Download Zahn, M., Impact Charging of an Isolated Cylinder with Skewed Field and Flow, Journal of Electrostatics 5, 85-99, Sept. 1978

Journal of Electrostatics, 5 (1978) 85--99
© Elsevier Scientific Publishing Company, A m s t e r d a m -- Printed in The Netherlands
85
Paper No. 2-A
IMPACT CHARGING OF AN ISOLATED CYLINDER WITH SKEWED FIELD AND FLOW
MARKUS ZAHN
Department
of Electrical
Gainesville,
Engineering,
Florida
University
of Florida,
32611
ABSTRACT
The charging of a cylinder due to impacting charged particulate
when the imposed electric
In a quasi-static
field and inviscid
limit and neglecting
is analyzed
fluid flow are in arbitrary directions.
self-field
effects the combined
flow stream function is a constant along the particle
trajectories.
field and
Separation
streamlines which pass through points where the total force on the particle is zero
divide those trajectories
travel past.
which deposit
charge onto the cylinder
from those which
The cases of co-linear and crossed field and flow are particularly
examined.
I. INTRODUCTION
Electrofluidized
beds
(1-3) represent a new class of precipitation
where small charged particulates
are collected on larger particles.
collecting area of fluidized particles
that of a plate collector
distributed
in a conventional
results in a better collection efficiency,
and a smaller size device.
Furthermore,
The effective
over a volume is much larger than
electrostatic
precipitator.
This
the ability to collect submicron particles,
the agglomerated
can themselves be used to collect additional
charged particulate
particulate.
Past analysis has been based on the Whipple and Chalmers model
applied to the charging of raindrops,
devices
(4), originally
where an applied electric field uniform at
infinity is in the same or opposite direction to an imposed uniform flow field.
Any charged particulate
injected at infinity is propelled along the force lines.
If a force line is incident upon an object,
If the object is isolated,
the particulate
is deposited
there.
the charge build-up tends to diminish further charge
collection due to Coulombic repulsion,
until it reaches a saturation value of charge
where all particle lines pass around the object so that the charge on the body
remains constant thereafter.
The Whipple and Chalmers model has been used with the limiting cases of viscous
dominated
(inertia-less)
charging characteristics
and inviscid
(inertia dominated)
flows around a sphere.
for either flow are the same because for both cases the
normal component of the fluid velocity on the sphere must be zero
(3).
The
86
Because the flow and electric
depends on two spherical
field are in the same direction the problem only
coordinates
allowing the definition of a stream-function.
If the field and flow are not in the same direction a stream-function
defined because the solutions now depend on all three coordinates.
cannot be
This problem
is avoided in the case of impacting charge onto a cylinder with the electric field
and flow at arbitrary angles as the solution remains two-dimensional
dependence, on the axial coordinate.
We choose to only consider
flow to avoid the paradox concerning an inertia-less
cylinder
(5).
with no
inviscid
fluid
flow around an infinitely long
However we suspect that the results are representative
of any laminar
flow around the cylinder as was found for spheres, because the boundary condition
of zero normal fluid velocity at the cylinder wall is true for any fluid properties.
II. GOVERNING EQUATIONS AND APPROXIMATIONS
The infinitely long perfectly
conducting
cylinder of radius R in Fig.
instantaneously
carries a uniformly distributed
unit length I.
The imposed electric
(I)
surface charge with charge per
field is at an angle %
o
to the imposed velocity
field
lim
r ÷ ~
v = V i = Vo(ir cos 8-[ 8 sin 0)
o y
(I)
= Eo(iy cos 8 o + ~ x sin Co) = Eo(ir
= cos (e - eo)- ~0 sin (O - 8o))
Charged particulate
injected at infinity with density Po and mobility b results
in drift and convection currents with density
= 0(bE + ~)
(2)
In the absence of further charge generation or recombination,
charge conservation
requires
~p
V • 3+~-f = 0
(3)
while the electric field E, charge density p, and permittivity
e are related
through Gauss's law
v
•
~ = £c
(4)
Fluid incompressibility
also requires
that
V • v = 0
(5)
Substituting
eqn.
