Before applying model with detailed flow field to calculate the transfer function, the user has to spend lots of time on calculating the velocity profile of the field. Moreover, because the velocity profile is dependent on the rotation speed, flow rate and geometry of the APM, we have to calculate the velocity profile for each different operating condition. After that, we have to spend more time on calculating the transfer function. To simplifying the calculation process, the study develops the fitting model based on the numerical results to calculate transfer function in more efficient manner. Moreover, the study also applied numerical results to develop the modified Ehara model, which can calculate the transfer function as analytical solution. The two methods are described in following sections.
Fitting Model
The study builds up a fitting model, which is based on the numerical results simulated by the model with extended domain and detailed flow field. Gaussian distribution, as described in Eq. (42), is applied to fit the transfer function simulated by the developed numerical swirl model. Particle classified by the APM is considered to be spherical and singly charged, so the specific mass of particle can be easily converted to the diameter dp.
dΩAPM�dp, ωλc, V� = σ√2πX exp �−(V−V2σ2c)2� dV (42)
In Eq. (42), ΩAPM�dp, ωλc, V� is the transfer function of the particle, whose diameter is dp, passing through the APM operated with rotation speed ωλc and voltage V. ωλc is the rotation speed determined based on the size of center particle (dp,c) and the chosen λc (Eq. (4)), and V is the voltage applied to the APM. is the standard deviation of the voltage range that enable particle with diameter dp to pass through the APM without being removed when
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the rotation speed is fixed at ωλc. The value of is obtained by trail and errors until the maximum difference between the fitted ΩAPM�dp, ωλc, V� and numerical ΩAPM�dp, ωλc, V� is minimized. Moreover, X is the correction factor, which not only normalizes the Gaussian distribution but also makes the maximum height of the Gaussian distribution equal to the maximum height of the numerical transfer function. Vc is the center voltage, derived from the rotation speed ωλc through equation. ΩAPM�dpc, ωλc, Vc� is the maximum transfer function when the APM is operated with ωλc and Vc. If the terms of σ and X are known, we can calculate the transfer function through Eq. (42) without additional numerical calculation.
Table 4 The results of fitting numerical transfer function with Gaussian distribution.
Gaussian Distribution Fits Numerical Models
In this section, the study chose the case of applying the APM-3600 to measure the mass distribution of particles with 0.22 of λc and 1 lpm of flow rate. Gaussian distribution is applied to fit seven different numerical transfer functions, which are simulated for particles with diameter 20 nm, 30.6 nm, 51 nm, 100 nm, 208 nm, 479 nm, 791 nm respectively. The
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parameters, σ and X, is determined by trail and errors to minimize the maximum difference between the numerical transfer function and fitting Gaussian distribution. The obtained and X for each transfer function are listed in table 4. T.F. shown in table 4 is the abbreviation of the transfer function. The and X shown in table 4 are further fitted to predict the σ and X for size of particle which is not converted by the table. The fitting equation of and X are shown as below. The parameters, A,B,C,D,E, shown in Eq.
(43)~(48) are listed in table 5 respectively. To remain the accuracy of the prediction, the digits of parameters after decimal point should not be rounded off.
Fitted σ:
For 208 nm > dp > 17 nm,
σ = A × e�B×dp�+ C × e�D×dp�+ E. (43) For 791 nm > dp > 208 nm,
σ = A × e�B×dp�+ C × e�D×dp�+ E. (44) For dp > 791 nm,
σ = 0.139 × Vc. (45)
Fitted X:
For 208 nm > dp > 17 nm,
X = A × e�B×dp�+ C × e�D×dp�+ E. (46) For 791 nm > dp > 208 nm,
X = A × e�B×dp�+ C × e�D×dp�+ E. (47) For dp > 840 nm
X = 239.431 + 0.3505126 × (dp− 791). (48)
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Table 5 The parameters of equations which are applied to fitted the obtained and X.
