Up to now, several models were developed. Some of the models have been compared with experimental data (Ehara et al., 1996, Olfert et al., 2006, Lall et al., 2009, Tajima et al., 2011); however, no model has accurately agreed with experimental data for nanoparticles even if the effects of Brownian motion were considered in the model.
Nout(V) = ∫ Nin(S)ΩAPM(S, V)dS (6)
Ehara et al., (1996) calculated the number concentration of monodisperse particles passing through the APM (Eq. (6)). In Eq. (6), the particle concentration at the APM outlet, denoted as Nout(V), is the function of voltage. The particle concentration at the APM inlet was considered the function of the specific mass (denoted as Nin(S)) which was assumed to be proportional to the function, and the transfer function of the APM was denoted as the function of the specific mass and voltage (denoted as ΩAPM(S,V)). The rotation speed of the APM was fixed, while the voltage of the APM was shifted to scan the specific mass distribution of the particles. The theoretical relative particle concentration, which is the ratio of the total particle concentration at the APM outlet to that at the APM inlet, was calculated with different voltage and compared with the experimental one. Good agreement of the comparisons between the experimental data (monodisperse 309 nm PSL) and the simulated results showed the validity of the theoretical model.
Since the theoretical model developed by Ehara et al., (1996) neglected the Brownian motion of particles, the model is not suitable to nanoparticles. Tajima et al., (2011) applied
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the model disregarding the Brownian motion of particles to simulate the APM response spectra which is same as the relative particle concentration calculated in Ehara et al., (1996).
The flow field of the model was assumed to be parabolic. Different to Ehara et al., (1996), Tajima et al., (2011) considered the size distribution of monodisperse PSL at the APM inlet more carefully as described in Eq. (7). The size distribution of the particles at the inlet of Differential Mobility Analyzer (DMA) were considered the Gaussian distribution (denoted as N0(dp)) based on the mean and standard deviation of size of size standard PSL. Moreover, the particles classified by the DMA (Nin(dp)) is considered the product of the N0(dp) and the transfer function of the DMA (denoted as ΩDMA). VDMA is the voltage applied to the DMA.
The rotation speed of the APM was fixed based on the specific λc, while the voltage was shifted to scan the specific mass distribution of monodisperse PSL, and the normalized particle concentration were calculated and compared with the experimental one.
Nin�dp� = N0�dp� × ΩDMA�dp, VDMA� (7)
Fig. 3. Theoretical and experimental normalized particle concentration. (Tajima et al., 2011)
Fig. 3 showed the results of comparisons presented in Tajima et al., (2011). The number on each curve was the size of the monodisperse PSL. The thick grey lines were the simulated results, and the thin black lines fitted the experimental data (points) with the least
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square fitting method. Differences between the simulated results and experimental data became significant for PSL less than 100 nm. Tajima et al., (2011) concluded that the differences (overestimations) were caused by diffusion loss.
Lall et al., (2009) applied the MC method (Hagwood et al., 1995) to calculate the APM transfer function with the assumption of the parabolic flow field, and they also calculated the particle concentration at APM outlet with the manner similar to the manner did in Tajima et al., (2011) as described in Eq. (6) and (7). Different from Tajima et al., (2011), Lall et al., (2009) considered the N0 as the constant function and applied the triangular function to the transfer function of the DMA. Comparing to experimental data, Lall et al., (2009) found the simulated results overestimated the penetration for nanoparticles (60 nm, 100 nm PSL) and submicron particles (300 nm PSL). Lall et al., (2009) concluded that was due to diffusion losses and transport losses.
Olfert et al., (2006) verified the model presented in Olfert and Collings (2005). Instead of the APM, the major objective of the study for Olfert and Collings (2005) is the Couette Centrifugal Particles Mass Analyzer (CPMA). Because the only difference between two instruments is the rotation speeds of the inner and outer cylinders, while the cylinders of the APM have the same rotation speed, the CPMA is very similar to the APM. The different rotation speed of cylinders of the CPMA was applied in order to achieve the stabler state of the classification (decrease the loss of particles during the classification). Since the APM is very similar to the CPMA, the diffusion model developed by Olfert and Collings (2005) not only available to the CPMA but also available to the APM. Olfert et al., (2006) compared the model of the CPMA with experimental data. The assumption of parabolic flow field was made in the model, and of assuming that particles at the APM inlet are strict monodisperse (particles are in same size). For 50 nm PSL, the diffusion model significantly overestimated the transfer function compared to the experimental data. Olfert et al., (2006) concluded that the overestimation was due to the particle diffusion. Because the model of the CPMA is
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very similar to the model of the APM, we consider that the result concluded for the CPMA in Olfert et al., (2006) would also be available to the APM.
In sum, for submicron particles, some models have been verified by the experimental data (Ehara et al., 1996, Tajima et al., 2011). For nanoparticles, however, no model has agreed well with experimental data. Table 1 summarizes the performance of previous transfer function models.
