- Email: [email protected]
Asymptotic solution of the aerosol general dynamic equation for small coagulation
Asymptotic Solution of the Aerosol General Dynamic Equation for Small Coagulation C H R I S T O D O U L O S PILINIS AND J O H N H. SEINFELD Department of Chemical Engineering, California Institute o f Technology, Pasadena, California 91125 Received March 24, 1986; accepted April 28, 1986 The aerosol general dynamic equation is solved in the limit of small coagulation. The solution includes particle growth by gas-phase diffusion, surface and volume reaction, particle production, and loss by deposition to surfaces. © 1987AcademicPress,Inc.
INTRODUCTION
The dynamic behavior of a spatially homogeneous aerosol undergoing particle growth, coagulation, augmentation by particle sources, and first order removal such as that by deposition onto surfaces is governed by the general dynamic equation (4, 9, 10)
On(D, t) Ot
×
00--D[I(D, On(D,t)] + D2F °
Yj~
fl[(D 3 - D'3) 1/3,D']n[(D 3 - D'3) 1/3, t]n(D', t)
×dD'-n(D,t)
( D 3 - D'3)1/3
f?
B(D,D')n(D',t)dD'
On(D,t)_ D 2 f D Ot 2 3D~ ×
a
+ S0(D, t) - a(D, t)n(D, t),
D + dD]; and a(D, t) is the first order coefficient for particle removal by deposition on container surfaces. When all the phenomena represented in Eq. [ 1] are important, Eq. [ 1] must generally be solved numerically (2, 5, 11). If particle growth or evaporation is not occurring, then I(D, t) = 0. Additionally, in the absence of sources or deposition, Eq. [1] reduces to the coagulation equation
[ 1]
where D is the particle diameter; Db and Da are the upper and lower limits on D, respectively; n(D, t) is the size distribution density function at time t, such that n(D, OdD is the number concentration of particles having diameters in the range [D, D + dD];/3(D, D') is the coagulation coefficient for particles with diameter D and D'; I(D, t) is the rate of change of particle diameter, dD/dt, due to gas-to-particle conversion by vapor condensation; S0(D, t) is the rate of particle production from external sources, such that So(D, OdD is the rate of introduction per unit volume in the system of particles having diameters in the range [D,
/3[(D 3 - D'3) 1/3 , D']n[(D 3 - D'3) 1/3, t]n(D',
t)
( D 3 - D,3)1/3
× d D ' - n(D, t)
f?
~(D, D')n(D', ODD'.
[2]
a
There exist a number of studies of analytical (1, 8, 13) and numerical (2, 5, 7, 11) solutions of Eq. [2]. On the other hand, in the absence of coagulation, one obtains the condensation equation
On(D, t) _ at
O OD [I(D, t)n(D, t)] + So(D, t) - a(D, t)n(D, t)
[31
which may be solved by the method of characteristics (3, 12).
472 0021-9797/87 $3.00 Copyright© 1987by AcademicPress,Inc. All rightsof reproductionin any formreserved.
Journal of Colloidand InterfaceScience, Vol. 115,No. 2, February 1987
SOLUTION OF THE AEROSOL EQUATION In a number of important aerosol processes, such as formation and growth of aerosols in atmospheric photochemical reactions, the dominant physical processes influencing the aerosol size distribution are those represented in Eq. [3], namely condensation, sources, and removal, while coagulation exerts only a minor influence on the dynamics of the distribution. This observation suggests that it might be useful to seek a solution of the general dynamic equation, Eq. [1], for the case in which the effect of coagulation is small relative to those of the other phenomena represented in the equation, but not small enough to be neglected. Such a solution is the objective of the present work. DIMENSIONLESS GENERAL DYNAMIC EQUATION It is useful to cast Eq. [ 1] in an appropriate dimensionless form (3), 0ff(Kn, 3") Or
- NokTX 2
Dn /(Kn, r) = ~b/(D, t) Dt 7~V~ and Kn is the Knudsen number, 2X/D; T is the absolute temperature; 4~ is a growth coefficient depending on the growth mechanism, 4) = NlVlD/X; D is the diffusivity of the vapor molecules; X is the mean free path; a(r) is the coefficient accounting for any temporal variation of the growth rate; N~, Vl are the gas phase concentration and the volume of the condensing molecules, respectively; No is the initial total particle number concentration; and ~ = viscosity. The dimensionless Brownian coagulation coefficient is (10) fl(Kni, Kn:) 2/3Q(PiKni + P:Knj) Q t- 7rA(PiKni + PjKnj) Q + (HI + H}) 1/2 2Q(Kn 3 + Kn3) 1/2
a(r)_ O [Kn2/(Kn, r)ff(Kn, r)l OKn
f 1 fKn + ({2 JK-u ¢/tKn', (Kn -3
-
-
473
with
Kn'-3)-1/31
1
1
Q = K-~n~-~ Knj X rT(Kn', r)rT[(Kn -3 - Kn'-3) -1/3, r] X [ (Kn-3
Kn 3/2 I'/ 2
Kn'-3~-1/3]4 -K--~ " ] d K n ' - •(Kn, r)
APi ~3 4
× afKn" K n ' fl(Kn' )r~(Kn"K~ -
-(kn
r)dKn'}
So(Kn, r) - a(Kn, r)ff(Kn, r),
[41
with initial condition
g n/: j
[ 8 p k T 11/2 A = [271r2rtXj Pi = 1 +Kni[1.257 +0.4 exp(-1.1/Kni)].
