In this section, the governing equation presented in chapter 3.1 is applied to build up the model. The calculation domain of the model is the classifying region of the APM and parabolic flow field is applied. The transfer functions simulated by the model are compared with ones simulated with previous models. The comparison is considered as the preliminary verification of our model. Based on the good agreements of the comparison, the model presented in the section is considered the representative of the previous models.
Calculation Domain
In Fig. 6, the dark orange area is the classifying region, which is the space between the inner and outer closely-spaced annular cylinders. Several studies considered the classifying region the calculation domain of their models (Hagwood et al., 1995, Ehara et al., 1996, Olfert and Collings 2005). To verify our model, the calculation domain of our model is defined to be the classifying region as previous studies did.
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Fig. 6. Caculation domain is the annular classifying region of the APM (dark orange area).
The area enclosed by thick red lines is the rotating region.
Flow Field
Because previous studies usually made the assumption of laminar parabolic flow field for their models (Hagwood et al., 1995, Ehara et al., 1996, Olfert and Collings 2005), we apply the same assumption of the flow field for the model. Eq. (32) is applied to describe the parabolic flow field in the classifying region of the APM.
u = uz(r) =32u� �1 − �rr−rc
2−r1�2� (32)
rc: The average of r2 and r1 (m), (r2+r1)/2
Eq. (33) describes the transfer function, ΩAPM, for particles with diameter dp. In eq. (33), the number concentration of the particles at the APM inlet Nin(dp , r) is considered homogeneous, while the number concentration of the particles at the APM outlet Nout(dp , r) is solved numerically by the SIMPLER algorithm (Semi-Implicit Method for Pressure-Linked Equations) (Patankar 1980, Lin et al., 2010).
22 normalized particle concentration, the ratio of particle concentration at the outlet to the inlet of the APM classifying region, at the position (r,z). Np∗(r, 0), the normalized particle concentration at the inlet of classifying region, is considered as 1. In addition, because particle contacting the walls in the classifying region is removed, particle concentration at the walls is considered zero (removed by the APM).
Np∗(r, 0) = 1 for r1<r<r2 (34) Np∗(r2, z) = Np∗(r1, z) = 0 (35)
Compared with Previous Studies
The transfer functions simulated with the model presented in this section are compared with ones simulated with three previous models respectively, which are the theoretical model developed by Ehara et al., (1996), the numerical models presented in Hagwood et al., (1995) and the diffusion model developed by Olfert and Collings (2005). To simplify the calculation, the particle is assumed spherical. The result of the comparisons is considered as the preliminary verification of the model.
For the comparisons of our model and the theoretical model (Ehara et al., 1996) and the numerical models (Hagwood et al., 1995), the parameters of the models are set to be same as the ones set in Hagwood et al., (1995) (table 3). For the comparison of our model and the diffusion model developed by Olfert and Collings (2005), the parameters of the models are set to be same as the ones used in Olfert and Collings (2005) (table 3). Because pressure and
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temperature applied in some previous studies were unknown, we assumed the atmospheric pressure and 25 ℃ for these parameters.
Table 3 Parameters presented in compared papers.
References
Fig. 7. (a) The relative width and (b) the maximum height of the transfer functions for different flow field applied to the Ehara model (Ehara et al., 1996).
It should be noted that Ehara et al., (1996) found that if the value of the λc is small (less than about 0.5), there is no significant difference between the transfer functions simulated by the Ehara model with uniform flow field and parabolic flow field (Fig. 7). The λc of the transfer function applied in the comparison of our model and Ehara model is about 0.044, which is much lower than the 0.5. Moreover, applying the uniform flow field makes the transfer function available to be solved as exact solution. Therefore, to simplify the calculation, we applied the assumption of the uniform flow field to the Ehara model.
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Fig. 8. The transfer function of comparing our model with (a) the theoretical model developed by Ehara et al., (1996), (b) the SDE model developed by Hagwood et al., (1995), (c) the diffusion model developed by Olfert and Collings (2005).
Fig. 8 shows the result of the comparisons. In Fig. 8(a), the dashed black line is obtained with our model, while the dashed red line is obtained with the theoretical model developed by Ehara et al., (1996). Since Ehara model neglected the Brownian motion, we zero the diffusivity of our model for making the comparison (D=0). The result showed good agreement between two models. In Fig. 8(b), the transfer function obtained with our model
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(dashed red line) agrees very well with one simulated with the SDE model (dashed black line).
The slight difference would be due to different principles or governing equations of the models. The SDE model is based on the escaped probability of particles whereas our model is based on the convection-diffusion equation. Fig. 8(c) shows the comparison of our model and the diffusion model developed by Olfert and Collings (2005). The transfer function simulated with our model is denoted as dashed red line, while the one simulated with the diffusion model is denoted as dashed black line.
In sum, the transfer functions simulated by our model agreed well with several models which were presented in previous studies. The good agreement is considered as the preliminary verification of the model. The verified model is considered as the representative of the previous models, whose calculation domain is the classifying region of the APM and the flow field is assumed parabolic.