Table 1 The summary of the performance of previous models
3 Numerical Method
A 2-D numerical model developed by our laboratory is applied to simulate the transfer function of the APM. The preliminary verification of the model is conducted with comparing the simulated transfer function with ones done by previous models with simple calculation domain (the classifying region of the APM) and assumption of parabolic flow field. After the preliminary verification, the model is further improved by extending calculation domain from classifying region to whole region in the APM and by considering detailed flow field based on the Navier-Stokes equations. The improved model is used to compare with the experimental data shown in Tajima et al., (2011) as the advanced verification.
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3.1 2-D Numerical Model Governing Equation
Fig. 4. Scheme of the APM and the flux of particles induced in the APM.
Fig. 4 shows the principle of the model applied in the model. Charged particles were introduced in the classifying region of the APM (the space between the closely-spaced annular cylinders). Particles passing through the region are classified by the centrifugal force Fc and electrostatic force Fe. In Fig. 4, L is the length of the APM. Three directions of flux in the classifying region are considered. First one is the flux induced by the carrier gas. Particles move with the direction of flow in the APM. Second one is for the particles of which Fe is greater than Fc, the flux toward the inner cylinders is induced. Third one is for particles of which Fe is smaller than Fc, the flux flowing toward the outer cylinders is induced.
The flux describe the particles which are classified by the APM.
The governing equation applied in the model is based on the convection-diffusion equation. The general equation of the convection-diffusion equation is
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∂Np
∂t = ∇ ∙ �D∇Np� − ∇ ∙ �u�⃑Np�. (8)
Np: Number concentration of particles in the APM (#/m3).
u�⃑: Velocity of the aerosol flow passing through the APM (m/s).
It is considered that there is no source, sink or chemical reaction in the APM classifying region. The study considers that the flow in the APM is isothermal and steady. Since the Mach number of the flow is much less than 0.5 (ex: 0.0046 for the APM-3601 or 0.015 for the APM-3600), the carrier gas (ex: air) is considered as incompressible fluid. In addition, the classification of the APM is assumed to be steady (∂N∂tp=0). Because centrifugal force and electrostatic force do not change in θ direction, the particle motion in θ direction and the flow field in θ direction are neglected in the model (uθ=0). Finally, the governing equation of 2-D model for the transfer function is
∂�ur+(uc−ue)Np�
∂r +∂�u∂zzNp�= D �1r∂r∂ �r∂N∂rp� +∂∂z2N2p�. (9)
ur: Velocity of flow in r direction (m/s) uz: Velocity of flow in z direction (m/s)
uc: Velocity of particle flow induced by centrifugal force (m/s).
ue: Velocity of particle flow induced by electric force (m/s).
r: Distance between the aerosol and the axis of the APM (m).
Eq. 9 is further rewritten with the detailed description of ue and uc, as described in Eq. (10) and (11), to be Eq. (12).
14 Er: Strength of electric field (N/C)
Zp: Electrical mobility of aerosol (m2/Volt．s)
Several dimensionless parameters are applied to obtain the dimensionless form of Eq. (12).
These parameters are listed in Eq. (13) to Eq. (20) respectively.
Np∗ =NNp
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Nin: Particle concentration at the APM inlet (#/m3)
Zp,c: Electric mobility of center particles (m2/Volt．s) (Eq. (21)) Bc: Mobility of center particle (m/N．s)
dp,c: Diameter of center particle (m) (Eq. (22)) C(dp,c): Cunningham slip correction factor (Eq. (23))
Zp,c = qBc = ne3πµdC�dp,c�
The 4δ shown in the denominator of Eq. (16) and (17) is the characteristic length of the APM. In the study, the hydraulic diameter of the classifying region (Dh) is considered the characteristic length (Eq. 24). In Eq. (24), A is the cross section area of the classifying region of the APM (m2), and P is the wet perimeter, the sum of the circumferences of inner and outer radius of the classifying space (m).
Dh = 4AP =4π�r2π(r22−r12�
2+r1) = 2(r2− r1) = 4δ (24)
With Eq. (13)~(20), the dimensionless form of governing equation is obtained as
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∂��τcτ∗ω2r∗4δ��N0Np∗��
4δ ∂r∗ −∂��Zp∗Zp,c4δ ∂r4δr∗K ��N∗ 0Np∗��+∂�(u�u4δ ∂rr∗)�N0∗Np∗��+∂�(u�u4δ ∂zz∗)�N∗0Np∗��
= DcD∗��4δr1∗�4δ ∂r∂ ∗�4δr∗ N4δ ∂r0∂N∗p∗� +(4δ)N02∂∂z2N∗2p∗�. (25)
Eq. (25) can be further rewritten to be Eq. (26).
