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Inertia effect on deposition of charged particles in a parallel-plate channel
Powder
Technology,
Inertia Effect R. Y. CHEN,
N.
J. Institute
(Received
34
(1983) 249 - 253
on Deposition
H. E. PAWEL
of Charged
Particles in a Parallel-Plate
channel
and W_ C. CHEN
of Technology.
March 23,1981;
249
Newark,
NJ
07102
(U.S.A.)
in revised form July 2, 1982)
SUMMARY The effect of inertia forces on the deposition of highly charged fine particles of sizes less than 20 pm in a parallel-plate channel has been analyzed numerically for uniform and fully-developed flows. The deposition is primarily due to space charge alone and the image and gravity forces are not included in the analysis. A charge-inertia parameter n (ranging from 0 to 1) was defined to characterize the flow deposition phenomena. It was found that for n less than 0-l the effect of inertia forces may be neglected and that the fraction of deposition near the entrance of the channel deviates substantially from the result neglecting the inertia effect_
INTRODUCTION When charged particles suspended in gas flow through a parallel-plate channel, deposition may take place on the wall. Theoretical investigations on deposition in channels have been carried out by Yu Cl], Chen 121, Ingham [3], and recently by Chen and Gelber [4]. In these analyses the continuity equation for the particle phase and the Poisson equation for the electrostatic field intensity were employed with the boundarylayer type assurnptionHowever, the axial diffusive term in the continuity equation was neglected and, thus, the axial velocity of the fluid was considered
to be the same as that of the particle. That is to say, the particle inertia has been completely neglected. The particle penetration through the channel at any given crosssection in these analyses was obtained by integration of the product of the fluid velocity uf and the particulate mass density p 0032-5910l33/0000-00001$30.00
with respect to the cros§ional area A (i-e_ Jufp cIA)_ The purpose of this analysis is to study the effect of particle inertia on the deposition of aerosol in a parallel-plate channel_ Uniform as well as fully-developed laminar flow of fluid suspended with highly charged particles in a parallel-plate channel is analyzed using a trajectory method.
ANALYSIS The equations of motion for a particle suspended in laminar two-dimensional flow are du rndt
dv
mdt
=6npa(uf-u)+B,
= 6xpa (vf-v)
(1) + B,-
(2)
where m is the particle mass, u and v are the particle velocities in the x- and y-direction respectively and t is the time. B, and BY are body forces in the x- and y-direction respectively, p is the dynamic viscosity of the fluid, and a is the particle radius- Sub-
script f is used to identify the fluid properties_ Take the x-direction as vertically downward_ Then the body force in the x-direction may be due to gravity when the flow is in a vertical channel. It may also include the force due to electrostatic field intensity in the x-direction_ The body force in the ydirection, on the other hand, may be due to gravity for a horizontal channel_ It may also include forces due to electrostatic field intensity in the y-direction or an image force of the charge on a particle when boundaries are grounded conductors_ @ Elsevier Sequoia/Printed in The Netherlands
-250 In order to investigate and assess clearly how the inertia force affects the deposition, the body force considered in this study will be that due to electrostatic field intensity alone_ This situation may arise when very fine particles, suspended in gas, flow in a vertical channel and also in a horizontal channel. For flow in a vertical channel, the body forces are B, = mg and B,. = qe. In these ezpressions the axial component (i.e. x-component directed downward) of electrostatic field intensity has been neglected. To neglect the gravity force in a vertical channel we must have 67r~au,+ mg where u,, is the mean velocity at the inlet of the channel. For flow in a horizontal channel, the body forces are = qe - mg. Clearly to neglect B,=OandB, the gravity force mg in a horizontal channel, qe > mg must be satisfied_ The electrostatic fieid intensity is governed by ae -=-
ay
p9
(3)
cm
where p is the particle density (particle mass concentration), c is the permittivity of the free space and m is the mass of a particle_ In eqn. (3), the electrostatic field intensity in the axial direction (x-direction) has been neglected. To estimate the magnitude of the electrostatic field intensity at the wall, y = h, where h is the half-width of the channel, we may assume a uniform particle density and. integrate eqn. (3) to obtain e =pqh/cm. For inlet uniform particle density of po, the electrostatic field intensity at the wall is eX = poghlcm. To neglect the gravity effect in the analysis, we must have poq2/cm S- mg or poq2h/cm2F > u,, where F = 6xpafm is the inverse of the transition time and cE = g/F is the terminal velocity.
