Brownian coagulation of fractal aggregates

~ Aero~'olS¢i. "Vbl.31, Suppl. I, pp. $574 $575~ 2000

Pergamon www.elsevier.com/locate/jaerosci

Session 6D - Particle formation and coagulation B R O W N I A N C O A G U L A T I O N OF F R A C T A L A G G R E G A T E S Margaritis Kostoglou and Athanasios G. Konstandopoulos*

Chemical Process Engineering Research Institute, Aerosol & Particle Technology Laboratory PO Box 361, Thermi 57001 Thessaloniki, Greece

KEYWORDS Brownian coagulation, fractal aggregates, multivariate population dynamics

INTRODUCTION Until recently, particle volume has been used as the sole independent variable in the general dynamic equation (GDE) for aerosol populations. This approach may be sufficient if the particle morphology is not taken into account (assuming spherical shape for the particles) and all the phenomena under consideration depend only on particle volume e.g. [1]. In this case all geometrical particle properties (for example its surface area) can be uniquely in aerosol reactor modelling determined from the particle volume. Lately the consideration of sintering effects (a phenomenon which acts on particle area keeping volume unchanged) led to the inclusion of the particle area as a second independent variable in the GDE [2] enabling a description of the particle morphology through the independent variables of volume and area. In this approach, employed in several studies, one-way coupling between the particle morphology and the coagulation dynamics has been assumed. The effect of the evolving particle morphology on the size distribution dynamics is currently unknown, even in the absence of sintering effects. Therefore in principle the coagulation kernel and accordingly the coagulation dynamics for aggregates should be dependent on the evolving particle morphology, as well. To fill this gap an approach which introduces thefractal dimension of the aggregates as an independent variable has been recently developed [3]. In the present work the formalism is applied to describe fractal aggregate coagulation in the free molecular and continuum regimes. In its present version the formalism can straightforwardly include the simultaneous effect of a class of restructuring mechanisms that leave the aggregate area invariant e.g. Brownian (thermal) restructuring, capillary-force induced restructuring, etc.

METHODS The two variable population balance that describe aerosol size and morphology evolution due to Brownian coagulation and invariant-area restructuring is 0f(x, D,t) 1 ~ D.,~ Din. 0t -- 2 J f f K ( x - y ' y ; D ' ' D 2 ) 8 ( D - c ( x ' y ' D ' ' D 2 ) ) f ( y ' D , , t ) f ( x - y,D 2,t)dydD,dD 2 0 Dmin Dmi .

- Do~x 0G(D,x)f(x, D,t) f(x,D,t)f S K(x,y;D, ,D)f(y,D. ,t)dydD~ - L 0 Dml" 0D The above equation has been rendered dimensionless employing standard procedures for the nondimensionalization of the population balance [1]. The reference size used for normalization is the arithmetic mean size of the initial size distribution, f(x,D,t) is the number density distribution of particles with respect to size x (taken as the dimensionless aggregate volume) and fractal dimension D of the aggregate, K(x, y; D~,D2) is the coagulation rate between two particles with descriptors (x, Dr) and (y,D2) respectively, G(x, D) is the rate of fractal dimension variation due to Brownian restructuring and c(x, y, D1, D2) is the fractal dimension of a new particle produced by the coagulation of two particles with descriptors (x, Dr) and (y, D2) respectively. The dimensionless number L provides a measure of the restructuring rate with respect to the coagulation rate. The numerical solution of the multivariate population balance with conventional techniques (e.g. sectional or moment methods) can be very demanding (for two independent variables) to almost intractable (for more independent variables) due to the exponential increase of the computational effort with the number of independent variables. For this

*Corresponding author. Fax: + 30 31 498-190. E-mail: [email protected] $574

Abstracts of the 2000 European Aerosol Conference

$575

reason a Monte Carlo methology is developed for the particular problem, based on the modem constant number, event driven approach [4]. RESULTS-CONCLUSIONS To illustrate the capability of the new approach to describe the evolution of aggregate morphology, we solve the bivariate-(x, D) population balance equation with a Brownian coagulation kernel. The limiting cases of c o n t i n u u m and f r e e m o l e c u l a r coagulation are examined in the absence of aggregate restructuring. The kernels for coagulation of aggregates with the same fractal dimension have been extended here to describe coagulation between aggregates with different fractal dimensions while the constitutive law c(x, y, D1, D2) has been described previously [3]. The population balance is solved for an initial population of equal sized aggregates each consisting of N,, monomers with D=2.5. Figure 1 shows the evolution of the mean fractal dimension Dm of the aggregate population for two initial N,, values. The tendency of aggregates with larger No to retain their initial fractal dimension is apparent. The reduction of the fractal dimension is faster for the case of continuum coagulation, despite the fact that the number of collisions that take place at the same value of dimensionless time, is smaller than the free molecular case. This is likely due to the bias towards coagulation between particles with very different size (as well as fractal dimensions) exhibited by the continuum kernel. Figure 2 shows the evolution of the total particle number for several cases. Interestingly enough, for the continuum kernel the results are insensitive to No (difference between No=100 and No=1000 is smaller than 0.5%) so only one curve is shown. To confirm this insensitivity, the result of the self similarity solution [1] for nonfractal aggregates is also shown. The relative difference of the self-similarity solution with the fractal coagulation result is below 5%. On the contrary, in the free molecular case, the coagulation dynamics as measured by the total aggregate number, are seen to depend very sensitively on No. The present approach provides a more complete description of aerosol aggregate population dynamics and can form the basis for the design of processes where aerosol aggregates with desired morphology (as measured by their fractal dimension) are produced.

2.6

_

i

~,l.,~

I

i

I,I,TI,

I

,

II1~.,[

,

100

,,I,,,.

2.5 2.4

\

10 -1

\

2.3 N 10 z

2.2

D m

2.1 free tool. N o = 1 0 0 free tool. N o = 1 0 0 0

2

continuum No=100

I

I I I

1,9

.....

1.8

....... i ........ I ........ i ...... ,:

0.1

1

10 .3

-

t

100

1000

. . . . .

10-4

i

1

I\

continuum

------

continuum No=1000I

10

-

I

free tool N o = 1 0 0 free tool. N o = l O 0 0 1

I

cont. self-sim.

~lIH.I

r

I IIIIII}

10

I

100

I

IILI

1000

t

Figure 1. Evolution of the mean fractal dimension Dm of Figure 2. Evolution of the total particle number N for the aggregate population for two values No of monomers the two modes of coagulation and two values of No. Also in the initial aggregates, the self-similarity solution [1] result for the continuum case is shown. AKNOWLEDGEMENTS - This work was supported in part by the Hellenic General Secretariat for Research and Technology through Grant OPRT-ll No. 127 and the EU Commission through contract BRST-CT98-5537. We thank Prof. D. E. Rosner (Yale U.) for helpful discussions and correspondence.

REFERENCES 1. 2. 3. 4.

Friedlander, S.K. (1977) Smoke, Dust and Haze. Fundamentals o f aerosol behavior. Wiley, New York Koch, W. and Friedlander, S.K. (1990) J. Colloid. Int. Sei. 140, p. 419. Kostoglou, M. and Konstandopoulos A.G. (1999) J. Aerosol Sci. 30, SI, pS447 Smith, M. and Matsukas, T. (1998) Chem. Engng. Sei. 53, p1777