~P + (bE + ~ )
8t
(2)
into eqn.
2
" Vp + bp__@_= 0
E
(3) and using
The chain rule of differentiation
dp(x,y,z,t)
eqn.
(4) and
eqn.
(5)
yields
(6)
operating on the charge density
= ~ t dt + ~3-~xdx + ~-P-dy
3y
+ ~-~z dz = ~~P at + (d~ • V)p
(7)
g?
shows that along those lines
d~ = ~ + bE
(8)
dt
eqn.
(6) can be rewritten as an ordinary differential
2
d--e=!e+
dt
equation
(d~ . v)p = - h °
~t
dt
(9)
E
with solution
P
p =
d~
on ~ = bE
o
1 + b p?
+ v
(i0)
to )
(t -
C
where Po is the charge density at some time to.
Although eqn.
the charge density decreases with time it is necessary
find where the charge is.
of convection
concentric
are necessary
(6).
solve Poisson's
problem,
or
numerical
techniques
equation for the electric
field is much larger than the self-fields
generated by the charged particulate,
we can neglect
these self-fields
and
write Gauss's law as
v • ~ ~ o
(ll)
In this limit eqn. (9) is also modified
to
d_p_p~ 0 ÷ p = Po
dt
(12)
as the square of the charge density term on the right in eqn.
self-field
generated by the charge density.
charge cloud keeps a constant
with time because
repel!).
(8) to
in the absence
the charged particulate.
if the imposed electric
approximately
eqn.
geometries
coaxial cylinders,
For a two or three dimensional
to self-consistently
field which propels
However,
to integrate
This is easy in one dimensional
(~ = 0) such as between parallel planes,
spheres
(i0) tells us how
(9) is due to the
In the solution of eqn.
(i0), a
charge as it convects but its density decreases
the self-field
tends to make the cloud expand
If we neglect self-field
effects,
this expansion
(like charges
is negligible
for
short times where
bp°(t - to) << 1
(13)
C
At the cylinder's
surface the normal component of fluid velocity
while the perfectly conducting cylinder with uniformly distributed
requires
the electric
which satisfies
field to be normally incident.
these boundary conditions
~
1
v = V[(l
~
E =
- ~l-z-) c o s
e ir -
(i +
is zero
surface charge
The electric field and flow
and approximations
are
- -
~7-) sin e ie]
.
(14)
r
1
[(i + ~)r cos
i__)
(0-Co) + ~-- ] i r + (i - ~2
sin (8 - 80)i 0
(15)
88
w h e r e w e normalize all q u a n t i t i e s as
-
r
~
[,
r V = v°
=
1
~
~--'
=
o
~=__~
2~E
o
R
'
E
bE
v_ = v
'
o
t
(16)
o
V--
o
'
R
Because the flow and field have zero d i v e r g e n c e it is convenient to define
stream functions as
v-
=
1 ~_~
~ ~8
r
-
~2
÷ ~ = - (r +i) sin 8
ir + ~ i [ e
8r
i
i
(17)
r
+ ~
i°
f =-
(~2+I) sin (e-e) - ~e
(18)
w h i c h a u t o m a t i c a l l y s a t i s f y the zero d i v e r g e n c e conditions of eqn.
(5) and eqn.
(Ii).
U s i n g eqn.
d~
+ V~
-- m
dt
r
r ~d0
=
dt
r
~e
(17) and eqn.
i -~
= -- -~ ~O
+v@
= ~~
~r
(18) in eqn.
(8) yields
(~ + ~ )
(19)
(X + ~)
(20)
Since X and ~ are functions of r, O, and t, we w r i t e their total d i f f e r e n t i a l
as
~ ~
d(~+b =7~r
~ (~+~)d~ + ~~
In eqn.