To investigate the magnitude of the various forces (i.e. diffusive, charge, gravity and inertia) in a flow, the following dimensionless parameters are presented and compared. ~ =
poq2h’
y
1
diffusive charge force
cm ‘g
gravity force
Poq2h cm2F2
-
us _
gravity force
u.
viscous force
(Poq’hlcm)
= (charge force/viscous eter)
force)
6npah (inertia param-
The charge-inertia parameter n is the product of the charge-flow parameter and the inertia parameter (or Stokes number)_ This parameter is to be introduced in this analysis later. As an example, consider a particle concentration of 10’ particles per 1 cm3 and a charge electron density of 1 electron per l-18 X lo-i0 cm2 (i-e_ 29 electrons on a 0.33 pm particle as was given in reference [3]). Other variables are assumed to be u. = 30 cm/s, h = 2 cm, pressure = 1 atm, temperature = 25 OC, and particle specific gravity of 1. The results are listed in Table l_ At 10 lun diameter size, the charge per unit mass is 8.1 X 10m6 C/g which is of the same order of magnitude as those measured by Cheng and Soo [5]. If the surface charge density is halved, the values of 6 and St will remain the same, but the values of CY,y and n will be one-quarter of the original values_ For flow in a channel, the diffusive force may be neglected when the dimensionless
Comparison ofdimensionlessparameters Particle diameter&n
a
Y
10006
R
0.33 26 10 20
5_19E4* 7.01E7 5.6739 4.38310 7.01Ell
264 1600 4810 8010 16100
0.01 0.38 3.41 9-47 37.9
l-293-8 0.0001 O-026 0.33 10.6
Stokesnumber(=10-4
mu0 x-
(67wauo)
TABLE1
*5.19E4 = 5.19 X104,**St=
force
poQ2h =
g=mg=__ 6npau, n -
charge force
=
4cm2FD
m2 s-l)
1ooost**
251
parameter QI is greater than 50 [3]. Clearly, the diffusive forces for all cases listed in Table 1 may be neglected. For flow in a vertical parallel-plate channel, the gravity effect becomes important when the velocity ratio 6 = us/u0 is greater than 0.02. For a horizontal channel flow, the gravity effect may be neglected when the ratio of the electrostatic force to the gravity force ris greater than 100. Inspection of Table 1 shows that for highly charged particles of 10 to 20 m diameter, the inertia effect (n = 0.33 and 10.6) may become importent, while the gravity effect may be neglected (y > 100) It is interesting to note that when diffusion is the dominant force on deposition, a p-Aicle as small as 1 pm may have significant gravity effect on the deposition. The particle density p for uniform and fully-developed flow is governed by the continuity equation L(-=-aP
a (qPe/6wa
a_x
)
(4)
aY
In eqn. (4) the diffusive force and the mass flux in the x-direct.ion p(v- - ZQ) have been neglected. Equations (3) and (4) are approximate expressions and are used only for the determination of the electrostatic field intensity e which appears in eqn_ (2)_ The governing eqns. (1) through (4) are non-dimensionalized with the following dimensionless parameters:
pq2
n=
X=
cm2F2
XPC12 cm2Fu,
- tpq= cm’F
T=
ffr_
j-J=
R=
u0
2 PO
emc hPoS
EC
The governing
equations
become
dE -
dY
=R
(7)
(S) The boundary and initial conditions in dimensionless forms are, for eqn. (3) and ? eqn- (4), R=landE=Y
atX=OandO<
Y<
and for eqn. (1) and eqn. (2) X=0, dx
-
dT
Y=
=
Y,,,E=
Uf(O,Yo)
Y,,
and g
= 0
atT=O_
In this analysis, the inlet. particle density R is assumed to be uniform_ The channel walls are assumed to be grounded conductors and due to symmetry the electrostatic field intensity E is zero along the centerline Y = 0 which results in E = Y at the inlet plane_ The particle under consideration is initially located at Y = Y. with its axial velocity. dX/dT equal to that of t-he local f’luid velocity Ut(O, Yo) and with zero transverse velocity dYjdT = O_ The velocity profile for the fluid is assumed to be either uniform U, = 1 or fully-deseloped profile U, = 1 - I”_ Starting with an initial location of a particle at the inlet plane, the trajectory of the particle was calculated by integrating eqn. (5) and eqn. (6) with the Runge-Kutta method, and eqn. (7) and eqn. (S) with backward difference forms using 41 nodal points between Y = 0 (the centerline) and 1 (the wall). The time increment was chosen in such a way that AY < 0.003 and 14 dXid771 < 0.005. .A particle is considered to have deposited on the wall as soon as it reaches the wall at (X, 1). Since all of the particles t.hat entered the inlet plane between Y = Y. and Y = 1 would also have deposited on the wall between X = 0 and X, the fraction of deposition at X is calculated from 1
