In addition, according to the velocity profile of the detailed flow filed, we find that recirculation flow occurs in the classifying region (the region marked by green rectangular Fig.
17). Fig. 18 shows detailed velocity profile and the streamlines of the region. For the still APM (0 rpm), the flow filed of the region is very similar to the parabolic flow field. In contrast, for the rotating APM, the counterclockwise recirculation flow appears in the region.
The scale of the recirculation flow increases with greater rotation speed due to the stronger forced vortex. On the other hand, there is no recirculation flow in parabolic flow field because the flow in the region has no velocity in r direction. Hence, we conclude that the enhanced losses of transfer functions as shown in Fig. 15 are due to the enhanced convection-diffusion deposition caused by the recirculation flow, which cannot be considered with the parabolic flow filed.
4.3 APM Response Spectra
The APM response spectra or normalized particle concentration are the ratios of particle concentration at the APM outlet to particle concentration at the APM inlet. The section applies the transfer functions calculated with our fitting model and modified Ehara model to calculate the APM response spectra respectively. Furthermore, the response spectra are compared to the experimental ones presented in Tajima et al., (2011) as verification.
Response Spectra
Fig. 19 shows the scheme of DMA-APM measurement system to which Tajima et al., (2011) used measure the experimental response spectra. Size standard particles (Duke Inc.) are neutralized before enter DMA. Then, the particles are monodispersed by the DMA and classified by the APM. The experimental response spectra are calculated based on the particle concentration measured by the Condensed Particle Counter (CPC). The system can
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be expressed mathematically as described in previous studies (Ehara et al., 1996, Lall et al., 2009, Tajima et al., 2011). The neutralizer is not included in the calculation because it is assumed that size distribution of the size standard particles is so narrow (Duke Inc.) that it is not affected by the neutralization. Eq. (65) and (66) describe the particles classified by the DMA-APM system in the specific mass form and particle size form respectively.
Fig. 19. The scheme of DMA-APM measurement system
Nout(V) = ∫ N0�dp�ΩDMA�dp, VDMA�ΩAPM(S, ωλc, V) dS (65) Nout(V) = ∫ N0�dp�ΩDMA�dp, VDMA�ΩAPM�dp, ωλc, V� ddp (66)
In Eq. (65) or Eq. (66), N0�dp� is the size standard particle (PSL), which is considered Gaussian distribution with standard deviation and mean diameter provided by the manufacturer of the PSL (JSR Corp.). Before entering the APM, particles are classified by the DMA so the size distribution becomes sharper (monodisperse). We applied the theoretical transfer function of the DMA (Stolzenburg and McMurry 2008), denoted as ΩDMA�dp, VDMA�, to describe the classification of the DMA. VDMA is the voltage applied to the DMA, and the dp is the chosen size of the monodisperse particles. Then, monodisperse particles are classified by the APM. Similarly, we describe the classification of the APM with the transfer function ΩAPM�dp, ωλc, V�, which are calculated with our fitting model and modified Ehara model respectively. For example, if we are going to measure the mass distribution of 30.6nm monodisperse particles with 0.22 of λc, the ωλc can be calculated based
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on the chosen λc (described by Eq. (4)) and the relaxation time of the 30.6 nm particle. Then, the rotation speed is fixed at ωλc, while the voltage V is shifted to scan or measure the specific mass distribution (or mass distribution) of the particles. Then, particles passing through the DMA-APM measuring system is described by Nout(V), which is the number concentration of particles at the outlet of the APM. Finally, the response spectra, which are the function of V, can be calculated with Eq. (67). The calculated response spectra are compared to experimental one presented in Tajima et al., (2011).
Response Spectra(V) =∫ N Nout(V)
0�dp�ΩDMA�dp,VDMA�ddp= ∫ NNout(V)
in�dp�ddp (67)
It should noted that the calculation presented in the section is only available for spherical particles because their specific mass S and diameter dp can be converted to each other easily.
If particles are non-spherical (ex: carbon nanotube), the calculation presented in the study would be inadequate, which is not discussed in the thesis. The thesis prefers the size of the particles (Eq. 66) because it is more intuitive to describe a small particle with its size. The calculated response spectra are shown in Fig. 20 ~ 22, and each set of points in the figures are the experimental response spectra presented in Tajima et al., (2011).