r~(Kn, 0) = f ( K n ) , where ff(Kn, r) -
A2p213/2]_ - 2 Kni
Xn(D, t) NoKn 2
S'o(Kn, r) - So(D, t)X3 DNoKn 2 a(D, I)X2 a(Kn, r) - - D
[51
ASYMPTOTIC SOLUTION OF THE GENERAL DYNAMIC EQUATION FOR SMALL COAGULATION The coagulation terms in Eq. [4] are multiplied by the parameter ~. This parameter depends on the conditions in the system and the total particle concentration. If ~ is small, then the coagulation contribution to the evolution of ff(Kn, r) is correspondingly small. Journal of Colloid and Interface Science, Vol. 115,No. 2, February1987
474
PILINIS AND SEINFELD
In the case of small ~ we can express the solution of Eq. [4] in terms of a regular perturbation series (6),
f
--
i-1
Knafl(Kn, Kn')[ Z ni_p_l(Kn, r) 'dKnb
p=0
× tio(Kn', r)]dKn'
oo
ff(Kn, r) = Y. ff,-(Kn, r)(k
[6]
and where Kn" = (Kn -3 - Kn'-3) -1/3. We note that for each i, Gi(Kn, r) is a known function of Kn and r, since it depends only on lower orders of tii(Kn, r). As a result, the system of equations [7] and [8] can be solved sequentially for successively higher values of i by the method of characteristics. The characteristic equations for the system [7] and [8]
i=0
Substituting Eq. [6] into Eq. [4] and equating terms containing like powers of ( we obtain the following system of equations O& 0 _ 20-7 = ~(r) ~ [n0(Kn, r)Kn I(Kn)] - ~%(Kn, r) - a(Kn, r)tTo(Kn, r) OrTi
0
- a(r) ~nn
are
[71
-dKn -dr
2[~,(Kn, r)Kn I(Kn)]
a(r)KnZ[(Kn)
dffo = [ ~(r) d(Kn2/(Kn))
dr
+ G~(Kn, r) - a(Kn, r)tii(Kn, r) i = 1,2 . . . . .
[11]
dKn
a(Kn, r) ]
× ffo(Kn, r) - ~¢o(Kn, r)
[8]
[12]
[ 13]
subject to
subject to ffo(Kn, 0) = f ( K n ) tii(Kn,0):0,
tio(Kn, 0) = f ( K n )
[91
i = 1,2,3 . . . . ,
[10]
d(Kn2/(Kn))
dni
-d-(: [~(~) ~nn
where ! f Kn
[ 141
and
~(Kn,~)]
×ffi(Kn, r)+Gi(Kn, r)
/Kn"\ 4
Gi(Kn, r) = 2 dK~bfl(Kn',Kn")~-~ )
[15]
subject to
i-1
ffi(Kn,0)=0
X [ ~ tii_p_l(Kn", r)no(Kn', r)ldKn'
i = 1,2 . . . . .
[16]
Integrating Eqs. [13] and [15] along the characteristic curves of Eq. [ 12] we obtain
p=O
8o(Kn, r)
f (Kn(O))Kn2(O)[(Kn(O)) - H(r - ro)f i'~fo(Kn,r')Kn2[(Kn)exp[f i'a(Kn,r")dr "]dr' - -
7
]
and
fii(Kn, r) =
, Kn2/(Kn)exp[fo a(Kn, r')dr']
Journal of Colloid and Interface Science, Vol. 115, No. 2, February 1987
,
[18]
475
SOLUTION OF THE AEROSOL EQUATION
where ro is the time at which the particles are introduced into the system, and {;
H(x)=
f°r
x>0
for
x<0
TABLE I Estimates of Maximum Dimensionlessand Real Time Correspondingto InitialAerosolNumber Concentrations for Which AsymptoticSolution Is Expectedto be Valid
The zeroth-order solution, ffo(Kn, r), accounts only for the condensation, sources, and removal mechanisms in shaping the size distribution. The effect of coagulation appears then in the first order term ffl(Kn, r). As r increases, the effect of coagultion increases relative to those of the other principal processes, so the perturbation solution is expected to hold for values of r that satisfy the constraint,
~-f~[ fr~aB(Kn(r'),Kn')~o(Kn',r')dKn']dr' 2 dr0 [dKnb < 1.
[19]
For a smooth zeroth-order solution Eq. [19] can be reduced to max(fl(Kn, Kn'))
J~oN(r')dr'<1,
[19a]
where N(r) is the dimensionless total particle number concentration, N(r) = In the initial value problem N(r) < 1, since particles are consumed due to deposition and coagulation. Therefore Eq. [ 19a] reduces to
N(r)/No.
½~r max(/3(Kn, Kn')) < 1.
