ELECTRIFICATION BY LIQUID DIELECTRIC FLOW
by
STEVEN MARC GASWORTH
S.B., Massachusetts Institute of Technology
(1976)
S.M., Massachusetts Institute of Technology
(1981)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
(February, 1985)
0
Massachusetts Institute of Technology
Signature of Author
Department-'of Electrical tgineering
and Computer Science
January
Certified by
I
\,
.
J
v
31, 1985
--
James R. Melcher
Thesis Supervisor
by,,
Certified
7
\- -
.Markus Zahn
-Thesis Supervisor
Accepted by
,,
Arthur C. Smith
Chairman, Departmental Committee on Graduate Students
INSTITUTE
lASSACHUSETTSF
OFTECHNOLOGY
APR 01 1985
L1DRPIES
ARCHIVES
2
ELECTRIFICATION BY LIQUID DIELECTRIC FLOW
by
STEVEN MARC GASWORTH
Submitted to the Department of Electrical Engineering and
Computer Science on January 31, 1985 in partial fulfillment of
the requirements
for the Degree
of Doctor
of Philosophy
ABSTRACT
The charging of insulating surfaces by the flow of semi-insulating liquids is investigated with a view towards identifying ways to forestall
insulation failure in liquid-cooled power distribution equipment. Both
fully developed and fully mixed turbulent flows are considered, the former as a model for the flows in insulating tubes, the latter approximating conditions in certain practical elements such as pumps and expansion
regions. Models are developed for the space-time evolution of charge and
electrical stress that emphasize the interplay between electrokinetic
charge generation, convective transport, ion diffusion and self-precipitation, and conduction driven by the generated fields themselves.
Experiments are designed to allow measurement of the currents influent
to the insulating tube or expansion region, and either the accumulating
surface charge or the generated electric field. Apart from the imposed
flow conditions, the liquid conductivity is controlled by a commercial
antistatic agent, while the external configuration of conductors is prescribed with an awareness of the contribution of external image charges
to the generated fields. Results indicate that significant electrical
stresses stem from the transfer of a net charge to the flow upstream of
the insulating element, and migration of the entrained ions in a space
charge field that is predominantly normal to the the insulating surface.
Leverage over the flow-induced stresses is afforded by the external conductor configuration, control of bulk and surface conductivities, and
control of the influent convection current with a properly designed expansion region inserted just upstream of the insulating element.
Physicochemical features of the liquid-insulating solid interface are
probed in two supplementary experiments that make use of a multi-phase
helical winding around a section of an insulating tube. In the first
experiment the flow is imposed and half of the phases are excited with a
sinusoidal potential. Convective displacement of the resulting standingwave perturbation in the entrained volume charge density is detected as
an imbalance in the currents carrying perturbation image charge to two
of the unexcited phases. The response is interpreted in terms of the undisturbed convection current, the surface conductivity, and the conductivity gradient at the interface. In the second experiment, the flow is
induced by a traveling-wave excitation of the winding with the objective
of investigating the charge injection process at the same interface.
Thesis
Title:
Thesis
Title:
Supervisor: James R. Melcher
Stratton Professor of Electrical Engineering and Physics
Supervisor: Markus Zahn
Associate Professor of Electrical Engineering
3
ACKNOWLEDGEMENTS
With the fresh perspective that comes from extending my contacts outside
of the Continuum Electromechanics Lab, I am struck by the uniqueness and
depth of the education I've received, all the more so because so much of
it has been shaped by one man, Professor James Melcher. To his credit as
an educator, the form of this thesis is as much a product of the careful
grounding in fundamentals he has provided over the years, beginning long
before this project commenced, as of discussions specific to the thesis.
The balance of the thesis committee included Professors Thomas Lee, Ain
Sonin and Markus Zahn. Their participation in discussions and encouragement are deeply appreciated. Professor Sonin's influence was especially
strong,.through both his insightful comments on the present work, and
his own previous publications in the field.
Through MIT's undergraduate research opportunities program, several students participated in various aspects of the project. Of these, three
will find their efforts represented here: Yana Kisler, who assisted with
the streaming current measurements of Sec. 2.2.3, Steven Lin, who collected essentially all of the traveling-wave pumping data of Sec. B.2,
and Lim Nguyen, who obtained the conductivity data of Fig. 3.6.
Despite a remarkable range of projects in the Continuum Electromechanics
Lab there is a sense of unity, traceable to the attention paid to fundamental aspects and to a recognition that all of these projects impact on
our standard of living. That such a sense is maintained in an academic
setting that characteristically emphasizes individual efforts is a credit to the students and faculty of the lab. Only it remains a mystery how
our resident computer expert, Eliot Frank, finds enough time for his own
thesis; his efforts have greatly simplified the task of preparing documents like the present one.
There can be no understudy for the role played by family and friends.
Joel Adler and my father, Kenneth, are two who took the initiative when
they saw ways to help.
The stresses that accompany a thesis are shared by the spouse of a married student, and Debbie somehow bore her share patiently while doing so
much to lessen mine with her advice and encouragement. The result is as
much a joint effort as my department will allow me to admit before revising the name on the diploma.
This work was sponsored by the Electric Power Research Institute, under
contract no. RP1536-7.
4
TABLE OF CONTENTS
Abstract
2
Acknowledgements
3
List of Figures
8
List of Tables
13
1
14
2
Introduction
1.1
Engineering Motivation
14
1.2
Flow-Induced Electric Field Generation
16
1.3
Scientific Motivation
20
1.4
Overview
22
Background and Preliminary Experiments
26
2.1
Classification of Tubes as Conducting or Insulating
26
2.2
Streaming Currents in Conducting Systems
29
2.2.1
Fully Developed Volume Charge Density
31
2.2.2
Spatial Development of the Volume Charge Density
36
2.2.3
Preliminary Experiments
38
Streaming Currents in Insulating Tubes
42
2.3.1
Early Work
43
2.3.2
Steady State
48
2.3.3
Transient
53
2.3.4
Flow-Induced Failure
59
2.3
2.3.5--Summary
64
5
3
Flow-Induced Charging of Thin Insulating Tubes:
Migration Limit
3.1
Introduction
65
3.2
Migration Model
68
3.2.1
Assumptions
70
3.2.2
Conservation Laws
73
3.2.3
External Fields
75
3.2.4
Surface Charge Relaxation
77
3.2.5
Charging Transient
80
3.2.6
Steady State Solution
84
3.3
3.4
4
65
Experiments
86
3.3.1
Arrangement and Materials
86
3.3.2
Close Fitting Sleeve
90
3.3.3
Capped Cylinder
93
Discussion
Charge Trapping and Generation in Fully Mixed Flows
96
98
4.1
Introduction
98
4.2
Charge Distributions
99
4.2.1
Assumptions
99
4.2.2
Volume Charge
101
4.2.3
Surface Charge in Insulating Expansions
105
4.3
4.4
4.5
Experiments
108
4.3.1
Arrangement and Procedure
108
4.3.2
Results
111
Criteria for Trapping and Generation
115
4.4.1
Conditions on the Sherwood Number
115
4.4.2
Mass Transfer Correlations
117
Discussion
120
6
5
Flow-Induced Charging of Thin Insulating Tubes:
Diffusion Effects
5.1
Introduction
122
5.2
Experiments
123
5.2.1
Arrangement and Procedure
124
5.2.2
Results
126
5.2.3
Preliminary Interpretation of the
Temporal Transient
136
5.3
5.4
5.5
6
122
Model for the Temporal Transient
138
5.3.1
Volume Charge Distribution
138
5.3.2
Axial Electric Field
146
5.3.3
Conservation and Relaxation of Surface Charge
148
5.3.4
Charging Transient and Electrical Stress
150
5.3.5
Terminal Currents
151
Comments on the Model
152
5.4.1
The Surface Conductivity
152
5.4.2
The Continuum Approximation
153
Engineering Implications
154
5.5.1
Charging Transient with Capped Cylinder
156
5.5.2
Range of Validity of the Migration Model
156
Standing-Wave Interaction with Imposed Flow
158
6.1
Introduction
158
6.2
Experiments
160
6.2.1
Arrangement
160
6.2.2
Experimental Results
161
6.2.3
Interpretation
164
6.3
6.4
Model
164
6.3.1
Assumptions
165
6.3.2
Turbulent Flow Model
167
6.3.3
Conductivity Gradient
168
6.3.4
Perturbation Analysis
170
6.3.5
Diffusion of Perturbation Charge
Discussion
.176
179
7
7
182
Engineering Implications
7.1.1
Dependence of Ultimate Electrical Stress
on System Parameters
182
7.1.2
Control of Ultimate Electrical Stress
186
188
Chemical Equilibrium
Appendix A
A.l
Introduction
188
A.2
The Chemical Equilibrium Assumption
189
A.3
Fully Developed Electrical Conditions
192
A.4
Chemical Equilibrium in Inhomogeneous Systems
195
A.5
Physical Significance of the Rate Constants
200
A.6
Limiting Recombination Rate
205
Appendix B
Traveling-Wave Pumping of Low Viscosity
Liquids
214
B.1
Introduction
214
B.2
Experimental Arrangement and Preliminary Results
216
B.3
Model Outline and Preliminary Numerical Results
218
Notation
227
References
235
8
LIST OF FIGURES
figure
page
1.1
Basic processes in (a) Van De Graaff generator,
where various processes are identifiable with
physically distinct regions, and (b) section
of insulating tube, where processes overlap.
18
1.2
Definition of electric field components at the insulating
tube wall. Turbulent velocity profile is shown.
18
2.1
Illustrative external conductor configuration. The
axis of the insulating tube coincides with that of a
grounded cylindrical conductor. Conducting discs truncate
both the cylinder and the insulating tube which conveys the
liquid between metal tubes of the same diameter. The
liquid and the free space region outside the tube have
permittivities
and co respectively.
28
2.2
Section of tube showing generators of closed
cylindrical control surfaces S1 and S2 that are
aligned with the tube axis. The curved part of
28
S 1 coincides
with the outer
surface
of the tube
and lies within the perfectly insulating gas. The
curved part of S2 coincides with the inner surface,
and encloses only volume charge. Mean turbulent
velocity profile is shown.
2.3
Fully developed charge density profiles in turbulent
flow (from reference 22). (a) S/Xm >> 1, (b) S/Xm << 1.
The Debye length Xt is given by Eq. 2.9 with the molecular diffusivity Dm replaced by the eddy
diffusivity
2.4
33
Dt of Eq. 6.9..
Experimental arrangement for measuring screaming
currents. Nitrogen gas pressurizes a stainless steel
reservoir to drive liquid from left to right. All
tubes are stainless steel joined together by Teflon
unions. Inside radius and length of the test section
are 1.3 mm and 1.14 m. Electrometers E monitor
currents generated in the test section and collected
by the stainless steel receiver.
40
9
figure
page
2.5
Streaming current data for Freon-113 doped with
DCA-48. Solid curves are currents measured by
electrometers in Pig. 2.4. Broken curves are
predicted values of Ito) and I (L) from
Measured
Eqs. 2.11 and 2.15 using P = 3. c,/b.
currents are negative, but plotted as positive.
41
2.6
Experimental arrangement of Keller and Hoelscher.
A plasticized polyvinyl chloride (PVC) test section
communicates with a reservoir and receiver through
unplasticized polyethylene (PE) tubes. The Faraday
cage and receiver have capacitances Cf and Cr to
their respective grounded enclosures.
46
2.7
Experimental arrangement of Carruthers and Marsh,
and Shafer et al. Carruthers and Marsh used a plain
PTFE test section and measured currents I1 and 13. ..
Shafer et al measured I1 and 12 and used a PTFE
test section with a conductive inner lining.
46
2.8
Experimental results of Gibson. (a) Effluent current.
(b) Surface charge density on polyethylene tube.
54
3.1
Experimental arrangement with close fitting coaxial
conducting sleeve. For clarity the length of insulating
tube is exaggerated relative to that of the sleeve.
67
3.2
Experimental arrangement with capped cylinder.
67
3.3
Definition of electric field components at the insulating
wall. Cylindrical coordinate system is defined, and typical
velocity and volume charge density profiles are shown.