d’Y
DEPOSITION
(5)
n aTZ d’Y ndT2
=E-
d$
(6)
1
= l-
f
URdY
YO
where U is the given inlet particle velocity (ie_ Uf) and R is the inlet particle density (ie_ R = 1)
252 RESULTS
AND
DISCUSSIONS
In this analysis, the particle velocity and the fluid velocity at the inlet plane were assumed to be identical, i-e_ U= U, and V = V, = 0 at X = 0. The electrostatic field intensity E was based on the approximate particle continuity eqn. (8) which neglected the axial flux of the flow by letting U = U,. The gravity effect in the equations of motion for the particle has also been neglected_ It is, therefore, necessary to limit our analysis on flow with small inertia effect such that n<
1.
Figures 1 and 2 present the deposition for n = 0 through 1 in a channel at X < 1 03
a
Fig_ l_ Deposition at small axial distance in a channel for uniform velocity profile with dYfdT = 0 at X = 0.
for uniform velocity profile and parabolic (or fully-developed) velocity profile respectively_ The charge-inertia parameter, n = (charge force/viscous force)(inertia parameter), is the product of the charge effect and the inertia effect, while the dimensionless axial distance, X = (r/-k) (charge force/viscous force), is the product of distance and charge force. It is seen that these figures depict the effect of inertia on the deposition for any given charge on a particle. Figures 1 and 2 show that the deposition decreases with increasing inertia. For n = 0 (i-e_ no inertia effect), the deposition becomes equal to the result obtained from the solution of eqn. (7) and eqn_ (8)_ The deviation in deposition between n = 0 and n > 0 increases with X at small X and then continues to decrease as X is increased. The maximum deviation at n = 0.1 for the uniform flow is O-07 and that for the parabolic velocity profile is O-075_ The maximum deviation at n = O-05 for both velocity profiles is 0.04 at X = O-2. The deposition at large X is shown in Fig_ 3 for both uniform and parabolic velocity profiles. As is seen from the figure, at large X the deviation in deposition between n = 0 and n > 0 continues to decrease as X is increased_ Since the fraction of penetration, which is one minus the deposition, for the urnform velocity profile at n = 0 is l/(1 + X), use of 1 +X as the abscissa will result in a straight line in the figure_ It is seen from
.
am
,
I.0
.*
.
. . . .
.
. mo
ID
_
Fig. 2. Deposition at small axial distance in a channel for parabolic velocity profile with dYfdT = 0 at X = 0.
Fig. 3. Penetration x = 0_
in a chakzl
with dY/dT
= 0 at
253
Fig 3 that for X > 2 there is an insignificant difference in deposition for n < 0.1. Since X = (xlh)njSt > 2, we have x/h > 2 St/n_ Under the conditions given in Table 1 the axial distances for X = 2 is x/h < O-04_ This implies that, for all practical purposes, the effect of inertia may be neglected when n < 0-l and that under highly charged conditions most deposition takes place near the entrance of the channel. The above results are based on the initial condition that dY/dT = 0. This is to say that the particle has not exposed itself to the space charge before entering the channel. In many practical situations, the fine particles entering the channel may have been already exposed to the space for a sufficient length of time to acquire some velocity in the y-directionFigures 4 and 5 are the results based on dY/dT = E(0, Ya) = Y,-, at X = O_ For a para(14
001
.