[19b]
For temperature T = 300 K and pressure p = 1 arm, in air, X ~ 0.065/zm and ~ ~ 1.8 × 10 -5 kg m -1 s-1. For a diffusion coefficient for typical condensible species in air, D ~ 10-5 m 2 s-1 and ~ is of order ~ -~ 10 -26 No, where No is in m -3. For particles in the range 0.110 u m the m a x i m u m value of the dimensionless coagulation coefficient is of the order max(/3(Kn, Kn')) ~ 102. Using these values for the various parameters and Eq. [ 19b], and upper limit of r for which the perturbation solution gives accurate results can be estimated. Table I shows rmax, as well as the corresponding tmax, in hours, for different values of N0. Physically, the reason for this behavior
No (m-3)
r~
t~ (h)
10 102 l04 106 10s 101o 1012
1023 1022 1020 1018 1016 1014 1012
101° 109 107 105 103 101 10-I
can be understood by considering a system in which condensation and coagulation are the only processes occurring. As time progresses, while condensation is primarily shaping the distribution, coagulation is slowly reducing the number of particles and hence the surface area available for condensation. Eventually a point is reached at which the effect of coagulation relative to that of condensation is no longer small. The smaller the value of ~, the longer the perturbation solution of Eq. [6] remains valid. EVOLUTION OF A MONODISPERSE AEROSOL BY GROWTH AND COAGULATION
A general expression for the dimensionless condensation growth law [ that incorporates diffusion-, surface reaction-, and volume reaction-limited growth is (12) [ = crKn~,
[20]
where 3' = 1 diffusion-control 3' = 0 surface reaction control 3" = - 1 volume reaction control and where a can be assumed independent of time r. W i t h / g i v e n by Eq. [20] the characteristic growth curves given by the solution of Eq. [ 12] are Journal of Colloid and Interface Science, Vol. 115,No. 2, February 1987
476
PILINIS
AND
Kn(~) [o-(3`+ 1)(r- ~o) =
+Kn0-o)-(~+l)] -1/¢~+1) 3 ` ~ - 1 Kn(ro)e -+-~°)
3,=-1,
[21]
SEINFELD
have Kn(ro) > 0, i.e., follow real characteristics. We now consider an initially monodisperse aerosol, ri(Kn, 0) = 6(Kn - Kn*), [23]
[22]
where Kn(ro) is the K n u d s e n n u m b e r of the particles at ~o. We note that in Eq. [21] only particles with the Knudsen n u m b e r satisfying
Kn(O < [o-(3`+ 1)(r- ~o)1-1/(~+1>
that undergoes growth and coagulation. Thus, particle sources and deposition losses are neglected. Using Eq. [17], for 3' 4: - 1, we obtain fio(Kn, ~-) = [ 1 - o-(-y+ 1)7-Kn~/+l] -('Y+z)/('y+I) × 6[(Kn -(~+1) - trey + l)r) -1/(~+1) - Kn*]. [24]
Since 1 ¢":~
[Kn"\ 4
61(Kn, ~) = 2 J K ¢~(Kn',Kn")[.-:---/~o(Kn', O~o(Kn",OdKn' \
nb
r~n /
f
Kna
/3(Kn, Kn')fio(Kn, r)fio(Kn', r)dKn',
- - dKnb
we obtain 21/3Kn) + 1,3
_
]-1/(3,+ 1) _ K i l l
/
+'
X H ( K n - 2-1/3[Kn *-(~+D + o-(T + 1)r] -1/(v+1)) - [1 - ~r(3, + 1)rKneY+l)] -(~+2)/(~'+1) ×
¢~(Kn(r'), [Kn *-(~+1) + ~(3' + 1)r]-l/(~'+D)dr'~[(Kn-(v+l) - o(3' -t- 1)r) -1/(~+1) - Kn*].
[251
Therefore, the dimensionless size distribution function is given to first order by
[fo
fi(Kn, r) = 1 - ~
/3(Kn0"), [Kn *-(v+l) + ~(3` + 1)'c]-l/(*+l)d'c'
]
× [1 - ~r(3`+ 1)'rKn('l+l)] -e~+2)/(7+1)X 6[(Kn -e~+l) - a(3` + 1)'r) -1/(~'+D - K n * ] ]-- 1/(3'+1)
× H ( K n - 2-1/3[Kn *-e~+l) + o-(3`+ 1)r]-1/(~'÷1)). If ~ = 0, Eq. [26] reduces to the solution for the size distribution function in the presence of pure growth. In that case, with the initial distribution given by Eq. [23] the aerosol remains monodisperse with time. Journal of Colloid and Interface Science, Vol. 115, No. 2, February 1987
[26]
The dimensionless total particle n u m b e r concentration, N(z) = N('r)/No, is given by =
fi(Kn, r ) d K n . OKn b
[27]
477
SOLUTION OF THE AEROSOL EQUATION
In general the integral in Eq. [27] m u s t be evaluated numerically. In the special case of fl(Kn', Kn") -= 2 C the integral can be evaluated analytically to give
10 NO COAGULATION ......
COAGULATION ~" = 10-4
2-= 103
0.8
e = 10-5 %. 0.6
N0-) = 1 - ~C~- + O(~2).
[28]
T h e exact result for the total n u m b e r in the case of a constant coagulation coefficient is 1
N0-) - - 1 + ~C~-"
i~ O4 O.2
~
~
_
~
[29] 2
Equation [28] is just the first two t e r m s in the Taylor series expansion o f Eq. [29]. EVOLUTION OF A LOG-NORMALLY DISTRIBUTED AEROSOL BY GROWTH, DEPOSITION, AND COAGULATION
3
4
5
Kn -t
FIG. 2. Normalized aerosol size distribution function at a dimensionless time of z = 1 0 3 in the cases of no coagulation and small coagulation (~ = 10-4). Deposition occurring at a dimensionless rate of a = 10-3. T w o cases will be considered:
We now consider the case of an initially logn o r m a l l y distributed aerosol, 1
r~(Kn, 0) =
(ii) Growth, coagulation, and deposition, with a = 10 -3.
2~-~ K n In % Xexp[
(i) G r o w t h and coagulation without deposition, i.e., a = 0.0;
ln2(Kng/Kn)-] 21nZrg j ,
[30]
with Kng = 0.65 and ~g = 1.4, evolving by growth, as described in Eq. [21] with 3' = 1, i.e., diffusion-limited growth, deposition with a = constant and coagulation with ~ = 10 -4.
It is c u s t o m a r y to plot the data so that the particle size increases along the x axis. Therefore plots of the dimensionless v o l u m e distribution ff(Kn -1, r) and dimensionless v o l u m e distribution l?(Kn -l, T) versus K n -1 will be presented. In t e r m of the previously defined distribution, fi(Kn -1, r) = - K n 2 t i ( K n , r)
I.O
47r I?(Kn -1, T) = -~- Kn-3ff(Kn -1, z).
