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Analytical solution to change in size distribution of polydisperse particles in closed chamber due to diffusion and sedimentation
Atmospheric Environment 36 (2002) 5459–5467
Analytical solution to change in size distribution of polydisperse particles in closed chamber due to diffusion and sedimentation S.H. Park, K.W. Lee* Department of Environmental Science and Engineering, Kwangju Institute of Science and Technology, 1 Oryong-dong, Buk-gu, Gwangju 500-712, South Korea Received 17 December 2001; received in revised form 16 May 2002; accepted 23 May 2002
Abstract Analytical solutions to the change in particle size distribution due to deposition in a closed chamber are given for different size ranges. The solutions correlate the size distribution of aerosol particles with a log-normal function. The derived solutions consist of three parameters of the log-normal size distribution. They are compared with the exact solution and the advantage of the new solutions is discussed. The monodisperse model for deposition problem is examined and is shown to cause significant errors except only in the intermediate size range where the size-dependency of the deposition rate is weak. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Deposition; Aerosol; Moment method; Indoor particles; Wall loss
1. Introduction Deposition loss of aerosol particles suspended in a closed chamber is of great importance in various applications including nuclear reactor safety analysis and indoor air pollution mitigation. For example, deposition onto indoor surfaces is an important removal mechanism for particulate indoor air pollutants such as tobacco smoke particles. In some cases where sizing of highly concentrated aerosol particles is required, dilution chambers are frequently used for substantial reduction of particle concentration before measurement. During dilution, however, particles are removed by deposition to the walls of the dilution chamber due to several mechanisms such as diffusion, thermophoresis, electrostatic forces, and gravitational settling, making the measured particle size distributions different from the true ones. *Corresponding author. Tel.: +82-629702438e2432; fax: +82-62-970-2434. E-mail address: [email protected] (K.W. Lee).
There have been several theoretical studies for the deposition rate of monodisperse suspended particles in a vessel (Corner and Pendlebury, 1951; Fuchs, 1964; Crump and Seinfeld, 1981). Theoretically the decay of the particle number concentration of monodisperse aerosols is exponential and a function of particle size and vessel dimensions. In practice, however, the aerosols are not monodisperse but polydisperse. The change in the particle size distribution of polydisperse particles due to deposition can be obtained by directly solving the deposition equation where the deposition coefficient is a function of particle size. This solution, however, is often not practical because it does not provide explicit information on the time change of the polydispersity, the mean particle size, or the total particle concentration. In the present study, an alternative solution is derived employing the moment method of a log-normal size distribution function. The deposition coefficient recently developed is used. The moment method is based on the assumption that the particle size distribution can be represented by a time-dependent log-normal distribution function. This
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Nomenclature
nðxÞ
constant appearing in Eq. (10) ðm2ðn1Þ=n s1 Þ area of side wall of chamber ðm2 Þ constant appearing in Eq. (10) ðm2 s1 Þ slip correction factor (dimensionless no:) ventilation rate ðs1 Þ Brownian diffusion coefficient of particles ðm2 s1 Þ De turbulent diffusion coefficient of particles ðm2 s1 Þ g gravity constant ðm s2 Þ H height of the chamber (m) kB Boltzmann constant ðkg m2 s2 K1 Þ ke constant related to the turbulent energy dissipation rate ðs1 Þ k unit vector in the vertical direction L characteristic size of the flow (m) L2 turbulent boundary layer thickness (m) Mk kth moment of particle size distribution ðmk3 Þ mk ðr; tÞ kth moment distribution density function ðmk4 Þ n constant appearing in Eq. (1) (dimensionless no:) N total number concentration of particles ðm3 Þ nðr; tÞ particle size distribution density function ðm4 Þ
r Re rg s S t T ut V x y z b l m r sg
A As B C cv D
method has been widely applied in simulation of aerosol dynamics (Clark, 1976; Lee et al., 1984; Whitby, 1985; Pratsinis, 1988) because it has the advantage of simplicity while providing important information on the change of size distribution of aerosol (Pratsinis et al., 1986). The solution derived in this study provides the three particle size distribution parameters of a log-normal distribution as explicit functions of time. It is compared with the exact solution and the advantage of the new solution is discussed.
2. Deposition coefficient In the absence of electrical and thermal forces, diffusion to surfaces and gravitational sedimentation are the most important deposition mechanisms. Theoretical evaluation of the deposition rate of aerosols due to Brownian and turbulent diffusion accompanying gravitational sedimentation in a rectangular chamber was developed by Corner and Pendlebury (1951). This work was extended by Crump and Seinfeld (1981) to evaluate the particle loss in an arbitrary shaped closed chamber.
unit outward vector normal to the chamber surface particle radius (m) Reynolds number (dimensionless no:) geometric mean particle radius (m) 2ðn1Þ=n rg ðm2ðn1Þ=n Þ total surface area of the chamber ðm2 Þ time (s) absolute temperature (K) terminal particle settling velocity ðm s1 Þ volume of the chamber ðm3 Þ distance from the wall (m) expð2 ln2 s2 g Þ (dimensionless no:) expf2ðn1Þ ln2 sg g (dimensionless no:) n2 deposition coefficient ðs1 Þ mean free path of gas molecules (m) gas viscosity ðkg m1 s1 Þ particle density ðkg m3 Þ geometric standard deviation of particle radius (dimensionless no:)
Subscript 0 refers to initial condition d diffusion-dominant size range g gravitation-dominant size range
According to the theory of Crump and Seinfeld (1981), the deposition coefficient for a chamber of arbitrary shape is represented by the following equation: bðrÞ ¼
1 V
Z
ut nðxÞ k dAðxÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ; ð1Þ n S expfput nðxÞ k=n sinðp=nÞ ke Dn1 g 1
where r is the particle radius, V is volume of the chamber, S denotes the surface of the chamber, ut is the terminal particle settling velocity, nðxÞ is the unit outward vector normal to the surface, k is the unit vector in the vertical direction, dAðxÞ is the differential area element on S; D is the Brownian diffusion coefficient, and ke is the coefficient of the turbulent diffusivity which can be evaluated from the turbulent energy dissipation rate (e.g. see Corner and Pendlebury, 1951). In the above equation, the turbulent diffusion coefficient was approximated to ke xn where x denotes distance from the wall and n is an experimentally fitted parameter. Crump and Seinfeld (1981) initially suggested that n ¼ 2 and it was validated by a later experiment (Crump et al., 1983). Many other researchers, however, have reported higher values ranging 2.5–2.8 (Okuyama et al.,
S.H. Park, K.W. Lee / Atmospheric Environment 36 (2002) 5459–5467
1986; Holub et al., 1988; Van Dingenen et al., 1989; Cheng, 1997; Park et al., 2001). This difference in the exponent can cause problems because ke is evaluated from the turbulent energy dissipation rate and hence always has a unit of s1 which is true only if n ¼ 2: In order to eliminate this problem, Benes and Holub (1996) proposed a new expression for the turbulent diffusion coefficient based on the rules of dimensional analysis as follows: n x De ¼ ke L22 ; ð2Þ L2 where L2 is the turbulent boundary layer thickness. Only an estimate of the order of magnitude is known for this quantity as the process of diffusion near the wall is not yet well understood (Fuchs, 1964). The upper bound for L2 is given by the size of the turbulent boundary layer (Schlichting, 1960): L L2 ¼ pffiffiffiffiffiffi; Re