69
3.4
Cross sectional view of capped cylinder and insulating
tube with control surfaces S and S 2 defined. For
clarity the tube radius (a) is exaggerated relative to
the radius (R) of the external conductor.
69
10
page
figure
3.5
Experimental test facility. Flow path is completed by
experiments shown in Figs. 3.1 and 3.2. All tubes and
beakers are stainless steel. Not shown are a dedicated
three-terminal conductivity cell on the floor of the beaker
in the reservoir, a PTFE tube through which spent liquid is
returned to the reservoir, and sections of aluminum screen
"wrapped" around the tubes and grounded to provide
electrostatic shielding.
87
3.6
C2 C% 3 F 3 conductivity measured at the indicated frequencies
as a function of additive concentration. All conductivities
reported in the text are measured at 200.0 cps.
87
Experimental results with close fitting sleeve configuration
92
3.7
of Fig.
3.8
3.1.
Conditions
are given
in Table
3.2 as run 36.
Experimental results with capped cylinder configuration
of Fig.
3.2.
Conditions
are given
in Table
92
3.3 as run 60.
4.1
Insulating expansion with internal volume V, bounded by
the surface S. At the outer surface of the insulating wall is
a grounded conducting layer. Two types of leakage channels
are shown: the dead-end type containing stationary liquid in
contact with the conducting layer, and those carrying the
influent and effluent liquid streanis.
100
4.2
Three-port insulating expansion corresponding to expansion
110
B in Table 4.1.
4.3
Experimental arrangement showing three-port expansion,
electrometers E and charge amplifier C. The stainless steel
tubes are electrically isolated from the expansion or its
conducting enclosure.
110
4.4
Experimental results with liquid-filled insulating
expansion. Qe is the net.charge within the expansion.
Conditions are given in Table 4.2 as run 34. Residence
time r = 2.5.s and relaxation time T =. 4. 5 s.
114
11
page
figure
4.5
Experimental results with conducting expansion packed with
steel wool. Conditions are given in Table 4.2 as run 155.
Residence
time
r = 3.2 s and relaxation
time T
114
= 2.8 s.
5.1
Experimental arrangement for studying the convection
current generated within an insulating tube. Electrometers
E monitor influent and effluent currents and currents from
individual sleeve segments.
125
5.2
Section of the insulating tube showing generators of the
closed cylindrical surfaces S1 and S2. Not shown is a
thin air gap between the outer surface of the insulating
wall and the conducting sleeve.
125
5.3
Experimental results with the arrangement of Fig 5.1.
128
Conditions
are given
in Table
5.1 as run 106.
5.4
Definitions of current differences whose ratio is
tabulated in Table 5.2. Flow is from left to right.
128
5.5
Temporal evolution of measured influent and effluent
currents. Conditions are given in Table 5.3 as run 141.
134
5.6
Measured steady state currents from sleeve segments (discontinuous bars) and corresponding net axial current (continuous
curve). Conditions are given in Table 5.3 as run 187.
134
5.7
Section of the insulating tube showing core region where
p = (z) and diffusion sublayer where
= (x,z).
Turbulent velocity profile is shown and coordinate systems are
defined. Diffusion sublayer is enlarged for clarity.
140
5.8
Section of the insulating tube showing typical profiles
of quasi-one-dimensional electric fields. Radial and axial
field components are drawn to different scales. The small
air gap between the outer surface of the tube wall and the
inner surface of the sleeve is neglected.
140
12
page
figure
6.1
An electrode structure on the outer surface of the
insulating wall imposes a potential distribution there
that approximates a standing-wave distribution. Typical
velocity and conductivity profiles are as shown.
159
6.2
Experimental configuration. Numbered circles identify
wires of the helical winding. L = 0.33 (m), X = 0.01 (m),
159
a = 1.3 (mm), w = 0.3 (mm) and R = 1.0 M.
6.3
Typical temporal response.
162
6.4
Frequency response, illustrating resonances that broaden
and shift towards higher frequencies with increasing flow
rate U. o = 400.0 pS/m.
162
6.5
Amplitude response.
peak
responses
Frequencies (f) correspond to
163
in Fig. 6.4.
6.6
Cross sectional view showing insulating wall and
typical velocity and charge density profiles. Coordinate
systems and surfaces (a - e) are defined.
163
6.7
Theoretical frequency response. Entrained volume charge
density is fixed while flow rate varies.
177
6.8
Theoretical frequency response. Flow rate is fixed while
entrained volume charge, and hence convection current, varies.
177
B.1
Experimental arrangement for traveling-wave pumping.
217
B.2
Typical
dependence
of displacement
d (Fig. B.1) on
217
traveling-wave amplitude and frequency.
B.3
Planar model with channel half-width (a) much greater
than excitation wavelength (X).
220
13
LIST OF TABLES
table
page
1.1
Cooling System Materials
23
2.1
Room Temperature Data for Hydrocarbon Liquids
44
2.2
Summary of Experiments with Insulating Tubes
45
3.1
Room Temperature Data for Freon TF
89
3.2
Experiments with Close Fitting Sleeve
91
3.3
Selected Experiments with Capped Cylinder
95
4.1
Descriptions of Experimental Expansions
109
4.2
Selected Experiments with Expansions
112
5.1
Evidence of Internal Charging Mode
127
5.2
Evidence of Long Development Length
132
5.3
Evidence of Evolving Surface Conductivity
135
5.4
Ratio of Debye Length to Mean Ion Separation
155
14
Chapter
1
INTRODUCTION
1.1
Engineering Motivation
A fundamental
from
understanding of the
the flow
of
semi-insulating
electrification phenomena resulting
liquids is
essential
to the
safe
operation of liquid-cooled power distribution equipment. In transformers
and high-voltage
direct-current (VDC)
manifested
build-up of
as a
allowed for
by design,
electric fields
with the prospect
originating in electrical discharge.
is the observation
these phenomena are
much greater
than those
of physical damage
or fire
What implicates the cooling system
that the field build-up is
initiated by the flow of
whether or not the equipment is energized.
the liquid coolant,
Without
flow rates and hence cooling capacity must
control over such phenomena,
be limited,
substations
and equipment power ratings will be compromised to a degree
that must be determined empirically on a case by case basis.
In transformers,
flow-induced fields stress the oil ducts whose oil-im-
pregnated paper surfaces tend to collect a net charge.
full scale systems
(1) typically record the dependence
charge density probes immersed in the oil,
Experiments with
of signals from
and of currents flowing from
terminals, on such parameters as oil temperature and flow rate.
Even if
such in situ measurements had clear interpretations the quantitative and
qualitative differences among reported measurements emphasize that their
applicabilty is limited to the system at hand.
the oil as the common
element that is most readily replaced,
to correlate flow-induced failure with,
oil
property.
A suggested
(2)
and trace
chemicals.
is
the so-called
some
"charging
separated as a sample volume flows
and which is found to be sensitive
While it
and seeks
and appropriately modify,
property
tendency," which measures the charge
through a paper filter,
One expedient recognizes
to handling
would be a welcome development to find
15
that
range of
an identifiable
existing units,
remain
and that this
no rational
failure in
tendency presages
charging
basis for
property can be controlled,
anticipating
there would
safe operation
of planned
units when increased heat transfer capacity is pursued.
The present investigation concerns the electric field build-up initiated
by the fully-developed turbulent flow of semi-insulating liquids through
insulating tubes.
configuration is of practical interest whenever
This
such as heat
the coolant must be transferred between metallic elements,
sinks,
The flow-induced
constrained to different potentials.
that are
field stresses the wall of
and failure of the insulation may
the tube,
take the form of a pin hole through the wall that allows an exchange between the liquid coolant and the gas that surrounds the tube.
Several
with
factors intensify
related
these problems
vast experience
electrification phenomena
despite the
acquired
in other
industrial
contexts, notably petroleum processing (3,4) and fueling operations (5).
First,
there
because of
has been
loads.
a trend
For such
they must
pressures to use
generated power
towards locating
installations to be
be increasingly compact,
more efficiently,
power substations
acceptable near
fire-resistant,
closer to
populated areas,
and
benign to the
environment, with significant demands upon the capacity and integrity of
the cooling system a direct
result.
Second, because of the sensitivity
of electrification phenomena to materials and flow conditions, empirical
from a
guidelines evolved
review of superficially
related experiences
will be of limited value for purposes of design.
Third, the remedial
measures that have been explored
in other contexts
are not immediately transferable to the high voltage environment. Active
systems (6), employing a charge injector and charge density monitor in a
feedback
arrangement
to eliminate
any
net
charge
from the
liquid, are too cumbersome for compact high voltage equipment.
active systems,
flow elements
.
,
passive charge
further downstream.
injectors (7) are intended
However, these schemes
flowing
Like the
to protect
require the
16
field
grounded points
generated at
threshold value that
points.
Both the
is likely to drift as
dielectric strength
transfer characteristics
of the
flow to
inserted in the
exceed a
contaminants collect on the
of the
insulation and
the heat
metallic surfaces in contact
with the
liquid are also likely to be sensitive to residues. t'us, the antistatic
agents (8,9) successfully added to petroleum products and jet fuels must
be
applied conservatively
effects are fully
controlled
in
high
appreciated.
charge
leakage path
voltage systems
Bonding (10),
by
until their
to provide a
which aims
constraining
the
potential of
conductor in contact with the insulating tube along its length,
neutralizing a liquid stream is the relaxation volume,
volume as
point is
that without
there is
no basis
a
defeats
Outwardly the simplest approach to
the insulation function of the tube.
increases with
side
whose efficiency
discussed in Chapter 4 below.
an understanding of the
for limiting this
The essential
electrification process
volume other than
the contraints
imposed by the available space.
efforts to
Traditionally,
voltage equipment
operating conditions
maintain safe
have centered on the identification
in high
of mechanisms of
electrical breakdown and factors which control dielectric strength. Such
efforts support
objective will
with applied potentials,
designs consistent
not be achieved unless
flow-induced electrical stresses
can be anticipated and controlled as well.
basic understanding of the
an objective
of the
that confronts
but their
While helping to establish a
flow-induced stresses in a specific context,
present research is
to encourage
the electrification problem as an
informed design
integral step towards
meeting heat transfer specifications.
1.2
Flow-Induced Electric Field Generation
The source
of the
electric field that
is not attributable
to applied
potentials is a distribution of net charge that develops when the liquid
flow
separates
charges of
opposite
sign.
Petroleum processing
and
17
aircraft fueling are
examples in which the liquid
There the absence of
partially filled tank.
into a
flow allows the
carries a net charge
a significant mean
liquid volume and generate
charge to accumulate in the
an electric field in the vapor-filled space above the liquid (11,12,13).
works to neglect the surface charge accumulation
It is typical of these
However, where the mean flow is signifi-
at the liquid-vapor interface.
cant,
as in
that makes
the insulating tube,
the dominant
it is
a surface charge distribution
generated field.
contribution to the
Thus a
in space and
specific objective here is a description of the evolution,
time, of the surface charge on the insulating tube and of the associated
electrical stress.
Whether
the accumulating
despite differences
charge is
a surface
or volume
in materials and configurations,
electric fields by mechanical action
four basic processes:
density, and
the generation of
can always be analyzed in terms of
charge generation,
transport,
accumulation, and
leakage. A simple context for illustrating these processes is the Van de
Fig. l.la.
Graaff generator (14) shown in
the bottom and
to
the metal
imparted to the moving belt
A net charge is generated at
dome
at
the top.
generates an electric field that
region occupied by the belt.
Charge
which transports the charge
accumulates on
the dome
and
by design is predominantly outside the
A resistive divider running the length of
the belt both supports a leakage current from the dome, and ensures that
the
field associated
transport process.
the dome
charge in transit
will not
oppose the
If the dielectric strength of the medium surrounding
permits, the
leakage current
with the
maximum generated field
from the dome
is that which
equal to the convection
drives a
current carried
upward by the belt.