NO COAGULATION ......
COAGIJLATION ~" = I0-4
0.8
T = 103 =
0.6
~K 0.4 0.2
,
2
5
4
I
5
Kn-I
FIG. 1. Normalized aerosol size distribution function at a dimensionless time of T = 103 in the cases of no coagulation and small coagulation (~ = 10-4). No deposition (~ = 0).
Figure 1 shows the n u m b e r distribution for r = 103, for Case (i) with a n d without coagulation. C o m p a r i s o n between the two curves in Fig. 1 shows that the n u m b e r o f small particles has been decreased, while the n u m b e r o f larger particles has been increased; c o m p a r e this to the case in which coagulation is neglected. Figure 2 shows exactly the same situation for Case (ii). C o m p a r i s o n between the two curves in Fig. 2 shows that the coagulation process is not as important as in Case (i). Since a substantial n u m b e r of small particles is cons u m e d due to deposition on surfaces, fewer Journal of Colloid and lnterface Science, Vol.115,No. 2, February1987
478
PILINIS
AND
can participate in the coagulation process, with the immediate result being a decrease in the importance of coagulation on the evolution of the number density function. We expect that the particles will shift to higher diameters due to the growth and coagulation processes. Figures 3 and 4 show this shift as a function of dimensionless time for Cases (i) and (ii), respectively. We find in Fig. 4 that a steady-state condition for large particles, Kn -1 > 3, is achieved. Thus the flux of particles to the large particle domain balances the loss from that domain due to deposition. The volume distribution for Case (i) is shown in Fig. 5. Figure 5 shows the development of a sharp peak, which is the result of the large number of small particles growing to an appreciable size. CONCLUSIONS A solution of the dimensionless aerosol general dynamic equation in the limit of small coagulation has been obtained. The expressions for the particle sources, removal, and growth mechanisms can be quite general. The case of initially monodisperse aerosol undergoing growth and coagulation was tested. Comparison between the solution via pertur-
SEINFELD
1.0 - =
10- 4
a = 10- 3
0,8
~o.o
/
../~.o
'~
T=5Xl0 2
T = I03
o.z j/J
- ~ : l . ~ , o~ = x 03
0
,
0
FIG.
2
4. Normalized
I
4 Kn -1 aerosol
,
,
,
,
6
size distribution
I
8
function
at
a series of times in the case of small coagulation (~ = 10-4).
Deposition occurring at a dimensionless rate o f a = 10 -3.
bation series and the analytical solution for the total particle number evolution shows that the series describes accurately the fate of the aerosol, the accuracy depending on the number of terms of the series evaluated. The evolution of a log-normally distributed aerosol by growth, deposition, and coagulation was also presented and in which our solution predicts the "expected" aerosol behavior for
60
1.0
= 10-4
= i0 -4
a:0
a=O 0.8 0
T : 2 x 103
T = 5 x 102
40 O.E
T = IO~
~" ~'~ i-d 0.4
=
2- = 1,5 x 103
-p
T = 1.5 × 103 x 103
T = I0 3
'~ I> 20
P = 5 x io 2
0.2
2
4
, , , I .... 6
I
8
Ka-1
FIG.3. Normalizedaerosolsizedistributionfunctionat
0
r,=
0
I 2
4 Kn-I
6
,
T
,
,
I
8
FIG.5. Normalizedaerosolvolumedistributionfunction
a series of times in the case of small coagulation (~
at a series of times in the case of small coagulation ( (
= 10-4). N o d e p o s i t i o n ( a = 0).
= 10-4). N o d e p o s i t i o n ( ~ = 0).
Journal of Colloid and InterfaceScience, VoL 115, No. 2, February 1987
SOLUTION OF THE AEROSOL EQUATION
the number and volume density functions. A steady-state phenomenon for large particles that is the consequence of the competition between growth and coagulation on the one hand, and deposition on the other, is demonstrated by the solution. The solution presented here can provide a framework for the modeling of situations in which coagulation is expected to exert only a minor influence on the aerosol size distribution. ACKNOWLEDGMENT This work was supported by National Science Foundation Grant ATM-8503103. REFERENCES 1. Drake, R. L., in "Topics in Current Aerosol Research," Part 2 (G. M. Hidy and J. R. Brock, Eds.), p. 203. Pergamon, Elmsford, NY, 1972. 2. Gelbard, F., and Seinfeld, J. H., J. Comp. Physiol. 28, 357 (1978).
479
3. Gelbard, F., and Seinfeld, J. H., J. Colloid Interface Sci. 68, 173 (1979). 4. Gelbard, F., and Seinfeld, J. H., J. Colloid Interface Sci. 68, 363 (1979). 5. Gelbard, F., Tambour, Y., and Seinfeld, J. H., J. Colloid Interface Sei. 76, 541 (1980). 6. Kevorkian, J., and Cole, J. T. "Perturbation Methods in Applied Mathematics," Springer-Verlag, New York/Berlin, 1981. 7. Middleton, P., and Brock, J., J. Colloid Interface Sci. 54, 249 (1976). 8. Peterson, T. W., Getbard, F., and Seinfeld, J. H., J. Colloid Interface Sci. 63, 426 (1978). 9. Ramabhadran, T. E., Peterson, T. W., and Seinfeld, J. H., AIChE J. 22, 840 (1976). 10. Seinfeld, J. H., "Atmospheric Chemistry and Physics of Air Pollution," Wiley, New York, 1986. 11. Warren, D. R., and Seinfeld, J. H., AerosolSci. Technol. 4, 31 (1985). 12. Williams, M. M. R., Y. Colloid Interface Sci. 93, 252 (1983). 13. Williams, M. M. R., J. Colloidlnterface Sci. 101, 19 (1984).
