Deposition of Charged Aerosol Particles Flowing through Parallel Plates t

Deposition of unipolarly charged aerosol particles flowing through parallel plates has been investigated both theoretically and experimentally. The equation of motion for the particles was solved numerically in consideration of the Coulomb force, image force, particle-inertia force, and fluid-velocity profile. Then the deposition efficiency was calculated based on the limiting trajectories of the particles. It was found that the calculated deposition efficiency, assuming a laminar flow, was a little smaller than the corresponding analytical solution obtained for the plug flow distribution. It was also found that the effect of the image force and particle inertia force was negligible in the experimental range. The experimental deposition efficiencies for charged fly-ash particles were well explained in so far as the actual velocity distributions were taken into consideration.


Introduction
The deposition of charged aerosol particles suspended in the air flow is enhanced by the effect of electric field caused by the particles themselves. This effect, called the space-charge effect, may be applicable to practical use such as dust collection. The reduction of the collection efficiency due to back discharge in an electrostatic precipitator has become a serious problem in recent years. This problem may be successfully solved and the collection efficiency will become predictable to considerable extent when the dust particles are precharged and then captured with the aid of the space-charge effect.
The deposition of monodispersed, charged particles on the circular tube due to the spacecharge effect has been studied analytically by Wilson to) and experimentally by Kasper6), Adachi et al. 1 ), and the author et al. 7 ) These experimental results are found to have reasonable agreement with the analytical solution derived by Wilson. Adachi et al. have also studied the effect of image force. For the deposition of charged aerosol particles in parallel channel flow, the effects of space charge and thermal diffusion have been studied by Ingham 5 ) and Chen 2 ), the effect of image force by Yu et al. 11 • 12 ), and the studies on the gravity effect has been reported elsewhere 3 • 9 ). Unfortunately, however, there have been a very few reports which analyze these problems synthetically.
In this paper, the deposition efficiency of unipolarly charged aerosol particles flowing in a parallel channel was theoretically determined based on the critical particle trajectory, which is obtained through numerical integration of the equation of motion for the particles. In the numerical calculation, the effects of space charge, image force, particle-inertia force, and air-velocity profile were taken into considerations. Further, the deposition of unipolarly charged fly-ash particles was experimentally investigated, and the results were compared with the theoretical predictions.

Theory
Assuming a plug flow, a parallel plate model of L in length, 2Z 1 in width, and 2X 1 in space between the plates is considered with the coordinate system illustrated in Fig. 1. When unipolarly charged aerosol particles flow in to the model channel, the X-component of the intensity of electric field at the position X 0 is given by the following equation: (l) where N 0 is the particle number per unit space volume, q is the electric charge on a particle and e 0 is the air permittivity. The Y-and Zcomponents of the electrostatic field are negligible in the following analysis. Also, the electric force on a particle is given by (2) If the particles are assumed to be monodisperse, the Coulomb force Fe on a particle is unchanged throughout the movement. For this case, X-component of the particle velocity is determined by the force balance between the force Fe and the fluid resistance given by the Stokes law, as follows: The parameter a in Eq. (3) is defined by the following equation: where f..L is the air viscosity, DP is the particle 18 diameter and C is the Cunningham's slip correction factor. Thus the displacement of a particle in the X-direction is where T is the average residence time. The deposition efficiency TJ is defined as By use of the following definitions: where C 0 is the mass concentration of the particles, mP is the particle mass, q is the electric charge on a unit mass of the particles, Uy is the average air velocity, and P P is the particle density.
Although the particles are assumed to move along the plate (in the Y-direction) with the average air flow velocity, the effect of the actual velocity profile should be taken into consideration in a practical use. In addition, the inertia force and the image force should also be considered for relatively coarser parti-des and for low particle concentration, respectively.
Assuming that inlet air flow is fully developed so as to form a laminar velocity profile and the gravity effect is negligible, the equations of motion for particles are: (8) for Y-component where Fe is the Coulomb-force given by Eq. (2) and F; is the image force given by the following equation: If the air flow is assumed to be laminar, the air flow velocity is expressed as for X-component Ux = 0 (11) for Y-component (12) By substituting Eqs. (11) and (12) into Eqs. (8) and (9) respectively, the following non-dimensional equations will be obtained; where x = X/X 1 , y = Y/X1 and t = Dy T/X1. The Coulomb-force parameter Ke and the image-force parameter K; may be defined by the following non-dimensional forms; Using the above parameters, Eq. (13) is rewritten in non-dimensional form as follows: The numerical integration of Eqs. (14) and (18) in combination yields the particle trajectories. Considering the particle which travels from the initial point X 0 to the boundary point (X = X 1 ) at the outlet of the parallel channel (Y = L), the deposition efficiency can be determined by the following equation; In this study, the numerical integration based on the Runge-Kutta-Merson method was applied to Eqs. ( 14) and (18) for various initial conditions in order to obtain the deposition efficiency by use of Eq. (19).
Equations (14) and (18) show that the deposition efficiency 11 is a function of P, Ke, K;, L, and X 1 • Figure 2 shows a typical set of correlations between 11 and Ke when K; = 0 and L/X 1 = 100. All the curves using the parameter P indicate that 11 increases with an increase in Ke, approaching the dotted line as P decreases. This dotted line represents the ,.. parameter under the conditions of P=O and L/X 1 = 100, where 71Ki=o represents the deposition efficiency for Ki = 0. The ratio 71/TIK·=o increases with increasing the ratio KJ Ke: This means that the influence of image force may be conspicuous with decreasing the particle number concentration and/ or the space between the plates in consideration of Eq. (20).