In Fig. 20, the response spectra are calculated with the transfer function done with our fitting model. The calculated response spectra agree very well with the experimental ones.
The height difference between the calculated transfer function and experimental transfer functions are less than 6% in normalized particle concentration (response spectra). The results show the validity of the fitting model.
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Fig. 20. The APM response spectra calculated with fitting model. (λC=0.22)
In Fig. 21 and 22, the response spectra are calculated with both modified Ehara model and Ehara model for λc=0.2 and λc=0.49 respectively. For the modified Ehara model, the calculated response spectra of both nanoparticles and submicron particles are very close to the experimental ones. For the case of λc=0.22 and λc=0.49, the maximum height difference between the calculated and experimental response spectra is less than 5% and 10% in response spectra respectively. For Ehara model, the calculated response spectra of submicron particles agree well with experimental ones, whereas the response spectra of nanoparticles overestimate the experimental ones significantly. For the case of λc=0.22 and λc=0.49, the maximum difference between the calculated and experimental response spectra is about 20% and 17% in response spectra respectively. The significant overestimations is due to Brownian motion of particle is neglected by Ehara model. After the Ehara model is modified by the modified Gromley and Kennedy equation, the overestimation of response spectra of nanoparticles is improved significantly.
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Fig. 21. The response spectra calculated with the modified Ehara model. (λC=0.22)
Fig. 22. The response spectra calculated with the modified Ehara model. (λC=0.49)
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Table 7 and table 8 summarize the accuracy of the tested models. For submicron particles, all the calculated response spectra agree well with the experimental ones for both operating conditions (λc=0.22 and λc=0.49). In contrast, for nanoparticles, the response spectra calculated with Ehara model overestimate the experimental ones, while both the fitting model and the modified Ehara model still agree very well with the experimental data of both nanoparticles and submicron particles. The differences the height of the response spectra between the calculated results and the experimental data are less than about 10%, which shows the validity of the fitting model and the modified Ehara model.
Table 7 The difference between the heights of the calculated response spectra and experimental response spectra (λc=0.22).
λc=0.22
PSL APM-3600 Difference in Height of the Reponse Spectra dp (nm) rpm Ehara Model Fitting Model Modified Ehara Model
Table 8 The difference between the heights of the calculated response spectra and experimental response spectra (λc=0.49).
λc=0.49
PSL APM-3600 Difference in Height of the Reponse Spectra dp (nm) rpm Ehara Model Modified Ehara Model
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Notice and Restrictions of the Models
This paragraph describes the notice to ensure the validity of the simplified models as well as the restriction of the models. As mentioned in chapter 3.4, the parameters applied to the fitting model are dependent on the model of the APM applied in the experiment, the value of λc, and the flow rate in the APM. The parameters of the fitting model presented in the thesis are only available to the APM-3600 which is operated with 0.22 of λc and 1lpm of flow rate in the APM. If λc or the flow rate is changed, user has to produce another set of numerical transfer functions based on the whole domain of the APM and detailed flow field to update the parameters of the fitting model. Fortunately, the flow rate of the APM-3600 and APM-3601 are usually set at 1 lpm and 0.3 lpm respectively, so the parameters would be mainly based on the value of λc. When applying the modified Ehara model, the calibrating factor K presented in the thesis, which depends on the geometry of the APM, is only available for the APM-3600 operated with 1 lpm of flow rate. If one applied different model of the APM or different flow rate, the factor K should be newly modified with corresponding numerical results which are calculated by our detailed numerical model, as we did in Fig. 12.
Compared to the fitting model, we don’t have to change K factor of the modified Ehara model when we operate the APM with various λc. The response spectra presented in the thesis is calculated with ideal size distribution of monodisperse particles. The ideal assumption may lead the discrepancy between the calculated results and experimental ones. For example, it is found that the calculated response spectra are narrower than the experimental one, which is probability due to the size distributions of particles applied in the calculation (ideal sharp distribution) are different to the ones actually presented in the experiments (actual distribution). Another example is that the calculation is based on the assumption that all the particles passing through the APM are singly charged. In experiments, some of particles would be multiply charged, which will lead more particles loss in the APM and results in the
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lower experimental response spectra compared to the experimental ones. Despite of the fact that the calculated response spectra may be different to the experimental one, some of the problem could be solved or eased with more detailed consideration (ex: applied experimental size distribution of particles for response spectra), the calculated response spectra still can be the references for the researchers.