Journal of Colloid and Interface Science, V ol. 1 t 5, No. 2, February 1987
INTRODUCTION
The dynamic behavior of a spatially homogeneous aerosol undergoing particle growth, coagulation, augmentation by particle sources, and first order removal such as that by deposition onto surfaces is governed by the general dynamic equation (4, 9, 10)
On(D, t) Ot
×
00--D[I(D, On(D,t)] + D2F °
Yj~
fl[(D 3 - D'3) 1/3,D']n[(D 3 - D'3) 1/3, t]n(D', t)
×dD'-n(D,t)
( D 3 - D'3)1/3
f?
B(D,D')n(D',t)dD'
On(D,t)_ D 2 f D Ot 2 3D~ ×
a
+ S0(D, t) - a(D, t)n(D, t),
D + dD]; and a(D, t) is the first order coefficient for particle removal by deposition on container surfaces. When all the phenomena represented in Eq. [ 1] are important, Eq. [ 1] must generally be solved numerically (2, 5, 11). If particle growth or evaporation is not occurring, then I(D, t) = 0. Additionally, in the absence of sources or deposition, Eq. [1] reduces to the coagulation equation
[ 1]
where D is the particle diameter; Db and Da are the upper and lower limits on D, respectively; n(D, t) is the size distribution density function at time t, such that n(D, OdD is the number concentration of particles having diameters in the range [D, D + dD];/3(D, D') is the coagulation coefficient for particles with diameter D and D'; I(D, t) is the rate of change of particle diameter, dD/dt, due to gas-to-particle conversion by vapor condensation; S0(D, t) is the rate of particle production from external sources, such that So(D, OdD is the rate of introduction per unit volume in the system of particles having diameters in the range [D,
/3[(D 3 - D'3) 1/3 , D']n[(D 3 - D'3) 1/3, t]n(D',
t)
( D 3 - D,3)1/3
× d D ' - n(D, t)
f?
~(D, D')n(D', ODD'.
[2]
a
There exist a number of studies of analytical (1, 8, 13) and numerical (2, 5, 7, 11) solutions of Eq. [2]. On the other hand, in the absence of coagulation, one obtains the condensation equation
On(D, t) _ at
O OD [I(D, t)n(D, t)] + So(D, t) - a(D, t)n(D, t)
[31
which may be solved by the method of characteristics (3, 12).
472 0021-9797/87 $3.00 Copyright© 1987by AcademicPress,Inc. All rightsof reproductionin any formreserved.
Journal of Colloidand InterfaceScience, Vol. 115,No. 2, February 1987
SOLUTION OF THE AEROSOL EQUATION In a number of important aerosol processes, such as formation and growth of aerosols in atmospheric photochemical reactions, the dominant physical processes influencing the aerosol size distribution are those represented in Eq. [3], namely condensation, sources, and removal, while coagulation exerts only a minor influence on the dynamics of the distribution. This observation suggests that it might be useful to seek a solution of the general dynamic equation, Eq. [1], for the case in which the effect of coagulation is small relative to those of the other phenomena represented in the equation, but not small enough to be neglected. Such a solution is the objective of the present work. DIMENSIONLESS GENERAL DYNAMIC EQUATION It is useful to cast Eq. [ 1] in an appropriate dimensionless form (3), 0ff(Kn, 3") Or
- NokTX 2
Dn /(Kn, r) = ~b/(D, t) Dt 7~V~ and Kn is the Knudsen number, 2X/D; T is the absolute temperature; 4~ is a growth coefficient depending on the growth mechanism, 4) = NlVlD/X; D is the diffusivity of the vapor molecules; X is the mean free path; a(r) is the coefficient accounting for any temporal variation of the growth rate; N~, Vl are the gas phase concentration and the volume of the condensing molecules, respectively; No is the initial total particle number concentration; and ~ = viscosity. The dimensionless Brownian coagulation coefficient is (10) fl(Kni, Kn:) 2/3Q(PiKni + P:Knj) Q t- 7rA(PiKni + PjKnj) Q + (HI + H}) 1/2 2Q(Kn 3 + Kn3) 1/2
a(r)_ O [Kn2/(Kn, r)ff(Kn, r)l OKn
f 1 fKn + ({2 JK-u ¢/tKn', (Kn -3
-
-
473
with
Kn'-3)-1/31
1
1
Q = K-~n~-~ Knj X rT(Kn', r)rT[(Kn -3 - Kn'-3) -1/3, r] X [ (Kn-3
Kn 3/2 I'/ 2
Kn'-3~-1/3]4 -K--~ " ] d K n ' - •(Kn, r)
APi ~3 4
× afKn" K n ' fl(Kn' )r~(Kn"K~ -
-(kn
r)dKn'}
So(Kn, r) - a(Kn, r)ff(Kn, r),
[41
with initial condition
g n/: j
[ 8 p k T 11/2 A = [271r2rtXj Pi = 1 +Kni[1.257 +0.4 exp(-1.1/Kni)].
