Page 133 - Air pollution and greenhouse gases from basic concepts to engineering applications for air emission control

4.4 Particle Coagulation 107

o 1 Z V Z 1
ð
ð
ð
fv; tÞ ¼ Ku; v uÞfu; tÞfv u; tÞdu fv; tÞ Ku; vÞfu; tÞdu ð4:35Þ
ð
ð
ð
ð
ot 2
0 0
where K(u, v) specifies the collision rate between particles of volume u and v.f is
the size distribution density function.
It is assumed in conventional aerosol analysis that particles attach to each other
upon collision, therefore, the collision rate can also be considered as the coagulation
rate.
Because of the dynamic change of the size distribution, the Smoluchowski [23]
equation can only be solved numerically. However, analytical solutions can be
obtained for simple cases with assumptions.
4.4.1 Monodisperse Aerosol Coagulation

The analysis is simplified if we only consider monodispere coagulation, where the
particle diameters are within a narrow range. Smoluchowski [23] developed a
model for Brownian coagulation of monodisperse particles in the continuum
regime. It is applicable to particles with sizes in a narrow range, say
1\d pA =d pB \1:25. The corresponding Brownian monodisperse coagulation effi-
cient for the continuum regime (Kn ≪ 1) and free molecule regime (Kn ≫ 1) are
simplified as

8kT3l continuum regime; Kn 1

K ¼ p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð4:36Þ
9:8 kTq d p free molecule regime, Kn 1
p
where k is Boltzmann’s constant and l is the dynamic viscosity of the carrier gas.
Otto et al. [19] introduced an analytical model for the regime in between, but its
form is complicated.
Integration of Smoluchowski equation using the initial condition of Nðt ¼ 0Þ¼
N 0 leads to the solution for the total particle concentration as a function of time and
the coagulation coefficient above.

2N 0
NtðÞ ¼ ð4:37Þ
2 þ KN 0 t
3
where N 0 is the initial particle concentration (#/m ).
By defining the characteristic dimensionless time
2
s ¼ ð4:38Þ
KN 0