(~+f)dO + ~
(×+~)d~
(21)
(17) X is only a function of r and 8 but not time.
In eqn.
(18) the im-
posed field c o n t r i b u t i o n is also not a f u n c t i o n of time but I changes w i t h time as
the charged p a r t i c u l a t e collects on the cylinder w i t h rate
di
--=
dt
I
"
'
I =
I
2 p bE R
oo
(22)
w h e r e I is the total current incident on the cylinder.
If I changes slowly compared to the transit time of charged p a r t i c u l a t e
injection point to the cylinder,
time-independent electric field.
This allows us to neglect the time d e p e n d e n c e
of Z so that the last term on the right in eqn.
(19) and eqn.
(20) in eqn.
from the
each packet of injected charge sees an e s s e n t i a l l y
(21) is zero.
T h e n using eqn.
(21) w e find that the sum of s t r e a m - f u n c t i o n s
is constant
a l o n g a flow trajectory
dr
d(X+E) = 0 + X + Z = constant o n - ~ = E + v
dt
(23)
so that the governing e q u a t i o n of the charge trajectories is
}2[V sin 8 + sin(8-8
o
)] + r[%8-c] + [sin(8-8 ) - V sin 8] : 0
o
(24)
89
where c is a constant
of a given curve found by specifying
a single point
(r,8)
w h i c h the curve passes through.
Defining
the coefficients
= [V sin 8 + sln(8-8o) ]
= V sin 8 - sin(8 - 8o)
eqn.
~2
(24) is rewritten as
+ ~
_ ~ = 0
(26)
w h i c h has solutions
-
--=
r =
.
2A
-
+
2A
_
(27)
A
The necessary
trajectory
computations in plotting r vs. 8 from eqn. (27) for a given
~
constant c are easily performed by a pocket calculator.
Two conditions
are necessary
for the trajectories
of eqn.
(27) to deposit particulate
onto the
cylinder:
(i)
The flux of ions must be directed
cylinder,
dr/dt
I~=l < 0.
zero because of the rigid wall boundary,
that the radial component of electric
If this condition
(Er(r=l)
radially
inward at the surface of the
Since the radial component
of flow velocity
this condition
field be negative
is equivalent to requiring
~
~ ~
at r = i, Er(r=l) < 0.
is not met the trajectory which emanates
> 0) from the cylinder carries
is already
radially outwards
no charge because no charge is injected
at r = i.
(2)
Even if Er(r=l) < 0, the trajectory carries charge only if the streamline
~
starts ar r = = w h e r e charged particulate is injected.
We will find cases where
characteristic
trajectories
III.
trajectories
start and terminate on the cylinder,
but these
have zero charge because they do not begin in a charged region at
COLIN-EAR FIELD AND FLOW
Only that section of the cylinder w h i c h has a negative
component
can
(but not necessarily
2 cos 8 + k < 0 ÷ 8
~ 8 ~ 2~ - 8
S
Thus,
does) collect
(8
S
= cos
-1
-
~ - 2 the entire cylinder can collect charge.
be collected
surface collects
"window" becomes
)
(28)
on the cylinder while if
Between these limits,
charge.
over the lower half of the cylinder
charge this angular
field
S
if k ~ 2 no further charge is collected
a fraction of the cylinder's
radial electric
charge
I~I ! 2,
When ~ = 0, charge can
(z/2 E 8 x 3/2 z).
For positive
smaller being zero when ~ ~ 2 while for
90
n e g a t i v e charge the w i n d o w increases b e i n g 2~ w h e n I ! -2.
(I)
V h 0
When the electric field and flow are in the same direction,
s e p a r a t i o n streamlines are incident onto the cylinder at angles @
and
s
trajectories o u t s i d e this plume
2~ - @ as shown in Figure 2a.
Characteristic
s
miss the cylinder carrying their charge out to infinity w h i l e those trajectories
b e t w e e n the s e p a r a t i o n streamlines intersect
the cylinder d e p o s i t i n g their charge.
