- Email: [email protected]
Continuum field-diffusion theory for bipolar charging of aerosols
J. Aerosol Sci.. Vol. 14, N o 4. pp 541-556, 1983. Printed in Greal Britain
0021-8502/83 $3.00+0.00 ~ 1983 Pergamon Press Ltd.
C O N T I N U U M F I E L D - D I F F U S I O N THEORY FOR BIPOLAR CHARGING OF AEROSOLS ROBERT A. FJELD Department of Environmental Systems Engineering, Clemson University, Clemson, SC 29631, U.S.A.
RANDALLO. GAt~NTr* Department of Nuclear Engineering
and ANDREW R. MCFARLAND Department of Civil Engineering, Texas A&M University, College Station, TX 77843, U.S.A. (First received 9 September 1982 and in revised form 29 November 1982) A h s t r a c t ~ o n t i n u u m regime field-diffusion theory for the acquisition of charge by particles exposed to bipolar ions is developed. An approximate numerical solution to the governing equations is described which, in the limit as the external field approaches zero, yields results within 1 ~ of exact solutions. The numerical method is used to calculate particle charge as a function of time, external electric field strength, ratio of positive to negative ion conductivity, particle radius and particle dielectric constant. With increasing time, charge is predicted to approach a steady state value asymptotically. For a given conductivity ratio, steady state charge is predicted to increase almost linearly with increasing external electric field. For a given charging time and field strength, charge asymptotically approaches unipolar levels with increasing conductivity ratio. Predictions based on theory generally show good agreement with experiment, typically being within + 10--15~. The exception to this is for intermediate conductivity ratios, where theory exceeds experiment by as much as 30",.
INTRODUCTION For the continuum regime (i.e. K n ~ 1, where K n = 2 / a , 2 = ionic mean free path, a = particle radius) there are two classical models for predicting the acquisition of electrical charge by a particle simultaneously exposed to positive and negative ions. These may conveniently be termed continuum field charging theory and continuum diffusion charging theory. In continuum field theory charging is conceptualized as being due to the electrical transport of ions along field lines which intersect the particle surface, and diffusion is neglected. This model was originally derived for unipolar ions, but was extended to the bipolar case by Pauthenier et al. (1956) and Gunn (1956). In continuum diffusion theory, charging is based on the diffusive transport of ions due to the ion density gradient at the particle surface. This requires knowledge of the spatial dependence of ion densities in the vicinity of the particle. In the classical formulations of bipolar diffusion charging theory (Lissowski, 1940; Fuchs, 1947 and 1963; Bricard, 1949; Gunn, 1954), the effect of particle charge on ion density profiles is considered, but the effect of an external field is not. Thus, in field theory, diffusion is neglected; and in diffusion theory, the external field is neglected. Recently, results of bipolar charging experiments have been compared to predictions of these classical theories (Fjeld et al., 1981; Gauntt, 1982). When data for small particles in low fields are extrapolated to zero field strength, the extrapolated charge shows good agreement with diffusion theory. As particle size and electric field strength increase, the data tend to approach field theory predictions. However, for small particles (0.1/~m < a < 0.5 #m) charged in the presence of moderate fields (50 < E < 300 kV/m), measured charge is significantly greater than either field or diffusion theory predictions alone. This suggests that continuum theory for bipolar charging of small particles in moderate applied fields must *Present address: Sandia Laboratories, Albuquerque, NM 87115, U.S.A. 541
542
ROB/oR! A FJ'~!t[) cl ~/
account for the combined effects ot diffusion and of the external field. These obserx a ~ , ~ : consistent with unipolar results, where the charge acquired by small particles it: ~c~,-, io'~ fields agrees well with diffusion theory predictions (Liu and Pui, 1977~, the charge acquired b~ small particles in moderate fields is substantially greater than predictions of both field thcor~ [Smith et al., 1978) and diffusion theory, and the charge acquired by large partich:,~ ~;~ moderate to high fields agrees with field theory (White, 1963). For the case of small particles and moderate fields, a combined field-diffusion model (Pluvinage, 1946) is lbund I lLit~ and Kapadia, 1978) to compare well with experiment. In the present paper, the combined field-diffusion charging model is generalized t¢,r the case of bipolar charging. Since the ion continuity equations are not amenable to analvtica~ solution when an external field is included, a numerical approach is used. The numerica~ solution is documented and compared to special cases for which analytical solutior'~,~ arc available. Results of the bipolar charging calculations are presented and compared witln experimental data. THEORY
To model bipolar charge acquisition it is necessary to describe the transport ot" positive and negative ions in the vicinity of a particle. In the continuum regime, ion transport is determined by the following equation: j
where
±(r)
- D -*VN -* {rl _+ /a -+ E ( r ) N
=
-*(r),
t 1!
j -+(r) = ion current density N +-(r) = ion density E(r) = electric field strength D ~ = ion diffusion coefficient # -" = ion electric mobility r = position vector.
The ( + ) and ( - ) signs in equation (1) refer to positive and negative ions, respectively. Consider now a particle of radius a, exposed to positive and negative ions in the presence of an external electric field (Fig. 1). Positive and negative ion densities in the bulk of the gas volume are equal to N~ and N O ' and, in general, N~ 4= N o. The particle causes a perturbation of the ion densities within its immediate vicinity; with the affected region being typically within a distance of five to fifty particle radii, depending on the particle charge and external field strength. Within this region, ion sources (due for example, to gas breakdown or ionizing radiation) and ion sinks (such as recombination) are neglected, and the ion continuity equation may be written as V.j-*(r) = 0,
i2)
D ~-V2N ~- (r)q:p-+V.E(r)N-~(r) = 0.
t3)
or
---~
(r,~
--,-
E.
Fig. 1. Coordinate system utilized tbr developing theory for charging of particle of radius, a. m an external electric field. The external electric field is of uniform strength, Eo, except in the immediate vicinity of the particle.
Continuum bipolar charging
543
The boundary conditions are N ± (a) = 0
(4a)
lim N ± (r) = N o.
(4b)
Equations (2) and (3) apply only to the immediate vicinity of the particle, and not to the bulk of the gas volume, where there may be significant sources and sinks. Neglecting space charge, employing the Einstein relation, p / D = q ~ / k T (where qe is the elementary charge unit, k is Bottzmann's constant, and T is absolute temperature}, and using E(r) = - V 0 ( r ) (where 0 is the electric potential), equation (3) can be written as V 2 N ± (r) -+- ~T V0(r)-VN ±(r) = 0.
(5)
The electric potential possesses azimuthal symmetry and, in terms of the spherical coordinate system of Fig. 1, is given by (Reitz and Milford, 1967) K - 1 aSEo (o(r, O) = - E o r
cos 0 + K + ~
nq~
r 2 cos 0 + 47re~'
(6)
where K = particle dielectric constant e = permittivity n = number of charges on the particle. Under the assumption that charge is a continuous variable, the particle charging rate is now given by dt
J + (a) - j - (a)]" d A , , (7)
=-ff[j;(a)-j;(o)]dA.
Since ion densities vanish at the particle surface, the radial current densities j~(a) are determined from the ion density gradient at the particle surface, yielding d~- =
D÷
-- D -
dr
dA.
dr
r=a
(8)
r=a
It is convenient to express the ion continuity and particle charging equations in terms of spherical coordinates and the following dimensionless parameters: u ± =N±/No
co = a E o q e / k T
~
x = r/a
ct = (K - 1)/(K + 2)
v=
r = qela+N~ t/e
nq~/4n~akT
7=p+N~/p-N~.
The time parameter, z, is defined in terms of the dominant ion species, which is exclusively positive in the work reported here. Substituting equation (6) into equation (5) and converting to dimensionless variables yields t~2U -+
1 O2u ±
?x 2 + x 2 ~c30
~ e [ x 2v +~--x q:~cocosO
+ 0cos,
(
(
2~t)l 1+~- 3 (9)
544
,3,, ~ I E L I ) t't a t
ROBERT
The boundary conditions are u :~( 1 ) = 0 lim u z:(x)= 1. x-4 t
The charging rate, equation (8), can also be put in dimensionless form, viz.,
dv lj'~ .... dr
2
du"
ldu-
I~
~= o
7
dx
] sin0d0,
tl!)
The expression on the right hand side of equation (11) is a function of charge. Thus, charge as a function of time is obtained by the following, r =
f" dr' . o dv'/dr
~12)
When written for ions of a single sign, equations (9), (11), and (12) are equivalent to the equations for unipolar field-diffusion charging given by Liu and Kapadia, although the polar angle, 0, and the dimensionless charging time, r, have been defined differently herein: The angle 0 is the supplement of the polar angle defined by Liu and Kapadia, and the time parameter, 3, as defined here is larger than the time parameter of Liu and Kapadia by a factor of four. The remaining dimensionless parameters are numerically equal to the corresponding parameters defined by Liu and Kapadia.
CHARGING CALCULATIONS The determination of particle charge as a function of time requires that (1) the two equations represented by equation (9) be solved and positive and negative ion density gradients at the particle surface be determined, (2) the surface integral indicated in equation (11) be evaluated to calculate the charging rate and (3) the integral of equation (12) be solved. The key step in this calculational procedure is determination of the ion density gradients. For the special case of co = 0, the equations can be solved analytically and u z (x) can be differentiated and evaluated at x = 1. The results are du* dx
du dx X=I
= 1
du ~
ve-
dx
1 -e-"
du
__ dx
I
b
[
v= 0
il3a)
v> 0
t13b)
X=|
ve ~
........ e ~ - 1"
(13cl
x=l
Note that if v is replaced by - v in equation (13b), the complementary equation, (13c), is obtained. Physically, this means the positive ion density profile in the vicinity of a particle with positive charge is identical to the negative ion density profile in the vicinity of a particle with negative charge of the same magnitude, and vice versa. For ~,J ~ O, an analytical solution is not available and a finite difference method is utilized to solve the ion continuity equations and to determine density gradients at the particle surface. Implementation of the numerical method requires that the exact boundary condition represented by equation (lOb) be replaced by an approximate boundary condition in which
Continuum bipolar charging
545
the ion density becomes unity at some finite value of x. This approximate condition is u -+(x,.) = 1.