* 1 ,+,z*!
01
x
10
Fig. 4. Deposition at srnall axial distance in a channel with dYldT = Yo at X = 0. bolic velocity profile, the effect of inertia on the deposition is less than that shown in Fig. 1 through Fig. 3. For a uniform velocity profile, the axial velocity and the transverse velocity of a particle remain constant throughout the flowfield. Consequently, the deposition is. independent of n and is equal to the deposition calculated from 1 - l/(1 + X). Based on dY/dT = 0 at X = 0, the distances required for a particle to gain dY/dT = 0.9Yc
10 i-x
1.0 Fig. 5_ Penetration x = 0.
in a channel with dY/dT
= Y0 a:
for n = O-01 and O-1 are O-023 and 0.37 respectively_ For n = I, the maximum dY/dT that a p&icle can attain is only 0_59Y,. Therefore, a reasonable initial condition for particles with n = 1 should be dY1dT-E 0_59Y,_ To investigate the effect of charge on the deposition, one should keep in mind that the dimensionless axial distance X is the product of the charge effect and the axial distance_ When the charge is increased by a factor of 2 the dimensionless axial disincreased by a factor of 2 i)_ This-will give higher deposicharges on a In conclusion, the analysis has shown that for highly charged particles entering a parallelplate channel with uniform particle density, the deposition decreases with increasing inertia and that the effects of inertia may be neglected when n < 0-L
REFERENCES
1 C. P_ Yu. I Aerosol
Sci.. 8 (1977) 237 - 241. 2 R. Y. Chen, J_ Aerosol Sci_, 9 (19’75) 253 - 260. 3 D. B. Ingham, J. Aerosol Sci. II (1979) 47 - 52. 4 R Y_ Chen and %I_ IV_ Gelber, Powder TechnoL. 2S (1981) 229 - 234. 5 L. Cheng and S. L. Soo, J_ AppZ. Phys_. 41 (1970) 585.
Technology,
Inertia Effect R. Y. CHEN,
N.
J. Institute
(Received
34
(1983) 249 - 253
on Deposition
H. E. PAWEL
of Charged
Particles in a Parallel-Plate
channel
and W_ C. CHEN
of Technology.
March 23,1981;
249
Newark,
NJ
07102
(U.S.A.)
in revised form July 2, 1982)
SUMMARY The effect of inertia forces on the deposition of highly charged fine particles of sizes less than 20 pm in a parallel-plate channel has been analyzed numerically for uniform and fully-developed flows. The deposition is primarily due to space charge alone and the image and gravity forces are not included in the analysis. A charge-inertia parameter n (ranging from 0 to 1) was defined to characterize the flow deposition phenomena. It was found that for n less than 0-l the effect of inertia forces may be neglected and that the fraction of deposition near the entrance of the channel deviates substantially from the result neglecting the inertia effect_
INTRODUCTION When charged particles suspended in gas flow through a parallel-plate channel, deposition may take place on the wall. Theoretical investigations on deposition in channels have been carried out by Yu Cl], Chen 121, Ingham [3], and recently by Chen and Gelber [4]. In these analyses the continuity equation for the particle phase and the Poisson equation for the electrostatic field intensity were employed with the boundarylayer type assurnptionHowever, the axial diffusive term in the continuity equation was neglected and, thus, the axial velocity of the fluid was considered
to be the same as that of the particle. That is to say, the particle inertia has been completely neglected. The particle penetration through the channel at any given crosssection in these analyses was obtained by integration of the product of the fluid velocity uf and the particulate mass density p 0032-5910l33/0000-00001$30.00
with respect to the cros§ional area A (i-e_ Jufp cIA)_ The purpose of this analysis is to study the effect of particle inertia on the deposition of aerosol in a parallel-plate channel_ Uniform as well as fully-developed laminar flow of fluid suspended with highly charged particles in a parallel-plate channel is analyzed using a trajectory method.