That the Van de Graaff
much
in common
is
generator and the liquid circulation system have
emphasized
most graphically
by
Boumans (15)
who
constructed a high voltage generator with the liquid playing the role of
the belt, and a closed metal sphere playing that of the dome. What makes
the Van de Graaff generator a helpful, though less literal, analogue for
18
potential-constrained
J
/)
7-
U
accumulation
insulating
tube
leakage
transport
rt,
ion,
I
generation
J
---
/
(a)
Fig. 1.1
& transpo rt
TT
u
(b)
Basic processes in (a) Van De Graaff generator, where various
processes are identifiable with physically distinct regions,
and (b) section of insulating tube, where processes overlap.
(-.--7--~~
_
external conductor
o
0
o0 .
E
E'tE
Ei t
E
r
z
-
z
Fig. 1.2
Definition of electric field components at the insulating tube
wall. Turbulent velocity profile is shown.
19
present
purposes is
the clear
spatial
separation of
the four
basic
processes (Fig. l.la).
The same processes underlie the
charging of the
insulating
here they cannot
with physically
tube,
distinct regions
but
(Fig.
.lb).
be identified
It is to contend
with this complication
that two modes of charging, distinguished by the origin of the collected
charge, are defined.
In the external mode the
collected charge origi-
nates in flow elements situated upstream of the insulating tube.
internal mode,
which may operate simultaneously,
itself supplies
a net
charge to the
In the
the insulating tube
flow while retaining
the counter
charge which generates an electrical stress.
Much as in the Van de Graaff generator,
the external mode is character-
ized by charge generation and accumulation in separate regions. However,
charge generation
and transport occur
are intimately linked
the diffuse
virtually
in a common region
to the mean liquid motion.
part of the
The flow that shears
charge double layer that
all liquid-solid
interfaces also
because both
is characteristic of
entrains the
double layer
charge and gives rise to a convection or streaming current. This current
can be continuous if there
is a mechanism for replacing the charge that
is swept away by the flow.
Thus, charge generation is likely to be most
significant where the flow
is bounded by a potential-constrained metal-
lic surface that supports a charge transfer reaction.
Charge transport and accumulation overlap as well,
this time in the in-
sulating tube where the potential is free to rise. In the external mode,
liquid enters
Gauss' law,
the tube carrying
a net charge that,
in accordance with
sets up a radial electric field at the inner surface of the
tube wall, denoted E
in Fig. 1.2.
The entrained charge migrates in its
own space charge field to the inner surface of the wall where it accumulates as a surface density.
cally neutral
liquid enters
The internal mode is revealed when electrithe insulating tube.
charge field absent near the tube entrance,
Now with
the space
a rate of change of surface
charge there points to the role of the radial diffusion current.
',. ' .
20
components both normal (EW) and
through the slightly
a current
drives
tangential (Ez ) to the inner surface of
charge.
configuration of
leakage process
This
and perhaps
conducting liquid,
that tends to leak away the collected
along the liquid-solid interface,
surface
the latter
the insulation,
former stresses
While the
tube wall.
the
the generated field generally includes
As indicated in Fig. 1.2,
quid.
presence of ions in the li-
these processes depend on the
cause all of
and accumulation be-
leakage overlaps charge transport
Finally, charge
conductors in the
controlled in
is
region outside of the
the
part by
tube because
they support image charges that contribute to the generated field. As in
the
Graaff,
Van de
insulation fails
either the
build-up until
continues to
accumulation processes
or the leakage and
It is to ensure that conduction rather than dielectric
reach a balance.
strength
generated field
the
ultimate
determines the
stress
that proper
design aims
to
enhance charge leakage while attenuating charge accumulation.
Scientific Motivation
1.3
field generation in
model for the flow-induced electric
A quantitative
material drawn from a
calls for the integration of
the insulating tube
range of disciplines, and provides an opportunity to address issues that
have import beyond the immediate application.
A
double layer in
equilibrium charge
interface vanishes by virtue of
migration
in the
studies (16,17)
in classical electrokinetic
central concept
space
charge
which the net current
is the
normal to an
a balance between ion diffusion and ion
field of
ions themselves.
the
Here,
diffusion originates in the thermal motion of the ions, and the familiar
Debye length
(see Sec. 2.2.1 below) characterizes the
diffuse part
of the
surface that
bounds the
particle,
.....
.
a net charge exists.
electrolyte is that
the electrophoretic velocity
are of interest;
....
double layer where
thickness of the
or the
of a
When the
submicron colloidal
sedimentation potential
if it is that of a capillary, the electro-osmotic flow
.,
_.
21
or
the
streaming potential
equilibrium ion
be
distribution in
applications must
cooling
may
observed.
A description
the highly turbulent flows
take
account of
distribution defines,
for a
diffusivity
This equilibrium
a
given flow condition,
the
typical of
the turbulent
profile that is the province of mass transfer studies.
of
fully developed
convection or streaming current that is characteristic of a flow element
in the sense of being independent of conditions further upstream.
diffusion and
migration accounts for
the interface and the flowing
normal to the interface.
charge field,
imposed
field
while
is
one
and hence a nonvanishing current
liquid,
in
the
makes
need
to
the liquid
charge is transferred across
here the ions migrate in a
electrodialysis
What
the
reaction in
ionization
charge between
The parallels with other convective diffusion
dominates.
nonequilibrium
a transfer of net
weakened by the fact that
processes are not
space
an imbalance between
generation and accumulation proceed,
Where charge
(18),
ion
distribution
specify the
bulk and
for example,
of
rates
an
now
both
a
an
reaction if
an electrode
the liquid-solid interface (see Sec. 2.2.2
below). Unfortunately, even when the chemical identities of the ions are
known,
as when an antistatic agent supplies their majority,
processes
remain
poorly
understood
in
the
these rate
semi-insulating
liquids
typical of high voltage applications.
The leakage process also accounts for ties to apparently unrelated areas
because apart
of research,
potential paths
from the liquid
along which charge
bulk there are
may leak from the
two other
inner surface of
the insulating wall. The first is a thin region of enhanced conductivity
at that
surface that
supports a surface
time if
in
the presence
of a
This surface conductivity may itself
field (Ez in Fig. 1.2).
tangential
evolve with
current
it stems from
the adsorption of
a surface-active
substance from the liquid. The second path is the bulk of the insulating
wall
into which charge may be injected under the influence of the normal
component of
the generated field
(Ew in Fig. 1.,2). The same injection
process that may compromise the charge retention capacity of an electret
22
will be beneficial here if it helps relieve the electrical stress within
the wall.
1.4
Overview
The guiding
objective
is to lend a rational
quid circulation system in
an insulating
technical importance,
in Table 1.1,
(19).
to the design
of a li-
which the electrical stresses induced within
the flow must
tube by
basis
be controlled.
the focus is on the
Because
of their
specific materials indicated
which are typical of those selected for current equipment
However,
applicable,
to ensure
that
design
the four basic processes
terms of standard material
are more
guidelines
broadly
of Sec. 1.2 must be elucidated in
parameters.
To ensure that these guidelines
are easily applied, these processes must be integrated into a picture of
the field generation
process that is flexible enough
to make clear the
implications for a general system configuration.
To
help outline
essential
Chapter 2
issues,
organizes
a review
of
previous work on the flow of semi-insulating liquids through tubes under
section headings that divide the work along conceptual lines rather than
by author.
tube as
That chapter begins
conducting or insulating,
cation hinges
on the conductivity
discusses work
on conducting
help clarify the behavior
data with
with
the
for classifying the
and finding that a
of the liquid as
useful classifi-
well.
Section 2.2
liquid-tube systems in terms
that should
of practical charge generating elements.
the liquid fluorocarbon
comparison
carbons.
by seeking a basis
previous
Section 2.3 critiques
in Table 1.1 are
experimental
New
presented here for
work with
liquid
hydro-
the previous work on insulating liquid-
tube systems to set the stage for the present contribution.
Chapter 3 gives
charging mode
iments.
The
quantitative expression to the picture
introduced in Sec. 1.2,
model presented in
and
of the external
describes supporting exper-
this chapter regards ion
migration as
the dominant contribution to the normal current at the inner surface of
23
TABLE 1.1
Cooling System Materials
Element
Materials
Trade Name and
Manufacturer
Properties
Liquid
trichlorotrifluoroethane
Freon 113
Dupont
high dielectric strength,
good heat transfer
properties, compatibility
with other materials
(C2C9 3 F3 )
Insulating
ethylenetetrafluoroethylene
copolymer
Tefzel
Dupont
chemically inert, good
mechanical properties,
retains dry surface after
exposure to moisture
Additive
polymeric
amine salt
DCA-48
Mobil
maintains desired
Freon conductivity
over extended periods
Metal
Sections
aluminum,
copper,
stainless
steel
compatible with Freon 113
SF6
high dielectric strength,
nonpolar (so does not
enhance ionization when
dissolved in Freon)
Tube
..
Gas
Insulation
24
the tube wall,
and leads to a description
is appealing because
of the small set of
to limit that set by
Appendix A helps
of the field generation that
material parameters involved.
offering a nonspecific treatment
the liquid bulk.
of ionic reaction rates in
It is left to Chapter 5 to
does not contribute
of conditions for which diffusion
define the range
significantly to the normal current,
range encompasses conditions of
and to Chapter 7 to show that this
practical
interest.
The supporting
experiments exploit configurations that are not immediately identifiable
with components of an actual
have clear
system,
but for which measured quantities
significance and bear direct comparison
to theoretical pre-
dictions.
Whereas the
fully developed turbulent flow,
tube supports a
the liquid is
considers flow elements in which
that the
fully mixed in the sense
wipe
obscure the mean flow and
turbulent motions that
The random
charge is uniformly distributed.
entrained volume
Chapter 4
out gradients in
the charge density may be due to obstructions in the flow or to external
energy sources,
sidered
as
in
in detail in
liquid Jet
a pump.
In the expansion element
this chapter,
it is the
that is presumed responsible.
An
energy of
that is conthe influent
expansion with insulating
walls permits investigation of the basic processes in a context that is
simpler than that of the
are now separated.
tube because charge transport
and accumulation
Practical interest in both insulating
and
conducting
expansions stems from their potential application as flow elements to be
inserted
immediately upstream
of an insulating
tube to
attenuate the
influent convection current. A theme of this chapter is the relationship
between expansion dimensions and its effectiveness in
this
role.
Experiments reported in Chapter 5 reveal the limitations of the model of
Chapter 3
by incorporating
an
expansion
large enough
to render
the
liquid influent to the insulating tube essentially electrically neutral.
The
diffusion
component of
interface that was left out
model.
Because
the revised
the
current
normal
of that model is
model
to the
liquid-solid
included here in a revised
introduces
an additional
boundary
25
condition
that
brings
in
phenomenological material
parameters,
its
practical utility is likely to be compromised. Nevertheless, it provides
a basis
for comparing
the contributions of
modes (see Sec. 1.2 above),
the internal
and external
well as a framework into which a physi-
as
cally motivated boundary condition may eventually be incorporated.
It is
because of complications
other
methods are
whose role
in the
inherent in the use
of interest for
leakage process was
described in Chapters 3 and 5,
rate of the surface charge
tivity without
surface
of the
noted in Sec. 1.3.
One method,
relaxation that ensues when flow through the
Chapter 6 introduces a device that pro-
to continuously monitor changes in
interrupting the flow.
insulating
tube is
and simultaneously induces
within the
flowing liquid to
subject of
Appendix B is
the surface conduc-
A helical winding on
excited
potential,
tion is
surface conductivity
relates the surface conductivity to the
insulating tube is interrupted.
vides the means
measuring the
of electrodes that
with a
and detects
standing wave
interface.
the inverse interaction where
forces when the same
with a traveling wave of potential,
of
perturbation charge
reveal conditions at the
induced by electrical
the outer
The
liquid convec-
winding is excited
again with the objective of probing
the interface.
Finally,
seeks
Chapter 7
to limit
is directed
primarily to
flow-induced electrical
equipment capacity.
Here, the objective
the design
stresses
engineer who
without compromising
is to translate the models and
results of the previous chapters into concrete design suggestions.