(14)
The particle affects ion densities in its immediate vicinity because it is a sink for ions and because it perturbs the electric field. I f the influence of the particle on ion densities is significant at x,., use of the approximate b o u n d a r y condition, equation (14), can cause errors in calculations o f the ion density gradients at the particle surface. The effect of Xm on the ion density gradient can be determined exactly for ¢o = 0. The results, which are obtained by solving equation (9) subject to the boundary conditions represented by equations (10a) and (14), are du + dx
-
dudx
X=I
du + dx
=
xm x m- 1
v
=
(15a)
0
X=I
t =
ve-
e - ~ / ~ . - e -~
v> 0
(15b)
x=t
dudx
ve v = e~_e~/~ .
(15c)
x=l
These equations reduce to the exact forms, equations (13a--c), as xm ~ oo. Shown in Fig. 2 is the surface ion density gradient as a function o f x m for co = 0. T w o analytical solutions are displayed. The solid line is for the exact boundary condition, equation (10b), and the dashed line is for the approximate boundary condition, equation (14). The
•~ 10.0
I
r
I
1
I
c
='=;6
.9
1/=;2 ii
x I--
I/=0
-
IO
l--z ~J . . . . . . .
>I-
--EXACT
z 010 14J ¢3 Z O
....
=/='-2
BOUNDARY CONDITION
APPROXIMATE BOUNDARY CONDITION
% %
t
~,=*_6 0.01
I 3
I 5
Xm-OUTER
I IO
[ 20
l 40
B O U N D A R Y (dimensionless)
Fig. 2. Surface ion d e n s i t y g r a d i e n t s for exact a n d a p p r o x i m a t e b o u n d a r y c o n d i t i o n s . These c a l c u l a t i o n s are for to = 0.
546
ROBERT "~ [:JELD
et
al
a p p r o x i m a t i o n asymptotically approaches the exact solution as x,, increases. F or ~ ~. ~:he a p p r o x i m a t i o n is within 11 "o o f the exact solution at xm = 1 0 a n d 3 0° at x,, = 40. For I~,',t'~:,ir~ the vicinity o f a particle o f like charge, the a p p r o x i m a t e solution approaches the exact solution m o r e slowly; for v = 6, it is 8017o high at x~, = 10 and 13°o high at x,, = 40. i h i s occurs because the radius within which the particle perturbs ions o f like sign becomes larger as the charge increases. The opposite is true for ions in the vicinity o f a particle o f opposite charge. In this case the radius at which ion concentrations are perturbed by the panicle becomes smaller, and the a p p r o x i m a t i o n a p p r o a c h e s the exact solution more rapid[3 F r o m the preceding discussion it is evident that, at least for small co, x~ must be large. However, it is also necessary for ion densities to be calculated close to the particle surtace, i.e, at x close to one, This IS to obtain an accurate value o f the density gradient. These are conflicting requirements for a finite difference solution with a given n u m b e r of grias, and special measures are taken in an attempt to satisfy them. First, the following variable transformation is applied to equation (gj, " = ln(xi.
, L6)
The t r a n s f o r m e d equation is
a~u ~ a~u ± au ~ ~3z--~- + ~
+ c~z[_ 1 q: v e - : q: co cosO(e~ + 2ae -2:)
~u[ cos0 ] + c ~ 0 1 ~ n - - ~ + c o s i n 0 ( e ~ - 0 c e -2z) = 0 .
]
c17)
The resulting finite difference grid has a c o n s t a n t mesh spacing in z coordinates, but has mesh spacings in x coordinates which increase exponentially with increasing x. The result is a m u c h finer mesh near x = 1, where it is needed, than can be achieved without the transformation. In addition the solution is carried out in two steps. In the first step, a relatively coarse mesh is utilized with the first two radii being located at x = 1.0 (the surface o f the particle) and x = 1.1. I o n densities obtained at x = 1.1 then become the outer b o u n d a r y condition for a second step in which ion densities are determined between x = 1.1 and the particle surface; the first two radii n o w being located at x = 1.0 a n d x = 1.01. This proves to be an efficient and accurate means for achieving the necessary detail close to the particle while also allowing xm to be large. The finite difference equations formed from equation (17) are solved by the Gauss-Seidel M e t h o d (Ferziger, 1981) with successive overrelaxation or underrelaxation as needed for convergence. The relaxation parameter is varied f r o m 1.5 for small co to 0.8 for large co. The finite difference grid is formed by 41 concentric circles (0 < zi < lnx,,) and 19 radial lines (0 < 0i < 180°). A l t h o u g h it is f o u n d that a 21 x 11 grid gives almost identical results as the 41 x 19 grid (being within 0.1 oj~;for co = 1 and v = 0), the smaller mesh spacing with the latter allows the solution to converge at higher values o f co. As co increases, x m must be reduced for convergence. T h e values ofxm are as follows: for co < 0.5, x~, = 40; for 0.5 < co < 3, x,, = 30; for 3 < co < 7, x,. = 20; and for co = 10, x,. = 5. F o r large co, it is found that the value o f x,. does not have a significant effect on ion density gradients. However, for small co the approximate b o u n d a r y condition impacts calculations even with x,, = 40. An analytical correction, derived from equations (13-15) and rigorous only for co = 0, is thus applied to the numerical charging rate for co < 0.1. T o assess the validity a n d accuracy o f the numerical method, comparisons are made with analytical solutions which are available for co = 0 and with the numerical results o f Liu and Kapadia for unipolar charging. In Fig. 3, numerical charge calculations are given for ~,~ < 1, r = 1 and 100, and ~, = 3 and ~c. The numerical results obtained here for co = 0 a r e within 1 " o o f the exact solution. Further, it is seen f r o m the figure that the numerical results at eJ > 0 extrapolate to the correct value at co = 0. Liu and Kapadia's numerical result at co = 0.01 is consistent with the exact solution for r = l, but is approximately 30 °0 high for r = 100. The difference is likely due to their relatively large mesh spacing (Ax = 0.1667). The large mesh
C o n t i n u u m bipolar c h a r g i n g
547
spacing results in an underestimate of the charging rate at small v. A likely reason for this is as follows. At small v the ion density profile near the surface of the particle is concave downward and a finite difference approximation to the derivative is an underestimate. At high v the converse is true and, for small to, the effect is exacerbated by the approximate boundary condition (Liu and Kapadia utilize x,, = 7) as shown in Fig. 2. Nonetheless, the numerical results of Liu and Kapadia differ from the unipolar calculations reported here by less than 30~o for 1 < 09 < 10 and 10 < r < 100. Numerical ion density profiles for to = 3.0 and v = 0 and 10 are presented in Fig. 4. Ion densities are given for three polar angles: 0 °, 90 °, and 180 °. Analytical profiles for co = 0, which are independent of the polar angle, are included as dashed lines in Fig. 4. For v = 0 (Fig. 4a) the external field is seen to increase positive ion densities in the vicinity of the particle for large values of 0, where the field is directed radially inward (refer to Fig. 1) and to decrease densities for 0 near zero where the field is directed radially outward. An analogous effect is observed for negative ions (Fig. 4c), except that densities are high for small 0 and vice versa. For v 4= 0, the electric field has two components; one due to the presence of the particle in an otherwise uniform external field and the other due to the presence of charge on the particle. When the charge is positive, positive ion densities are diminished near the particle and, correspondingly, negative ion densities are increased, as shown in Figs. 4b and 4d. The result is a monotonically decreasing charging rate by positive ions and a monotonically increasing charging rate by negative ions as v increases. Thus, as in the case of pure diffusion charging, a particle exposed to bipolar ions in the presence of an external electric field accumulates charge until the rate of charging by negative
I00
~'~r'~
~ ~-~'~'~")":(13 IT: IO0
c
,_~ 50 W ~e T W-J n0.~
~s < i
/ i
O
I
i Jlt f
~ "~ 7= 0O 1
0~ Z~)"- 3
-- - - A - -
--APPROXIMATE APPROXIMATE
0 ,
00
"
EXACT
0
SOLUTION, THIS WORK SOLUTION,
LIU a KAPADIA
SOLUTION
0.5 - ELECTRIC
FIELD
1.0 (dimemionless)
Fig. 3. N u m e r i c a l solution for v vs co at small to. T h e exact solution at co = 0 is presented for c o m p a r i s o n . T h e s e c a l c u l a t i o n s are for = = 1.0.
54g
ROBER
I A.
el a /
~JELD
0,8 /
/
II!
0
~ -
).0 0,8 el
~. 0,6 o .~
0.4
o
E
1"
~
~°>//.-04 /i p ~~os,,,v~ 0,2 _ / / . " 0.6
)ll
/
I
I
0"8o~'I"--
.
/
/
I
I
- -
w . ~O
~ ' - -~..
f
.-
F,OSmVE
I
:i
-'7~
e.,o-
/
I
i(b)
I
v=o --~.o~
.-""
~
~
/-;
i
0.2
"¢1
~ ' ~
0
,_
J--~--"f'-~
I
~ - ~ , o
i
)-
P"
LO
7' W
().8
7
0.6 OA
//,.:o..._. [e) /
/I //. ."/ - ' "..-
~
o.,.~
i
......
<
..,.,vE
--~.o -4
0
,
I0 08 i
C).g
0.4
V
NEGATIVE
V-IO
0.2
----
0
1
i
OJ= 0
- - w-3,O
I
I
I
I
I
2
4
6
8
tO
20
x- RADIAL DISTANCE (dimensionless)
Fig. 4. Representative positive and negative ion density profiles. These calculations are tot • = 1.0
I° I
'1
i ~
'
'
I
'
,o. Lo
--),=cO
'1
j ?'-300
i
7, ioo !
oI
L
L
, ----
L
L
I
1
L
J
Io ]o.o "r-CHARGING TIME (dimensionless)
a
I
i
Ioo.o
Fig. 5, Charge as a function of charging time for (t) = l.O, :x = 1.0, and various conductivity ratios.
Continuum bipolar charging
549
ions equals the rate o f charging by positive ions. When this occurs, the particle acquires a steady state level o f charge as shown in Fig. 5, where charge is plotted as a function of time for = 1.0, ~ = 1.0, and various values o f the conductivity ratio. Steady state charge increases with increasing conductivity ratio and with increasing field strength (Fig. 6). The dependence on field strength is almost linear, with slight deviations from linearity at small o~. The effect of particle dielectric constant is also illustrated in Fig. 6. The charging results for ~t = 1.0 are presented in tabular form in Appendix I.