The streamlines emanating from the cylinder o u t s i d e the w i n d o w for charge collection
191 < @
do not carry any charge as they did not start in a region of charge.
s
The charging current is found by r e f l e c t i n g the s e p a r a t i o n streamline b a c k
to y . . . .
constant ~
c
s
= 2 sin 0
s
From eqn. (27) w i t h O = 0 at r = 1 for
o
of the s e p a r a t i o n streamline is
s
IiI!
2, the trajectory
+ X0
(29)
At @ = ~, r = ~ its d i s t a n c e x
from the y axis is
~*
x = r sin 9
lim
+ ~
(30)
so that from eqn.
~*
X
CS
- i~
(24) w i t h @
= 0 w e have
o
2 sin @ S + l(Os-~)
~+l
(31)
~+i
The current per unit length at y = - ~ b e t w e e n the s e p a r a t i o n s t r e a m l i n e s is
the current incident u p o n the cylinder
*
I = 2Po(Vo+bEo]X._
I
+ i = (V+l)x*
= 2 sin 9
;
s
I =
2PobEo R
(32)
+ %(@ -~)
s
and is plotted in F i g u r e 3 versus I.
This charging current is independent of the
v e l o c i t y V for V ~ 0 b e c a u s e i n c r e a s i n g V increases the current d e n s i t y by the
same factor as the area b e t w e e n separation streamlines decreases.
> 2, { = O w h i l e
(2)
- 1 < V ~ 0
for ~ < -
2, ~ = -
For
~.
W h e n the field and flow are in o p p o s i t e d i r e c t i o n s w i t h the
charges still entering the system at y = - ~ some streamlines b e g i n and end on
the cylinder and thus carry no charge as in Figure 2b.
Critical points of zero
force (d#/dt=O) now exist outside the cylinder
+ E
r
= 0 ÷ V(r2-1) cos @ +
(r2+l) cos @ + lr = 0
r
(33)
~
v@ + E0 = 0 + V(r2+l) sin @ +
(r2-1) sin @ = 0
91
Solutions
to the pair of equations
in eqn.
(33) are
sin 8 = 0 ÷ 0 = O,
(34)
2 (V+I)
2 (V+I)
[
"] - [ I+V~
]
and
~ , l - V • cos Oc = - ~ k
(35)
This second critical point at angle 8 c only has real values outside the cylinder
when - 1 < V < 0 and is important
which intersect
for it separates
the cylinder and those which don't.
streamline with constant
C
w h i c h passes
By reflecting
the separation
through the critical point back to
c
from the y axis is
y = - o% its distance
regions between those streamlines
1/2
~*
x
Cc - ~
_ [4(I-V 2) - k2]
~+i
so that the charging
{
= (V+I)x*=
- k(~-Sc)
(36)
~+i
~
current
is
[4(l-v2) - x231/2 - X(~-oc) ]~l ~ 2i f ~-~2
0
t _~
w h i c h is plotted
(3)
~!-21-f[~-
in Figure 3.
V = - i If the imposed
opposite
(37)
~ m 21-~ ~V2
in direction
fluid flow velocity
to the electrical
for charges at infinity.
is equal in magnitude but
drift velocity,
All trajectories
there is no net motion
begin on the cylinder and thus carry
zero charge.
(4)
V < - 1
direction
If the flow velocity
to the electric
field,
is larger in magnitude
injected
In this flow regime there are two critical
(34) and shown in Figure 4.
~
Ccl = 0
Oc
in
points at 8 = 0 and O = ~ given by eqn.
Each separation
the critical points has a respective
but opposite
charges enter the System from y = + ~.
streamline w h i c h passes
trajectory
through
constant
= 0
(38)
Cc2 = ~
ec = ~
No streamlines
intersect
the cylinder
for k ~ 0 so that the charging current
92
is zero.
