE, D. D., DAVIS, E. J., KERKER, M., AND MATIJEVI(~, E., J. Colloid Interface Sci. 35, 490 (1971). 4. DAVIS, E. J., AND NICOLAON, G., J. Colloid Interface Sci. 37, 768 (1971). 5. NICOLAON,G., AND KERKER, M., J. Colloid Interface Sci. 43, 246 (1973). 6. SHARRIARI, S., AND GOODRICH, F. C., f . Colloid Inter:face Sci. 39, 305 (1972). 7. SHAHRIARI, S., AND GOODRICH, F. C., ft. Colloid Interface Sci. 39, 312 (1972). 8. REIss, H., J. Chem. Phys. 19, 482 (1951). 9. EINSTEIN,A., Ann. Physik. 19, 289 (1906). 10. Hsu, C. J., AIChE J. 17, 732 (1971). 11. DAVIS, E. J., Can. J. Chem. Eng. 51, 562 (1973). 12. JERRI, A. J., AND DAVIS, E. J., J. Eng. Math. 8, 1 (1974). 13. HID¥, G. M., ANDBROCK,J. R., "The Dynamics of Aerocolloidal Systems." Pergamon Press, New York, 1970. 14. FucHs, N. A., AND SUTUGIN,A. G., "Highly Dispersed Aerosols." Ann Arbor Science Publishers, Ann Arbor London, 1970. 15. FUCHs, N. A., Proceedings of the 7th International Conference on Condensation and Ice Nuclei, p. 10, 1969. 16. SAHNI,D., J. Nucl. Energy 20, 915 (1966). 17. LIAO, S. C., Ph.D. Thesis, Clarkson College of Technology, 1974. 18. ABRA~aOWlTZ,M., AND S~EGUN,I. A., "Handbook of Mathematical Functions." National Bureau of Standards, Washington, DC, 1964. 19. ESPENSCHEID,W. F., KERKER, M., ANDMATIJEVI(~, E., J. Phys. Chem. 68, 3093 (1964).

Journal of Colloid and Interface Science, Vo!, 50, No, 3, March 1975