32 a
=i.o
e = 0.43
(K=CO) (K =2,5)
2B
S-
24
g .E_ /
w
2O f t
ilZ w l---
~" = 3 0 0 12
>Q ~ if)
1
/
n-"
bm
i
I
~
s
~ ~
~
I
~
~
s
J
e
i
0
I
2 ','-
3
4
ELECTRIC
5
6
7
8
9
I0
FIELD STRENTH (dlmen$ionlelli)
Fig. 6. Steady state charge as a function of electric field strength for ct = 1.0 and 0.43 and various conductivity ratios.
COMPARISONS
WITH EXPERIMENTAL
DATA
Experimental measurements o f bipolar charge acquisition are described in a thesis by Gauntt (1982). In these experiments, highly monodisperse aerosols o f polystyrene are exposed to counter currents o f positive and negative ions in the presence o f an external electric field. Particle diameters range from 0.3 to 1.1 # m and the external field is varied between 5 and 300 k V / m . The mean aerosol charge is measured by mobility analysis. These data are compared with continuum field-diffusion theory calculations in Figs. 7-9. In each o f these figures the experimental variables are expressed in terms o f the dimensionless parameters defined in the theory section. The number o f charge measurements for a given set of experimental conditions varies from one to about ten. When three or more measurements are available, the mean and experimental standard deviation are displayed in the graphs. The average o f two observations and single observations are represented by a symbol without error bars. Mean charge is given as a function o f electric field strength for various values of the conductivity ratio and z = 11 and 34 in Figs. 7a and b, respectively. The agreement between theory and experiment is generally very good. The largest differences occur for 7 = 10 and r = 11. where experiment exceeds theory by as much as 20 ° o. There also appears to be some deviation for co > 3 at ~, = 3 and Z = 1 1, where experiment is somewhat (10"~o) lower than
550
ROBER i : \
i
12
f:JELD ~t ,.~.
"T-~
r
I
/
l~
--r----]
y. GO I
/
SYMBOL
Op (F,m)
"
o
o.3,
~o
-
~
j
-
~
o.~6
~
TI-IEORY
w
:
~
o.so ~
r /
i
/
,,
-~ 7.10
~'
t
O~
I-- 4
O
,
2,
@ o
O
I I
~
_~
,,
I 2
I 3
I 4
I 5
I 6
I 7
l
w-ELECTRIC FIELD STRI~TH (dimeMieel~s) o, r - l l
IO
I
I
I T, 34
|YM~L
=
I
I
Do (~ m)
0 O
A8
I
0.31 0.50 0.76
5
)" ~
THEORY
"o bJ n..
~4
~J -
~ o
u
o
'
X"
I0
r
v
~2 i 0
i
0.2
I
I
b
Fig. 7. C o m p a r i s o n
I
t
0.4 0.6 0,8 1.0 w-ELECTRIC FIELD STRENTH (dimen$ionk~ls)
I 1.2
r-34
of bipolar continuum field,diffusion theory calculations with experimental datac h a r g e vs field s t r e n g t h .
theory. Otherwise, the differences are small. In addition to showing the degree of agreement between continuum theory and experiment, these graphs of dimensionless charge expressed as a function of dimensionless parameters also serve to illustrate the self-consistency of the charging data. Time dependent data for 7 - 3, 10, and ~ are compared with theory in Figs. 8a and b for ~,~) = 0.48 and 1.07, respectively. The steady state charge levels predicted by theory for 7 < ;< are clearly supported by the data. Agreement between theory and experiment is very good,
Continuum bipolar charging
'
,
'
'
i
'
,
,
r
551 ,
I
r
F
c~• 0 48 ( Dp" 0 50, E = 50 kV/m) SYMBOL IC
..Z
/.
3 Jo
CO THEORY (a=O
,~
--
._8
7" = CO
~e Qa .E
(J 4
),=lO 2 c~
0
,
,
,
I
i
I
t
I
I0
Ol
L
rJrJ
"y=3
,
I
l
I
I
I Ol0
r-CHARGING TIME
]
I
i
I0000
I000 (dimenaionless)
o to = 0 4 8
(Dp'l.l, E" 5OkV/m)
w=lO7
SYMBOL
__.Z
,~
I0
3
y=CO
/
T H E O R Y 1o •
- -
~8
¢: 6 z u
~4
y,3
,
0.1
,
,
,
i
,
,
,
I.O
,I
,
,
IQO r-CHARGING
TIME
b.
~
,
, i tOQO
,
,'oooo
(dimensionless)
• I.O7
Fig. 8. Comparison of bipolar continuum field-diffusion theory calculations with experimental datacharge vs time.
generally being within 15 5o. The decrease in measured charge at large z for y = 3 is felt to be an experimental anomaly (Gauntt, 1982). Other than this difference, theory and experiment basically agree within the bounds o f experimental uncertainty, estimated to be less than + 15°o (Gauntt, 1982). Up to this point, the comparisons have been limited to 7 = 3, 10 and ~ . Presented in Figs. 9a and b is charge versus conductivity ratio for ~ = 0.48 and r = 11 and 34. For r = 11, theory and experiment are in good agreement. For r = 34, theory and experiment are in good agreement for ~, < 30 and 7 = ~ . However, there is a significant difference for the
552
ROBERT a
F~ELt) e~ j i
o
/
w'048
(Dp" 0 5~m, Eo' 50kV/m)
~" 0429
0 EXPERIMENT --THEORY
. . . . . ,0. . .
,~, ' ' ':o~ ' ' 7 " ION CONOUCTWITY RATIO (k~*N*//~'N ") , W~0411
O T'll
[
m
*
I
e
',~oo
r , 34 w ' O . q (Op. OS/,~m.Eo. SOkV/m) = . 04,29 0 EXPE~M(NT
i
rO
7-ION CONOUCTMTY RATIO (/~'N'//4"N') r,34~
~,04e
F
!
~,, .L 34
* 0,?4 (D~*O.~kAm, (*SOkV/ml a o 0.4IS
i
(XPERIM|NT
d i
--THEORY
//
ICe
FO0
I000
I0.000
)'-ION CONDUCTIVITY RATIO (/,L*N'//~'N') c T,34,
~lOT4
Fig. 9. Comparison of bipolar continuum field-diffusion theory calculations with experimental datacharge vs conductivity ratio.
i n t e r m e d i a t e c o n d u c t i v i t y ratios, i.e. 30 < ~ < ~ , w h e r e m e a s u r e m e n t s are a p p r o x i m a t e l y 30 ~ b e l o w theory. O t h e r cases for w h i c h d a t a exist [~o = 0.74 a n d r = 34 (Fig. 9c); vJ = 0.30 a n d ~ = 11; co = 0.30 a n d z = 34] s h o w the s a m e trends; i.e. g o o d a g r e e m e n t at r = 11 a n d at z = 34 for s m a l l 7 a n d for " / = ~ , but less satisfactory a g r e e m e n t at r = 34 for i n t e r m e d i a t e 7.
Continuum bipolar charging
553
The discrepancy between theory and experiment at intermediate ratios is also noted in the prior paper on bipolar diffusion charging (Fjeld et al., 1981). It is uncertain whether this is due to an experimental problem at these ratios or a limitation of the theory.
SUMMARY
AND
CONCLUSIONS
A continuum regime theory for the acquisition of charge by a particle simultaneously exposed to positive and negative ions in the presence of an external electric field is presented. The theory is an extension to the bipolar case of the combined field-diffusion charging model of Pluvinage and of Liu and Kapadia. Approximate numerical solutions are developed which, in the limit as the external field approaches zero, yield results within 1 ~ of the exact solution. For the unipolar case differences are observed between these results and the results of Liu and Kapadia. These differences are probably due to their use of a larger mesh spacing in the radial direction near the particle surface and a smaller maximum radius. Charge is calculated as a function of time for dimensionless electric fields from zero to ten and ratios of positive to negative ion conductivity from one to infinity. After sufficient time charge is predicted to saturate at steady state levels. For a given conductivity ratio, steady state charge is predicted to increase almost linearly with increasing external field. For a given charging time and field strength, charge asymptotically approaches unipolar levels with increasing current conductivity ratio. Theoretical predictions are compared with experimental measurements for polystyrene aerosols in the 0.3 to 1.1/zm diameter size range. The data are supportive of the continuum field-diffusion description of bipolar charge acquisition, typically being within + 10-15 ~o of theory. The major exception to this is for intermediate current density ratios (I0 < ~ < ~), where theory exceeds experiment by as much as 30 ~. Acknowledgements--This work received support from the National Science Foundation under Grant ENG7701130. The Texas Engineering Experiment Station, and Clemson University.
REFERENCES Bricard, J. (1949) J. geophys. Res. 54, 39. Ferziger, J. H. (1981) Numerical Methods For Engineering Application, John Wiley, New York. Field, R. A., Gauntt, R. O. and McFarland, A. R. (1981) J. Colloid Interface Sci. 83, 82. Fuchs, N. A. (1963) Geof. Pura Appl. 56, 185. Fuchs, N. A. (1947) Izv. Akad. Nauk. SSSR Ser. Geor. Geofiz. 11, 341. Gauntt, R. O. (1982) PhD Dissertation, Texas A&M University. Gunn, R. (1954) J. Meteorol. 11, 339. Gunn, R. (1956) J. Meteorol. 13, 283. Lissowski, P. (1940) Acta Physiochim. URSS 13, 157. Liu, B. Y. H. and Kapadia, A. (1978) J. Aerosol Sci. 9, 227. Liu, B. Y. H. and Pui, D. Y. H. (1977) J. Colloid Interface Sci. ~ , 142. Pauthenier, M., Cochet, R. and Dupuy, J. (1956) Comptes Rendu 243, 1606. Pluvinage, P. (1946) Ann. Geophys. 2, 31. Reitz, J. R. and Milford, F. J. (1967) Foundations of Electromagnetic Theory, Addison-Wesley, Reading, MA. Smith, W. B., Felix, L. G., Hussey, D. H. and Pontius, D. H. (1978) J. Aerosol Sci. 9, 101. White, H. J. (1963) Industrial Electrostatic Precipitation, Addison-Wesley, Reading, MA.