For I < 0, each streamline
~
-*
Ccl
xI =
x~
= 0
~+1
;
reflected
Cc2
~
O+l
~+l
-* ~ *I) = - Cc2
~
(x2-x
independent
:
-
IV.
in Figure
CROSSED
3 for various
(40)
the charged
field
particulate
= r , 8 = 8
C
+ E
are summarized
values
in Figure
5, and the charging
of I and V.
FIELD AND FLOW
Again it is important
r
is
of the flow velocity.
When the electric
v
current
~
All the flow and field regimes
current
is at distance
(39)
from the y axis so that the charging
=-(V+I)
back to infinity
~
is perpendicular
at infinity
starts
to find the critical
from eqn.
(14) and eqn.
to the velocity
flow field,
out at angle O = ~ + y where
points
(0 =~/2),
o
tan y = I/V.
of zero force at
(15)
C
r
= 0 + V(r2-1)
cos 8 + (r2+l)
v e + -E0 = 0 ÷ - -V(r2+l)
- sin e +
which yields
tan O
the equivalent
sin @ + Ir = 0
-(r2-1) cos e
-~
(41)
(42)
0
set of equations
(6-i)
c
(43)
V(~+I)
84 _ ~63 _ 262 [i +--a(V2-1)
] -~6+i
= 0
~2+i
where
B = ~2
c
;
~ =
(44)
~2+i
The four roots of eqn.
61,2 = ~i { ~2
(43) are B I, 1/61 , 62 , I/B 2 where
+ Q +_ [(~ + Q)2 _ 411/2 ~
(45)
with
_)2
4~ 71/2
Q = [(~ + z - - ~ + i j
As shown in Figure
streamlines
intersect
constant
passes
C
c
angle 0cy I.
(46)
6 for I = - i, all the charge contained
the cylinder.
through
The first separation
the critical
Its perpendicular
distance
between
streamline
point and intersects
two separation
with trajectory
the cylinder
at
X*c from the line at angle 8 = - (T-y)
98
at r = o~ is
+ Y,(~-y)
C
X~*
=
c
lim(r sin(8+~-T))
= -
C
r + ~
O -~
4 ~2+I
The second separation
radial electric
2 sin e
s
(47)
(w - y)
streamline
intersects
the cylinder
at angle 8
s
w h e r e the
field is zero
(48)
+ ~ = 0
~
Its distance x
from the line at angle 8 = w + y is
S
cs - ~(~+~)
~~
x
= lira r sin(0-(~+y))
s
=
(49)
~
4V2+I
r*y
8+~+y
Note that the line at 0 = w + y is the same line as O = - (w-y).
principal values of the trigonometric
functions
in determining
Using the
C
and C
C
us to keep
0 a continuous
function along a given trajectory.
requires
S
Thus along C
we
c
in the third and fourth quadrants while along C ,8
s
from ~.
must consider 0 as negative
is positive
increasing
Each of the trajectory
c
=
constants
(r - i__) ~ sin 8 - ~ +
c r
c
(rc
i___)
r
cos 0 c + X8 c
C
C
+ h0
S
S
The total charging
and is plotted
(50)
C
= - 2 cos 0
S
is
current
in Figure
is then
7.
The separation
There is a transition point when 8cy I = 8s as then Cs = ~c"
streamline
through the critical
point is then just tangent
entire surface area collects charge.
of charge collection
to the cylinder and the
More negative values of ~ have the region
bounded by both sides of this separation
streamline
= - ~w
The results of eqns.
(52)
(47) - (52) are only valid
the critical point appears
8
in the third quadrant
is in the fourth quadrant
S
eqn.
(48) become
so that
for negative
~.
When ~ > 0
(w < 8 < 3/2 ~) while the angle
(3/2 w < 8 < 2w) as in Figure 8 so that eqn.