ð3Þ
where L is some characteristic size of the flow and Re is the Reynolds number. In this study L is taken as 1 m similar to the choice made by Benes and Holub (1996) and Nomura et al. (1997). Repeating the asymptotical solution of Crump and Seinfeld (1981) with the turbulent diffusion coefficient of Eq. (2), the deposition coefficient is given for a chamber with a vertical side wall by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 nAs sinðp=nÞ n ke L2n 2 D bðrÞ ¼ 0pV 1 ut put B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC þ coth@ A; H 2n sinðp=nÞ n k L2n Dn1 e
ð4Þ
2
where As is the side wall area of the chamber and H is the height of the chamber. The b in Eq. (4) is a highly complicated nonlinear function of particle size, and this makes it impossible to apply the moment method to the deposition problem. Therefore, we need some approximations for Eq. (4) to apply the moment method. Park and Lee (2000) used the property of the hyperbolic trigonometric function to modify the deposition coefficient into a simpler form. Repeating that procedure, one can derive the following deposition coefficient accounting for turbulent and Brownian diffusion and gravitational sedimentation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nS sinðp=nÞ n ke L22n Dn1 ut b¼ þ ; ð5Þ pV H where S is the total surface area of the chamber. In Eq. (5), D and ut depend on the particle size as follows: D¼
kB TC ; 6pmr
ð6Þ
ut ¼
5461
2r2 rgC ; 9m
ð7Þ
where kB is the Boltzmann constant, T is the absolute temperature, m is the gas viscosity, r is the particle density, g is the gravity constant, and C is the slip correction factor which is represented as (Allen and Raabe, 1985):
l 0:999r C ¼ 1 þ 1:142 þ 0:558 exp ; ð8Þ r l where l is the mean-free path length of the gas molecules. Eq. (5) combined with Eqs. (6)–(8) still has quite complicated functional form making the application of the moment method difficult. Noting that, however, the first term on the right-hand side of Eq. (5) is dominant for small particles and the second term is for large particles, we can apply more approximations for the slip correction factor in Eq. (8). For a particle with its radius much smaller than l; we use the following approximation: 1:7l CD for r5l: r
ð9Þ
On the other hand, for a particle with its radius much larger than l; the slip correction factor can be neglected. Applying these two approximations to the first and second terms of Eq. (5), respectively, Eq. (5) is rewritten as the following: bðrÞ ¼ bd ðrÞ þ bg ðrÞ ¼ A r2ðn1Þ=n þ B r2 ;
ð10Þ
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 n 2n 1:7lkB T ðn1Þ=n @nS sinðp=nÞ ke L2 A A¼ 6pm pV B¼
and
2rg : 9mH
The subscripts d and g in this equation refer to the diffusion term and the gravitation term, respectively. Eq. (10) has appropriate functional form for the moment method. Fig. 1 shows the comparison of Eq. (10) with the more accurate deposition coefficient represented by Eq. (4) for n ¼ 2:6: A rectangular chamber with the dimension of 3 m 5 m 2:5 m and ke ¼ 36 s1 were arbitrarily used for the calculation. It is shown from this figure that Eq. (10) is a reasonably good approximation of Eq. (4). Thus, Eq. (10) will be used in this study as an approximate deposition coefficient of particles in a closed chamber to derive new analytical solutions. Eq. (4) is utilized for the calculation of the exact solution that is used for the validation of the new solutions.
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where k is an arbitrary real number. Among the moments, M0 represents the total number concentration of particles ð¼ NÞ: We will develop analytical solutions to aerosol deposition problem for gravitation-dominant size range first and then for diffusion-dominant size range.
-2
10
-3
accurate coefficient: Eq. (4)
10
(s-1)
approximate coefficient: Eq. (10) -4
10
3.1. Gravitation-dominant size range
-5
10
-6
10
0.01
0.1
1
10
r (µm) Fig. 1. Comparison of the approximate deposition coefficient as a function of particle size with a more accurate coefficient.
3. Derivation of analytical solutions and discussion The change in the size distribution of particles by simultaneous Brownian and turbulent diffusion and gravitational sedimentation is represented by the following equation: qnðr; tÞ ¼ fbðrÞ þ cv gnðr; tÞ; qt
ð11Þ
where nðr; tÞ is the particle size distribution function at time t; bðrÞ is the deposition coefficient of particles with radius r; and cv is the ventilation rate which is assumed to be independent of particle size. From the physical insight, ventilation does not affect the change in the mean particle size or the particle size distribution width. Only the total concentration is decreased due to ventilation by a factor of expðcv tÞ: Therefore, ventilation is not accounted for any longer in this study, but only deposition is considered. In the moment method, the particle size distribution is represented with a time-dependent log-normal function. In general this approach has been widely employed for representing the size distribution of particles both experimentally and theoretically. The log-normal size distribution function for particles whose radius is r is written as " # 1 NðtÞ ln2 fr=rg ðtÞg nðr; tÞ ¼ pffiffiffiffiffiffi exp ; ð12Þ r 2p ln sg ðtÞ 2 ln2 sg ðtÞ where NðtÞ is the total number concentration of particles, rg ðtÞ is the geometric number mean particle radius, and sg ðtÞ is the geometric standard deviation of particle radius. The kth moment of size distribution function nðr; tÞ is defined as Z N Mk ¼ rk nðr; tÞ dr; ð13Þ 0
For relatively large particles, the diffusion term in Eq. (10) may be neglected. By substituting Eq. (10), with neglect of the first term on the right-hand side, into Eq. (11), multiplying the equation by rk ; and integrating from 0 to N; one can obtain the following equation: dMk ¼ BMkþ2 : ð14Þ dt The properties of a log-normal distribution function are such that the following equation holds for any kth moment: 2 k 2 k ln sg : ð15Þ Mk ¼ Nrg exp 2 Substituting Eq. (15) into Eq. (14) and rewriting the equation for k ¼ 0; 2; and 2, we have dN ¼ BNr2g y; dt
ð16Þ
dðNr2 g yÞ ¼ BN; dt
ð17Þ
dðNr2g yÞ ¼ BNr4g y4 ; ð18Þ dt where y ¼ expð2 ln2 sg Þ: Substituting Eq. (16) into Eqs. (17) and (18), we have 2r3 g y
drg dy ¼ Bðy2 1Þ; þ r2 g dt dt
ð19Þ
drg dy ¼ By2 ðy2 1Þ: þ r2 ð20Þ g dt dt Rewriting Eq. (19) and (20) in terms of rg and y; we have 2r3 g y
Br3g ðy4 1Þ drg ; ¼ 4y dt
ð21Þ
Br2g ðy2 1Þ2 dy : ð22Þ ¼ 2 dt Eliminating dt from Eqs. (21) and (22) and integrating the resulting equation, the following equation is obtained: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rg y0 ðy2 1Þ ; ð23Þ ¼ rg0 yðy20 1Þ where rg0 and y0 are the initial values of rg and y; respectively. Substituting Eq. (23) into Eq. (22) and
S.H. Park, K.W. Lee / Atmospheric Environment 36 (2002) 5459–5467
3 mmÞ; is also shown in this figure. The monodisperse model is shown to fail to predict the number concentration decay. Initially, the monodisperse model underestimates the deposition rate due to the neglect of the polydispersity effect. As time elapses, the mean size of polydisperse particles decreases owing to faster settling of larger particles and it results in decrease of the mean settling rate. The monodisperse model which cannot reflect this change, however, overestimates the particle loss rate. Fig. 2 dramatically demonstrates that the monodisperse model should not used to predict the particle concentration change in the gravitation-dominant range. Fig. 3 shows that the whole particle size distribution is also predicted well by the derived solution.