ANALYSIS The equations of motion for a particle suspended in laminar two-dimensional flow are du rndt
dv
mdt
=6npa(uf-u)+B,
= 6xpa (vf-v)
(1) + B,-
(2)
where m is the particle mass, u and v are the particle velocities in the x- and y-direction respectively and t is the time. B, and BY are body forces in the x- and y-direction respectively, p is the dynamic viscosity of the fluid, and a is the particle radius- Sub-
script f is used to identify the fluid properties_ Take the x-direction as vertically downward_ Then the body force in the x-direction may be due to gravity when the flow is in a vertical channel. It may also include the force due to electrostatic field intensity in the x-direction_ The body force in the ydirection, on the other hand, may be due to gravity for a horizontal channel_ It may also include forces due to electrostatic field intensity in the y-direction or an image force of the charge on a particle when boundaries are grounded conductors_ @ Elsevier Sequoia/Printed in The Netherlands
-250 In order to investigate and assess clearly how the inertia force affects the deposition, the body force considered in this study will be that due to electrostatic field intensity alone_ This situation may arise when very fine particles, suspended in gas, flow in a vertical channel and also in a horizontal channel. For flow in a vertical channel, the body forces are B, = mg and B,. = qe. In these ezpressions the axial component (i.e. x-component directed downward) of electrostatic field intensity has been neglected. To neglect the gravity force in a vertical channel we must have 67r~au,+ mg where u,, is the mean velocity at the inlet of the channel. For flow in a horizontal channel, the body forces are = qe - mg. Clearly to neglect B,=OandB, the gravity force mg in a horizontal channel, qe > mg must be satisfied_ The electrostatic fieid intensity is governed by ae -=-
ay
p9
(3)
cm
where p is the particle density (particle mass concentration), c is the permittivity of the free space and m is the mass of a particle_ In eqn. (3), the electrostatic field intensity in the axial direction (x-direction) has been neglected. To estimate the magnitude of the electrostatic field intensity at the wall, y = h, where h is the half-width of the channel, we may assume a uniform particle density and. integrate eqn. (3) to obtain e =pqh/cm. For inlet uniform particle density of po, the electrostatic field intensity at the wall is eX = poghlcm. To neglect the gravity effect in the analysis, we must have poq2/cm S- mg or poq2h/cm2F > u,, where F = 6xpafm is the inverse of the transition time and cE = g/F is the terminal velocity.
To investigate the magnitude of the various forces (i.e. diffusive, charge, gravity and inertia) in a flow, the following dimensionless parameters are presented and compared. ~ =
poq2h’
y
1
diffusive charge force
cm ‘g
gravity force
Poq2h cm2F2
-
us _
gravity force
u.
viscous force
(Poq’hlcm)
= (charge force/viscous eter)
force)
6npah (inertia param-
The charge-inertia parameter n is the product of the charge-flow parameter and the inertia parameter (or Stokes number)_ This parameter is to be introduced in this analysis later. As an example, consider a particle concentration of 10’ particles per 1 cm3 and a charge electron density of 1 electron per l-18 X lo-i0 cm2 (i-e_ 29 electrons on a 0.33 pm particle as was given in reference [3]). Other variables are assumed to be u. = 30 cm/s, h = 2 cm, pressure = 1 atm, temperature = 25 OC, and particle specific gravity of 1. The results are listed in Table l_ At 10 lun diameter size, the charge per unit mass is 8.1 X 10m6 C/g which is of the same order of magnitude as those measured by Cheng and Soo [5]. If the surface charge density is halved, the values of 6 and St will remain the same, but the values of CY,y and n will be one-quarter of the original values_ For flow in a channel, the diffusive force may be neglected when the dimensionless
