Turbulent Brownian Coagulation Model

The 16 th IASTED International Conference on Modelling and Simulation 2005

SIMPLE MODEL FOR TURBULENT BROWNIAN COAGULATION OF POLYDISPERSE AEROSOLS

by Ana Teresa Celada Murillo Alejandro Salcido González Instituto de Investigaciones Eléctricas, México

TOPICS •

INTRODUCTION

•

MEAN FIELD COAGULATION MODEL

•

RESULTS

•

CONCLUSIONS

INTRODUCTION AEROSOL Suspension of fine particles, solid or liquid, in a gas Nuclei mode

Environmental impacts: • Adverse healt effects

(0.001– 0.1 µm) Fine

• Weather changes

Accumulation mode

• Degradation of visibility

(0.1- 2.5 µm)

• Precursors of acid rain Coarse

( > 2.5 µm )

INTRODUCTION diffusion nucleation

coagulation

sedimentation condensation

COAGULATION Process by which some of the aerosol particles collide with each other and they join to form larger particles Brownian motion (~ 0.001 a 1 µm) Turbulent motion (~ 1 a 10 µm) Differential sedimentation ( > 10 µm )

Smoluchowsky (1917)

• Monodisperse size distribution • All the particles are spherical

dn k dt

=1 2

∞ ∑ β ij ni n j − ∑ i+ j = k i

β ik n i nk

• Only binary collisions • These collisions conserve mass and volume • All collisions produce coagulation

MEAN FIELD COAGULATION MODEL (MFC) - Only binary collisions, conserve mass and volume

- Polydisperse size distribution - Aerosol particles are spherical - All collisions

coagulation

200 150

βκ

βj

100

βi 50 0 0.001 0.01 0.02 0.03

0.04 0.08 0.16

0.32

0.64 1.28

Diameter intervals

2.56

5

10

=

βk

MEAN FIELD COAGULATION MODEL (MFC) • The probability that one collision takes place in the system. • The probability that such collision involves particles of particular intervals βi and βj

⎧⎪ Pij = K ( ri , Di , r j , D j ,...) ⎨ ⎪⎩

ni n j

if i = j

2n i n j

if i ≠ j

• The probability Qijk that such collision produces a particle in a given interval βk

MEAN FIELD COAGULATION MODEL (MFC) βj βi

dn k dt

= 1 2

M ∑ Pij Qij k i, j ≠ k

=

βk

M − ∑ P (1 − Q k ) ik ik i =1

MEAN FIELD COAGULATION MODEL (MFC)

K B = 4π ( ri + r j ) (Di + D j )

K = K B + KT

1/ 2

⎛ 8π ⎞ KT = ⎜ ⎟ ⎝ 3 ⎠

( ri + r j ) (w a 2 + w c 2 ) * ε c

RESULTS Simulation of brownian coagulation

Kim et al., 2003

Rooker and Davies, 1979

• NaCl aerosol

• CaCO3 aerosol

• Diameters: 0.050, 0.115 µm

• Diameter interval: 0.005 – 0.030 µm

• Duration: 1800 – 3000 s

• Duration: 1800 s

• Sample time: 300 – 350 s.

• Sample time: 300 - 400 s.

SIMULATION OF BROWNIAN COAGULATION Kim et al., 2003 0.050 µm

1 0.8

0.8

0.6

N/No

N/No

0.115 µm

1

0.6

0.4 0.4

0.2 0.2

0

0

0

1000

2000

3000

400

800

1200

TIME (s)

TIME (s)

Experimental data

MFC model - Brownian Kernel

1600

SIMULATION OF BROWNIAN COAGULATION Rooker and Davies, 1979 C90

1

1

0.9

0.8 N/No

N/No

C50

0.8

0.6

0.7

0.4

0.6

0.2 0

400

800 1200 TIME (s)

1600

2000

Experimental data

0

400

800 1200 TIME (s)

MFC model - Brownian Kernel

1600

2000

SIMULATION OF TURBULENT BROWNIAN COAGULATION Okuyama et al., 1977 • Tobacco smog aerosol • Constant temperature (25°C) and pressure. • Particle diameter: 0.700 µm • Stirr speed : 600 and 1800 rpm • Duration: 300 – 500 s

SIMULATION OF TURBULENT BROWNIAN COAGULATION 600 rpm ε = 30,000 cm2 s-3

1

1800 rpm

0.8

0.8

0.6

0.6

N/No

N/No

1

εc = 1.0 0.4

0.4

0.2

0.2

0

0

0

100

200

300

TIME (s)

Experimental data

400

500

ε = 810,000 cm2 s-3

εc = 3.5

0

100

200

300

TIME (s)

MFC model - Turbulent Kernel by Kruis and Kusters (1997)

CONCLUSIONS • The Mean Field Coagulation model (MFC) simulates, in a simple form, the coagulation of polydispersed aerosols. • The simulation experiments showed that the MFC model reproduces satisfactorily the experimental data of pure brownian and brownian turbulent coagulation. • The MFC model takes into account the polydisperse nature of the aerosol. It is simulated with a probability function, Qijk, that conserves the aerosol volume.

Thank you for your attention

REFERENCES • Kim D. S., Park S. H., Song Y. M., Kim D. H., Lee K. W. (2003).

Brownian coagulation of polydisperse aerosols in the transition regime. Journal of Aerosol Science, Vol. 34(7):859-868. • Kruis F. E., Kusters K. A. (1997). The collision rate of particles in turbulent flow. Chem. Eng. Comm., Vol. 158:201-230. • Okuyama K. Kousaka Y., Kida Yoshinori y Yoshida Tetsuo. (1977). Turbulent coagulation of aerosols in a stirred tank. Journal of Chemical Engineering of Japan, Vol. 10(2):142-147. • Rooker S. J., Davies C. N. (1979). Measurement of the

coagulation rate of a high Knudsen number aerosol with allowance for wall losses. Journal of Aerosol Science, Vol. 10:139-150.