This
calls for an assessment of the practical value of the migration model of
Chapter 3,
as well as an argument that results phrased in terms of spe-
cific experimental configurations have broader applicability.
-
26
Chapter
2
BACKGROUND AND PRELIMINARY EXPERIMENTS
and the
the liquid-solid interface.
can serve
so these
govern the
the insulating
for a description
of departure
as conducting or
of
basis for
Section 2.1 establishes a
liquid-tube combination
classifying a
Second, these
equations that
liquid are common to
as a point
accumulation process.
the charge
of the
specialized forms
distribution in the
volume charge
tube,
reflects a specific
the transverse boundary condition that
electrochemical attribute of
can turn
attention
profiles are known,
and diffusivity
directly to
profiles
First, because the fully devel-
process.
tion to the charge generation
oped velocity
context for an introduc-
the conducting tube a useful
Two factors make
insulating.
A
with flowing semi-insulating liquids is divided
review of previous work
along these lines in Secs. 2.2 and 2.3,
with some new streaming current
data included in the former section. These reviews emphasize themes that
could be the basis for critiques of papers not cited here.
Classification of Tubes as Conducting or Insulating
2.1
It is
the engineer's license
to their
to classify materials according
behavior in a particular context without regard for fundamental physical
In
differences.
the following two sections,
flow-induced currents are
reported for tubes of materials that cover both ends of the conductivity
spectrum.
the
Bowever, whether
rate process
that establishes
these
currents forms a significant part of the observations depends in part on
the tube's
conductivity.
electrical conditions
than that
long enough to
stationary
on a time scale
a steady hydrodynamic flow.
even shorter
In the highly
the electrical transient initiated by the flow can be
dominate the observations,
state never develops
the transient.
stationary
limit,
the highly conducting
may be established
needed to attain
insulating limit,
In
the extreme case the
and in
because electrical
failure terminates
27
raise the issue faced by
Intermediate conductivities
the electric cir-
cuit designer who, in considering the step response, would prefer not to
model rate processes that proceed quickly on the scale of the excitation
but must do so
rise time,
to verify the ordering of
times.
Here, the
role of the rise time is played by the liquid residence time in the tube
(L/U) which typifies the shortest time scale of interest, and it remains
to determine the time scale
of the electrical transient in terms of the
tube's conductivity.
The description of the electric field generation in Sec. 1.2 regards the
transient as complete when
found to
regard both the
cs
at
that in Chapter
the same time that
the inner surface of
as ohmic,
a description of
electrical quantities in terms of a set of Fourier modes,
n
=
-
each of which
n:
+ CoFn(R,a)
Cy + (2 ww/a)
with
in Chapter 3 of
A detailed treatment
shown in Fig. 2.1 leads to
the specific configuration
'n
governs the
respectively, and allow a uniform surface con-
their interface.
evolves on its own scale
3 the tran-
the tube wall.
tube material and the liquid
conductivities aw and a,
ductivity
is natural
be characterized by
surface charge from
relaxation of
For now,
So it
process.
the accumulation
sient is
the leakage process becomes competitive with
(2.1)
+ (2as/a)
where
2 ( I0 (kno)Kl(kn8) + I(knB)Kg(knm)
ak n I(k no)K0 (kn8) - I (knS)Ks(kn)
(2.2)
J)
and In and Kn are the nth order Bessel functions of the first and second
kinds, respectively,
and kn
nn/L is the wavenumber of the nth Fourier
mode. Evidently, the relaxation time is a heterogeneous one in the sense
that it depends on the
properties of more than one region as well as on
their dimensions. It is because the tube wall is regarded as too thin to
store
appreciable electric
wall does
not enter,
field energy that
and because dissipative
the permittivity
processes in
of the
the region
28
external
cond
insulating tube
2a
U
2R
Tit
meta
1-111,
LUUv
r
Fig. 2.1
C-----
L
>
_
Illustrative external conductor configuration. The axis of the
insulating tube coincides with that of a grounded cylindrical
conductor. Conducting discs truncate both the cylinder and the
insulating tube which conveys the liquid between metal tubes
of the same diameter. The liquid and the free space region
outside the tube have permittivities e, and so respectively.
K
Az-I
kr
-aw
-a
insulating
wall
Pig. 2.2
z
Section of tube showing generators of closed cylindrical control surfaces S1 and S2 that are aligned with the tube axis.
The curved part of S 1 coincides with the outer surface of the
tube and lies within the perfectly insulating gas. The curved
part of S 2 coincides with the inner surface, and encloses only
volume charge. Mean turbulent velocity profile is shown.
29
surrounding
the tube
are neglected
that a
gas conductivity
does not
appear. The geometric factor Fn proves to be positive-definite and a decreasing function of n,
series
and thus it is the longest time constant of the
1 that defines the duration of the transient.
For other
configurations,
principle by
the counterpartof Eq. 2.1 can be
the procedure outlined in Chapter 3.
conducting systems
defined as those for which
render T1 < O(L/U).
found in
Section 2.2 concerns
the combined conductivi-
Thus Varga's (20)
dielectric tubes
as conducting
on
the
the relatively
aqueous liquids involved.
In
practical terms the steady state develops
ties
classified
quickly
enough to
basis of
forestall significant
steady state charge generation
would be
conducting
charge accumulation,
while a
process emerges as the dominant feature.
Section 2.3 concerns insulating systems
for which the ordering of times
is reversed, and now the charge accumulation process will be significant
to an
extent determined in
part by the geometric factor
(Fn ) that re-
flects the configuration of external conductors.
2.2
Streaming Currents in Conducting Systems
The charge
all
double layer that
liquid-solid interfaces
resides in the liquid
consists of
dimensions bound to the interface
an
phase at virtually
inner layer
of molecular
by,a combination of specific and cou-
lomb forces, and an outer or diffuse layer held by coulomb forces alone.
Thus,
essential differences
between double
layers
in semi-insulating
liquids and those in the more extensively studied aqueous liquids should
be confined
to the
interface is
inner layer (21).
Another specific feature
the charge transfer reaction that
streaming
currents
through a
metal tube (21).
process that follows
observed
when
the
of the
sustains the continuous
semi-insulating
The introduction to the
depends only on the existence
liquid
flows
charge generation
of the double layer
and the availability of a charge transfer reaction, while their specific
attributes are reflected in phenomenological boundary conditions.
30
The streaming current generated in a fully developed flow is defined as
(2.3)
Is(z) - fP(r,z)vz(r)2rdr
where p,
charge density in the liquid, the
and a are the net volume
v
mean liquid velocity and the tube inside radius, respectively, and r and
z are coordinates in a cylindrical coordinate system.
inherent
and
in the measurement
prediction
To clarify issues
of I s it is necessary
to de-
fine a total axial current (I) carried by the tube, the liquid and their
the
fraction
and that
interface,
definition in
Sec. 2.1
(I*) carried
a
by the liquid
conducting system
bulk alone.
is characterized
By
by
quasistationary electrical conditions, so that
3
al
where
is
=
I
;
0
a+w
I
(pv
|
+
is the local conductivity,
the tube
wall thickness.
current traverses
curved part
the closed
is situated
net current
steady
Ez is the axial electric field and w
This is
just the
in the perfectly
traverse the
-2naJr(a,z)
;
statement that
cylindrical surface S1 in
S2 in
no net
Fig. 2.2 whose
insulating gas.
closed surface
encloses only volume charge, so that I(z)
aI
1=
(2.4)
Ez)2nrdr
Nor
does a
Fig. 2.2 which
satifies
a
IQ(z) - JJz(r,z)2nrdr
(2.5a)
where
and
Jr(r,z) are
current
respectively the
density in
(2.5b)
IEz(z))
Jz(r,z) = (P(r,z)vz(r) +
the liquid.
axial and
Of
radial components
course, Eqs. 2.4
and 2.5
of the
do not
preclude temporal variations on time scales longer thanr-l:(see Sec..2,1.
above) due to (say) variations
in either the upstream or the
transverse
31
boundary conditions. The definitions of I and I
the large Peclet number (
in view of
are two
conditions at
Hence, to determine the integrand in
modify it.
charge profile of which
only to specify the volume
Eq. 2.3, it remains
there
based on the tube length.
is imposed in the sense that electrical
Also, the mean velocity profile
forces are too weak to
LU/De)
neglect axial diffusion
the tube
p(r,z) that
developing profile
the
aspects:
profile p(r,M)
and the fully developed
inlet,
reflects
that prevails far downstream of the inlet.
2.2.1 Fully Developed Volume Charge Density
Although both the fully developed and the developing charge profiles are
governed
here
same basic equations,
by the
to emphasize
that treatment
Eq. 2.4 indicates
aE2E
rdr
the axial
Because
the tube
of a
fully developed,
(2.6)
field arises from
sources distributed
over lengths
radius, and is therefore essentially uniform
cross section,
independent of z.
are fully
less data
= 0
much greater than the tube
over
requires
that
+w
az
of the former
considered separately
volume charge density is
Where the
specific nature.
they are
Eq. 2.6 requires the
It follows that both Jz
developed as well,
axial field
to be
(Eq. 2.5b) and I1 (Eq. 2.5a)
and now to ensure a
steady state current
density in the liquid that is divergence free, the radial component must
satisfy
r d (rJr(r,,)] =
Jr(r)
with the integration constant set
= =
to zero to satisfy Eq. 2.5a.
(2.7)
For two
ionic species with valences of equal magnitude, this result implies only
that the radial flux densities
of the individual species are equal.
It
32
is left to Sec. A.3 to show that with the additional constraint that the
total solute concentration remain independent of z, these flux densities
vanish indentically:
r+(rc)
= 0
= r(rJw)
(2.8)
in a reaction that transfers a
Thus, neither ionic species participates
net charge across the
will be
interface,
an equilibrium one
reactions do
and the transverse boundary condition
not enter.
in the sense that rates
of charge transfer
Section A.3 also develops the
liquid bulk,
corollary that
chemical equilibrium prevails within the
so that the spe-
cies conservation equations specialized to the region of fully developed
electrical
conditions will
not call
for the
the ionization
rates of
reaction.
As shown in Sec. A.2, the chemical equilibrium condition allows the con-
density
Jr to
be
developed volume
radial current
diffusion to the vanishing
migration and
tributions of
expressed entirely
charge profile
in
(r,c).
terms of
the unknown
Abedian and Sonin
fully
(22),
who
summarize the related work prior to 1980, confine attention to the small
charge
density
limit
where
the
is
liquid conductivity
essentially
uniform and equal to its rest value. As a result the equation for p(r,0)
is a linear one,
facilitating the development
Their essential contribution is a
turbulent
diffusion.
diffusivity
in
diffusion sublayer thickness
AIm
consistent account of the role of the
balance
between radial
and
migration
Whether the turbulent diffusivity appreciably influences the
charge distribution
volume
the
of analytical solutions.
Xd
a
is
shown to
depend
on the
the
ratio of
(see Eq. 3.3 below) to the Debye length
(2.9)
Figure 2.3 illustrates that when the ratio 8/Xm is large, the net charge
33
-
-6
(a)
p (a,-)
p
I
Klm
!
(a-r) -
(b)
p(a,x)
I
p
I
m
X
-
t
> a
tt
<< a
-
I
-
-
(a--) -
Fig. 2.3
Fully
developed
(from reference
length
charge density
22).
(a) /m
t is given by Eq. 2.9
Dm replaced
profiles
>> 1, (b) 8/
with the
by the eddy diffusivity
in turbulent
m
<< 1.
flow
The Debye
molecular diffusivity
Dt of Eq. 6.9.
34
is confined to the laminar
region near the interface;
the original diffuse layer is
into the core.
increase in
exposed to the turbulent motions and drawn
Thus, if the interfacial value
the mean
when it is small
velocity yields a
p(a,w) remains fixed, an
decrease in this ratio
and an
increase in the streaming current given by Eq. 2.3.