integrating the resulting equation, one can obtain the solution for y: y20 1 ffi: y2 1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2Br2g0 y0 ðy20 1Þt
5463
ð24Þ
Substituting Eqs. (23) and (24) into Eq. (16) and integrating the resulting equation, one can obtain the following equation for the number concentration decay: 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 þ 2Br2g0 y0 ðy20 1Þt 1 N 5; ð25Þ ¼ exp4 N0 y20 1 where N0 is the initial value of N: Eqs. (23)–(25) represent the solutions to the three size distribution parameters of log-normal size distribution as a function of the initial conditions and time t: From the obtained solution, one can construct the whole particle size distribution using Eq. (12). Thus, the analytical solution to the deposition problem of polydisperse particles in the gravitation-dominant range has just been obtained. Eq. (25) gives an explicit expression for the total concentration decay of polydisperse particles which cannot be provided by the exact solution. This is the important advantage of the solution derived in this study. For y0 ¼ 1 ðsg0 ¼ 1Þ; Eq. (25) simply reduces to N=N0 ¼ expðBr2 tÞ which is the solution for the monodisperse particles. Fig. 2 compares the results calculated for the total particle concentration, the geometric mean radius, and the geometric standard deviation with the exact solution. For the initial condition, rg0 ¼ 3 mm and sg0 ¼ 1:5 were used. It is demonstrated in this figure that the particle size distribution parameters can be predicted well by the analytical solution derived in this study. Comparison with the monodisperse model, where every particle is assumed to have the same size as the geometric mean size of a polydisperse aerosol ðr ¼
3.2. Diffusion-dominant size range For relatively small particles, the gravitation term in Eq. (10) may be neglected. By substituting Eq. (10), with neglect of the second term on the right-hand side, into Eq. (11), multiplying the resulting equation by rk ; and integrating from 0 to N; one can obtain the following equation: dMk ¼ AMk2ðn1Þ=n : dt
ð26Þ
Substituting Eq. (15) into Eq. (26) and rewriting the equation for k ¼ 0; 2ðn 1Þ=n; and 2ðn 1Þ=n; the following three equations are obtained: dN ¼ ANs1 z; dt
ð27Þ
dðNs1 zÞ ¼ ANs2 z4 ; dt
ð28Þ
dðNszÞ ¼ AN; dt
ð29Þ 2
2ðn1Þ=n
where s ¼ rg and z ¼ expf2ðn1Þ ln2 sg g: A repetin2 tion of the previous process for this case gives the
1
1.0
1.5
exact solution this study monodisperse model
σg
1.4
0.8
σg
0.6
rg /rg0
N/N0
1.3 0.1
1.2 rg /rg0
0.4
1.1
N/N0
0.01 0
5000
10000
15000
0.2 20000
1.0
t (s)
Fig. 2. Comparison of changes in particle size distribution parameters in the gravitation-dominant range (rg0 ¼ 3 mm; sg0 ¼ 1:5).
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1.2 exact solution this study
1.0
t=0s
0.8
n(r)r/N0
700 s 0.6 2000 s
0.4
5000 s
0.2
0.0 0.1
1
10
r/rg0 Fig. 3. Comparison of change in particle size distribution in the gravitation-dominant range. Initial values are the same as those in Fig. 2.
1 exact solution this study monodisperse model
1.5
1.8
1.4
1.6
1.3
0.1
rg/rg0
10000
20000
1.4
1.2
1.2
1.1
1.0 70000
1.0
N/N0
0.01 0
σg
rg /rg0
N/N0
σg
2.0
30000
40000
50000
60000
t (s) Fig. 4. Comparison of changes in particle size distribution parameters in the diffusion-dominant range (rg0 ¼ 0:01 mm; sg0 ¼ 1:5).
following solutions: s zðz20 1Þ ; ¼ s0 z0 ðz2 1Þ
ð30Þ
z20 1 ; z2 1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1Þt z ðz 1 þ 2As1 0 0 0
ð31Þ
2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 1 þ 2As1 N 0 z0 ðz0 1Þt 1 5; ¼ exp4 N0 z20 1
ð32Þ
where s0 and z0 are the initial values of s and z; respectively. Eqs. (30)–(32) constitute the solution for the diffusion-dominant size range. Figs. 4 and 5 compares the results calculated for the size distribution parameters and the whole particle size distribution, respectively, with the exact solution for the diffusion-dominant range. For the initial condition, rg0 ¼ 0:01 mm and sg0 ¼ 1:5 were used. These figures show that the change in particle size distribution in the diffusion-dominant range can also be predicted well by the analytical solution derived in this study. It is noted that the monodisperse model ought not to be used for predicting the particle concentration change of polydisperse particles in the diffusion-dominant range either.
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1.2 exact solution this study
1.0
t=0s
n(r)r/N0
0.8 4000 s 0.6 10000 s
0.4
23000 s
0.2
0.0 0.1
1
10
r/rg0 Fig. 5. Comparison of change in particle size distribution in the diffusion-dominant range. Initial values are the same as those in Fig. 4.