p=3 0.07 0.13 0.23 0.33 0.41 0.49 0.76 1.00 1.11 1.11 1.11 1.11 1.I1 1.11 1.11 1.11 1. I I t.11
),=3 0.07 0.13
t 0.10 0.20 0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 100~00 200.00 400.00
r 0.10 0.20
i = 10 0.09 0.17
? = 10 0.09 0.17 0.32 0.46 0.59 0.70 1.15 1.67 1.95 2.11 2.21 2.33 2.33 2.33 2.33 2.33 2.33 2.33
),= 3{) 0.10 0.19
to = 0.t0
7=30 0.10 0.18 0.35 0.50 0.64 0.77 1.28 1.94 2.35 2.62 2.81 3.43 3.43 3.43 3.43 3.43 3.43 3.43
{o = 0.00
7 = 100 0.10 0.19
ct = 1.00
) ' = 100 0.10 0.19 0.36 0.51 0.66 0.79 1.33 2.05 2.51 2.85 3.11 3.84 4.36 4.63 4.63 4.63 4.63 4.63
- (K
)'=3(~) 0.10 0.19 0.36 0.52 0.66 0.80 1.34 2.08 2.56 2.92 3.20 4.05 4.80 5.15 5.36 5.49 5.72 5.72
CHARGING
No tie
(dimensionless time)
(dimensionless electric field)
{dimensionless charge)
BIPOLAR
UNIPOLAR 0.10 0.19
UNIPOLAR 0.10 0.19 0.36 0.52 0.67 0.80 1.35 2.09 2.59 2.96 3.25 4.17 5.07 5.59 5.95 6.22 7.06 7.87
: 0.10 0.20
t 0.10 0.20 0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 100,00 200.00 400.00
vii, w, ?, ~t
)~ = 3 0.09 0.16
?=3 0.07 0.13 0.23 0.33 0.42 0.49 0.76 1.00 1.12 1.12 1.12 !.12 1.12 1.12 1.12 1.12 1.12 1.12
I}/{K + 2) {dimensionless dielectric constant}
3 , : 10 0.11 0.22
; , = 10 0.09 0.17 0.32 0.46 0.59 0.71 1.16 1.68 1.96 2.12 2.22 2.34 2.34 2.34 2.34 2.34 2.34 2.34
/ = 31) 0.12 0.23
~2~:= 0.50
?=30 0.10 0.18 0.35 0.50 0.64 0.77 1.29 1.95 2.36 2.63 2.82 3.44 3.44 3.44 3.44 3.44 3.44 3.44
o~ = 0.01
CALCULATIONS
÷ i~, ,~'o .'It N o ( p o s i t i v e to n e g a t i v e i o n c o n d u c t i v i t y r a t i o )
r = q,.ll
7 = ii
'),=300 0.10 0.19
a = 1.00
nq~,4neakT
I:
t~ = a E o q ~ / k T
v -
APPENDIX
p = 100 0.12 0.24
a = 1.00
7 = 100 0A0 0.19 0.36 0.52 0.66 0.79 1.34 2.06 2.52 2.86 3.12 3.85 4.37 4.64 4.64 4.64 4.64 4.64
a = 1.00
i :-30(} 0.12 0.24
y=300 0.10 0.19 0.36 0.52 0.67 0.80 1.35 2.09 2.58 2.93 3.21 4.06 4.81 5.17 5.37 5.49 5.73 5.73
[iNIPOtAR 0.13 0.24
UNIPOLAR 0.10 0.19 0.36 0.52 0.67 0.80 1.36 2. ! 0 2.60 2.97 3.26 4.18 5.08 5.60 5.96 6.23 7.07 7.88
;o 2
©
0.33 0.47 0.60 0.72 1.19 1.73 2.02 2.19 2.42 2.42 2.42 2.42 2.42 2.42 2.42 2.42
7 = 10 0.14 0.27 0.51 0.73 0.93 !.11 1.82 2.64 3.07 3.31 3.45 3.64 3.64 3.64 3.64 3.64 3.64 3.64
0.24 0.34 0.43 0.51 0.79 1.17 !.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17
y=3 0.10 0.20 0.37 0.52 0.66 0.78 1.21 1.60 1.76 1.78 1.78 1.78 1.78 1.78 1.78 1.78 1.78 1.78
0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 I00.00 200.00 400.00
0.10 0.20 0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 100.00 200.00 400.00
7=30 0.15 0.29 0.55 0.79 1.01 1.21 2.02 3.05 3.68 4.10 4.39 5.04 5.28 5.28 5.28 5.28 5.28 5.28
o9= 1.00
0.36 0.51 0.66 0.79 1.32 2.00 2.42 2.70 2.91 3.38 3.55 3.55 3.55 3.55 3.55 3.55
) ' = 100 0.15 0.29 0.56 0.81 1.04 1.25 2.10 3.22 3.94 4.45 4.84 5.91 6.65 7.00 7.00 7.00 7.00 7.00
0.37 0.53 0.68 0.82 1.39 2.14 2.64 3.01 3.30 4.17 4.94 5.30 5.51 5.65 5.90 5.90
),=300 0.15 0.30 0.57 0.82 1.05 1.26 2.12 3.26 4.01 4.56 4.98 6.23 7.30 7.79 8.06 8.22 8.51 8.51
a = 1.00
0.37 0.53 0.68 0.81 1.37 2.11 2.59 2.94 3.20 3.95 4.49 4.71 4.78 4.78 4.78 4.78
UNIPOLAR 0.15 0.30 0.57 0.82 1.05 1.26 2.13 3.29 4.05 4.62 5.06 6.41 7.70 8.42 8.92 9.30 10.38 11.57
0.37 0.53 0.69 0.82 1.39 2.16 2.67 3.05 3.35 4.28 5.21 5.74 6.10 6.39 7.24 8.07
z 0.10 0.20 0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 100.00 200.00 400.00
0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 100.00 200.00 400.00
7=3 0.19 0.36 0.68 0.96 1.21 1.43 2.21 2.90 3.24 3.24 3.24 3.24 3.24 3.24 3.24 3.24 3.24 3.24
0.30 0.42 0.53 0.63 0.98 1.44 1.44 1.44 1.44 1.44 i.44 1.44 1.44 1.44 1.44 1.44
7 = 10 0.25 0.49 0.94 1.35 1.71 2.05 3.35 4.83 5.58 5.99 6.21 6.51 6.51 6.51 6.51 6.51 6.51 6.51
0.41 0.59 0.75 0.90 1.48 2.15 2.49 2.70 2.97 2.97 2.97 2.97 2.97 2.97 2.97 2.97
7=30 0.27 0.53 1.02 1.46 1.87 2.24 3.73 5.59 6.68 7.38 7.86 8.86 9.16 9.16 9.16 9.16 9.16 9.16
o9=3.00
0.45 0.64 0.82 0.98 1.64 2.48 3.00 3.34 3.59 4.14 4.35 4.35 4.35 4.35 4.35 4.35
7 = 100 0.28 0.55 1.04 1.50 1.92 2.31 3.87 5.88 7.13 8.00 8.64 10.32 11.33 11.72 11.72 11.72 11.72 11.72
a=l.00
0.46 0.66 0.84 1.01 1.71 2.62 3.21 3.63 3.96 4.86 5.50 5.72 5.82 5.82 5.82 5.82
7 = 300 0.28 0.55 1.05 1.51 1.94 2.33 3.91 5.96 7.27 8.19 8.89 10.86 12.39 13.04 13.38 13.57 13.83 13.83
0.46 0.66 0.85 1.02 1.72 2.65 3.27 3.72 4.07 5.13 6.05 6.48 6.72 6.88 7.15 7.15
UNIPOLAR 0.28 0.55 1.06 1.52 1.94 2.34 3.93 6.01 7.34 8.29 9.02 11.15 13.06 14.08 14.75 15.25 16.75 18.11
0.46 0.67 0.85 1.03 1.73 2.67 3.30 3.77 4.13 5.28 6.39 7.01 7.45 7.78 8.79 9.75
T.
"O O
¢.. e..,
O =1
r 0.10 0.20 0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 100.00 200.00 400.00
;¢=3 0.28 0.54 1.01 1.42 1.79 2.12 3.28 4.30 4.65 4.75 4.81 4.81 4.81 4.81 4.81 4.81 4.81 4.81
Appendix I (contd.)
),= 10 0.38 0.73 1.39 1.99 2.54 3.03 4.96 7.14 8.23 8.81 9.13 9,54 9.54 9.54 9.54 9.54 9.54 9.54
) ' = 30 0.40 0.79 1.51 2.16 2.76 3.31 5.51 8.23 9.81 10.81 11.48 12.81 13.18 13.18 13.18 13.18 13.18 13.18
to=5.0 7 = 100 0.41 0.81 1.55 2.22 2.84 3.41 5.72 8.66 10.46 11.69 12.58 14.83 16.10 16.44 16.52 16.52 16.52 16.52
~ = 1.00 7=300 0.42 0.81 1.56 2.24 2.86 3.44 5.78 8.78 10.66 11.96 12.93 15.57 17.53 18.30 18.68 18.89 19.14 19.14
UNIPOLAR 0.42 0.82 1.56 2.25 2.87 3.46 5.81 8.84 10.76 12.10 13.11 15.98 18.43 19.69 20.52 21.13 22.88 24.49
r 0.10 0.20 0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 100.00 200.00 400.00
), = 3 0.50 0.97 1.83 2.59 3.26 3.86 5.99 7.88 8.50 8.73 8.81 8.81 8.81 8.81 8.8 ! 8.81 8.81 8.81
), = 10 0.68 1.33 2.53 3.62 4.61 5.52 9.03 13.00 14.97 16.01 16.57 17.26 i 7.26 17.26 17.26 17.26 17.26 17.26
), = 30 0.73 1.43 2.74 3.93 5.02 6.02 10.03 14.96 17.78 19.53 20.68 22.85 23.38 23.38 23.38 23.38 23.38 23.38
t o = 10.0 ~, = 100 0.75 1.47 2.81 4.04 5.16 6.21 10.40 15.71 18.92 21.06 22.58 26.26 28. I0 28.49 28.58 28.58 28.58 28.58
a = 1.00 y = 300 0.76 1.48 2.83 4.07 5.21 6.26 10.51 15.94 19.27 21.53 23.18 27.51 30.40 31.42 31.88 32.11 32.34 32.34
IJNIPOLAR 0.76 1.48 2.84 4.08 5.23 6.28 10.56 16.05 19.45 21.78 23.50 28.19 31.87 33.66 34.81 35.64 37.91 39.84
0021-8502/83 $3.00+0.00 ~ 1983 Pergamon Press Ltd.