5 Conclusion
A 2-D numerical APM transfer function model is successfully constructed based on the governing equation of convection-diffusion equations and continuity equation. Three dimensionless numbers, which can cover ones presented in previous studies, are obtained from the governing equation of the model. These dimensionless numbers could be applied to characterize the performance of the APM.
Different calculation domains and flow fields applied to the model are discussed in the thesis. When the transfer function model is coupled with extended calculation domain (whole region in the APM) and detailed flow field, the accuracy of predicting the penetration of nanoparticles passing through the still APM is significantly increased compared to that coupled with classifying region and parabolic flow field. The maximum overestimations of the predictions are significantly reduced from 20% to 10%. The significant improvement shows that diffusion loss of particles occurs not only in the classifying region but also in the inlet and outlet paths leading to the classifying region. We concluded that the calculation domain of the transfer function model should be extended from classifying region to the whole region of the APM.
In addition, enhanced loss of particles is found when applying detailed flow field to the transfer function model. Under the similar λc, smaller nanoparticles have greater enhanced loss, while submicron particles have no enhanced loss. The study concludes that the
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enhanced is due to the recirculation flow found in the velocity profile of the detailed flow field, and the scale of recirculation flow is increased with higher rotation speed.
Two transfer function models, the fitting model and the modified Ehara model, are developed to calculate the transfer function in a more convient manner compared to the numerical model. The former is developed by fitting the numerical results of transfer functions considering the whole calculation domain and detailed flow field, while the latter is developed by modifying Ehara model using the modified Gormley and Kennedy equation which is based on numerical convection-diffusion particle loss. These models are applied to calculate the APM response spectra of the DMA-APM mass measurement system.
Compared to the experimental response spectra, the maximum inaccuracies of calculated response spectra are less than 10% in normalized particle concentration. The results show the validity of the transfer function models.
According to the results of the thesis, we expect that accurate real time mass distribution measurement of both nanoparticles and submicron particles can be realized in the future.
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Appendix A Some Properties of Previous Models
Transfer Function Model (Ehara et al., 1996) Based on 5 assumptions
a) The particles introduced in the classifying region of the APM rotated at the same angular velocity as the classifying region (annular cylinders).
b) The particle inertia, Brownian motion, the interaction between aerosol particles, and the image potential were neglected. The drag force is balanced by the electrostatic and centrifugal force
c) It is assumed that there was no flow in r and θ direction. The flow is in z direction only. Flow field in the classifying region was steady.
d) The Coriolis force was neglected due to the primary motion of the particles was parallel to the axis of rotation.
e) The distance between the inner and outer electrodes was assumed to be much smaller than their radii. (r2, r1>>r2-r1)
Transfer Function Model (Hagwood et al. 1995) Laminar flow, spherical particles, uniform density a) Stochastic Differential Equations(SDE)
Deriving PDE from the concept of escape probability of particle.
The Brownian motion in z direction was neglected (along the flow direction).
Solving PDE with a finite difference discretization along the r direction. (The complete discussions can be refer to Kahaner et al. (1989), solved by FORTRAN) b) Monte Carlo Approach (MC)
The SDE which govern the aerosol trajectory was a Langevin equation. A Monte Carlo method like that described in Risken (1984) was applied to solve the
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Langevin equation.
Considering the Brownian motion both in z and r direction; hence, it can consider the diffusion broadening effect and diffusion loss simultaneously.
MC Model needs more computation time compared to SDE Model.
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Appendix B The Geometry of Classifying Region of the APM Applied in Previous Studies
The geometry of classifying region of the APM applied in previous studies.
Geometry of Classifying Region of the APM Applied in Previous Studies
Geometry of APM Operating Condition DMA Aerosol