3.3. Intermediate size range For the intermediate size range, both terms in Eq. (10) should be accounted for during the derivation of the solution. Inclusion of both terms, however, makes it impossible to derive an analytical solution in a similar way with those used in the limiting size ranges. Therefore, we need here an appropriate approximation. First, let us define the kth moment distribution function as follows:
Eq. (36) is exact for monodisperse aerosols but it may cause some error for polydisperse aerosols. This error always results in underestimation of moment depletion. This will be discussed again later in this section. If Eq. (15) is written for k ¼ 0; 1; and 2, and subsequently solved for rg and sg in terms of M0 ; M1 ; and M2 ; we have M0 M2 ; ð37Þ ln2 sg ¼ ln M12
mk ðr; tÞ ¼ rk nðr; tÞ:
rg ¼
ð33Þ
The change of the moment distribution by deposition is then represented by the following equation from Eq. (11): qmk ðr; tÞ ¼ bðrÞmk ðr; tÞ: qt
ð34Þ
mk ðr; tÞ ¼ exp½fbd ðrÞ þ bg ðrÞgt : mk ðr; 0Þ
ð35Þ
From Eq. (35), and using an additional approximation, one can obtain the kth moment as a function of time as follows: RN mk ðr; 0Þexp½fbd ðrÞ þ bg ðrÞgt dr Mk ðtÞ RN ¼ 0 Mk ð0Þ 0 mk ðr; 0Þ dr RN RN m ðr; 0Þexpfb k d ðrÞtg dr 0 mk ðr; 0Þ expfbg ðrÞtg dr RN D 0 ½ 0 mk ðr; 0Þ dr 2 Mk ðtÞ Mk ðtÞ ¼ : ð36Þ Mk ð0Þd Mk ð0Þg
3=2
1=2
:
ð38Þ
M0 M2
From Eq. (36)–(38), the following three equations are easily derived: N N N ¼ ; ð39Þ N N N 0
Integrating Eq. (34), we have
M12
0 d
0 g
rg rg rg ¼ ; rg0 rg0 d rg0 g
ð40Þ
expðln2 sg Þ expðln2 sg Þ expðln2 sg Þ ¼ : expðln2 sg0 Þ expðln2 sg0 Þd expðln2 sg0 Þg
ð41Þ
Fig. 6 compares the results calculated for the three size distribution parameters for the intermediate size range with the exact solution. For the initial condition, rg0 ¼ 0:1 mm and sg0 ¼ 1:5 were used. In this case, the geometric mean radius and the geometric standard deviation are predicted well by the derived solution whereas the number concentration is poorly estimated. The large error for the number concentration stems
S.H. Park, K.W. Lee / Atmospheric Environment 36 (2002) 5459–5467 1 exact solution this study monodisperse model
0.1
1.5
1.4
1.4
1.3
1.3
rg /rg0
N/N0
σg
1.5
N/N0
σg
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1.2
1.2
1.1
1.1
1.0 5e+5
1.0
rg /rg0
0.01 0
1e+5
2e+5
3e+5
4e+5
t (s)
Fig. 6. Comparison of changes in particle size distribution parameters in the intermediate particle size range (rg0 ¼ 0:1 mm; sg0 ¼ 1:5).
1.2 exact solution this study
1.0
t=0s
σg0 = 1.5
n(r)r/N0
0.8
rg0 = 0.1 µm
4 5*10 s
0.6
5
10 s
0.4 5
2.3*10 s 0.2
0.0 0.1
1
10
r/rg0 Fig. 7. Comparison of change in particle size distribution in the intermediate particle size range. Initial values are the same as those in Fig. 6.
partially from the underestimation of the deposition rate in this size range as was shown in Fig. 1 and partially from the error due to the approximation used in Eq. (36). Note that the second type of error seldom affected sg and rg because the errors for each moment compensated one another in Eqs. (37) and (38). Another interesting point noted in this figure is that the monodisperse model predicts the particle concentration decay very accurately in the intermediate size range. It is because in this size range the change in the mean-particle size is relatively small and the size-dependency of the deposition rate is weak compared to that for the gravitation- or diffusion-dominant size range. Therefore, it is recommended to use the monodisperse model to predict the particle concentration change rather than the analytical solution derived in this study for the intermediate particle size range.
Fig. 7 shows a comparison of the particle size distribution change between the exact solution and the analytical solution derived in this study. For the number concentration decay, the monodisperse model was used. This figure shows that the particle size distribution can be predicted well by the derived solution associated with the monodisperse model in the intermediate size range. As was mentioned earlier, the analytical solution derived in this study is exact for monodisperse aerosols, but some error exists for polydisperse aerosols. This error mainly stems from the assumption of log-normal size distribution assumption. Therefore, the magnitude of the error depends on how much the size distribution deviates from the log-normal shape, and it is expected to increase with increasing geometric standard deviation sg : We have examined the accuracy of the solution
S.H. Park, K.W. Lee / Atmospheric Environment 36 (2002) 5459–5467
derived in this study with respect to the value of sg upto 1.8. As is shown in Figs. 3, 5 and 7, the particle size distribution shape was always almost log-normal, so the error of the derived solution was not so sensitive to sg :
4. Conclusions In this study, size distribution change of polydisperse indoor particles by gravitational sedimentation and turbulent and Brownian diffusion were investigated. Based on the assumption that the particle size distribution is represented by a time-dependent log-normal distribution, analytical solutions for the gravitation- and the diffusion-dominant size ranges were derived. The solutions were compared with the exact solution and good agreement was obtained. These new solutions have the advantage that the particle size distribution parameters are expressed as explicit functions of time. By the comparisons with the polydisperse models, it was shown that the monodisperse model should be used to predict the particle concentration decay of polydisperse particles neither for the gravitation- nor for the diffusiondominant size range. For the intermediate particle size range where neither the diffusion nor the sedimentation can be neglected, two limiting-size-range solutions were mechanistically combined. This type of approximation, however, was shown to cause severe error for the number concentration change. On the other hand, the monodisperse model gives very accurate prediction in this size range. Therefore, the analytical solution derived in this study is recommended to be used for the geometric mean particle size and for the geometric standard deviation while the monodisperse model is for the particle concentration in the intermediate particle size range.
Acknowledgements This work was supported by the Brain Korea 21 program from the Ministry of Education through the Graduate Program for Chemical and Environmental Engineering at Kwangju Institute of Science and Technology.
References Allen, M.D., Raabe, O.G., 1985. Slip correction measurements of spherical solid aerosol particles in an improved Millikan apparatus. Aerosol Science and Technology 4, 269–286.
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Benes, M., Holub, R.F., 1996. Aerosol wall deposition in enclosures investigated by means of a stagnant layer. Environment International 22, S883–S889. Cheng, Y.S., 1997. Wall deposition of radon progeny and particles in a spherical chamber. Aerosol Science and Technology 27, 131–146. Clark, T.L., 1976. Use of log-normal distributions for numerical calculations of condensation and collection. Journal of Atmospheric Science 33, 810–821. Corner, J., Pendlebury, E.D., 1951. The coagulation and deposition of a stirred aerosol. Proceedings of the Physics Society B 64, 645–654. Crump, J.G., Seinfeld, J.H., 1981. Turbulent deposition and gravitational sedimentation of an aerosol in a vessel of arbitrary shape. Journal of Aerosol Science 12, 405–415. Crump, J.G., Flagan, R.C., Seinfeld, J.H., 1983. Particle wall loss rates in vessels. Aerosol Science and Technology 2, 303–309. Fuchs, N.A., 1964. The Mechanics of Aerosols. Pergamon Press, New York. Holub, R.F., Raes, F., Van Dingenen, R., Vanmarcke, H., 1988. Deposition of aerosols and unattached radon daughters in different chambers: theory and experiment. Radiation Protection Dosimetry 24, 217–220. Lee, K.W., Chen, H., Gieseke, J.A., 1984. Log-normally preserving size distribution for Brownian coagulation in the free-molecule regime. Aerosol Science and Technology 3, 53–62. Nomura, Y., Hopke, P.K., Fitzgerald, B., Mesbah, B., 1997. Deposition of particles in a chamber as a function of ventilation rate. Aerosol Science and Technology 27, 62–72. Okuyama, K., Kousaka, Y., Yamamoto, S., Hosokawa, T., 1986. Particle loss of aerosols with particle diameters between 6 and 2000 nm in stirred tank. Journal of Colloids and Interface Science 110, 214–223. Park, S.H., Lee, K.W., 2000. Lognormal size distribution theory for deposition of polydisperse aerosol particles. Nuclear Science Engineering 135, 288–295. Park, S.H., Kim, H.O., Han, Y.T., Kwon, S.B., Lee, K.W., 2001. Wall loss rate of polydispersed aerosols. Aerosol Science and Technology 35, 710–717. Pratsinis, S.E., 1988. Simultaneous nucleation, condensation, and coagulation in aerosol reactors. Journal of Colloids and Interface Science 124, 416–427. Pratsinis, S.E., Kodas, T.T., Dudukovic, M.P., Friedlander, S.K., 1986. Aerosol reactor design: effect of reactor type and process parameters on product aerosol characteristics. Industrial and Engineering Chemistry, Process Design and Development 25, 634–642. Schlichting, H., 1960. Boundary Layer Theory. McGraw-Hill, New York. Van Dingenen, R., Raes, F., Vanmarcke, H., 1989. Molecule and aerosol particle wall losses in smog chambers made of glass. Journal of Aerosol Science 20, 113–122. Whitby, E.R., 1985. The modal aerosol dynamics model. Part I. Report to the U.S. Environmental Protection Agency, Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN.