C O N T I N U U M F I E L D - D I F F U S I O N THEORY FOR BIPOLAR CHARGING OF AEROSOLS ROBERT A. FJELD Department of Environmental Systems Engineering, Clemson University, Clemson, SC 29631, U.S.A.
RANDALLO. GAt~NTr* Department of Nuclear Engineering
and ANDREW R. MCFARLAND Department of Civil Engineering, Texas A&M University, College Station, TX 77843, U.S.A. (First received 9 September 1982 and in revised form 29 November 1982) A h s t r a c t ~ o n t i n u u m regime field-diffusion theory for the acquisition of charge by particles exposed to bipolar ions is developed. An approximate numerical solution to the governing equations is described which, in the limit as the external field approaches zero, yields results within 1 ~ of exact solutions. The numerical method is used to calculate particle charge as a function of time, external electric field strength, ratio of positive to negative ion conductivity, particle radius and particle dielectric constant. With increasing time, charge is predicted to approach a steady state value asymptotically. For a given conductivity ratio, steady state charge is predicted to increase almost linearly with increasing external electric field. For a given charging time and field strength, charge asymptotically approaches unipolar levels with increasing conductivity ratio. Predictions based on theory generally show good agreement with experiment, typically being within + 10--15~. The exception to this is for intermediate conductivity ratios, where theory exceeds experiment by as much as 30",.
INTRODUCTION For the continuum regime (i.e. K n ~ 1, where K n = 2 / a , 2 = ionic mean free path, a = particle radius) there are two classical models for predicting the acquisition of electrical charge by a particle simultaneously exposed to positive and negative ions. These may conveniently be termed continuum field charging theory and continuum diffusion charging theory. In continuum field theory charging is conceptualized as being due to the electrical transport of ions along field lines which intersect the particle surface, and diffusion is neglected. This model was originally derived for unipolar ions, but was extended to the bipolar case by Pauthenier et al. (1956) and Gunn (1956). In continuum diffusion theory, charging is based on the diffusive transport of ions due to the ion density gradient at the particle surface. This requires knowledge of the spatial dependence of ion densities in the vicinity of the particle. In the classical formulations of bipolar diffusion charging theory (Lissowski, 1940; Fuchs, 1947 and 1963; Bricard, 1949; Gunn, 1954), the effect of particle charge on ion density profiles is considered, but the effect of an external field is not. Thus, in field theory, diffusion is neglected; and in diffusion theory, the external field is neglected. Recently, results of bipolar charging experiments have been compared to predictions of these classical theories (Fjeld et al., 1981; Gauntt, 1982). When data for small particles in low fields are extrapolated to zero field strength, the extrapolated charge shows good agreement with diffusion theory. As particle size and electric field strength increase, the data tend to approach field theory predictions. However, for small particles (0.1/~m < a < 0.5 #m) charged in the presence of moderate fields (50 < E < 300 kV/m), measured charge is significantly greater than either field or diffusion theory predictions alone. This suggests that continuum theory for bipolar charging of small particles in moderate applied fields must *Present address: Sandia Laboratories, Albuquerque, NM 87115, U.S.A. 541
542
ROB/oR! A FJ'~!t[) cl ~/
account for the combined effects ot diffusion and of the external field. These obserx a ~ , ~ : consistent with unipolar results, where the charge acquired by small particles it: ~c~,-, io'~ fields agrees well with diffusion theory predictions (Liu and Pui, 1977~, the charge acquired b~ small particles in moderate fields is substantially greater than predictions of both field thcor~ [Smith et al., 1978) and diffusion theory, and the charge acquired by large partich:,~ ~;~ moderate to high fields agrees with field theory (White, 1963). For the case of small particles and moderate fields, a combined field-diffusion model (Pluvinage, 1946) is lbund I lLit~ and Kapadia, 1978) to compare well with experiment. In the present paper, the combined field-diffusion charging model is generalized t¢,r the case of bipolar charging. Since the ion continuity equations are not amenable to analvtica~ solution when an external field is included, a numerical approach is used. The numerica~ solution is documented and compared to special cases for which analytical solutior'~,~ arc available. Results of the bipolar charging calculations are presented and compared witln experimental data. THEORY
To model bipolar charge acquisition it is necessary to describe the transport ot" positive and negative ions in the vicinity of a particle. In the continuum regime, ion transport is determined by the following equation: j
where
±(r)
- D -*VN -* {rl _+ /a -+ E ( r ) N
=
-*(r),
t 1!
j -+(r) = ion current density N +-(r) = ion density E(r) = electric field strength D ~ = ion diffusion coefficient # -" = ion electric mobility r = position vector.
The ( + ) and ( - ) signs in equation (1) refer to positive and negative ions, respectively. Consider now a particle of radius a, exposed to positive and negative ions in the presence of an external electric field (Fig. 1). Positive and negative ion densities in the bulk of the gas volume are equal to N~ and N O ' and, in general, N~ 4= N o. The particle causes a perturbation of the ion densities within its immediate vicinity; with the affected region being typically within a distance of five to fifty particle radii, depending on the particle charge and external field strength. Within this region, ion sources (due for example, to gas breakdown or ionizing radiation) and ion sinks (such as recombination) are neglected, and the ion continuity equation may be written as V.j-*(r) = 0,
i2)
D ~-V2N ~- (r)q:p-+V.E(r)N-~(r) = 0.
t3)
or
---~
(r,~
--,-
E.
Fig. 1. Coordinate system utilized tbr developing theory for charging of particle of radius, a. m an external electric field. The external electric field is of uniform strength, Eo, except in the immediate vicinity of the particle.
Continuum bipolar charging
543
The boundary conditions are N ± (a) = 0
(4a)
lim N ± (r) = N o.
(4b)
Equations (2) and (3) apply only to the immediate vicinity of the particle, and not to the bulk of the gas volume, where there may be significant sources and sinks. Neglecting space charge, employing the Einstein relation, p / D = q ~ / k T (where qe is the elementary charge unit, k is Bottzmann's constant, and T is absolute temperature}, and using E(r) = - V 0 ( r ) (where 0 is the electric potential), equation (3) can be written as V 2 N ± (r) -+- ~T V0(r)-VN ±(r) = 0.
(5)
The electric potential possesses azimuthal symmetry and, in terms of the spherical coordinate system of Fig. 1, is given by (Reitz and Milford, 1967) K - 1 aSEo (o(r, O) = - E o r
cos 0 + K + ~
nq~
r 2 cos 0 + 47re~'
(6)
where K = particle dielectric constant e = permittivity n = number of charges on the particle. Under the assumption that charge is a continuous variable, the particle charging rate is now given by dt
J + (a) - j - (a)]" d A , , (7)
=-ff[j;(a)-j;(o)]dA.
Since ion densities vanish at the particle surface, the radial current densities j~(a) are determined from the ion density gradient at the particle surface, yielding d~- =
D÷
-- D -
dr
dA.
dr
r=a
(8)
r=a
It is convenient to express the ion continuity and particle charging equations in terms of spherical coordinates and the following dimensionless parameters: u ± =N±/No
co = a E o q e / k T
~
x = r/a
ct = (K - 1)/(K + 2)
v=
r = qela+N~ t/e
nq~/4n~akT
7=p+N~/p-N~.
The time parameter, z, is defined in terms of the dominant ion species, which is exclusively positive in the work reported here. Substituting equation (6) into equation (5) and converting to dimensionless variables yields t~2U -+
1 O2u ±
?x 2 + x 2 ~c30
~ e [ x 2v +~--x q:~cocosO
+ 0cos,
(
(
2~t)l 1+~- 3 (9)
544
,3,, ~ I E L I ) t't a t
ROBERT
The boundary conditions are u :~( 1 ) = 0 lim u z:(x)= 1. x-4 t
The charging rate, equation (8), can also be put in dimensionless form, viz.,
dv lj'~ .... dr
2
du"
ldu-
I~
~= o
7
dx
] sin0d0,
tl!)
The expression on the right hand side of equation (11) is a function of charge. Thus, charge as a function of time is obtained by the following, r =
f" dr' . o dv'/dr
~12)
When written for ions of a single sign, equations (9), (11), and (12) are equivalent to the equations for unipolar field-diffusion charging given by Liu and Kapadia, although the polar angle, 0, and the dimensionless charging time, r, have been defined differently herein: The angle 0 is the supplement of the polar angle defined by Liu and Kapadia, and the time parameter, 3, as defined here is larger than the time parameter of Liu and Kapadia by a factor of four. The remaining dimensionless parameters are numerically equal to the corresponding parameters defined by Liu and Kapadia.
CHARGING CALCULATIONS The determination of particle charge as a function of time requires that (1) the two equations represented by equation (9) be solved and positive and negative ion density gradients at the particle surface be determined, (2) the surface integral indicated in equation (11) be evaluated to calculate the charging rate and (3) the integral of equation (12) be solved. The key step in this calculational procedure is determination of the ion density gradients. For the special case of co = 0, the equations can be solved analytically and u z (x) can be differentiated and evaluated at x = 1. The results are du* dx
du dx X=I
= 1
du ~
ve-
dx
1 -e-"
du
__ dx
I
b
[
v= 0
il3a)
v> 0
t13b)
X=|
ve ~
........ e ~ - 1"
(13cl
x=l
Note that if v is replaced by - v in equation (13b), the complementary equation, (13c), is obtained. Physically, this means the positive ion density profile in the vicinity of a particle with positive charge is identical to the negative ion density profile in the vicinity of a particle with negative charge of the same magnitude, and vice versa. For ~,J ~ O, an analytical solution is not available and a finite difference method is utilized to solve the ion continuity equations and to determine density gradients at the particle surface. Implementation of the numerical method requires that the exact boundary condition represented by equation (lOb) be replaced by an approximate boundary condition in which
Continuum bipolar charging
545
the ion density becomes unity at some finite value of x. This approximate condition is u -+(x,.) = 1.