Fitted σ Fitted X
208 nm > dp > 17 nm 791 nm > dp > 208 nm 208 nm > dp 840 nm > dp > 208 nm A 17.9219737 A 83.6538647 A 77.0379673 A -6.9615277
B 0.0039508 B 0.0017356 B 0.0024631 B 0.0037867
C 2.9876668 C -11.6140818 C 10.0264075 C 169.7009613 D -0.058499 D 0.0031225 D -0.0259945 D 0.0014929 E -19.9238423 E -76.9483039 E -86.6056268 E -174.1644899
Fig. 10. Comparison between the numerical transfer function (solid lines) and the transfer function predicted by the fitting model (dashed lines)
To verify the fitting model, the transfer functions determined by the fitting model are compared with the fitted numerical transfer function. Fig. 10 shows the results of the
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comparison. Good agreements are obtained and considered the verification of the fitting model. It should be keep in mind that because the fitted numerical transfer functions are simulated for the APM-3600 which is operated with 0.22 of the λc and 1lpm of the flow rate, the parameters shown in this section is only available for the APM-3600 operating with the same condition.
Modified Ehara Model
The study develops a modified Ehara model to calculate the transfer function as exact solution with considering the effects of Brownion motion. The model combines the model developed by Ehara et al., (1996) and the modified Gormley and Kennedy equation, which is modified based on our numerical results.
Ehara et al., (1996) developed a model (Ehara model) to calculate the transfer function of particles classified by the APM based. With the assumption of uniform flow field in the classifying region of the APM, Ehara model can calculate the transfer function as exact solution. Moreover, when the λc is sufficient low, the transfer function calculated with the uniform flow field can be very similar to the one calculated with the parabolic flow field, which is more close to the real flow field (Fig. 7). Hence, Ehara model was available to calculate the transfer function as exact solution without considerable error for the APM operated with low value of λ.
Fig. 11 is the typical transfer function calculated with the Ehara model with the uniform flow field. In Fig. 11, the shape of the transfer function can be determined by four special specific masses S1+, S1-, S2+, S2-. These specific masses can be calculated with Eq. (49) and (50) (Ehara et al., 1996). Sc is the specific mass of center particle (Eq. (51)), which achieves force balance of centrifugal force and electrostatic force at the center position (r=rc) between the inner and the outer of the annular cylinders (classifying region).
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Fig. 11. Four particular specific masses
S2±≈ Sc�1 ± 2rδ
c� (49) S1±≈ Sc�1 ± 2 �rδ
c� coth �λ2c�� (50) Sc =r V
c2ω2ln�r2r1� (51)
Based on the specific mass S of particle, the transfer function ΩAPM(S) can be determined by three equations. If S ranges between S1− and S2−, ΩAPM (S) can be calculated with Eq.
(52). Similarly, if S ranges between S2− and S2+, ΩAPM (S) can be determined by Eq. (53).
If S ranges between S2+ ≤ S ≤ S1+, ΩAPM (S) can determined by Eq. (54). If S is out of these ranges, particle with such S will be completely removed by the APM.
For S1− ≤ S ≤ S2− (ρ0h = 1)
ΩAPM(S) =12�[1 − ρ(S)] + [1 + ρ(S)]e−λ� (52) For S2− ≤ S ≤ S2+
ΩAPM(S) = e−λ (53)
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For S2+ ≤ S ≤ S1+ (ρ0l = −1)
ΩAPM(S) =12�[1 + ρ(S)] + [1 − ρ(S)]e−λ� (54)
ρ(S) in Eq. (52)~(54) is the position of particles expressed in normalized coordinate as described by Eq. (55) and (56). r(S), derived from Eq. (1) is the position where centrifugal force and electrostatic force acting on particle of specific mass S are the same.