r~(Kn, 0) = f ( K n ) , where ff(Kn, r) -
A2p213/2]_ - 2 Kni
Xn(D, t) NoKn 2
S'o(Kn, r) - So(D, t)X3 DNoKn 2 a(D, I)X2 a(Kn, r) - - D
[51
ASYMPTOTIC SOLUTION OF THE GENERAL DYNAMIC EQUATION FOR SMALL COAGULATION The coagulation terms in Eq. [4] are multiplied by the parameter ~. This parameter depends on the conditions in the system and the total particle concentration. If ~ is small, then the coagulation contribution to the evolution of ff(Kn, r) is correspondingly small. Journal of Colloid and Interface Science, Vol. 115,No. 2, February1987
474
PILINIS AND SEINFELD
In the case of small ~ we can express the solution of Eq. [4] in terms of a regular perturbation series (6),
f
--
i-1
Knafl(Kn, Kn')[ Z ni_p_l(Kn, r) 'dKnb
p=0
× tio(Kn', r)]dKn'
oo
ff(Kn, r) = Y. ff,-(Kn, r)(k
[6]
and where Kn" = (Kn -3 - Kn'-3) -1/3. We note that for each i, Gi(Kn, r) is a known function of Kn and r, since it depends only on lower orders of tii(Kn, r). As a result, the system of equations [7] and [8] can be solved sequentially for successively higher values of i by the method of characteristics. The characteristic equations for the system [7] and [8]
i=0
Substituting Eq. [6] into Eq. [4] and equating terms containing like powers of ( we obtain the following system of equations O& 0 _ 20-7 = ~(r) ~ [n0(Kn, r)Kn I(Kn)] - ~%(Kn, r) - a(Kn, r)tTo(Kn, r) OrTi
0
- a(r) ~nn
are
[71
-dKn -dr
2[~,(Kn, r)Kn I(Kn)]
a(r)KnZ[(Kn)
dffo = [ ~(r) d(Kn2/(Kn))
dr
+ G~(Kn, r) - a(Kn, r)tii(Kn, r) i = 1,2 . . . . .
[11]
dKn
a(Kn, r) ]
× ffo(Kn, r) - ~¢o(Kn, r)
[8]
[12]
[ 13]
subject to
subject to ffo(Kn, 0) = f ( K n ) tii(Kn,0):0,
tio(Kn, 0) = f ( K n )
[91
i = 1,2,3 . . . . ,
[10]
d(Kn2/(Kn))
dni
-d-(: [~(~) ~nn
where ! f Kn
[ 141
and
~(Kn,~)]
×ffi(Kn, r)+Gi(Kn, r)
/Kn"\ 4
Gi(Kn, r) = 2 dK~bfl(Kn',Kn")~-~ )
[15]
subject to
i-1
ffi(Kn,0)=0
X [ ~ tii_p_l(Kn", r)no(Kn', r)ldKn'
i = 1,2 . . . . .
[16]
Integrating Eqs. [13] and [15] along the characteristic curves of Eq. [ 12] we obtain
p=O
8o(Kn, r)
f (Kn(O))Kn2(O)[(Kn(O)) - H(r - ro)f i'~fo(Kn,r')Kn2[(Kn)exp[f i'a(Kn,r")dr "]dr' - -
7
]
and
fii(Kn, r) =
, Kn2/(Kn)exp[fo a(Kn, r')dr']
Journal of Colloid and Interface Science, Vol. 115, No. 2, February 1987
,
[18]
475
SOLUTION OF THE AEROSOL EQUATION
where ro is the time at which the particles are introduced into the system, and {;
H(x)=
f°r
x>0
for
x<0
TABLE I Estimates of Maximum Dimensionlessand Real Time Correspondingto InitialAerosolNumber Concentrations for Which AsymptoticSolution Is Expectedto be Valid
The zeroth-order solution, ffo(Kn, r), accounts only for the condensation, sources, and removal mechanisms in shaping the size distribution. The effect of coagulation appears then in the first order term ffl(Kn, r). As r increases, the effect of coagultion increases relative to those of the other principal processes, so the perturbation solution is expected to hold for values of r that satisfy the constraint,
~-f~[ fr~aB(Kn(r'),Kn')~o(Kn',r')dKn']dr' 2 dr0 [dKnb < 1.
[19]
For a smooth zeroth-order solution Eq. [19] can be reduced to max(fl(Kn, Kn'))
J~oN(r')dr'<1,
[19a]
where N(r) is the dimensionless total particle number concentration, N(r) = In the initial value problem N(r) < 1, since particles are consumed due to deposition and coagulation. Therefore Eq. [ 19a] reduces to
N(r)/No.
½~r max(/3(Kn, Kn')) < 1.