Comparison ofdimensionlessparameters Particle diameter&n
a
Y
10006
R
0.33 26 10 20
5_19E4* 7.01E7 5.6739 4.38310 7.01Ell
264 1600 4810 8010 16100
0.01 0.38 3.41 9-47 37.9
l-293-8 0.0001 O-026 0.33 10.6
Stokesnumber(=10-4
mu0 x-
(67wauo)
TABLE1
*5.19E4 = 5.19 X104,**St=
force
poQ2h =
g=mg=__ 6npau, n -
charge force
=
4cm2FD
m2 s-l)
1ooost**
251
parameter QI is greater than 50 [3]. Clearly, the diffusive forces for all cases listed in Table 1 may be neglected. For flow in a vertical parallel-plate channel, the gravity effect becomes important when the velocity ratio 6 = us/u0 is greater than 0.02. For a horizontal channel flow, the gravity effect may be neglected when the ratio of the electrostatic force to the gravity force ris greater than 100. Inspection of Table 1 shows that for highly charged particles of 10 to 20 m diameter, the inertia effect (n = 0.33 and 10.6) may become importent, while the gravity effect may be neglected (y > 100) It is interesting to note that when diffusion is the dominant force on deposition, a p-Aicle as small as 1 pm may have significant gravity effect on the deposition. The particle density p for uniform and fully-developed flow is governed by the continuity equation L(-=-aP
a (qPe/6wa
a_x
)
(4)
aY
In eqn. (4) the diffusive force and the mass flux in the x-direct.ion p(v- - ZQ) have been neglected. Equations (3) and (4) are approximate expressions and are used only for the determination of the electrostatic field intensity e which appears in eqn_ (2)_ The governing eqns. (1) through (4) are non-dimensionalized with the following dimensionless parameters:
pq2
n=
X=
cm2F2
XPC12 cm2Fu,
- tpq= cm’F
T=
ffr_
j-J=
R=
u0
2 PO
emc hPoS
EC
The governing
equations
become
dE -
dY
=R
(7)
(S) The boundary and initial conditions in dimensionless forms are, for eqn. (3) and ? eqn- (4), R=landE=Y
atX=OandO<
Y<
and for eqn. (1) and eqn. (2) X=0, dx
-
dT
Y=
=
Y,,,E=
Uf(O,Yo)
Y,,
and g
= 0
atT=O_
In this analysis, the inlet. particle density R is assumed to be uniform_ The channel walls are assumed to be grounded conductors and due to symmetry the electrostatic field intensity E is zero along the centerline Y = 0 which results in E = Y at the inlet plane_ The particle under consideration is initially located at Y = Y. with its axial velocity. dX/dT equal to that of t-he local f’luid velocity Ut(O, Yo) and with zero transverse velocity dYjdT = O_ The velocity profile for the fluid is assumed to be either uniform U, = 1 or fully-deseloped profile U, = 1 - I”_ Starting with an initial location of a particle at the inlet plane, the trajectory of the particle was calculated by integrating eqn. (5) and eqn. (6) with the Runge-Kutta method, and eqn. (7) and eqn. (S) with backward difference forms using 41 nodal points between Y = 0 (the centerline) and 1 (the wall). The time increment was chosen in such a way that AY < 0.003 and 14 dXid771 < 0.005. .A particle is considered to have deposited on the wall as soon as it reaches the wall at (X, 1). Since all of the particles t.hat entered the inlet plane between Y = Y. and Y = 1 would also have deposited on the wall between X = 0 and X, the fraction of deposition at X is calculated from 1
d’Y
DEPOSITION
(5)
n aTZ d’Y ndT2
=E-
d$
(6)
1
= l-
f
URdY
YO
where U is the given inlet particle velocity (ie_ Uf) and R is the inlet particle density (ie_ R = 1)
252 RESULTS
AND
DISCUSSIONS
In this analysis, the particle velocity and the fluid velocity at the inlet plane were assumed to be identical, i-e_ U= U, and V = V, = 0 at X = 0. The electrostatic field intensity E was based on the approximate particle continuity eqn. (8) which neglected the axial flux of the flow by letting U = U,. The gravity effect in the equations of motion for the particle has also been neglected_ It is, therefore, necessary to limit our analysis on flow with small inertia effect such that n<
1.