Analytical solution to change in size distribution of polydisperse particles in closed chamber due to diffusion and sedimentation S.H. Park, K.W. Lee* Department of Environmental Science and Engineering, Kwangju Institute of Science and Technology, 1 Oryong-dong, Buk-gu, Gwangju 500-712, South Korea Received 17 December 2001; received in revised form 16 May 2002; accepted 23 May 2002
Abstract Analytical solutions to the change in particle size distribution due to deposition in a closed chamber are given for different size ranges. The solutions correlate the size distribution of aerosol particles with a log-normal function. The derived solutions consist of three parameters of the log-normal size distribution. They are compared with the exact solution and the advantage of the new solutions is discussed. The monodisperse model for deposition problem is examined and is shown to cause significant errors except only in the intermediate size range where the size-dependency of the deposition rate is weak. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Deposition; Aerosol; Moment method; Indoor particles; Wall loss
1. Introduction Deposition loss of aerosol particles suspended in a closed chamber is of great importance in various applications including nuclear reactor safety analysis and indoor air pollution mitigation. For example, deposition onto indoor surfaces is an important removal mechanism for particulate indoor air pollutants such as tobacco smoke particles. In some cases where sizing of highly concentrated aerosol particles is required, dilution chambers are frequently used for substantial reduction of particle concentration before measurement. During dilution, however, particles are removed by deposition to the walls of the dilution chamber due to several mechanisms such as diffusion, thermophoresis, electrostatic forces, and gravitational settling, making the measured particle size distributions different from the true ones. *Corresponding author. Tel.: +82-629702438e2432; fax: +82-62-970-2434. E-mail address: [email protected] (K.W. Lee).
There have been several theoretical studies for the deposition rate of monodisperse suspended particles in a vessel (Corner and Pendlebury, 1951; Fuchs, 1964; Crump and Seinfeld, 1981). Theoretically the decay of the particle number concentration of monodisperse aerosols is exponential and a function of particle size and vessel dimensions. In practice, however, the aerosols are not monodisperse but polydisperse. The change in the particle size distribution of polydisperse particles due to deposition can be obtained by directly solving the deposition equation where the deposition coefficient is a function of particle size. This solution, however, is often not practical because it does not provide explicit information on the time change of the polydispersity, the mean particle size, or the total particle concentration. In the present study, an alternative solution is derived employing the moment method of a log-normal size distribution function. The deposition coefficient recently developed is used. The moment method is based on the assumption that the particle size distribution can be represented by a time-dependent log-normal distribution function. This
1352-2310/02/$ - see front matter r 2002 Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 2 - 2 3 1 0 ( 0 2 ) 0 0 6 7 3 - 8
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Nomenclature
nðxÞ
constant appearing in Eq. (10) ðm2ðn1Þ=n s1 Þ area of side wall of chamber ðm2 Þ constant appearing in Eq. (10) ðm2 s1 Þ slip correction factor (dimensionless no:) ventilation rate ðs1 Þ Brownian diffusion coefficient of particles ðm2 s1 Þ De turbulent diffusion coefficient of particles ðm2 s1 Þ g gravity constant ðm s2 Þ H height of the chamber (m) kB Boltzmann constant ðkg m2 s2 K1 Þ ke constant related to the turbulent energy dissipation rate ðs1 Þ k unit vector in the vertical direction L characteristic size of the flow (m) L2 turbulent boundary layer thickness (m) Mk kth moment of particle size distribution ðmk3 Þ mk ðr; tÞ kth moment distribution density function ðmk4 Þ n constant appearing in Eq. (1) (dimensionless no:) N total number concentration of particles ðm3 Þ nðr; tÞ particle size distribution density function ðm4 Þ
r Re rg s S t T ut V x y z b l m r sg
A As B C cv D
method has been widely applied in simulation of aerosol dynamics (Clark, 1976; Lee et al., 1984; Whitby, 1985; Pratsinis, 1988) because it has the advantage of simplicity while providing important information on the change of size distribution of aerosol (Pratsinis et al., 1986). The solution derived in this study provides the three particle size distribution parameters of a log-normal distribution as explicit functions of time. It is compared with the exact solution and the advantage of the new solution is discussed.
2. Deposition coefficient In the absence of electrical and thermal forces, diffusion to surfaces and gravitational sedimentation are the most important deposition mechanisms. Theoretical evaluation of the deposition rate of aerosols due to Brownian and turbulent diffusion accompanying gravitational sedimentation in a rectangular chamber was developed by Corner and Pendlebury (1951). This work was extended by Crump and Seinfeld (1981) to evaluate the particle loss in an arbitrary shaped closed chamber.