(14)
The particle affects ion densities in its immediate vicinity because it is a sink for ions and because it perturbs the electric field. I f the influence of the particle on ion densities is significant at x,., use of the approximate b o u n d a r y condition, equation (14), can cause errors in calculations o f the ion density gradients at the particle surface. The effect of Xm on the ion density gradient can be determined exactly for ¢o = 0. The results, which are obtained by solving equation (9) subject to the boundary conditions represented by equations (10a) and (14), are du + dx
-
dudx
X=I
du + dx
=
xm x m- 1
v
=
(15a)
0
X=I
t =
ve-
e - ~ / ~ . - e -~
v> 0
(15b)
x=t
dudx
ve v = e~_e~/~ .
(15c)
x=l
These equations reduce to the exact forms, equations (13a--c), as xm ~ oo. Shown in Fig. 2 is the surface ion density gradient as a function o f x m for co = 0. T w o analytical solutions are displayed. The solid line is for the exact boundary condition, equation (10b), and the dashed line is for the approximate boundary condition, equation (14). The
•~ 10.0
I
r
I
1
I
c
='=;6
.9
1/=;2 ii
x I--
I/=0
-
IO
l--z ~J . . . . . . .
>I-
--EXACT
z 010 14J ¢3 Z O
....
=/='-2
BOUNDARY CONDITION
APPROXIMATE BOUNDARY CONDITION
% %
t
~,=*_6 0.01
I 3
I 5
Xm-OUTER
I IO
[ 20
l 40
B O U N D A R Y (dimensionless)
Fig. 2. Surface ion d e n s i t y g r a d i e n t s for exact a n d a p p r o x i m a t e b o u n d a r y c o n d i t i o n s . These c a l c u l a t i o n s are for to = 0.
546
ROBERT "~ [:JELD
et
al
a p p r o x i m a t i o n asymptotically approaches the exact solution as x,, increases. F or ~ ~. ~:he a p p r o x i m a t i o n is within 11 "o o f the exact solution at xm = 1 0 a n d 3 0° at x,, = 40. For I~,',t'~:,ir~ the vicinity o f a particle o f like charge, the a p p r o x i m a t e solution approaches the exact solution m o r e slowly; for v = 6, it is 8017o high at x~, = 10 and 13°o high at x,, = 40. i h i s occurs because the radius within which the particle perturbs ions o f like sign becomes larger as the charge increases. The opposite is true for ions in the vicinity o f a particle o f opposite charge. In this case the radius at which ion concentrations are perturbed by the panicle becomes smaller, and the a p p r o x i m a t i o n a p p r o a c h e s the exact solution more rapid[3 F r o m the preceding discussion it is evident that, at least for small co, x~ must be large. However, it is also necessary for ion densities to be calculated close to the particle surtace, i.e, at x close to one, This IS to obtain an accurate value o f the density gradient. These are conflicting requirements for a finite difference solution with a given n u m b e r of grias, and special measures are taken in an attempt to satisfy them. First, the following variable transformation is applied to equation (gj, " = ln(xi.
, L6)
The t r a n s f o r m e d equation is
a~u ~ a~u ± au ~ ~3z--~- + ~
+ c~z[_ 1 q: v e - : q: co cosO(e~ + 2ae -2:)
~u[ cos0 ] + c ~ 0 1 ~ n - - ~ + c o s i n 0 ( e ~ - 0 c e -2z) = 0 .
]
c17)
The resulting finite difference grid has a c o n s t a n t mesh spacing in z coordinates, but has mesh spacings in x coordinates which increase exponentially with increasing x. The result is a m u c h finer mesh near x = 1, where it is needed, than can be achieved without the transformation. In addition the solution is carried out in two steps. In the first step, a relatively coarse mesh is utilized with the first two radii being located at x = 1.0 (the surface o f the particle) and x = 1.1. I o n densities obtained at x = 1.1 then become the outer b o u n d a r y condition for a second step in which ion densities are determined between x = 1.1 and the particle surface; the first two radii n o w being located at x = 1.0 a n d x = 1.01. This proves to be an efficient and accurate means for achieving the necessary detail close to the particle while also allowing xm to be large. The finite difference equations formed from equation (17) are solved by the Gauss-Seidel M e t h o d (Ferziger, 1981) with successive overrelaxation or underrelaxation as needed for convergence. The relaxation parameter is varied f r o m 1.5 for small co to 0.8 for large co. The finite difference grid is formed by 41 concentric circles (0 < zi < lnx,,) and 19 radial lines (0 < 0i < 180°). A l t h o u g h it is f o u n d that a 21 x 11 grid gives almost identical results as the 41 x 19 grid (being within 0.1 oj~;for co = 1 and v = 0), the smaller mesh spacing with the latter allows the solution to converge at higher values o f co. As co increases, x m must be reduced for convergence. T h e values ofxm are as follows: for co < 0.5, x~, = 40; for 0.5 < co < 3, x,, = 30; for 3 < co < 7, x,. = 20; and for co = 10, x,. = 5. F o r large co, it is found that the value o f x,. does not have a significant effect on ion density gradients. However, for small co the approximate b o u n d a r y condition impacts calculations even with x,, = 40. An analytical correction, derived from equations (13-15) and rigorous only for co = 0, is thus applied to the numerical charging rate for co < 0.1. T o assess the validity a n d accuracy o f the numerical method, comparisons are made with analytical solutions which are available for co = 0 and with the numerical results o f Liu and Kapadia for unipolar charging. In Fig. 3, numerical charge calculations are given for ~,~ < 1, r = 1 and 100, and ~, = 3 and ~c. The numerical results obtained here for co = 0 a r e within 1 " o o f the exact solution. Further, it is seen f r o m the figure that the numerical results at eJ > 0 extrapolate to the correct value at co = 0. Liu and Kapadia's numerical result at co = 0.01 is consistent with the exact solution for r = l, but is approximately 30 °0 high for r = 100. The difference is likely due to their relatively large mesh spacing (Ax = 0.1667). The large mesh
C o n t i n u u m bipolar c h a r g i n g
547
spacing results in an underestimate of the charging rate at small v. A likely reason for this is as follows. At small v the ion density profile near the surface of the particle is concave downward and a finite difference approximation to the derivative is an underestimate. At high v the converse is true and, for small to, the effect is exacerbated by the approximate boundary condition (Liu and Kapadia utilize x,, = 7) as shown in Fig. 2. Nonetheless, the numerical results of Liu and Kapadia differ from the unipolar calculations reported here by less than 30~o for 1 < 09 < 10 and 10 < r < 100. Numerical ion density profiles for to = 3.0 and v = 0 and 10 are presented in Fig. 4. Ion densities are given for three polar angles: 0 °, 90 °, and 180 °. Analytical profiles for co = 0, which are independent of the polar angle, are included as dashed lines in Fig. 4. For v = 0 (Fig. 4a) the external field is seen to increase positive ion densities in the vicinity of the particle for large values of 0, where the field is directed radially inward (refer to Fig. 1) and to decrease densities for 0 near zero where the field is directed radially outward. An analogous effect is observed for negative ions (Fig. 4c), except that densities are high for small 0 and vice versa. For v 4= 0, the electric field has two components; one due to the presence of the particle in an otherwise uniform external field and the other due to the presence of charge on the particle. When the charge is positive, positive ion densities are diminished near the particle and, correspondingly, negative ion densities are increased, as shown in Figs. 4b and 4d. The result is a monotonically decreasing charging rate by positive ions and a monotonically increasing charging rate by negative ions as v increases. Thus, as in the case of pure diffusion charging, a particle exposed to bipolar ions in the presence of an external electric field accumulates charge until the rate of charging by negative
I00
~'~r'~
~ ~-~'~'~")":(13 IT: IO0
c
,_~ 50 W ~e T W-J n0.~
~s < i
/ i
O
I
i Jlt f
~ "~ 7= 0O 1
0~ Z~)"- 3
-- - - A - -
--APPROXIMATE APPROXIMATE
0 ,
00
"
EXACT
0
SOLUTION, THIS WORK SOLUTION,
LIU a KAPADIA
SOLUTION
0.5 - ELECTRIC
FIELD
1.0 (dimemionless)
Fig. 3. N u m e r i c a l solution for v vs co at small to. T h e exact solution at co = 0 is presented for c o m p a r i s o n . T h e s e c a l c u l a t i o n s are for = = 1.0.
54g
ROBER
I A.
el a /
~JELD
0,8 /
/
II!
0
~ -
).0 0,8 el
~. 0,6 o .~
0.4
o
E
1"
~
~°>//.-04 /i p ~~os,,,v~ 0,2 _ / / . " 0.6
)ll
/
I
I
0"8o~'I"--
.
/
/
I
I
- -
w . ~O
~ ' - -~..
f
.-
F,OSmVE
I
:i
-'7~
e.,o-
/
I
i(b)
I
v=o --~.o~
.-""
~
~
/-;
i
0.2
"¢1
~ ' ~
0
,_
J--~--"f'-~
I
~ - ~ , o
i
)-
P"
LO
7' W
().8
7
0.6 OA
//,.:o..._. [e) /
/I //. ."/ - ' "..-
~
o.,.~
i
......
<
..,.,vE
--~.o -4
0
,
I0 08 i
C).g
0.4
V
NEGATIVE
V-IO
0.2
----
0
1
i
OJ= 0
- - w-3,O
I
I
I
I
I
2
4
6
8
tO
20
x- RADIAL DISTANCE (dimensionless)
Fig. 4. Representative positive and negative ion density profiles. These calculations are tot • = 1.0
I° I
'1
i ~
'
'
I
'
,o. Lo
--),=cO
'1
j ?'-300
i
7, ioo !
oI
L
L
, ----
L
L
I
1
L
J
Io ]o.o "r-CHARGING TIME (dimensionless)
a
I
i
Ioo.o
Fig. 5, Charge as a function of charging time for (t) = l.O, :x = 1.0, and various conductivity ratios.
Continuum bipolar charging
549
ions equals the rate o f charging by positive ions. When this occurs, the particle acquires a steady state level o f charge as shown in Fig. 5, where charge is plotted as a function of time for = 1.0, ~ = 1.0, and various values o f the conductivity ratio. Steady state charge increases with increasing conductivity ratio and with increasing field strength (Fig. 6). The dependence on field strength is almost linear, with slight deviations from linearity at small o~. The effect of particle dielectric constant is also illustrated in Fig. 6. The charging results for ~t = 1.0 are presented in tabular form in Appendix I.