Figures 1 and 2 present the deposition for n = 0 through 1 in a channel at X < 1 03
a
Fig_ l_ Deposition at small axial distance in a channel for uniform velocity profile with dYfdT = 0 at X = 0.
for uniform velocity profile and parabolic (or fully-developed) velocity profile respectively_ The charge-inertia parameter, n = (charge force/viscous force)(inertia parameter), is the product of the charge effect and the inertia effect, while the dimensionless axial distance, X = (r/-k) (charge force/viscous force), is the product of distance and charge force. It is seen that these figures depict the effect of inertia on the deposition for any given charge on a particle. Figures 1 and 2 show that the deposition decreases with increasing inertia. For n = 0 (i-e_ no inertia effect), the deposition becomes equal to the result obtained from the solution of eqn. (7) and eqn_ (8)_ The deviation in deposition between n = 0 and n > 0 increases with X at small X and then continues to decrease as X is increased. The maximum deviation at n = 0.1 for the uniform flow is O-07 and that for the parabolic velocity profile is O-075_ The maximum deviation at n = O-05 for both velocity profiles is 0.04 at X = O-2. The deposition at large X is shown in Fig_ 3 for both uniform and parabolic velocity profiles. As is seen from the figure, at large X the deviation in deposition between n = 0 and n > 0 continues to decrease as X is increased_ Since the fraction of penetration, which is one minus the deposition, for the urnform velocity profile at n = 0 is l/(1 + X), use of 1 +X as the abscissa will result in a straight line in the figure_ It is seen from
.
am
,
I.0
.*
.
. . . .
.
. mo
ID
_
Fig. 2. Deposition at small axial distance in a channel for parabolic velocity profile with dYfdT = 0 at X = 0.
Fig. 3. Penetration x = 0_
in a chakzl
with dY/dT
= 0 at
253
Fig 3 that for X > 2 there is an insignificant difference in deposition for n < 0.1. Since X = (xlh)njSt > 2, we have x/h > 2 St/n_ Under the conditions given in Table 1 the axial distances for X = 2 is x/h < O-04_ This implies that, for all practical purposes, the effect of inertia may be neglected when n < 0-l and that under highly charged conditions most deposition takes place near the entrance of the channel. The above results are based on the initial condition that dY/dT = 0. This is to say that the particle has not exposed itself to the space charge before entering the channel. In many practical situations, the fine particles entering the channel may have been already exposed to the space for a sufficient length of time to acquire some velocity in the y-directionFigures 4 and 5 are the results based on dY/dT = E(0, Ya) = Y,-, at X = O_ For a para(14
001
.
* 1 ,+,z*!
01
x
10
Fig. 4. Deposition at srnall axial distance in a channel with dYldT = Yo at X = 0. bolic velocity profile, the effect of inertia on the deposition is less than that shown in Fig. 1 through Fig. 3. For a uniform velocity profile, the axial velocity and the transverse velocity of a particle remain constant throughout the flowfield. Consequently, the deposition is. independent of n and is equal to the deposition calculated from 1 - l/(1 + X). Based on dY/dT = 0 at X = 0, the distances required for a particle to gain dY/dT = 0.9Yc
10 i-x
1.0 Fig. 5_ Penetration x = 0.
in a channel with dY/dT
= Y0 a:
for n = O-01 and O-1 are O-023 and 0.37 respectively_ For n = I, the maximum dY/dT that a p&icle can attain is only 0_59Y,. Therefore, a reasonable initial condition for particles with n = 1 should be dY1dT-E 0_59Y,_ To investigate the effect of charge on the deposition, one should keep in mind that the dimensionless axial distance X is the product of the charge effect and the axial distance_ When the charge is increased by a factor of 2 the dimensionless axial disincreased by a factor of 2 i)_ This-will give higher deposicharges on a In conclusion, the analysis has shown that for highly charged particles entering a parallelplate channel with uniform particle density, the deposition decreases with increasing inertia and that the effects of inertia may be neglected when n < 0-L
REFERENCES
1 C. P_ Yu. I Aerosol
Sci.. 8 (1977) 237 - 241. 2 R. Y. Chen, J_ Aerosol Sci_, 9 (19’75) 253 - 260. 3 D. B. Ingham, J. Aerosol Sci. II (1979) 47 - 52. 4 R Y_ Chen and %I_ IV_ Gelber, Powder TechnoL. 2S (1981) 229 - 234. 5 L. Cheng and S. L. Soo, J_ AppZ. Phys_. 41 (1970) 585.