unit outward vector normal to the chamber surface particle radius (m) Reynolds number (dimensionless no:) geometric mean particle radius (m) 2ðn1Þ=n rg ðm2ðn1Þ=n Þ total surface area of the chamber ðm2 Þ time (s) absolute temperature (K) terminal particle settling velocity ðm s1 Þ volume of the chamber ðm3 Þ distance from the wall (m) expð2 ln2 s2 g Þ (dimensionless no:) expf2ðn1Þ ln2 sg g (dimensionless no:) n2 deposition coefficient ðs1 Þ mean free path of gas molecules (m) gas viscosity ðkg m1 s1 Þ particle density ðkg m3 Þ geometric standard deviation of particle radius (dimensionless no:)
Subscript 0 refers to initial condition d diffusion-dominant size range g gravitation-dominant size range
According to the theory of Crump and Seinfeld (1981), the deposition coefficient for a chamber of arbitrary shape is represented by the following equation: bðrÞ ¼
1 V
Z
ut nðxÞ k dAðxÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ; ð1Þ n S expfput nðxÞ k=n sinðp=nÞ ke Dn1 g 1
where r is the particle radius, V is volume of the chamber, S denotes the surface of the chamber, ut is the terminal particle settling velocity, nðxÞ is the unit outward vector normal to the surface, k is the unit vector in the vertical direction, dAðxÞ is the differential area element on S; D is the Brownian diffusion coefficient, and ke is the coefficient of the turbulent diffusivity which can be evaluated from the turbulent energy dissipation rate (e.g. see Corner and Pendlebury, 1951). In the above equation, the turbulent diffusion coefficient was approximated to ke xn where x denotes distance from the wall and n is an experimentally fitted parameter. Crump and Seinfeld (1981) initially suggested that n ¼ 2 and it was validated by a later experiment (Crump et al., 1983). Many other researchers, however, have reported higher values ranging 2.5–2.8 (Okuyama et al.,
S.H. Park, K.W. Lee / Atmospheric Environment 36 (2002) 5459–5467
1986; Holub et al., 1988; Van Dingenen et al., 1989; Cheng, 1997; Park et al., 2001). This difference in the exponent can cause problems because ke is evaluated from the turbulent energy dissipation rate and hence always has a unit of s1 which is true only if n ¼ 2: In order to eliminate this problem, Benes and Holub (1996) proposed a new expression for the turbulent diffusion coefficient based on the rules of dimensional analysis as follows: n x De ¼ ke L22 ; ð2Þ L2 where L2 is the turbulent boundary layer thickness. Only an estimate of the order of magnitude is known for this quantity as the process of diffusion near the wall is not yet well understood (Fuchs, 1964). The upper bound for L2 is given by the size of the turbulent boundary layer (Schlichting, 1960): L L2 ¼ pffiffiffiffiffiffi; Re
ð3Þ
where L is some characteristic size of the flow and Re is the Reynolds number. In this study L is taken as 1 m similar to the choice made by Benes and Holub (1996) and Nomura et al. (1997). Repeating the asymptotical solution of Crump and Seinfeld (1981) with the turbulent diffusion coefficient of Eq. (2), the deposition coefficient is given for a chamber with a vertical side wall by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 nAs sinðp=nÞ n ke L2n 2 D bðrÞ ¼ 0pV 1 ut put B qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC þ coth@ A; H 2n sinðp=nÞ n k L2n Dn1 e
ð4Þ
2
where As is the side wall area of the chamber and H is the height of the chamber. The b in Eq. (4) is a highly complicated nonlinear function of particle size, and this makes it impossible to apply the moment method to the deposition problem. Therefore, we need some approximations for Eq. (4) to apply the moment method. Park and Lee (2000) used the property of the hyperbolic trigonometric function to modify the deposition coefficient into a simpler form. Repeating that procedure, one can derive the following deposition coefficient accounting for turbulent and Brownian diffusion and gravitational sedimentation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nS sinðp=nÞ n ke L22n Dn1 ut b¼ þ ; ð5Þ pV H where S is the total surface area of the chamber. In Eq. (5), D and ut depend on the particle size as follows: D¼
kB TC ; 6pmr
ð6Þ
ut ¼
5461
2r2 rgC ; 9m
ð7Þ
where kB is the Boltzmann constant, T is the absolute temperature, m is the gas viscosity, r is the particle density, g is the gravity constant, and C is the slip correction factor which is represented as (Allen and Raabe, 1985):
l 0:999r C ¼ 1 þ 1:142 þ 0:558 exp ; ð8Þ r l where l is the mean-free path length of the gas molecules. Eq. (5) combined with Eqs. (6)–(8) still has quite complicated functional form making the application of the moment method difficult. Noting that, however, the first term on the right-hand side of Eq. (5) is dominant for small particles and the second term is for large particles, we can apply more approximations for the slip correction factor in Eq. (8). For a particle with its radius much smaller than l; we use the following approximation: 1:7l CD for r5l: r
ð9Þ
On the other hand, for a particle with its radius much larger than l; the slip correction factor can be neglected. Applying these two approximations to the first and second terms of Eq. (5), respectively, Eq. (5) is rewritten as the following: bðrÞ ¼ bd ðrÞ þ bg ðrÞ ¼ A r2ðn1Þ=n þ B r2 ;
ð10Þ
where qffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 n 2n 1:7lkB T ðn1Þ=n @nS sinðp=nÞ ke L2 A A¼ 6pm pV B¼
and
2rg : 9mH
The subscripts d and g in this equation refer to the diffusion term and the gravitation term, respectively. Eq. (10) has appropriate functional form for the moment method. Fig. 1 shows the comparison of Eq. (10) with the more accurate deposition coefficient represented by Eq. (4) for n ¼ 2:6: A rectangular chamber with the dimension of 3 m 5 m 2:5 m and ke ¼ 36 s1 were arbitrarily used for the calculation. It is shown from this figure that Eq. (10) is a reasonably good approximation of Eq. (4). Thus, Eq. (10) will be used in this study as an approximate deposition coefficient of particles in a closed chamber to derive new analytical solutions. Eq. (4) is utilized for the calculation of the exact solution that is used for the validation of the new solutions.
S.H. Park, K.W. Lee / Atmospheric Environment 36 (2002) 5459–5467
5462
where k is an arbitrary real number. Among the moments, M0 represents the total number concentration of particles ð¼ NÞ: We will develop analytical solutions to aerosol deposition problem for gravitation-dominant size range first and then for diffusion-dominant size range.
-2
10
-3
accurate coefficient: Eq. (4)
10
(s-1)
approximate coefficient: Eq. (10) -4
10
3.1. Gravitation-dominant size range
-5
10
-6
10
0.01
0.1
1
10
r (µm) Fig. 1. Comparison of the approximate deposition coefficient as a function of particle size with a more accurate coefficient.