32 a
=i.o
e = 0.43
(K=CO) (K =2,5)
2B
S-
24
g .E_ /
w
2O f t
ilZ w l---
~" = 3 0 0 12
>Q ~ if)
1
/
n-"
bm
i
I
~
s
~ ~
~
I
~
~
s
J
e
i
0
I
2 ','-
3
4
ELECTRIC
5
6
7
8
9
I0
FIELD STRENTH (dlmen$ionlelli)
Fig. 6. Steady state charge as a function of electric field strength for ct = 1.0 and 0.43 and various conductivity ratios.
COMPARISONS
WITH EXPERIMENTAL
DATA
Experimental measurements o f bipolar charge acquisition are described in a thesis by Gauntt (1982). In these experiments, highly monodisperse aerosols o f polystyrene are exposed to counter currents o f positive and negative ions in the presence o f an external electric field. Particle diameters range from 0.3 to 1.1 # m and the external field is varied between 5 and 300 k V / m . The mean aerosol charge is measured by mobility analysis. These data are compared with continuum field-diffusion theory calculations in Figs. 7-9. In each o f these figures the experimental variables are expressed in terms o f the dimensionless parameters defined in the theory section. The number o f charge measurements for a given set of experimental conditions varies from one to about ten. When three or more measurements are available, the mean and experimental standard deviation are displayed in the graphs. The average o f two observations and single observations are represented by a symbol without error bars. Mean charge is given as a function o f electric field strength for various values of the conductivity ratio and z = 11 and 34 in Figs. 7a and b, respectively. The agreement between theory and experiment is generally very good. The largest differences occur for 7 = 10 and r = 11. where experiment exceeds theory by as much as 20 ° o. There also appears to be some deviation for co > 3 at ~, = 3 and Z = 1 1, where experiment is somewhat (10"~o) lower than
550
ROBER i : \
i
12
f:JELD ~t ,.~.
"T-~
r
I
/
l~
--r----]
y. GO I
/
SYMBOL
Op (F,m)
"
o
o.3,
~o
-
~
j
-
~
o.~6
~
TI-IEORY
w
:
~
o.so ~
r /
i
/
,,
-~ 7.10
~'
t
O~
I-- 4
O
,
2,
@ o
O
I I
~
_~
,,
I 2
I 3
I 4
I 5
I 6
I 7
l
w-ELECTRIC FIELD STRI~TH (dimeMieel~s) o, r - l l
IO
I
I
I T, 34
|YM~L
=
I
I
Do (~ m)
0 O
A8
I
0.31 0.50 0.76
5
)" ~
THEORY
"o bJ n..
~4
~J -
~ o
u
o
'
X"
I0
r
v
~2 i 0
i
0.2
I
I
b
Fig. 7. C o m p a r i s o n
I
t
0.4 0.6 0,8 1.0 w-ELECTRIC FIELD STRENTH (dimen$ionk~ls)
I 1.2
r-34
of bipolar continuum field,diffusion theory calculations with experimental datac h a r g e vs field s t r e n g t h .
theory. Otherwise, the differences are small. In addition to showing the degree of agreement between continuum theory and experiment, these graphs of dimensionless charge expressed as a function of dimensionless parameters also serve to illustrate the self-consistency of the charging data. Time dependent data for 7 - 3, 10, and ~ are compared with theory in Figs. 8a and b for ~,~) = 0.48 and 1.07, respectively. The steady state charge levels predicted by theory for 7 < ;< are clearly supported by the data. Agreement between theory and experiment is very good,
Continuum bipolar charging
'
,
'
'
i
'
,
,
r
551 ,
I
r
F
c~• 0 48 ( Dp" 0 50, E = 50 kV/m) SYMBOL IC
..Z
/.
3 Jo
CO THEORY (a=O
,~
--
._8
7" = CO
~e Qa .E
(J 4
),=lO 2 c~
0
,
,
,
I
i
I
t
I
I0
Ol
L
rJrJ
"y=3
,
I
l
I
I
I Ol0
r-CHARGING TIME
]
I
i
I0000
I000 (dimenaionless)
o to = 0 4 8
(Dp'l.l, E" 5OkV/m)
w=lO7
SYMBOL
__.Z
,~
I0
3
y=CO
/
T H E O R Y 1o •
- -
~8
¢: 6 z u
~4
y,3
,
0.1
,
,
,
i
,
,
,
I.O
,I
,
,
IQO r-CHARGING
TIME
b.
~
,
, i tOQO
,
,'oooo
(dimensionless)
• I.O7
Fig. 8. Comparison of bipolar continuum field-diffusion theory calculations with experimental datacharge vs time.
generally being within 15 5o. The decrease in measured charge at large z for y = 3 is felt to be an experimental anomaly (Gauntt, 1982). Other than this difference, theory and experiment basically agree within the bounds o f experimental uncertainty, estimated to be less than + 15°o (Gauntt, 1982). Up to this point, the comparisons have been limited to 7 = 3, 10 and ~ . Presented in Figs. 9a and b is charge versus conductivity ratio for ~ = 0.48 and r = 11 and 34. For r = 11, theory and experiment are in good agreement. For r = 34, theory and experiment are in good agreement for ~, < 30 and 7 = ~ . However, there is a significant difference for the
552
ROBERT a
F~ELt) e~ j i
o
/
w'048
(Dp" 0 5~m, Eo' 50kV/m)
~" 0429
0 EXPERIMENT --THEORY
. . . . . ,0. . .
,~, ' ' ':o~ ' ' 7 " ION CONOUCTWITY RATIO (k~*N*//~'N ") , W~0411
O T'll
[
m
*
I
e
',~oo
r , 34 w ' O . q (Op. OS/,~m.Eo. SOkV/m) = . 04,29 0 EXPE~M(NT
i
rO
7-ION CONOUCTMTY RATIO (/~'N'//4"N') r,34~
~,04e
F
!
~,, .L 34
* 0,?4 (D~*O.~kAm, (*SOkV/ml a o 0.4IS
i
(XPERIM|NT
d i
--THEORY
//
ICe
FO0
I000
I0.000
)'-ION CONDUCTIVITY RATIO (/,L*N'//~'N') c T,34,
~lOT4
Fig. 9. Comparison of bipolar continuum field-diffusion theory calculations with experimental datacharge vs conductivity ratio.
i n t e r m e d i a t e c o n d u c t i v i t y ratios, i.e. 30 < ~ < ~ , w h e r e m e a s u r e m e n t s are a p p r o x i m a t e l y 30 ~ b e l o w theory. O t h e r cases for w h i c h d a t a exist [~o = 0.74 a n d r = 34 (Fig. 9c); vJ = 0.30 a n d ~ = 11; co = 0.30 a n d z = 34] s h o w the s a m e trends; i.e. g o o d a g r e e m e n t at r = 11 a n d at z = 34 for s m a l l 7 a n d for " / = ~ , but less satisfactory a g r e e m e n t at r = 34 for i n t e r m e d i a t e 7.
Continuum bipolar charging
553
The discrepancy between theory and experiment at intermediate ratios is also noted in the prior paper on bipolar diffusion charging (Fjeld et al., 1981). It is uncertain whether this is due to an experimental problem at these ratios or a limitation of the theory.
SUMMARY
AND
CONCLUSIONS
A continuum regime theory for the acquisition of charge by a particle simultaneously exposed to positive and negative ions in the presence of an external electric field is presented. The theory is an extension to the bipolar case of the combined field-diffusion charging model of Pluvinage and of Liu and Kapadia. Approximate numerical solutions are developed which, in the limit as the external field approaches zero, yield results within 1 ~ of the exact solution. For the unipolar case differences are observed between these results and the results of Liu and Kapadia. These differences are probably due to their use of a larger mesh spacing in the radial direction near the particle surface and a smaller maximum radius. Charge is calculated as a function of time for dimensionless electric fields from zero to ten and ratios of positive to negative ion conductivity from one to infinity. After sufficient time charge is predicted to saturate at steady state levels. For a given conductivity ratio, steady state charge is predicted to increase almost linearly with increasing external field. For a given charging time and field strength, charge asymptotically approaches unipolar levels with increasing current conductivity ratio. Theoretical predictions are compared with experimental measurements for polystyrene aerosols in the 0.3 to 1.1/zm diameter size range. The data are supportive of the continuum field-diffusion description of bipolar charge acquisition, typically being within + 10-15 ~o of theory. The major exception to this is for intermediate current density ratios (I0 < ~ < ~), where theory exceeds experiment by as much as 30 ~. Acknowledgements--This work received support from the National Science Foundation under Grant ENG7701130. The Texas Engineering Experiment Station, and Clemson University.
REFERENCES Bricard, J. (1949) J. geophys. Res. 54, 39. Ferziger, J. H. (1981) Numerical Methods For Engineering Application, John Wiley, New York. Field, R. A., Gauntt, R. O. and McFarland, A. R. (1981) J. Colloid Interface Sci. 83, 82. Fuchs, N. A. (1963) Geof. Pura Appl. 56, 185. Fuchs, N. A. (1947) Izv. Akad. Nauk. SSSR Ser. Geor. Geofiz. 11, 341. Gauntt, R. O. (1982) PhD Dissertation, Texas A&M University. Gunn, R. (1954) J. Meteorol. 11, 339. Gunn, R. (1956) J. Meteorol. 13, 283. Lissowski, P. (1940) Acta Physiochim. URSS 13, 157. Liu, B. Y. H. and Kapadia, A. (1978) J. Aerosol Sci. 9, 227. Liu, B. Y. H. and Pui, D. Y. H. (1977) J. Colloid Interface Sci. ~ , 142. Pauthenier, M., Cochet, R. and Dupuy, J. (1956) Comptes Rendu 243, 1606. Pluvinage, P. (1946) Ann. Geophys. 2, 31. Reitz, J. R. and Milford, F. J. (1967) Foundations of Electromagnetic Theory, Addison-Wesley, Reading, MA. Smith, W. B., Felix, L. G., Hussey, D. H. and Pontius, D. H. (1978) J. Aerosol Sci. 9, 101. White, H. J. (1963) Industrial Electrostatic Precipitation, Addison-Wesley, Reading, MA.