The charge
distributions in Fig. 2.3
are not fully determined
until a
boundary condition governing p(a,w) is specified. Abedian and Sonin postulate a linear
relation between the charge density
and radial current
at the interface
Jr(a,z)
where J
Jw
pwl
p
(2.10)
are empirical properties of the liquid, solid and ioniz-
and
able solute combination, but are independent of flow conditions. In view
of Eq. 2.7,
condition specializes to p(a,w)
this
=
Pw where electrical
conditions are fully developed, and leads to the analytical solution
.s
QPW
in
w(n 2
PmU2a2
= H
terms of
(S/Am)
sinh(S/Xm)
the shear
stress
at the
(s/xm)/sinh(8/Am)
1 + (aS/2xm2)
interface
Reynolds number Ry, the volume flow rate Q,
superficial
velocity U.
recognizing that
original
has
an
double layer.
unit area (~kmp w)
order of
magnitude determined
Then equating the total
to the product of the
°b
9.W2B
g P t0
.I2=-D
, 2
DM=0(
4
IN
by
also characterizesthe
and the diffuse layer capacity per unit area (g/Xm)
XTUPW )
the hydrodynamic
the mass density Pm and the
(by assumption) this parameter
stationary
charge per
P
T",
diffuse
zeta potential (%)
yields
(2.12)
il
35
with the final result
brings in the
exploiting Eq. 2.9,
ion mobility b),
the Einstein relation (which
and the understanding
that the thermal
voltage typifies the magnitude of the zeta potential. Nothing is implied
here about the sign of Pw which must be determined empirically.
owever,
the exchange current density Jw is a positive-definite quantity.
In
their theory
Pribylov and
for the
streaming current
Chernyi (23)
regard the bulk
species at the interface to
tions.
generated in
laminar flow
concentration of
one ionic
be constrained independently of flow condi-
Where electrical conditions are fully developed so that chemical
equilibrium prevails, this is tantamount to constraining the interfacial
concentration of the other species as well.
density
at the
interface
is
also
Thus, the net volume charge
independent
of
the flow,
and their
boundary condition acquires the same form as that of Abedian and Sonin.
Walmsley and Woodford (24) suggest the boundary condition
±
- Ks
+
rr(a,z) = K c
where c
are the concentrations of ionic species, the K
ality factors that
the
(2.13)
ions,
reflect the fraction of the
while the
concentrations
Ks
are
of the ions.
streaming current
proportional to
In
their
in both laminar
the equilibrium
radial position,
value of
A.2).
the
there is
terms.
to assuming
surface
of the fully developed
product' cc_
flows these
In addition, they
as independent
of
a uniform, density of
Consistent with Eq. 2.7 they:equate.
the two ion flux densities in Eq. 2.13.
and because
the respective
(24) and turbulent (25)
which is tantamount
the neutral solute (see Sec.
interface accessible to
treatments
authors specialize Eq. 2.13 by omitting the K
regard
are proportion-
no gradient and
But, because these are nonzero,
hence no radial flux
of neutral
solute, they implicitly allow a change in the total solute concentration
along the tube axis.
ductivity which,
This implies an axial variation of the liquid con-
however gradual, is inconsistent with the fully devel-
oped electrical condition.
36
2.2.2 Spatial Development of the Volume Charge Density
The
convection current
fully developed
value,
tube inlet
and thus
ap/az
generally differs
0 over a section
description of the developing profile
by more than the addition
be written
from its
of the tube
as the current tends towards :1s(m). However, as is now
near the inlet
shown, the
at the
p(r,z) is complicated
of an independent variable.
Equation 2.4 can
as
a+w
-
(PV + aIEze)2rrdr
Cf.
(2.14)
ow"Ez2nrdr
with surface conduction lumped into
the integral on the right.
With'E z
essentially uniform over the tube cross section, the relative conductivities determine
which conduction
term balances the
nonzero convection
term on the left. In the case of present interest the system is rendered
conducting (in the sense of
the liquid,
term on
so that typically
the right side
the left,
Sec. 2.1) by virtue of the tube rather than
0a
<
(2 w/a)aw.
Thus,
of Eq. 2.14 far exceeds the
and hence the left side must be nonzero.
tion the left
nonzero as
side is aIl/az,
well.
the magnitude
Now the
and Eq. 2.5a
rates of a
of the
conduction term on
Finally, by defini-
shows that Jr(a,z) must be
charge transfer reaction
at the
interface are needed, and with the argument of Sec. A.3 no longer applicable,
This
rates of
the bulk
reliance on
only at
ionization reaction must also
additional specific data,
which can
the expense of additional assumptions,
of the developing profile p(r,z)
be specified.
be circumvented
renders the description
and convection current Is(z) more ten-
tative than that of the fully developed current.
Abedian and
example:
Sonin's boundary
in the
fully
condition (Eq. 2.10) provides
developed region
only
Pw enters,
a concrete
but in
the
developing region where the wall current density Jr(a,z) is nonzero, the
"exchange current density" J
(which characterizes the rate of interfa-
cial charge transfer) must be specified as well.
assume that Jr(a,z)
remains much less than Jw'
However, these authors
which
is
tantamount to
37
regarding the
transport of reactants
rate limiting process,
P(a,z)
P
to and from the
interface as the
so that the boundary condition
again reduces to
Their assumption of a
in the sense
volume charge density that is small
perturbations of ionic concen-
that it results from small
trations from equilibrium
values obviates the need to
specify rates of
the ionization
A simple extension of their
solution for the
reaction.
developing convection current allows
for a finite inlet convection cur-
rent Is(a):
Is(z) = Is(0)exp(-z/d)
(l - exp(-z/d))
+ Is()
(2.15a)
where
-U 1 +
d
and
Is(0) is given
by
development length
Eq. 2.11.
d,
reflected
in
development unless
than
if the tube length
the fully developed current
the effluent
development described
In
Thus,
if L is less than or of the order
only
be
(2.15b)
-
current
Of course,
by Eq. 2.15 will be obscured
the latter is
will emerge,
of d will the inlet
Is(L).
L exceeds
the
current
while
Is(0)
the electrical
by the hydrodynamic
complete within a distance
much less
d.
his
treatment
Walmsley (26)
of
the
electrical development
retains the K
terms in
Eq. 2.13,
in
turbulent
flow
giving four empirical
parameters. He invokes the assumptions of local chemical equilibrium and
small
charge
K± are
density,
so that
only weakly dependent
shown that Eq. 2.13 reduces to
Sq. 2.10,
with
original
four.
the
Walmsley
a diffusion
case of K
= K
on surface ion concentrations,
or if the
it can be
a boundary condition of the same form as
two parameters
expressible
divides the
turbulent diffusion is assumed
and
in the special
flow
in terms
into a
of Walmsley's
core region
where
to render the ion distributions uniform,
sublayer where molecular
diffusion dominates;
but he
38
inconsistency in
introduces
an internal
respective
charge distributions.
sublayer as independent of
curvature of the
that
in the
consistent with the large radius of
radius,
and his assumption
sublayer relative to its thickness
net charge
imposed by a
is essentially
the radial field
in the
a net charge in the core by
However, he simultaneously precludes
core.
radial field
the
e regards
modeling the
process of
the
setting c+ = c_ there
when he splices the uniform core distributions to
the initially unknown
distributions in the sublayer.
Nevertheless, the
data is not for-
a portion of the experimental
apparent agreement with
tuitous,
but rather due to the lack of constraints on the empirical pa-
rameters.
Apart from the inconsistent assumptions, it reveals a lack of
predictive capability
that to achieve correlations,
values assigned to
these parameters ranged freely over several orders of magnitude.
Sonin (27)
Abedian and
(Eqs.
test their theory
2.11 and 2.15) against
published data, virtually all of which are for doped hydrocarbon liquids
in metallic tubes.
Blasius friction-factor
of work
characteristic
w in Eq. 2.11 is obtained from the
The shear stress
correlation (22).
with
Despite the
such liquids,
satisfactory agreement
discernible for a range of materials and conditions,
consistent with
value
Eq. 2.12.
consistent tendency of the
when 8/X m
exception to
An
scatter that is
is
with Pw assigned a
is a
the agreement
theory to overestimate the measured currents
1. The authors argue that under this condition a significant
>
fraction of the actual effluent convection current is lost to conduction
through
effluent
the
liquid
jet
back
upstream
to
the
potential-
constrained tube, a possibility discussed previously by Taylor (28).
2.2.3 Preliminary Experiments
Design guidelines
for the coolant circulation system
applicable unless
qualitative features
the various
candidate materials.
cannot be broadly
of the phenomena are
common to
The charge generation process
is one
basis for comparing the fluorocarbon-additive system of Table 1.1 to the
extensively
studied
hydrocarbon
systems. In the experimental arrangement
39
shown in Fig. 2.4 a test
to limit
section is fed by a tube of twice the diameter
the velocity and
upstream of the
hence the current generated
test section. The two electrometers agree closely whether the connection
at point
A is
made or broken,
indicating that essentially all
of the
convection current is generated in the test section. Rather than let the
liquid
et exiting
attempt is made
the test section flow freely
to avoid conduction in the
test section by channeling the
into the receiver,
an
effluent stream back to the
liquid directly into another tube of the
same diameter and held at virtually the same potential.
At issue is whether the streaming currents generated by the fluorocarbon
liquid parallel the behavior of the hydrocarbon liquids as summarized at
the end of the previous subsection. Figure 2.5 presents one third of the
collected data (for
the extremes of the range
numbers used) along
with the predictions of Eqs. 2.11 and 2.15.
former equation the
shear stress
tion-factor correlation (22).
data are for
w
of hydrodynamic Reynolds
In the
from the Blasius fric-
is obtained
The remaining two thirds of the collected
intermediate Reynolds numbers and reveal
Evident is some discrepancy for 8/m
the same trends.
> 1, suggesting that the attempt to
avoid back conduction was unsuccessful, perhaps due to the finite length
of the insulating union at
departures under
the end of the test section.
conditions where
not fully developed, that is,
There are also
the theoretical effluent
current is
where Is(L) differs from Is(w).
This may
be attributed to the more tentative nature of the theory for the spatial
development, which in this case apparently overestimates the development
length d
since the experimental
curve for Is(w).
curves tend to follow
the theoretical
Nevertheless, under conditions where a fully developed
effluent current is predicted,
the
of the theory applies,
that is, where the
more rigorous aspect
agreement is satisfactory with both sets of
theoretical curves based on a single value of Pw that is consistent with
Eq. 2.12.
Consistent with
metallic tubes,
the
much of the data for
convection currents
corresponding to negative values of p".
hydrocarbon liquids in
are observed to
be negative,
40
large diameter
\
tube
test section
t
reservoir
Fig. 2.4
I
TFE union
I
1
receiver
Experimental arrangement for measuring streaming currents.
Nitrogen gas pressurizes a stainless steel reservoir to drive
liquid from left to right. All tubes are stainless steel
joined together by Teflon unions. Inside radius and length of
the test section are 1.3 mm and 1.14 m. Electrometers E monitor currents generated in the test section and collected by
the stainless steel receiver.
41
-A
10 -
I
'
I
' Il
= 5900
R
Y
-9
t
10
I
f
()
j
(A)
-10
-11
10
/
Is (L)
\-
6/Xm = 0.21
0.68
2.14
d/L = 1.4
0.18
0.02
_-
I _
10
-11
10
-10
10
-9
at (S/m)
-8
R
Y
= 14200
0
-1
10( /
lo-lo
6/Axm=
0.10
d/L = 2.7
101
0.32
1.0
0.41
0.04
I
-11
lo-lo
10-9
ca (S/m) _
Fig. 2.5
Streaming current data for Freon-113 doped with DCA-48. Solid
curves are currents measured by electrometers in Fig. 2.4.
Broken curves are predicted values of Is(") and Is(L) from
Eqs. 2.11 and 2.15 using P = 3.0oz/b. Measured currents are
negative, but plotted as positive.