3. Derivation of analytical solutions and discussion The change in the size distribution of particles by simultaneous Brownian and turbulent diffusion and gravitational sedimentation is represented by the following equation: qnðr; tÞ ¼ fbðrÞ þ cv gnðr; tÞ; qt
ð11Þ
where nðr; tÞ is the particle size distribution function at time t; bðrÞ is the deposition coefficient of particles with radius r; and cv is the ventilation rate which is assumed to be independent of particle size. From the physical insight, ventilation does not affect the change in the mean particle size or the particle size distribution width. Only the total concentration is decreased due to ventilation by a factor of expðcv tÞ: Therefore, ventilation is not accounted for any longer in this study, but only deposition is considered. In the moment method, the particle size distribution is represented with a time-dependent log-normal function. In general this approach has been widely employed for representing the size distribution of particles both experimentally and theoretically. The log-normal size distribution function for particles whose radius is r is written as " # 1 NðtÞ ln2 fr=rg ðtÞg nðr; tÞ ¼ pffiffiffiffiffiffi exp ; ð12Þ r 2p ln sg ðtÞ 2 ln2 sg ðtÞ where NðtÞ is the total number concentration of particles, rg ðtÞ is the geometric number mean particle radius, and sg ðtÞ is the geometric standard deviation of particle radius. The kth moment of size distribution function nðr; tÞ is defined as Z N Mk ¼ rk nðr; tÞ dr; ð13Þ 0
For relatively large particles, the diffusion term in Eq. (10) may be neglected. By substituting Eq. (10), with neglect of the first term on the right-hand side, into Eq. (11), multiplying the equation by rk ; and integrating from 0 to N; one can obtain the following equation: dMk ¼ BMkþ2 : ð14Þ dt The properties of a log-normal distribution function are such that the following equation holds for any kth moment: 2 k 2 k ln sg : ð15Þ Mk ¼ Nrg exp 2 Substituting Eq. (15) into Eq. (14) and rewriting the equation for k ¼ 0; 2; and 2, we have dN ¼ BNr2g y; dt
ð16Þ
dðNr2 g yÞ ¼ BN; dt
ð17Þ
dðNr2g yÞ ¼ BNr4g y4 ; ð18Þ dt where y ¼ expð2 ln2 sg Þ: Substituting Eq. (16) into Eqs. (17) and (18), we have 2r3 g y
drg dy ¼ Bðy2 1Þ; þ r2 g dt dt
ð19Þ
drg dy ¼ By2 ðy2 1Þ: þ r2 ð20Þ g dt dt Rewriting Eq. (19) and (20) in terms of rg and y; we have 2r3 g y
Br3g ðy4 1Þ drg ; ¼ 4y dt
ð21Þ
Br2g ðy2 1Þ2 dy : ð22Þ ¼ 2 dt Eliminating dt from Eqs. (21) and (22) and integrating the resulting equation, the following equation is obtained: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rg y0 ðy2 1Þ ; ð23Þ ¼ rg0 yðy20 1Þ where rg0 and y0 are the initial values of rg and y; respectively. Substituting Eq. (23) into Eq. (22) and
S.H. Park, K.W. Lee / Atmospheric Environment 36 (2002) 5459–5467
3 mmÞ; is also shown in this figure. The monodisperse model is shown to fail to predict the number concentration decay. Initially, the monodisperse model underestimates the deposition rate due to the neglect of the polydispersity effect. As time elapses, the mean size of polydisperse particles decreases owing to faster settling of larger particles and it results in decrease of the mean settling rate. The monodisperse model which cannot reflect this change, however, overestimates the particle loss rate. Fig. 2 dramatically demonstrates that the monodisperse model should not used to predict the particle concentration change in the gravitation-dominant range. Fig. 3 shows that the whole particle size distribution is also predicted well by the derived solution.
integrating the resulting equation, one can obtain the solution for y: y20 1 ffi: y2 1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 2Br2g0 y0 ðy20 1Þt
5463
ð24Þ
Substituting Eqs. (23) and (24) into Eq. (16) and integrating the resulting equation, one can obtain the following equation for the number concentration decay: 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 þ 2Br2g0 y0 ðy20 1Þt 1 N 5; ð25Þ ¼ exp4 N0 y20 1 where N0 is the initial value of N: Eqs. (23)–(25) represent the solutions to the three size distribution parameters of log-normal size distribution as a function of the initial conditions and time t: From the obtained solution, one can construct the whole particle size distribution using Eq. (12). Thus, the analytical solution to the deposition problem of polydisperse particles in the gravitation-dominant range has just been obtained. Eq. (25) gives an explicit expression for the total concentration decay of polydisperse particles which cannot be provided by the exact solution. This is the important advantage of the solution derived in this study. For y0 ¼ 1 ðsg0 ¼ 1Þ; Eq. (25) simply reduces to N=N0 ¼ expðBr2 tÞ which is the solution for the monodisperse particles. Fig. 2 compares the results calculated for the total particle concentration, the geometric mean radius, and the geometric standard deviation with the exact solution. For the initial condition, rg0 ¼ 3 mm and sg0 ¼ 1:5 were used. It is demonstrated in this figure that the particle size distribution parameters can be predicted well by the analytical solution derived in this study. Comparison with the monodisperse model, where every particle is assumed to have the same size as the geometric mean size of a polydisperse aerosol ðr ¼
3.2. Diffusion-dominant size range For relatively small particles, the gravitation term in Eq. (10) may be neglected. By substituting Eq. (10), with neglect of the second term on the right-hand side, into Eq. (11), multiplying the resulting equation by rk ; and integrating from 0 to N; one can obtain the following equation: dMk ¼ AMk2ðn1Þ=n : dt
ð26Þ
Substituting Eq. (15) into Eq. (26) and rewriting the equation for k ¼ 0; 2ðn 1Þ=n; and 2ðn 1Þ=n; the following three equations are obtained: dN ¼ ANs1 z; dt
ð27Þ
dðNs1 zÞ ¼ ANs2 z4 ; dt
ð28Þ
dðNszÞ ¼ AN; dt
ð29Þ 2
2ðn1Þ=n
where s ¼ rg and z ¼ expf2ðn1Þ ln2 sg g: A repetin2 tion of the previous process for this case gives the
1
1.0
1.5
exact solution this study monodisperse model
σg
1.4
0.8
σg
0.6
rg /rg0
N/N0
1.3 0.1
1.2 rg /rg0
0.4
1.1
N/N0
0.01 0
5000
10000
15000
0.2 20000
1.0
t (s)
Fig. 2. Comparison of changes in particle size distribution parameters in the gravitation-dominant range (rg0 ¼ 3 mm; sg0 ¼ 1:5).
S.H. Park, K.W. Lee / Atmospheric Environment 36 (2002) 5459–5467
5464
1.2 exact solution this study
1.0
t=0s
0.8
n(r)r/N0
700 s 0.6 2000 s
0.4
5000 s
0.2
0.0 0.1
1
10
r/rg0 Fig. 3. Comparison of change in particle size distribution in the gravitation-dominant range. Initial values are the same as those in Fig. 2.
1 exact solution this study monodisperse model
1.5
1.8
1.4
1.6
1.3
0.1
rg/rg0
10000
20000
1.4
1.2
1.2
1.1
1.0 70000
1.0
N/N0
0.01 0
σg
rg /rg0
N/N0
σg
2.0
30000
40000
50000
60000
t (s) Fig. 4. Comparison of changes in particle size distribution parameters in the diffusion-dominant range (rg0 ¼ 0:01 mm; sg0 ¼ 1:5).
following solutions: s zðz20 1Þ ; ¼ s0 z0 ðz2 1Þ
ð30Þ
z20 1 ; z2 1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1Þt z ðz 1 þ 2As1 0 0 0
ð31Þ
2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 1 þ 2As1 N 0 z0 ðz0 1Þt 1 5; ¼ exp4 N0 z20 1
ð32Þ
where s0 and z0 are the initial values of s and z; respectively. Eqs. (30)–(32) constitute the solution for the diffusion-dominant size range. Figs. 4 and 5 compares the results calculated for the size distribution parameters and the whole particle size distribution, respectively, with the exact solution for the diffusion-dominant range. For the initial condition, rg0 ¼ 0:01 mm and sg0 ¼ 1:5 were used. These figures show that the change in particle size distribution in the diffusion-dominant range can also be predicted well by the analytical solution derived in this study. It is noted that the monodisperse model ought not to be used for predicting the particle concentration change of polydisperse particles in the diffusion-dominant range either.