ρ(S) =[r(S)−rδ c] (55) ζ =Lz (56)
r(S) = �Sω2Vln�r2r1� (57)
To apply the Ehara model with considering the effects of Brownian motion, the study applied Gormley-Kennedy equation (Eq. (58)~(60)) to consider the effects of the Brownian motion of particles on the transfer function. It is assumed that the diffusion loss of particles is independent to the classification of the APM. In other words, the study applies the product of ΩAPM (S) and PG&K to calculate the transfer function of nanoparticles. In the study, Ehara model modified by the diffusion loss equation is denoted as the Modified Ehara model, and the transfer function calculated with the modified Ehara model is denoted as Ω’APM(S).
µ =πDL(rQ�(r 2+r1)
2−r1)� (58) PG&𝐾 = 1 − 2.96µ23+ 0.4µ for μ<0.005 (59) PG&𝐾 = 0.910 exp(−7.54µ) + 0.0531exp(−85.7µ) for μ≧0.005 (60)
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D: The diffusivity or diffusion coefficient (m2/s) Q: The flow rate of the flow entering the APM (m3/s) PG&K: The penetration of particles passing through the APM
To verify Gormley and Kennedy equation, we applied the equation to predict the penetration of nanoparticles passing through the classifying region of the still APM (0 rpm, 0 volt). Then, the penetrations predicted are compared to the numerical ones simulated with our numerical model with extended domain and detailed flow field as reference. Because neither centrifugal force nor electrostatic force occurs in the still APM, it is considered that the penetration of nanoparticles is mainly due to the diffusion loss. The validity of the numerical penetrations has been checked with experimental data presented in Tajima et al., (2011), which will be mentioned in next chapter “4.1 Diffusion Loss Prediction”.
Fig. 12. The calculated penetration of particles passing through the still APM.
Fig. 12 shows the results of the simulation. The dashed red line is the penetration
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calculated with Gromley and Kennedy equation (L=0.25 m) and the solid blue line is the numerical penetration simulated with our numerical model. It is found that Gormley and Kennedy equation significantly overestimate the penetration of nanoparticles. The degree of the overestimation increases for smaller nanoparticles (increases from 0% to 20% or more on penetration). We conclude that the overestimation is due to Gormley and Kennedy equation consider the classifying region of the APM only, while the numerical model considers whole the region of the APM. More discussions are in chapter “4.1 Diffusion Loss Prediction”.
Instead of extending the calculation domain of Gormley and Kennedy equation (still not be accurate after the try), we apply another correction factor K to modify the PG&K directly as described in Eq. (61). The correction factor K is a function of particle size, which is described in Eq. (62) and (63). The function is obtained by fitting the difference between the PG&K and the numerical penetration for several sizes of nanoparticle. The penetrations calculated with the modified Gormley and Kennedy equation (solid red lines) agree very well with numerical penetration (Fig. 12). Hence, the modified Gromley and Kennedy equation is applied to modified Ehara model as described in Eq. (64).
P′G&𝐾= K × PG&𝐾 (61) For dp smaller than 100 nm or equal to 100 nm
K = −1.64 × exp�−0.14 × dp� − 0.36 × exp�−0.03 × dp� + 1.02 (62) For dp larger than 100 nm
K = 1 (63) Ω𝐴𝑃𝑀′ (𝑆) = ΩAPM(S) × P′G&𝐾 (64)
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Fig. 13. The transfer functions are calculated with Ehara model (dashed black line) and modified Ehara model (solid black line). The modified Gormley and Kennedy equation applied in modified Ehara model is dented as solid red line.
After modifying Gormley and Kennedy equation, The transfer function calculated with the modified Ehara model is compared with the one calculated with the Ehara model. Fig.
13 shows the result of the comparison. The dashed black line is the transfer function calculated with the Ehara model, and the solid black line is the transfer function calculated with the modified Ehara model. The height difference of the transfer functions is due to the modified Gormley and Kennedy equation (solid red line), which represents the diffusion loss of particles in the APM.
4 Results
After building up the models (the model with classifying region domain parabolic flow profile and the model with whole region domain and detailed flow profile), the models are compared to study the accuracy of the prediction. The effects of the extended calculation
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domain and the detailed flow field are considered by comparing the models with the experimental data.