42
2.3
Streaming Currents in Insulating Tubes
To profitably review the previous work while accomodating a diversity of
emphases and perspectives,
is an elucidation of the
two essential themes are defined. -The first
rate processes that are initiated by the flow,
and which ultimately compete to establish a steady state.
the internal charging mode (see
tube
acquires
a
net
convection current.
charge
Sec. 1.2 above) in which the insulating
in excess
significant,
this net charge must
diffuse part
of the
Qdl
out
wall before
The net diffuse charge (Qdl) is approximately
(2.16)
that can be attributed to
the flow turns
leaving the fixed
=
by sweeping
charge to
findlnew
With the magnitude of the net fixed charge
Gauss' law yields for the radial electric field
within the tube wall of permittivity
2nacwL
the original double layer
the double layer "inside-out"
diffuse charge,
also given by Eq. 2.16,
ri
along the tube
double layer capacity per unit area (c,/Am), the
images outside of the tube.
Jdl
t|
influent
to be practically
j
2naL
all of the
integrated
and the area of the tube wall (2aL):
The maximum stress
results when
the
greatly exceed that contained in the
original double layer
the product of the the
zeta potential ()
of
A simple calculation shows that,
commencement of the flow.
The second is
Xmrw
w
' 2. 0 x 1098
where the final approximate equality
(2.17)
is based on permittivities of,2e,V
, and a zeta potential equal to the thermal voltage
a diffusivity of 10
at room temperature.
Even for liquid conductivities as high as 1.0 IS/m
this, field
does not
insulation.
Thus,
approach
the
dielectric
strength of
evidence that a different mechanism
generate a significant stress in
practical
is available to
the absence of upstream charge sources
would be the observation that the net accumulated charge exceeds the in-
43
tegrated influent current by an amount much greater than Qdl:
II(0)
- I(L)dt
-J
II(U)ldt
IQdl
(2.18)
where the integration limit t is the duration of the experiment.
2.1 and 2.2 summarize the
'Tables
conditions of the previously reported experi-
ments that are the subject of the next four subsections.
2.3.1 Early Work
Keller and Boelscher (29),
who summarize the work prior to 1957, exper-
imented primarily with nominally pure n-heptane in plasticized polyvinyl
chloride
tubes,
and
sought to correlate
their observations
with the
Helmholtz-Smoluchowski equation (17) for the streaming current:
-2nae
where the
,v-8
-4s
aa
=
Is
r
RY
a2JQ
(2.19)
second equality incorporates Poiseuille's law,
brings in the
volume flow rate Q.
This
and the third
equation relates the streaming
current to the zeta potential under conditions typical of the relatively
conducting aqueous
are short.
liquids for which spatial
While the use
Eq. 2.19 to laminar
limit 8/Xm >) 1
of Poiseuille's law restricts the validity of
flows,
it can be adapted
(see Fig. 2.3a)
for the wall shear stress
shown in
flow.
The capacities
receiver,
and the
empirical correlation
of an equation that does not recognize
their experimental arrangement
reveals only the
with respect
net charge separated
to ground of
the Faraday
parallel resistances which were limited
the polystyrene supports,
of Sec. 2.1.
by inserting an
of the phenomena is
Fig. 2.6 which
to turbulent flow in the
w (22).
Consistent with the authors' use
distributed aspects
and temporal developments
by the
cage and
by those of
imply time constants that play the role of
l
Because these times are long compared with the duration of
44
TABLE 2.1
Room Temperature Data for Hydrocarbon Liquids
n-heptane
Properties
iso-octane*
(C 8 H1 8 )
(C 7 H1 6 )
·
·
l
11
11i
111
-
-
S
~~ ~ ~ ~ ~ ~
-
1.94
-4
3.86
x 10
683.7
2.09
2.
5.42 x 1
- 4
9.25 x 1
-4
771.
692.
1.2
1.35 x 13
748.7
x 10- 6
1.8 x 1
7.83
x 10
5.2 x 10- 8
3.7
x 10 - 8
2.2 x 19-8
1.5
nil
u
1.4
9.8
x 10 10
5.8
3.9 x 10
Sc
400.
5.65
m
x 10
x 1
9
800.
(CH3 )3 CCH2-CH:(CH
3 )2
(2,2,4-trimethylpentane)
t MIL-F-7024A, Type II (Shafer et al, (31))
(2.0 x 10-11)/n
I Dm = VTb
0.027b
@ 25
IC
x 10 10
2100.
*
§ b
kerosene
II
1.97
n
hydrocarbon
napthat
-6
x 10 8
4600.
45
TABLE 2.2
Summary of Experiments with InsulatinQ Tubes
Authors
Liquid
& Tube
a, (pS/m)
__
Keller &
Hoelscher
n-heptane
Carruthers
iso-octane
& Marsh
x1
(n)
Ry x 10i
-
8
s/m
II
3.0-?
?-90.
0.03-0.4
0.16-?
17.-362
6.8-32.
1.8
0.2-0.8
1.2-6.0
35.-78.
17.5
0.08-0.17
100
10,000
8.6
0.86
6.7
6.7
0.3
3.0
1.7
65.
0.06-. 55
0.28-0.45
7.2-36.
1.5-9.1
0.1-3.2
4.0-85.
13.
.07-1.5
PVC
PTFE
kerosene
rubber
Gibson
kerosene
PE
Gibbings
& Saluja
kerosene
Shafer
naptha
et al
Mason &
Rees
PTFE
PTFE
kerosene
1-450
HDPE
notes:
1. conductivities, tube diameters and velocities are those reported by
the authors
2. others parameters needed to calculate Debye lengths, Reynolds numbers
and diffusion sublayer thicknesses are obtained from Table 2.1.
PVC
46
r
U
Faraday cage
Fig. 2.6
\
receiver
polystyrene
support
Experimental arrangement of Keller and Hoelscher. A plasticized polyvinyl chloride (PVC) test section communicates with
a reservoir and receiver through unplasticized polyethylene
(PE) tubes. The Faraday cage and receiver have capacitances Cf
and Cr to their respective grounded enclosures.
tubular electrode
f
1 for
r et al
U
d
-Fig. 2.7
receiver
Experimental arrangement of Carruthers and Marsh, and Shafer
et al. Carruthers and Marsh used a plain PTFE test section and
measured currents I and I3.
Shafer et al measured I and 12
and used a PFE tesE section with a conductive inner lining.
47
the experiments,
the measured rates
receiver potentials
the liquid.
reversed,
of change of the
ought to reflect
(In conducting systems
leakage is
only the convection of
liquid to
instantaneously on par
enter the tube
charge by
for which this ordering of times is
with convection,
resulting potentials should be time-independent.)
of the
Faraday cage and
and the
The reported tendency
wall and extract the
plasticizer was
manifested in preliminary tests as an increase in the rates of change of
these potentials,
rial.
and a reduction
in the resistivity of the tube mate-
The relatively small contribution of the polyethylene sections to
the measured potentials was attributed to the absence of plasticizer.
Consistent with
potentials
the long
observed
in
time constants,
the
formal
the
rates of change
experiments
were
of the
essentially
independent of time, although the rates increased slightly near the ends
of some runs.
The authors attributed these departures
of
a redistribution
change to
receiver,
which they
measurements;
they
erroneously believed
did not
plasticizer might be
of
change
suggest
responsible.
multiplied
indicating that all
of charge within
by
the
that
Thus,
cage and
be reflected
in the
gradual extraction
of
Nevertheless, the two measured rates
respective capacities
of the charge collected in
in the Faraday cage.
the Faraday
would
the
from fixed rates
remained
equal,
the receiver originated
the streaming current issuing from the test
section can be calculated from
der
dsf
Is(L)
This current
=
Cf
dt
=
Cr
(2.20)
dt
was independent of
tube length as
L ranged from 6
to 18
feet, and a linearly increasing function of hydrodynamic Reynolds number
over a range that encompassed both laminar and turbulent flow.
The
authors claim
satisfactory
agreement between
Eqs. 2.19 and
2.20
provided that the zeta potential is regarded as proportional to Reynolds
number, with
% E
.019 when Ry = 100.
However, their
claim is compro-
48
mised by more than this expedient. They assert that the observed dependence of streaming current on tube radius raised to the +1.7 power (based
on
streaming current
a single
r a Ry.
consistent with Eq. 2.19 when
fixed the
Reynolds number
measurement
in each
of two
tubes) is
While it is unclear whether they
or the volume
flow rate for
these measure-
ments, in neither case is the predicted dependence on radius to the +2.0
power as stated. In fact, with . a Ry, Eq. 2.19 indicates an exponent of
-1.0
if Ry is fixed or -3.0
if Q is fixed.
Enough information is provided to
consistent with
vinyl
Eq. 2.20,
plasticizer in
and
with
I(0) = 0 to be
Eq. 2.18 is satisfied.
charge generation process within
which
the liquid.
distributed between
wall would then
layer;
it can be shown that
a significant
chloride tube,
cage is
double
part of original
in the diffuse
This implies
estimate from Eq. 2.16 the net charge
the authors attribute
the poly-
to the
presence of
ow the charge
left behind in
the Faraday
the inner and
outer surfaces of
depend on the extent to
the tube
which the PVC tube is rendered
conducting by the extraction process.
2.3.2 Steady State
Although by the definition of
an observable
an insulating system (see Sec. 2.1 above)
transient precedes the steady
the latter remains
state,
valid ground for seeking insights.
Rutgers et al (30) outline a theory
for the steady state distributions
of volume charge and potential along
the tube
which Carruthers
axis,
explain their
and Marsh (10) elaborate
experimental results.
Shafer et al (31)
As discussed
in order to
in Sec. 2.3.4 below,
also invoke the (elaborated) theory
to interpret ex-
periments.
Rutgers et al combine
integral forms of
charge conservation statement,
both
Gauss' law and
a steady state
written for the control surface S2
in Fig. 2.2. In the latter statement the radial diffusion current at the
interface is assumed
to remain what was for
the liquid at rest,
inde-
49
pendent of flow conditions and the local volume charge distribution. :The
resulting ordinary
differential equation governs the
eraged over the tube cross section p(z,w),
volume charge av-
and has the solution subject
= 0,
to p(0,.)
p(z,=) =
(0,o,)(1 - exp(-z/LRe))
(2.21a)
Q2
(2.21b)
where
(o
,M)
=
ra 2 L
;
Re
"I
L/u
=
Here Re is the electric Reynolds number based on the mean ion velocity
which bears
an unspecifed relation
to the mean liquid
velocity.
With
the authors' assumption that the radial diffusion current equals that in
the stationary double layer,
the fully developed
is just that which generates
a radial migration
tude but opposite sign.
Thus,
charge density
(-,0)
current of equal magni-
(w,w) is also independent of flow condi-
tions, and can be identified with Qdl of Eq. 2.16.
With the axial
field Ez(z,w) assumed uniform over
the cross section of
the tube, Eq. 2.4 has the integral
I(z,-) =
where I
a2 (P(,)
is the axial current
substitute Eq. 2.21 and
the two
flows
out
freely at
over the
tube (see
the
carried by the tube wall.
end
of
by Rutgers et al.
the tube
then
It remains to
with one of
If the liquid
Iw(L,u) = 0 is
the
Here the authors count on Ez to vanish uniformly
the end of the tube,
implicitly denying the
conduction current in the effluent
Sec. 2.2.3 above).
tube are
(2.22)
resolve the integration constant C
cross section at
possibility of a
= C
+
boundary conditions suggested
recommended condition.
of the
E(z,))
Ez(z,)
+
liquid back to the
Alternatively, if electrodes
constrained in potential
then the
at each end
integral condition
50
Jo Ezdz = (i) - (L) applies, and now Iw(z,-) must be expressed in
terms of the axial field and the conductivity of the tube. While neither
boundary condition
tubes
and
the
latter
traditional modes
glass tubes,
with
of experimenting
metal
condition with
the former
associate
the authors
conductivity,
of this
applicability on the range
depends for its
reflecting
what
were
the
Finally, the
with these materials.
potential distribution along
the tube is obtained from
a line integral
of the axial electric field,
which follows from Eq. 2.22 once the inte-
gration constant is determined.