3-1 Experimental apparatus and method
The outline of the experimental arrangement   during I min operation with a range of I 0 to 400 mg/s. Figure 6 shows the outlook of the test plates with 39 em height, 18 em width, and I m length in which several steel plates were arranged in parallel and mounted with PVC hold- ing spacers on top and bottom sides. The space between each test plate was to be varied with the spacer thickness. The flow after the charging zone was enlarged toward the deposition zone and the disk of 4 em diameter with the attitude perpendicular to the air flow was mounted at the center of the enlarged duct. Electric charges on aerosol particles were measured with a Faraday cage and an electric meter (made by Takeda Riken Co., Ltd.) as illustrated by the dotted line in Fig. 5. In addition, the size distribution of aerosol particles was determined with a cascade impactor (made by Nippon Kagaku Kogyo Co., Ltd.).

2 Experimental results and discussion
The particle inertia parameter P and the image force parameter Ki were observed to be up to IQ-2 and IQ-6 , respectively. Thus, the effects of such parameters would be negligible in the numerical calculation. When it is assumed that P = 0 and K; = 0, Ke multiplied by L/X 1 gives the following new parameter cxr which is independent of the space between the plates. It has been recognized from the laboratory deposition experiment that the deposition efficiency may depend upon the plate material. In this work, we employed aluminum as well as PVC to examine the spacer effect. The result suggests the effect of the spacer material would be negligible as shown in Fig. 7 where the symbol • represents the measured value which was obtained by use of aluminum spacers.
It should be noticed that the actual velocity distribution of air flow would differ from the assumed one such as laminar or plug flow discussed above because of a lack of flow uniformity caused by the duct enlargement.
One of the velocity profiles at the inlet of the test plates measured by a wire anemometer is typically illustrated in Fig. 8. This figure, which is based on the non-dimensionalized coordinate system that corresponds to each inlet section in Figs. 1 and 6, implies that the flow velocity might be maximized at the center and the backward flow due to flow separation might take place near the wall. These phenomena are caused by the influence of duct enlargement, and they should be taken into consideration in a practical use.
The deposition efficiencies obtained from the modified flow distribution and the experimental flow distribution based on the actual profile indicated in Fig. 8 are also shown in The experimental relationship between deposition efficiency and plate space 2X 1 at ar = 4.2 is shown in Fig. 9, where the solid line represents a theoretical solution based on the assumption of the laminar flow. The deposition efficiency was found to be independent of the plate space except that 2X 1 = 1.2 em. The deposition efficiency at 2X 1 = 1.2 em is close to the theoretical solution, and this agreement might result from the flow rectification effect of the parallel plates.
It is well-known that electric discharge takes place in atmosphere when the electric field of more than 30kV /em is applied. If this discharge occurs between the aerosol particles and the channel wall, the deposition efficiency may be reduced because of immediate disappearance of the electrostatic force on the particle. The intensity of electric field formed by charged particles in the channel is given by Eq. (1) when the space charge density is held constant at the inlet. This intensity increases to the maximum value very near the wall, and thus its maximum value will be in proportion to the plate space. In this work, this intensity was found to be up to 10 kV /em when no steel plates were inserted (in other words, only the outer case of the test box shown in Fig. 6 was used).
Although it was ascertained that this intensity was smaller than the critical value for initiating atmospheric discharge, experimental studies were carried out on the distributions of particle charge and particle concentration of charged aerosols at the inlet of the test plates by use of a Faraday cage. The particle charge distributions at the inlet of the channel along the X (width) and Z Dimentionless coordinate XI X, [ -] and (b) respectively. The symbols 0 and 6. in the figure denote the sampling in forward and in backward directions, respectively. With the sampling in each direction for the region where flow separation took place, it was found that the particle charge distribution in the chamber was nearly uniform. Consequently, it is reasonably considered that no electric discharge between particles and a wall of the equipment might take place. Figures 11 (a) and (b) indicate the distributions of particle concentration in X and Z directions, respectively. The curve represents the velocity profile at the inlet of the parallel plate chamber. Although the particle concentration slightly increased near the wall, the correlation between particle concentration and velocity profile was not found.
It this work, the influences of deposition by gravity and by diffusion calculated from the equation derived by Marcus 8 ) were found to be up to 1% and up to 0.01 %, respectively, showing that the influences might be reasonably neglected.

Conclusion
The deposition of unipolarly charged aerosol particles flowing through parallel plates were investigated both theoretically and experimentally.
Summarizing the results of this work: (1) The deposition efficiency increased with increasing the Coumb-force parameter and the theoretical solution based on the assumption of laminar flow was slightly smaller than that for plug flow. When the image force and particle-inertia force were negligibly small, the deposition efficiency was expressed as a function of the unique parameter cxr which did not depend on the space between the parallel plates. (2) The deposition efficiency decreased with an increase in particle-inertia parameter P. The influence of the inertia was, however, reasonably negligible if P was less than 0.1. ( 4) The experimental deposition efficiency was slightly smaller than the theoretical one obtained by the assumption of laminar flow. This fact could be explained to some extent by considering the actual flow distribution. (5) The distributions of particle concentration and particle charge were found to be nearly uniform, and the electric discharge was not observed.