[19b]
For temperature T = 300 K and pressure p = 1 arm, in air, X ~ 0.065/zm and ~ ~ 1.8 × 10 -5 kg m -1 s-1. For a diffusion coefficient for typical condensible species in air, D ~ 10-5 m 2 s-1 and ~ is of order ~ -~ 10 -26 No, where No is in m -3. For particles in the range 0.110 u m the m a x i m u m value of the dimensionless coagulation coefficient is of the order max(/3(Kn, Kn')) ~ 102. Using these values for the various parameters and Eq. [ 19b], and upper limit of r for which the perturbation solution gives accurate results can be estimated. Table I shows rmax, as well as the corresponding tmax, in hours, for different values of N0. Physically, the reason for this behavior
No (m-3)
r~
t~ (h)
10 102 l04 106 10s 101o 1012
1023 1022 1020 1018 1016 1014 1012
101° 109 107 105 103 101 10-I
can be understood by considering a system in which condensation and coagulation are the only processes occurring. As time progresses, while condensation is primarily shaping the distribution, coagulation is slowly reducing the number of particles and hence the surface area available for condensation. Eventually a point is reached at which the effect of coagulation relative to that of condensation is no longer small. The smaller the value of ~, the longer the perturbation solution of Eq. [6] remains valid. EVOLUTION OF A MONODISPERSE AEROSOL BY GROWTH AND COAGULATION
A general expression for the dimensionless condensation growth law [ that incorporates diffusion-, surface reaction-, and volume reaction-limited growth is (12) [ = crKn~,
[20]
where 3' = 1 diffusion-control 3' = 0 surface reaction control 3" = - 1 volume reaction control and where a can be assumed independent of time r. W i t h / g i v e n by Eq. [20] the characteristic growth curves given by the solution of Eq. [ 12] are Journal of Colloid and Interface Science, Vol. 115,No. 2, February 1987
476
PILINIS
AND
Kn(~) [o-(3`+ 1)(r- ~o) =
+Kn0-o)-(~+l)] -1/¢~+1) 3 ` ~ - 1 Kn(ro)e -+-~°)
3,=-1,
[21]
SEINFELD
have Kn(ro) > 0, i.e., follow real characteristics. We now consider an initially monodisperse aerosol, ri(Kn, 0) = 6(Kn - Kn*), [23]
[22]
where Kn(ro) is the K n u d s e n n u m b e r of the particles at ~o. We note that in Eq. [21] only particles with the Knudsen n u m b e r satisfying
Kn(O < [o-(3`+ 1)(r- ~o)1-1/(~+1>
that undergoes growth and coagulation. Thus, particle sources and deposition losses are neglected. Using Eq. [17], for 3' 4: - 1, we obtain fio(Kn, ~-) = [ 1 - o-(-y+ 1)7-Kn~/+l] -('Y+z)/('y+I) × 6[(Kn -(~+1) - trey + l)r) -1/(~+1) - Kn*]. [24]
Since 1 ¢":~
[Kn"\ 4
61(Kn, ~) = 2 J K ¢~(Kn',Kn")[.-:---/~o(Kn', O~o(Kn",OdKn' \
nb
r~n /
f
Kna
/3(Kn, Kn')fio(Kn, r)fio(Kn', r)dKn',
- - dKnb
we obtain 21/3Kn) + 1,3
_
]-1/(3,+ 1) _ K i l l
/
+'
X H ( K n - 2-1/3[Kn *-(~+D + o-(T + 1)r] -1/(v+1)) - [1 - ~r(3, + 1)rKneY+l)] -(~+2)/(~'+1) ×
¢~(Kn(r'), [Kn *-(~+1) + ~(3' + 1)r]-l/(~'+D)dr'~[(Kn-(v+l) - o(3' -t- 1)r) -1/(~+1) - Kn*].
[251
Therefore, the dimensionless size distribution function is given to first order by
[fo
fi(Kn, r) = 1 - ~
/3(Kn0"), [Kn *-(v+l) + ~(3` + 1)'c]-l/(*+l)d'c'
]
× [1 - ~r(3`+ 1)'rKn('l+l)] -e~+2)/(7+1)X 6[(Kn -e~+l) - a(3` + 1)'r) -1/(~'+D - K n * ] ]-- 1/(3'+1)
× H ( K n - 2-1/3[Kn *-e~+l) + o-(3`+ 1)r]-1/(~'÷1)). If ~ = 0, Eq. [26] reduces to the solution for the size distribution function in the presence of pure growth. In that case, with the initial distribution given by Eq. [23] the aerosol remains monodisperse with time. Journal of Colloid and Interface Science, Vol. 115, No. 2, February 1987
[26]
The dimensionless total particle n u m b e r concentration, N(z) = N('r)/No, is given by =
fi(Kn, r ) d K n . OKn b
[27]
477
SOLUTION OF THE AEROSOL EQUATION
In general the integral in Eq. [27] m u s t be evaluated numerically. In the special case of fl(Kn', Kn") -= 2 C the integral can be evaluated analytically to give
10 NO COAGULATION ......
COAGULATION ~" = 10-4
2-= 103
0.8
e = 10-5 %. 0.6
N0-) = 1 - ~C~- + O(~2).
[28]
T h e exact result for the total n u m b e r in the case of a constant coagulation coefficient is 1
N0-) - - 1 + ~C~-"
i~ O4 O.2
~
~
_
~
[29] 2
Equation [28] is just the first two t e r m s in the Taylor series expansion o f Eq. [29]. EVOLUTION OF A LOG-NORMALLY DISTRIBUTED AEROSOL BY GROWTH, DEPOSITION, AND COAGULATION
3
4
5
Kn -t
FIG. 2. Normalized aerosol size distribution function at a dimensionless time of z = 1 0 3 in the cases of no coagulation and small coagulation (~ = 10-4). Deposition occurring at a dimensionless rate of a = 10-3. T w o cases will be considered:
We now consider the case of an initially logn o r m a l l y distributed aerosol, 1
r~(Kn, 0) =
(ii) Growth, coagulation, and deposition, with a = 10 -3.
2~-~ K n In % Xexp[
(i) G r o w t h and coagulation without deposition, i.e., a = 0.0;
ln2(Kng/Kn)-] 21nZrg j ,
[30]
with Kng = 0.65 and ~g = 1.4, evolving by growth, as described in Eq. [21] with 3' = 1, i.e., diffusion-limited growth, deposition with a = constant and coagulation with ~ = 10 -4.
It is c u s t o m a r y to plot the data so that the particle size increases along the x axis. Therefore plots of the dimensionless v o l u m e distribution ff(Kn -1, r) and dimensionless v o l u m e distribution l?(Kn -l, T) versus K n -1 will be presented. In t e r m of the previously defined distribution, fi(Kn -1, r) = - K n 2 t i ( K n , r)
I.O
47r I?(Kn -1, T) = -~- Kn-3ff(Kn -1, z).
NO COAGULATION ......