Because it calls for the mean ion velocity which is not directly measurable, and because the assumed
is physically unrealistic,
radial diffusion current at the interface
the theory of Rutgers et al remains unsatis-
factory. Making no attempt to rectify these difficulties, Carruthers and
by restating the
Marsh begin
influent convection
current.
theory,
for a finite
this time allowing
Their counterpart to
the finite value of P(0,a) in a straightforward way,
Eq. 2.21a brings in
and is substituted
into Eq. 2.22 to yield
(2.23a)
(1 - exp(-z/LRe))
I(z,-) = Is(0,c)exp(-z/LRe) +
+ na2 OdrEz(z,c) + I(z,)
= C
where
Is(
a2P(,)
)
The first
term on
(2.23b)
the right in
Eq. 2.23a accounts for
the convection
current'generated upstream of the tube; the last three correspond to the
one
two conduction
convection and
length
L
is
sufficiently
dominates the second,
can be
waived.
short
terms
that the
in Eq. 2.22.
first
term
When the
in
tube
Eq. 2.23a
the objections to Rutger's treatment of diffusion
Moreover, in
view of the rigorous
steady state theory
51
summarized by Eqs. 2.11 and 2.15, the neglect of radial diffusion in the
first
(2
is
term
under
justifiable
conditions
for
which
U
U
and
<< 1.
/aS)
Carruthers and Marsh do verify the condition on L by observing
In fact,
small effluent current that results
the relatively
current
generating
filter is
momentarily
when their upstream
bypassed.
The entries
in
Table 2.2 indicate that their experimental conditions entailed turbulent
flows with the volume charge density distributed as in Fig. 2.3b.
current carried by the core of the flow the
with most of the convection
authors are justified in replacing
the mean ion velocity by that of the
on the Debye length and diffusion
Finally, the above condition
liquid.
be shown to hold,
sublayer thickness can
so
that Eq. 2.23 should bear
their experimental results once the
comparison to
Thus,
conduction terms are
resolved.
Unfortunately, the authors invite skepticism even before reporting their
experiments,
by seeking
expressions for
explicit
principle cannot be isolated experimentally.
tion
in Eq. 2.23,
constant C
quantities that
in
In evaluating the integra-
they confine
attention to
the integral
condition that goes with constrained potentials at the ends of the tube.
expressions for
the
tube wall.
the total axial
conduction
current
currents in
currents carried by the
their plan
However,
effluent
measured
they write separate
terms evaluated on this basis,
With the conduction
the
is
compare the
to
liquid
former with
because
ill-founded
and the
liquid and by
wall
it
requires
to channel
the
the
through
different electrometers from a point where the potential is constrained.
In their experiments,
both copper
measured
tivity.
duction
and PTFE tubes
the
The
current
Carruthers and Marsh conveyed
current ratio
iso-octane through
in the arrangement shown
I3/I
1
as
a
function of
in Fig. 2.7,
liquid
and
conduc-
range of liquid conductivities was such that the.axial con-.
was carried entirely by the tube, in the case of copper,
and by the liquid, in the case of PTPE.
Despite the expected scatter, a.
52
trend towards higher current ratios
is
in the data.
This
confirmation
of the
I~(L,)/Is(,-),
liquid
authors' claim of satisfactory
the basis for the
theory which
that
at the
with the PTFE tube can be discerned
predicts
is, the ratio
outlet
to the
an increase
of the total axial
convection
decreasing tube conductivity.
in the
current at
ratio
current in the
the inlet,
with
The authors also performed full scale ex-
Deriments with aviation kerosene flowing through rubber hoses, but found
it
necessary to
adjust the measured
conductivity by
a multiplicative
factor of 0.08 to make results conform to the theory.
The prediction of
PTFE tube is
a higher value of the
easily explained.
the axial conduction
ratio
In the steady
right
decreases with
in Eq. 2.23,
=
convection at z
conduction
reflects
axial
component is
"fraction
component
the
convection component.
carried entirely
of
first term
will tend
With
unrelaxed."
In
to supplement
liquid, and
With the
hence I(L,w)
PTFE tube,
and now
the net
I,(L,0) is
the conduction component.
explanation by labeling the ratio
charge
on the
the copper tube, the
by the liquid,
magnitude because it includes
authors obscure this
render the net axial
accordance with the
excluded from
Eq. 2.4 requires
Then because the convection
and oppose it at z = 0.
only the
larger in
z in
the conduction
L
current is
state,
current to be distributed to
current at z = L the same as that at z = 0.
component
(L,w)/Is(0,w) with the
a review
The
in question the
article,
(32)
Gibson
perpetuates the misnomer by asserting that insulating tubes constitute a
greater
hazard for
receiving
vessels further
downstream because
the
higher ratio implies a "decrease in charge relaxation." What they failed
to
recognize is
that the
ratio is
higher by
virtue of
a conduction
current, not an increase in the effluent convection current.
That the
tube
data didn't reveal
is likely
due to
possibility discussed
attributed to
electrode
a
combination of
two factors.
earlier that a portion of
the liquid
that
more clearly a higher ratio
constrains
is diverted from
the
potential
with the PTFE
First,
the conduction current
the effluent stream
at
is the
z = L.
Second,
by the
is
an
53
indication from their measurements of
that the steady
found that
of
the potential along the PTFE tube
had never developed.
state modeled by the theory
discharges prevented the potential measured
the tube
from exceeding
roughly
modification of Rutgers' theory.
2.5% of
They
at the midpoint
that predicted
by their
If these discharges contributed to the
leakage process, the observed steady currents were not the result of the
balance
between
axial convection
and
conduction
that underlies
the
theory.
2.3.3 Transient
Keller and Hoelscher confine attention to such early times that only the
convection current is revealed, while the steady state theory of Rutgers
et al can only apply after long enough times that conduction and convection are
already in
balance.
Bridging the temporal gap
extremes is the transient where conduction is important,
between these
though not yet
sufficient to offset the rate of increase of surface charge. Both Gibson
(32)
and Gibbings
and Saluja (33)
explicitly consider
the transient,
though neither do so in a satisfactory way.
in (32),
In unpublished work summarized
Gibson and co-workers investi-
gate the effects induced by kerosene flowing through a polyethylene tube
that directly connects a conducting reservoir and receiver. The effluent
current and the surface charge
are shown in Fig. 2.8.
density on the tube for two typical runs
Gibson does not report the influent current, nor
does he indicated where the surface charge density is measured, what the
external
conductor
configuration
is,
or
whether
the
reservoir
is
grounded.
Gibson offers the following
interpretation of the results in Fig. 2.8a.
For the low conductivity liquid,
surface cannot migrate
the
progress
of
the "charge that separates to the pipe
quickly along the pipe and
further charge
separation."
its presence impedes
While for the high conduc-
tivity liquid, "the layer of conducting liquid adjacent to the pipe wall
54
(a)
2.0-
t
1.5 -
I (L, t)
(nA)
1.0 -
r = 102 pS/m
0.5 -
-
-
X~~~~~~~~~~~~~~~~~~~
|
100
50
time (sec)
.
b)
200
c
t
= 102 pS/m
150
(nC/m2 )
100
50
I
50
100
150
200
time (sec) ---
Fig. 2.8
Experimental results
of Gibson. (a) Effluent
face charge density on polyethylene tube.
current.
(b) Sur-
55
may
be considered
accumulation at the
charge
a leakage
to provide
Be
rapid decay of charge in
flow conditions
so reduce
then suggests
"confirm this general picture," noting in
quid after the flow has ceased,
liquid under
earth and
liquid-solid interface."
that the results in Fig. 2.8b
particular the more
path to
the high conductivity li-
but ignoring the fact that for the same
the linear increase
of charge
with time
contradicts his account of the constant effluent current.
In fact,
shown that Gibson's interpretation has no basis
it is readily
in the data as presented.
A charge conservation statement written for a
surface that encloses the entire tube is
where
over
is the charge
surface charge
given data.
by the
density reported
then the right
averaged value,
approximates this
fully specified
per unit inside surface area of the tube averaged
If the
that area.
(2.24)
+ I(L,t)
I(0,t) = (2aL)
For
the high
side of
in Fig. 2.8b
Eq. 2.24 is
conductivity liquid
Eq. 2.24 yields
I(O,t)
4.3 x 1
-
10
For the low conductivity liquid,
the given curves are regarded as expo-
nential with time constant
(sec), and thus
i(0,t)
(2.5 x 10-1l) +
3
(32.0
x
10
10
)exp(-t/)
Because the estimated influent currents are in each case apparently comparable to the effluent currents in Fig. 2.8a,
it is inconceivable that
the latter can be explained without reference to the former..Even if the
signs of the
quantities on the right in
Eq. 2.24 are misrepresented by
56
because for both runs the magnitude of
Fig. 2.8 this conclusion stands,
the
charge by at
The
exceeds that
effluent current
based on
are
Table 2.2
possible
(10 m/s)
the net
again indicating significant influent currents.
least 50%,
entries in
change of
rate of
of the
maximum liquid
velocity
data
given
If the
apparatus.
Gibson's
with
the
was
obtained at this velocity, then an additional factor behind the qualitawith the high and low conductivity liquids
tive differences in the runs
may have been their correspondence to Figs. 2.3a and 2.3b, respectively.
Gibbings and
quantitative theory for the transients in the
and present a
of Gibson,
similar to that
Saluja performed experiments on apparatus
In the experiments,
effluent current and in the surface charge density.
undoped kerosene
from a stainless
was withdrawn
steel reservoir
by a
straight section of PTFE tube and conveyed to a stainless steel receiver
into
which the
authors measured
The
freely.
liquid flowed
only the
in the liquid at the tube exit,
effluent current and the potential
the
and the latter by
former with an electrometer attached to the receiver,
A monotonic decrease
means of a probe inserted
in the effluent stream.
with time of the effluent
current was observed under both turbulent and
laminar flow conditions.
In the former case, a plot of the logarithm of
against time
curve which
they interpreted
piecewise linear
authors a
suggested to the
the current
as a succession of
exponential segments,
each implying a longer time constant than the preceding segment. On this
extrapolated the ultimate
basis they
case,
following
current to zero.
transient,
an erratic
the effluent
In
the laminar
current abruptly
settled to a finite asymptote within one hour.
The authors do not state whether the reservoir was grounded. Nor do they
either
the
influent current
or the
accumulating
attempt
to measure
charge,
and thus the comment on Gibson's work applies here as well: the
effluent
current alone
is
insufficient
basis for
a comparison
with
theory. Nevertheless, it is instructive to outline the theory because it
differs on a fundamental level
from the picture introduced in Secs. 1.2
and 1.3 above, and detailed below in Chapters 3 and 5.
57
Gibbings and Saluja's theory attributes
charge density
between
at the inner
"charging" and
density.
surface of the tube wall
"rejection"
The charging component is
current at
a rate of change of the surface
components of
the radial
current
an adaptation of the initial normal
a weakly ionized gas,
a wall bounding
to a competition
which arises because
the mean electron velocity exceeds that of the positive ions (34).
For a
binary electrolyte, this component acquires the form (5)
3zRT2
ch
4i
-
where R, T, and
(2.25)
(ctb+ - c_b_j
i are respectively the gas constant,
ture, and ionic mean free path, while c
the two ionic species with
tion component is
absolute tempera-
are the molar concentrations of
valencies ±z i
and mobilities b.
proportional to the product of
The rejec-
the liquid conductiv-
ity and the local surface charge density d
Jrej
-- ko
-
k kziF(c+b+
+ cb_)
where F is Faraday's constant
ing the
radial component
(2.26)
and k is a "positive coefficient express-
of the field
at the wall resulting
surface charge deposited on the wall per unit area."
from the
Neglecting conduc-
tion through the wall or along the liquid-solid interface,
conservation
of surface charge is expressed by
at
(az
"To simplify
t )
the discussion,"
negative valency
adjacent to
with c+ set to zero.
both time and
= Jch + re
(2.27)
the authors
the wall," and
"consider only
In addition,
ions of one
apply Eqs. 2.25
and 2.26
c_ is approximated as independent of
position along the tube,
so
that subject to the initial
condition of vanishing surface charge Eq. 2.27 has a solution
58
a(z,t) =
t(3 - exp(t))
;
%
(-Zic_b_kF)
(2.28)
and
RTIjexp(ot)
Jr(a,z,t) = (-zic-b-)
that is
independent of z.
remain zero,
(2.29)
Finally, the influent current
is assumed to
and thus the effluent current follows from Eqs. 2.24, 2.27
and 2.29 as
;
I(L,t) = I(L,0)exp(at)
ments,
the
(36):
ions
authors restate
deposit on the
first.
are
The
a hypothesis
different
wall in batches,
from an
valencies
with ions
are
then accounted
for
by the
the valence.
earlier publication
present,
and
these
of higher valency deposited
progressively longer time constants
reciprocal of
of exponential seg-
turbulent flow as a succession
of several
(2.30)
supports their interpretation of the decaying
To argue that this result
effluent current in
RT
I(L,O) = (2raL)(zic b_)
dependence
of successive segments
of 1/a
Objections to this
(Eq. 2.28) on
hypothesis and
the
to the
description of the data as a succession of exponential curves are raised
in the discussion
following
Of immediate concern
charging process.
presentation
is whether the theory captures
Two obvious
than
neglecting
(36).
substance.
positive
ions
the physics of the
difficulties are perhaps more
First,
while
is
assuming
the clear
that the
issues of
inconsistency
influent
in
current
vanishes. Second, there is nothing apparent in the theory that restricts
it to turbulent flow,
the effluent
yet it cannot account for the finite asymptote of
current observed in laminar flow.
that the finite
The authors recognized
asymptote implies a leakage process,
but
they did not
incorporate one in the theory. Nor did the finite asymptote lead them to
question whether the influent current was actually negligible.