τcω2 ∂�τ∗(r∂r∗)�N∗ p∗��−(4δ)Zp,cK2∂�Zp
Three dimensionless numbers are found from Eq. (27) as described in Eq. (28)~(30).
β1 = 4δτu�cω2 (28) β2 =4δu� ln�Zp,cVr2
r1� (29) β3 =4δu�Dc =Pe1 (30)
The β1 includes the rotation speed of the APM, the average velocity of the carrier gas, and the relaxation time of the center particle, which is almost same as the dimensionless number λ found by Ehara et al., (1996) (Eq. (4)). The β2 considers the electric mobility of the center
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particle, the strength of the electric field, and the average velocity of the carrier gas. β1 and β2 are dependent to each other through the force balance of Fc and Fe of the center particle. The β3 includes the diffusivity of the center particle. It is considered the reciprocal Peclet number (denoted as Pe). Greater value of β3 represents the stronger effects of the Brownian motion. Furthermore, it is found that the ratio of β1 to β3 is just equal to eight times of the dimensionless number η (Eq. (4)), which is described in Eq. 31. Since the β1 and β1/β3 are similar to the λcand η respectively, the properties of λc and η should be also applicable to the β1 and β1/β3. For example, the similarity rule found by Ehara et al., (1996) described that when the λc of the transfer function were similar, the height and shape of the transfer functions were similar too. The rule should be available on the β1 too. Another example is that Olfert et al., (2006) mentioned that the effects of the diffusion are important when the absolute value of ηc is less than 10 or the absolute value of β1/β3 is less than 80 for the simulated APM. In sum, three dimensionless numbers are derived from governing equation, and the dimensionless numbers can cover ones presented in previous studies.
β1
β3 =(4δ)D2τcω2
c = 8 × �2δ2Dτcω2
c � = 8 ηc (31)
In sum, the governing equation has been developed based on the convection-diffusion equation. The governing equation will be applied to study the transfer function of nanoparticle and submicron particle of the APM. Moreover, three dimensionless numbers (β1, β2 and β3) are found. The β1 is related to the rotation speed, the β2 is related to the voltage, and the β3 is related to the diffusivity of the particles. The obtained dimensionless numbers can be similar to the ones presented in previous studies (ex: λc, ηc); hence, the characteristics of the λc and ηc should be also available to our dimensionless numbers β1 and β1/β3.
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Dimensionless Numbers for two Different APM models
To compare the performance between different APMs (Model 3600, Model 3601, Kanomax Japan Inc.), dimensionless numbers β1, β2 and β3 are applied to characterized their performances. The geometry and the performance of the APMs are listed in table 2. The shape of particle is considered spherical; hence, the mass of particle can be easily converted to size with known particle density (ex: 1.05 g/cm3).
Table 2 The geometry and performance of the APMs (Kanomax Inc.) APM Model dimensionless numbers of the APM-3600, while the red lines are the values of the dimensionless numbers of the APM-3601. In Fig. 5(a) and 5(b), the dashed lines indicate the maximum values of the dimensionless numbers for each size of particles, and the solid lines indicate the minimum values of the dimensionless numbers.
In Fig. 5(a), the range of the β1 of the APM-3600 is wider than that of the APM-3601.
The available maximum rotation speed of the APM-3601 (14000 rpm) is higher than that of the APM-3600 (9500 rpm), yet the APM3600 can perform with wider range of the β1 than the APM-3600 due to different size and geometry of the classifying regions. For example, when the rotation speed are the same, the radius of classifying region of the APM-3600 is longer than that of the APM-3601, it makes the APM-3600 has stronger centrifugal force compared
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to the APM-3601. Moreover, when the APMs operated with same flow rate (0.3 lpm), the slower average velocity u� of carrier gas makes the APM-3600 has larger β1 compared to the APM-3600.
Fig. 5. The ranges of the dimensionless numbers for APM-3600 and APM-3601
In Fig. 5(b), although the available range of voltage of both APMs is the same, the β2 of the APM-3600 can be higher than that of the APM-3601. We conclude that it is due to slower u� of the APM-3600 compared to that of the APM-3601 when the flow rates of both APMs are the same. Hence, it is concluded that the APM-3600 can perform with the higher β2 compared with the APM-3601. Fig. 5(c) and 5(d) show the relationship between the size of the particles and the β3and Pe respectively. The β3 of the APM-3601 are lower than of the
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APM-3600. It is because that the gap between the outer and inner radius of classification space of the APM-3601 is narrower than that of the APM-3600. The narrower gap makes the flow velocity of the APM-3601 higher than of the APM-3600; hence, the retention time (diffusion time) of particles passing through the APM-3601 is decreased.
In sum, the APM-3600 can operate with wilder range of the β1 and higher β2 compared to the APM-3601, while the APM-3601 is more suitable to operate with smaller nanoparticles (less diffusion loss) compared to the APM-3600.