COAGIJLATION ~" = I0-4
0.8
T = 103 =
0.6
~K 0.4 0.2
,
2
5
4
I
5
Kn-I
FIG. 1. Normalized aerosol size distribution function at a dimensionless time of T = 103 in the cases of no coagulation and small coagulation (~ = 10-4). No deposition (~ = 0).
Figure 1 shows the n u m b e r distribution for r = 103, for Case (i) with a n d without coagulation. C o m p a r i s o n between the two curves in Fig. 1 shows that the n u m b e r o f small particles has been decreased, while the n u m b e r o f larger particles has been increased; c o m p a r e this to the case in which coagulation is neglected. Figure 2 shows exactly the same situation for Case (ii). C o m p a r i s o n between the two curves in Fig. 2 shows that the coagulation process is not as important as in Case (i). Since a substantial n u m b e r of small particles is cons u m e d due to deposition on surfaces, fewer Journal of Colloid and lnterface Science, Vol.115,No. 2, February1987
478
PILINIS
AND
can participate in the coagulation process, with the immediate result being a decrease in the importance of coagulation on the evolution of the number density function. We expect that the particles will shift to higher diameters due to the growth and coagulation processes. Figures 3 and 4 show this shift as a function of dimensionless time for Cases (i) and (ii), respectively. We find in Fig. 4 that a steady-state condition for large particles, Kn -1 > 3, is achieved. Thus the flux of particles to the large particle domain balances the loss from that domain due to deposition. The volume distribution for Case (i) is shown in Fig. 5. Figure 5 shows the development of a sharp peak, which is the result of the large number of small particles growing to an appreciable size. CONCLUSIONS A solution of the dimensionless aerosol general dynamic equation in the limit of small coagulation has been obtained. The expressions for the particle sources, removal, and growth mechanisms can be quite general. The case of initially monodisperse aerosol undergoing growth and coagulation was tested. Comparison between the solution via pertur-
SEINFELD
1.0 - =
10- 4
a = 10- 3
0,8
~o.o
/
../~.o
'~
T=5Xl0 2
T = I03
o.z j/J
- ~ : l . ~ , o~ = x 03
0
,
0
FIG.
2
4. Normalized
I
4 Kn -1 aerosol
,
,
,
,
6
size distribution
I
8
function
at
a series of times in the case of small coagulation (~ = 10-4).
Deposition occurring at a dimensionless rate o f a = 10 -3.
bation series and the analytical solution for the total particle number evolution shows that the series describes accurately the fate of the aerosol, the accuracy depending on the number of terms of the series evaluated. The evolution of a log-normally distributed aerosol by growth, deposition, and coagulation was also presented and in which our solution predicts the "expected" aerosol behavior for
60
1.0
= 10-4
= i0 -4
a:0
a=O 0.8 0
T : 2 x 103
T = 5 x 102
40 O.E
T = IO~
~" ~'~ i-d 0.4
=
2- = 1,5 x 103
-p
T = 1.5 × 103 x 103
T = I0 3
'~ I> 20
P = 5 x io 2
0.2
2
4
, , , I .... 6
I
8
Ka-1
FIG.3. Normalizedaerosolsizedistributionfunctionat
0
r,=
0
I 2
4 Kn-I
6
,
T
,
,
I
8
FIG.5. Normalizedaerosolvolumedistributionfunction
a series of times in the case of small coagulation (~
at a series of times in the case of small coagulation ( (
= 10-4). N o d e p o s i t i o n ( a = 0).
= 10-4). N o d e p o s i t i o n ( ~ = 0).
Journal of Colloid and InterfaceScience, VoL 115, No. 2, February 1987
SOLUTION OF THE AEROSOL EQUATION
the number and volume density functions. A steady-state phenomenon for large particles that is the consequence of the competition between growth and coagulation on the one hand, and deposition on the other, is demonstrated by the solution. The solution presented here can provide a framework for the modeling of situations in which coagulation is expected to exert only a minor influence on the aerosol size distribution. ACKNOWLEDGMENT This work was supported by National Science Foundation Grant ATM-8503103. REFERENCES 1. Drake, R. L., in "Topics in Current Aerosol Research," Part 2 (G. M. Hidy and J. R. Brock, Eds.), p. 203. Pergamon, Elmsford, NY, 1972. 2. Gelbard, F., and Seinfeld, J. H., J. Comp. Physiol. 28, 357 (1978).
479
3. Gelbard, F., and Seinfeld, J. H., J. Colloid Interface Sci. 68, 173 (1979). 4. Gelbard, F., and Seinfeld, J. H., J. Colloid Interface Sci. 68, 363 (1979). 5. Gelbard, F., Tambour, Y., and Seinfeld, J. H., J. Colloid Interface Sei. 76, 541 (1980). 6. Kevorkian, J., and Cole, J. T. "Perturbation Methods in Applied Mathematics," Springer-Verlag, New York/Berlin, 1981. 7. Middleton, P., and Brock, J., J. Colloid Interface Sci. 54, 249 (1976). 8. Peterson, T. W., Getbard, F., and Seinfeld, J. H., J. Colloid Interface Sci. 63, 426 (1978). 9. Ramabhadran, T. E., Peterson, T. W., and Seinfeld, J. H., AIChE J. 22, 840 (1976). 10. Seinfeld, J. H., "Atmospheric Chemistry and Physics of Air Pollution," Wiley, New York, 1986. 11. Warren, D. R., and Seinfeld, J. H., AerosolSci. Technol. 4, 31 (1985). 12. Williams, M. M. R., Y. Colloid Interface Sci. 93, 252 (1983). 13. Williams, M. M. R., J. Colloidlnterface Sci. 101, 19 (1984).
Journal of Colloid and Interface Science, V ol. 1 t 5, No. 2, February 1987