(47) and
94
~,
~
X
=
+ ~(~+~)
C
c
V~2+I
.,
c
(53)
- ~ (~+~)
S
X
=
--
s
/ ~ 2+ 1
giving
the charging current
~=~
+
as
=
(54)
As Soon as ~ reaches the value where C
= C
C
(at some value less than 2), the
S
current is zero for larger values of %.
A summary of the streamlines for various Z and V are shown in Figure 8.
ACKNOWLEDGMENTS
This work was performed while on sabbatical leave at the Department of Electrical
Engineering, Massachusetts Institute of Technology, Cambridge, Mass. 02139 with the
support of a Faculty Development Grant from the University of Florida, a National
Science Foundation Faculty Fellowship, and National Science Foundation Grant ENG
76-06605.
Stimulating conversations with Professor J. R. Melcher are gratefully
acknowledged.
REFERENCES
1 K. Zahedi and J. R. Melcher, Electrofluidized Beds in the Filtration of a Submioron
Aerosol, J. APCA, 26, No. 4, (1976) 345-352.
2 T. W. Johnson and J. R. Melcher, Electromechanics of Electrofluidized Beds, I&EC
Fund., 14, No. 3, (1975) 146-153.
3 J. R. Melcher and K. S. Sachar, Final Report:
Charged Droplet Scrubbing of
Submicron Particulate, Contract No. 68-02-0250, EPA, Research Triangle Park, N.C.,
1974.
4 F. J. W. Whipple and J. A. Chalmers, On Wilaon's Theory of the Collection of
Charge by Falling Drops, Quart. J, of the Roy. Met. Soc. 70 (1944) 103-119.
5 H. Lamb, Hydrodynamics, 1932, Dover, N.Y., pp. 614-616.
6 M. Zahn, Transient Drift Dominated Conduction in Dielectrics, IEEE Trans. on
Elec. Insul., EI-12, No. 2, (1977) 176-190.
95
+
+
÷
+
+
+
Coulombslmefer
////////////////
,njl~densi
ctedoftyCh°rgepo
] v+bE
~; Vor,; Vo~r co, o -to ,,n O]
lira E= Eo[TrcoS(~)-Oo)-Tesin{8-8o)]
r-,~
Figure 1
An imposed electric field is at an angle 8o to an imposed velocity
field incident upon an infinitely
R instantaneously
long perfectly conducting
cylinder of radius
carrying a charge per unit length ~.
Y
8
y/
~-Criticof Point
V+bE =0
Y
cc
;c
=-.5, ~'= J,Oo = o
FiBure 2
co-linear.
= i,[=-4eo--O
Separation streamlines when the electric and velocity fields are
Field and flow in the (a) same or (b) opposite directions with
charges entering
the system at y = - ~.
part of the cylinder's
surface.
Charge is deposited
on the darkened
96
-2
Figure
3
co-linear
2
i"
Charging
current
field and flow.
~ 0 or V ! - i.
I versus
cylinder charge per unit length ~ for
The current does not depend on the velocity
In between,
if
- i < V < 0, the current varies smoothly
between these two limits.
Ccl
X2
CriticalPoint
O, X = - ID, V = -1.5
Critical
Figure 4
Separation
oppositely
directed with charge entering
Point
streamlines w h e n the electric and velocity
the system at y = + ~.
fields are
Summary of the streamlines for co-linear field and flow as a
function of charge ~ and flow velocity V.
Fisure 5
--I
98
Critical Point (r c, e c)
Cc
XC
L,s
Figure 6
y
Separation streamlines when the electric and velocity fields are
perpendicular.
~
6-
\
I
\
\
%
\
_
-2
Figure 7
~,
V=OD
2
Charging current I versus ~ for crossed field and flow.
current only slightly depends on the flow velocity.
charge, the current varies linearly with ~.
The
For large enough negative
99
//
V=O
e8o=90
°
~' = I
X=2
,,,,\
X
=
I
k
k
-,,,,
>,=-2
Figure 8
of I and V.
Summary of the streamlines for crossed field and flow as a function