p=3 0.07 0.13 0.23 0.33 0.41 0.49 0.76 1.00 1.11 1.11 1.11 1.11 1.I1 1.11 1.11 1.11 1. I I t.11
),=3 0.07 0.13
t 0.10 0.20 0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 100~00 200.00 400.00
r 0.10 0.20
i = 10 0.09 0.17
? = 10 0.09 0.17 0.32 0.46 0.59 0.70 1.15 1.67 1.95 2.11 2.21 2.33 2.33 2.33 2.33 2.33 2.33 2.33
),= 3{) 0.10 0.19
to = 0.t0
7=30 0.10 0.18 0.35 0.50 0.64 0.77 1.28 1.94 2.35 2.62 2.81 3.43 3.43 3.43 3.43 3.43 3.43 3.43
{o = 0.00
7 = 100 0.10 0.19
ct = 1.00
) ' = 100 0.10 0.19 0.36 0.51 0.66 0.79 1.33 2.05 2.51 2.85 3.11 3.84 4.36 4.63 4.63 4.63 4.63 4.63
- (K
)'=3(~) 0.10 0.19 0.36 0.52 0.66 0.80 1.34 2.08 2.56 2.92 3.20 4.05 4.80 5.15 5.36 5.49 5.72 5.72
CHARGING
No tie
(dimensionless time)
(dimensionless electric field)
{dimensionless charge)
BIPOLAR
UNIPOLAR 0.10 0.19
UNIPOLAR 0.10 0.19 0.36 0.52 0.67 0.80 1.35 2.09 2.59 2.96 3.25 4.17 5.07 5.59 5.95 6.22 7.06 7.87
: 0.10 0.20
t 0.10 0.20 0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 100,00 200.00 400.00
vii, w, ?, ~t
)~ = 3 0.09 0.16
?=3 0.07 0.13 0.23 0.33 0.42 0.49 0.76 1.00 1.12 1.12 1.12 !.12 1.12 1.12 1.12 1.12 1.12 1.12
I}/{K + 2) {dimensionless dielectric constant}
3 , : 10 0.11 0.22
; , = 10 0.09 0.17 0.32 0.46 0.59 0.71 1.16 1.68 1.96 2.12 2.22 2.34 2.34 2.34 2.34 2.34 2.34 2.34
/ = 31) 0.12 0.23
~2~:= 0.50
?=30 0.10 0.18 0.35 0.50 0.64 0.77 1.29 1.95 2.36 2.63 2.82 3.44 3.44 3.44 3.44 3.44 3.44 3.44
o~ = 0.01
CALCULATIONS
÷ i~, ,~'o .'It N o ( p o s i t i v e to n e g a t i v e i o n c o n d u c t i v i t y r a t i o )
r = q,.ll
7 = ii
'),=300 0.10 0.19
a = 1.00
nq~,4neakT
I:
t~ = a E o q ~ / k T
v -
APPENDIX
p = 100 0.12 0.24
a = 1.00
7 = 100 0A0 0.19 0.36 0.52 0.66 0.79 1.34 2.06 2.52 2.86 3.12 3.85 4.37 4.64 4.64 4.64 4.64 4.64
a = 1.00
i :-30(} 0.12 0.24
y=300 0.10 0.19 0.36 0.52 0.67 0.80 1.35 2.09 2.58 2.93 3.21 4.06 4.81 5.17 5.37 5.49 5.73 5.73
[iNIPOtAR 0.13 0.24
UNIPOLAR 0.10 0.19 0.36 0.52 0.67 0.80 1.36 2. ! 0 2.60 2.97 3.26 4.18 5.08 5.60 5.96 6.23 7.07 7.88
;o 2
©
0.33 0.47 0.60 0.72 1.19 1.73 2.02 2.19 2.42 2.42 2.42 2.42 2.42 2.42 2.42 2.42
7 = 10 0.14 0.27 0.51 0.73 0.93 !.11 1.82 2.64 3.07 3.31 3.45 3.64 3.64 3.64 3.64 3.64 3.64 3.64
0.24 0.34 0.43 0.51 0.79 1.17 !.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17 1.17
y=3 0.10 0.20 0.37 0.52 0.66 0.78 1.21 1.60 1.76 1.78 1.78 1.78 1.78 1.78 1.78 1.78 1.78 1.78
0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 I00.00 200.00 400.00
0.10 0.20 0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 100.00 200.00 400.00
7=30 0.15 0.29 0.55 0.79 1.01 1.21 2.02 3.05 3.68 4.10 4.39 5.04 5.28 5.28 5.28 5.28 5.28 5.28
o9= 1.00
0.36 0.51 0.66 0.79 1.32 2.00 2.42 2.70 2.91 3.38 3.55 3.55 3.55 3.55 3.55 3.55
) ' = 100 0.15 0.29 0.56 0.81 1.04 1.25 2.10 3.22 3.94 4.45 4.84 5.91 6.65 7.00 7.00 7.00 7.00 7.00
0.37 0.53 0.68 0.82 1.39 2.14 2.64 3.01 3.30 4.17 4.94 5.30 5.51 5.65 5.90 5.90
),=300 0.15 0.30 0.57 0.82 1.05 1.26 2.12 3.26 4.01 4.56 4.98 6.23 7.30 7.79 8.06 8.22 8.51 8.51
a = 1.00
0.37 0.53 0.68 0.81 1.37 2.11 2.59 2.94 3.20 3.95 4.49 4.71 4.78 4.78 4.78 4.78
UNIPOLAR 0.15 0.30 0.57 0.82 1.05 1.26 2.13 3.29 4.05 4.62 5.06 6.41 7.70 8.42 8.92 9.30 10.38 11.57
0.37 0.53 0.69 0.82 1.39 2.16 2.67 3.05 3.35 4.28 5.21 5.74 6.10 6.39 7.24 8.07
z 0.10 0.20 0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 100.00 200.00 400.00
0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 100.00 200.00 400.00
7=3 0.19 0.36 0.68 0.96 1.21 1.43 2.21 2.90 3.24 3.24 3.24 3.24 3.24 3.24 3.24 3.24 3.24 3.24
0.30 0.42 0.53 0.63 0.98 1.44 1.44 1.44 1.44 1.44 i.44 1.44 1.44 1.44 1.44 1.44
7 = 10 0.25 0.49 0.94 1.35 1.71 2.05 3.35 4.83 5.58 5.99 6.21 6.51 6.51 6.51 6.51 6.51 6.51 6.51
0.41 0.59 0.75 0.90 1.48 2.15 2.49 2.70 2.97 2.97 2.97 2.97 2.97 2.97 2.97 2.97
7=30 0.27 0.53 1.02 1.46 1.87 2.24 3.73 5.59 6.68 7.38 7.86 8.86 9.16 9.16 9.16 9.16 9.16 9.16
o9=3.00
0.45 0.64 0.82 0.98 1.64 2.48 3.00 3.34 3.59 4.14 4.35 4.35 4.35 4.35 4.35 4.35
7 = 100 0.28 0.55 1.04 1.50 1.92 2.31 3.87 5.88 7.13 8.00 8.64 10.32 11.33 11.72 11.72 11.72 11.72 11.72
a=l.00
0.46 0.66 0.84 1.01 1.71 2.62 3.21 3.63 3.96 4.86 5.50 5.72 5.82 5.82 5.82 5.82
7 = 300 0.28 0.55 1.05 1.51 1.94 2.33 3.91 5.96 7.27 8.19 8.89 10.86 12.39 13.04 13.38 13.57 13.83 13.83
0.46 0.66 0.85 1.02 1.72 2.65 3.27 3.72 4.07 5.13 6.05 6.48 6.72 6.88 7.15 7.15
UNIPOLAR 0.28 0.55 1.06 1.52 1.94 2.34 3.93 6.01 7.34 8.29 9.02 11.15 13.06 14.08 14.75 15.25 16.75 18.11
0.46 0.67 0.85 1.03 1.73 2.67 3.30 3.77 4.13 5.28 6.39 7.01 7.45 7.78 8.79 9.75
T.
"O O
¢.. e..,
O =1
r 0.10 0.20 0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 100.00 200.00 400.00
;¢=3 0.28 0.54 1.01 1.42 1.79 2.12 3.28 4.30 4.65 4.75 4.81 4.81 4.81 4.81 4.81 4.81 4.81 4.81
Appendix I (contd.)
),= 10 0.38 0.73 1.39 1.99 2.54 3.03 4.96 7.14 8.23 8.81 9.13 9,54 9.54 9.54 9.54 9.54 9.54 9.54
) ' = 30 0.40 0.79 1.51 2.16 2.76 3.31 5.51 8.23 9.81 10.81 11.48 12.81 13.18 13.18 13.18 13.18 13.18 13.18
to=5.0 7 = 100 0.41 0.81 1.55 2.22 2.84 3.41 5.72 8.66 10.46 11.69 12.58 14.83 16.10 16.44 16.52 16.52 16.52 16.52
~ = 1.00 7=300 0.42 0.81 1.56 2.24 2.86 3.44 5.78 8.78 10.66 11.96 12.93 15.57 17.53 18.30 18.68 18.89 19.14 19.14
UNIPOLAR 0.42 0.82 1.56 2.25 2.87 3.46 5.81 8.84 10.76 12.10 13.11 15.98 18.43 19.69 20.52 21.13 22.88 24.49
r 0.10 0.20 0.40 0.60 0.80 1.00 2.00 4.00 6.00 8.00 10.00 20.00 40.00 60.00 80.00 100.00 200.00 400.00
), = 3 0.50 0.97 1.83 2.59 3.26 3.86 5.99 7.88 8.50 8.73 8.81 8.81 8.81 8.81 8.8 ! 8.81 8.81 8.81
), = 10 0.68 1.33 2.53 3.62 4.61 5.52 9.03 13.00 14.97 16.01 16.57 17.26 i 7.26 17.26 17.26 17.26 17.26 17.26
), = 30 0.73 1.43 2.74 3.93 5.02 6.02 10.03 14.96 17.78 19.53 20.68 22.85 23.38 23.38 23.38 23.38 23.38 23.38
t o = 10.0 ~, = 100 0.75 1.47 2.81 4.04 5.16 6.21 10.40 15.71 18.92 21.06 22.58 26.26 28. I0 28.49 28.58 28.58 28.58 28.58
a = 1.00 y = 300 0.76 1.48 2.83 4.07 5.21 6.26 10.51 15.94 19.27 21.53 23.18 27.51 30.40 31.42 31.88 32.11 32.34 32.34
IJNIPOLAR 0.76 1.48 2.84 4.08 5.23 6.28 10.56 16.05 19.45 21.78 23.50 28.19 31.87 33.66 34.81 35.64 37.91 39.84