S.H. Park, K.W. Lee / Atmospheric Environment 36 (2002) 5459–5467
5465
1.2 exact solution this study
1.0
t=0s
n(r)r/N0
0.8 4000 s 0.6 10000 s
0.4
23000 s
0.2
0.0 0.1
1
10
r/rg0 Fig. 5. Comparison of change in particle size distribution in the diffusion-dominant range. Initial values are the same as those in Fig. 4.
3.3. Intermediate size range For the intermediate size range, both terms in Eq. (10) should be accounted for during the derivation of the solution. Inclusion of both terms, however, makes it impossible to derive an analytical solution in a similar way with those used in the limiting size ranges. Therefore, we need here an appropriate approximation. First, let us define the kth moment distribution function as follows:
Eq. (36) is exact for monodisperse aerosols but it may cause some error for polydisperse aerosols. This error always results in underestimation of moment depletion. This will be discussed again later in this section. If Eq. (15) is written for k ¼ 0; 1; and 2, and subsequently solved for rg and sg in terms of M0 ; M1 ; and M2 ; we have M0 M2 ; ð37Þ ln2 sg ¼ ln M12
mk ðr; tÞ ¼ rk nðr; tÞ:
rg ¼
ð33Þ
The change of the moment distribution by deposition is then represented by the following equation from Eq. (11): qmk ðr; tÞ ¼ bðrÞmk ðr; tÞ: qt
ð34Þ
mk ðr; tÞ ¼ exp½fbd ðrÞ þ bg ðrÞgt : mk ðr; 0Þ
ð35Þ
From Eq. (35), and using an additional approximation, one can obtain the kth moment as a function of time as follows: RN mk ðr; 0Þexp½fbd ðrÞ þ bg ðrÞgt dr Mk ðtÞ RN ¼ 0 Mk ð0Þ 0 mk ðr; 0Þ dr RN RN m ðr; 0Þexpfb k d ðrÞtg dr 0 mk ðr; 0Þ expfbg ðrÞtg dr RN D 0 ½ 0 mk ðr; 0Þ dr 2 Mk ðtÞ Mk ðtÞ ¼ : ð36Þ Mk ð0Þd Mk ð0Þg
3=2
1=2
:
ð38Þ
M0 M2
From Eq. (36)–(38), the following three equations are easily derived: N N N ¼ ; ð39Þ N N N 0
Integrating Eq. (34), we have
M12
0 d
0 g
rg rg rg ¼ ; rg0 rg0 d rg0 g
ð40Þ
expðln2 sg Þ expðln2 sg Þ expðln2 sg Þ ¼ : expðln2 sg0 Þ expðln2 sg0 Þd expðln2 sg0 Þg
ð41Þ
Fig. 6 compares the results calculated for the three size distribution parameters for the intermediate size range with the exact solution. For the initial condition, rg0 ¼ 0:1 mm and sg0 ¼ 1:5 were used. In this case, the geometric mean radius and the geometric standard deviation are predicted well by the derived solution whereas the number concentration is poorly estimated. The large error for the number concentration stems
S.H. Park, K.W. Lee / Atmospheric Environment 36 (2002) 5459–5467 1 exact solution this study monodisperse model
0.1
1.5
1.4
1.4
1.3
1.3
rg /rg0
N/N0
σg
1.5
N/N0
σg
5466
1.2
1.2
1.1
1.1
1.0 5e+5
1.0
rg /rg0
0.01 0
1e+5
2e+5
3e+5
4e+5
t (s)
Fig. 6. Comparison of changes in particle size distribution parameters in the intermediate particle size range (rg0 ¼ 0:1 mm; sg0 ¼ 1:5).
1.2 exact solution this study
1.0
t=0s
σg0 = 1.5
n(r)r/N0
0.8
rg0 = 0.1 µm
4 5*10 s
0.6
5
10 s
0.4 5
2.3*10 s 0.2
0.0 0.1
1
10
r/rg0 Fig. 7. Comparison of change in particle size distribution in the intermediate particle size range. Initial values are the same as those in Fig. 6.
partially from the underestimation of the deposition rate in this size range as was shown in Fig. 1 and partially from the error due to the approximation used in Eq. (36). Note that the second type of error seldom affected sg and rg because the errors for each moment compensated one another in Eqs. (37) and (38). Another interesting point noted in this figure is that the monodisperse model predicts the particle concentration decay very accurately in the intermediate size range. It is because in this size range the change in the mean-particle size is relatively small and the size-dependency of the deposition rate is weak compared to that for the gravitation- or diffusion-dominant size range. Therefore, it is recommended to use the monodisperse model to predict the particle concentration change rather than the analytical solution derived in this study for the intermediate particle size range.
Fig. 7 shows a comparison of the particle size distribution change between the exact solution and the analytical solution derived in this study. For the number concentration decay, the monodisperse model was used. This figure shows that the particle size distribution can be predicted well by the derived solution associated with the monodisperse model in the intermediate size range. As was mentioned earlier, the analytical solution derived in this study is exact for monodisperse aerosols, but some error exists for polydisperse aerosols. This error mainly stems from the assumption of log-normal size distribution assumption. Therefore, the magnitude of the error depends on how much the size distribution deviates from the log-normal shape, and it is expected to increase with increasing geometric standard deviation sg : We have examined the accuracy of the solution
S.H. Park, K.W. Lee / Atmospheric Environment 36 (2002) 5459–5467
derived in this study with respect to the value of sg upto 1.8. As is shown in Figs. 3, 5 and 7, the particle size distribution shape was always almost log-normal, so the error of the derived solution was not so sensitive to sg :
4. Conclusions In this study, size distribution change of polydisperse indoor particles by gravitational sedimentation and turbulent and Brownian diffusion were investigated. Based on the assumption that the particle size distribution is represented by a time-dependent log-normal distribution, analytical solutions for the gravitation- and the diffusion-dominant size ranges were derived. The solutions were compared with the exact solution and good agreement was obtained. These new solutions have the advantage that the particle size distribution parameters are expressed as explicit functions of time. By the comparisons with the polydisperse models, it was shown that the monodisperse model should be used to predict the particle concentration decay of polydisperse particles neither for the gravitation- nor for the diffusiondominant size range. For the intermediate particle size range where neither the diffusion nor the sedimentation can be neglected, two limiting-size-range solutions were mechanistically combined. This type of approximation, however, was shown to cause severe error for the number concentration change. On the other hand, the monodisperse model gives very accurate prediction in this size range. Therefore, the analytical solution derived in this study is recommended to be used for the geometric mean particle size and for the geometric standard deviation while the monodisperse model is for the particle concentration in the intermediate particle size range.
Acknowledgements This work was supported by the Brain Korea 21 program from the Ministry of Education through the Graduate Program for Chemical and Environmental Engineering at Kwangju Institute of Science and Technology.
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