59
more substantive criticisms reveal the
But two
essential inadequacy of
the theory. First, the postulated rejection current arises from a radial
electric field that is proportional
and independent of
to the local surface charge density
the local volume charge density.
That Gauss' law is
writing its integral form for the surface
generally violated is seen by
S 2 in FPig. 2.2
Er(az)
-
=
;
z
=
iF(c
- c_)
(2.31)
which relates the radial field at the interface to both the local volume
charge
and
the axial
which
from distributed
arises
sources,
the charging
Second, because
external conductors.
those on
including
field
current density of Eq. 2.25 is generally finite even when the net volume
charge
vanishes,
the postulated transient can be
filling the tube with stationary
before the flow commences.
of the flow
and can proceed to completion
liquid,
If the
theory is to imply that coLmnencement
will initiate a new transient,
liquid must differ
(c±) in
the influent
liquid.
Because there is nothing
trations to flow conditions,
initiated simply by
then the ion concentrations
from those in
the stationary
in the theory to relate these concen-
it fails to account for the characteristic
feature of observed electrification phenomena,
namely, their initiation
by the flow.
2.3.4 Flow-Induced Failure
Both Shafer et
al (31) and Mason and
Rees (37) identified experimental
conditions that produced pin-hole failures in insulating tubes,
as well
as conditions under which the responsible discharges could be suppressed
or rendered less energetic.
Mason and Rees measured the charge transferred by discharges between the
surface of a high density polyethylene tube with an outer diameter of 63
mm and a movable grounded brass sphere of 28 mm diameter. The discharges
were induced by circulating kerosene
around a closed loop with a filter
60
and filter
bypass line
relaxation tank
situated
just upstream of
just downstream
upstream of
charge to the flow.
the polyethylene tube,
to avoid recirculating charge.
the filter was
thought to
A pump
contribute negligible
The 15 foot length of tube was supported at regular
intervals by a frame made
of the tube material,
and was arranged,
cording to the authors' schematic, in a zig-zag pattern.
tivity was
and a
controlled by
clay filtering and an
ac-
Liquid conduc-
unspecified antistatic
agent over the range indicated in Table 2.2.
A field
mill
howed
that the potential
with time following commencement of
after about ten
ground
by the
consistent with
In this state equal
filter and
returned to ground
Eq. 2.24 if
relaxation tank,
these currents represent
respectively the
time
tube wall
conductivity reported by
resided on
the tube.
about twenty
the authors,
the outer surface
distribution
was nonuniform
currents were drawn from
by the
effluent currents in
of
surface increased
the flow and reached a steady state
minutes.
influent and
of the
at the tube
A bulk
seconds,
charge relaxation
based
on
the wall
suggests that accumulated charge
of the tube.
with sign
The
steady state potential
reversals
cumference and
along the axis,
arrangement of
external conductors and partly
about the
perhaps due partly
tube cir-
to the uncontrolled
to undetected discharges
during the transient.
In this steady state,
discharges
Droach of the brass sphere
with its radius of curvature just under half
that of the outer tube surface.
surface and the sphere was
0.7
to 7.0
sphere
permitting
The charge transferred between the tube
insensitive to their spacing in the range of
diameters.
(~2 pS/m) discharges
of both signs were induced by the ap-
At
the escape
were energetic
of
liquid.
the
lowest
liquid
conductivities
enough to puncture the
With the
objective of
tube wall,
identifying
conditions that would limit the tranferred charge below the threshold of
75.0 nC which
the authors considered
influence of liquid conductivity,
incendive,
they investigated the
conductive coatings on the outer tube
surface, and the spacing of grounded tube supports.
61
transferred charge decreased linearly
that the
Their finding
with in-
does not have a clear-cut interpretation.
creasing liquid conductivity,
While enough information is provided to show that the tube wall contribresistance per unit axial length
uted a larger
than the liquid so that
the latter controlled the leakage process, the effects of liquid conductivity on
charge separation in
the upstream filter were
not reported.
The application of a conductive coating to the outer surface of the tube
was found to increase
if
was isolated or
the coating
coating creates
current.
or decrease the transferred charge, respectively,
"hotspots" by
In the latter, the
the coating undercuts
the
decreasing
the
axial conduction
resistance per unit length attributable to
and liquid, thus enhancing
The linear decrease
spacing of
supports is at
shorting out the local
that of the tube wall
leakage process.
former case,
In the
continuous.
charge with
(and hence increasing
of transferred
number of)
least qualitatively consistent with the
grounded tube
notion that the
steady state stress is reduced by enhancing the leakage process.
Based on these findings the authors offer specific recommendations,
caution that
they apply only to their
but
specific experimental configura-
tion. Perhaps the most significant observation was the pin-hole puncture
produced in
a tube
wall having a
time that the discharge may
the tube.
short enough bulk
charge relaxation
have been initiated at the outer surface of
This would suggest that to avoid insulation failure it is not
enough to relieve the stress within the wall.
Shafer et al provide a vivid description of pin-hole failures induced in
plain tetrafluoroethylene (TFE) tubes by the flow of a white hydrocarbon
naptha.
With a
Reynolds number
minutes of the
liquid
conductivity
of 45,000 a
start of the flow that
continued,
tube was used.
or the
hydrodynamic
about 4
left multiple pin-hole punctures
These pin holes became enlarged if
phenomenon could be reproduced
In contrast to the later
bulk charge relaxation
and a
visible discharge appeared within
through which the liquid could seep.
the flow
of 165 pS/m
if a fresh
study of Mason and Rees,
time of the tube wall
the
far exceeded the duration
62
of the experiment.
"paths" branching
entire surface,
Inspection of the inner surface of the tube revealed
out from each
suggesting that
pin hole and covering
practically the
the discharge originated at
the inner
surface and engaged essentially all of the accumulated surface charge.
The
the Reynolds number
significance of
determined in preliminary
in a
concentration.
those shown in
of the tube to flow
rate and additive
The authors observed "bell"
curves similar to
Fig. 2.5 for the streaming current,
currents generated by the filter
insulating tube alone.
results of a "charging
tendency" test (like that mentioned in Sec. 1.1 above,
ating behavior of
noted that the
and
easily exceeded those generated by the
They also found that the
capillary in place of the
conductivity were
experiments that related the charge generated
paper filter just upstream
type and
and liquid
but with a steel
filter) bore no relation to the charge gener-
their filter in situ,
confirming that
such tests do
not reflect an intrinsic property of the liquid-additive combination.
In a second set of experiments, with essentially the same arrangement as
that of Carruthers and Marsh
shown in Fig. 2.7,
the maximum steady state potential
Shafer et al
measured
developed along the tube under vari-
Unlike Carruthers and Marsh, however, Shafer et al
ous flow conditions.
achieved a steady state without discharges by using TFE tubes with inner
liners rendered
the dominant
trolled
sufficiently conducting by carbon
axial conduction path.
independently of
particles to provide
Thus, the leakage
the liquid conductivity,
process is con-
but because
of the
nonlinear current-voltage characteristics of the liners, their effective
conductance varies with each run.
support the view of the
Despite this complication the results
steady state as the result of a balance between
charge generation and leakage by showing that the steady state potential
increases with the
former and decreases with the
latter.
contrast to the outer coatings used by Mason and Rees,
Note that in
the inner liners
always enhance leakage regardless of the conductivity of the wall bulk.
63
A third set
of experiments,
intended to
measure the
apparatus in Fig. 2.7,
also with the
ratio of the
effluent to
was
influent convection
currents under conditions where the entrained charge originates entirely
upstream of the tube. The authors' implicit assumption that the effluent
current 13 was
entirely convective was likely based
that the large
conductance of the tube liner
cant conduction
on the expectation
would preclude a signifi-
The effluent
component in the liquid.
current was not
measured directly,
but rather
influent current I1
and the current 12 drawn from the tubular electrodes
at each end
of the test section which
inner liner.
rents,
inferred from the difference between the
There
is no mention
probably because the large
presumably made contact with the
of a transient in
conductance
the measured cur-
of the liner rendered
1
(see Sec. 2.1 above) sufficiently short.
Shafer et
al obtained a satisfactory correlation
on the right in Eq. 2.23a.
represented by the first term
and the model
between these results
For the measured ratio
I2
Il
11
- I3
I3
I
Il
that model predicts
Is(L)
1 -
The conditions under which this
steady state
above.
theory of
only over
ductivity,
with u
¢~/bl
(2.32)
represents a limit of the more rigorous
the entry
the identification
part of the
Re
Abedian and Sonin
In view of Fig. 2.3,
indicates that
;
1 - exp(-l/Re)
u
are discussed
in the last column of Table 2.2
U made by
the authors
experimental range of flow rate
U over the
in Sec. 2.3.2
remainder.
is valid
and liquid con-
This may help explain their
finding that Eq. 2.32 tends to slightly underestimate the measured ratio
(I2/I1),
a tendency which the authors tentatively attributed to ambigu-.
64
ous dc liquid conductivity measurements in which steady applied voltages
did not yield steady currents.
2.3.5 Summary
Perhaps more unfortunate than the paucity of work addressed to the flowinduced failure
of insulating
tubes are the limited
perspectives that
characterize the existing work. The experiments reviewed above typically
measure "terminal" currents
from which the net charge
on an insulating
tube may be inferred, but they fail to capture the distributed character
of the phenomena. Nor do the previous models reflect the interplay among
the four basic processes outlined in Sec. 1.2.
al
decouple the
radial diffusion
conditions (Sec. 2.3.2),
Saluja may
For example,
current at the
tube wall
Rutgers et
from flow
while the transient postulated by Gibbings and
proceed to completion
even before the flow
commences (Sec.
2.3.3). The need remains for controlled experiments that support a selfconsistent account of the space-time evolution of the flow-induced electrical stress within the insulating tube wall.
On the positive side, two results from the previous work are germane-because they are suggestive of the practical value of the model introduced
in the next
chapter.
the insulating
That model attributes the
tube wall exclusively
charge accumulating on
to sources upstream of
the tube,
and regards the radial current density at the wall as being dominated by
migration. The first germane result addresses an issue raised in connection with Eq. 2.18: in each experiment that produced discharges, including those of Carruthers and Marsh,
Mason and Rees, and Shafer et al, an
essential element was an upstream charge generator,
The second such result,
al with Eq. 2.32,
typically a filter.
the reasonable correlation secured by Shafer et
encourages
the expectation that practical situations
exist that are adequately represented
by a model that neglects the dif-
fusion component in the radial current density.
65
Chapter
3
FLOW-INDUCED CHARGING OF THIN INSULATING TUBES:
MIGRATION LIMIT
3.1
Introduction
The limiting condition that underlies
flow-induced
electric field
the model of this chapter for the
insulating tube
is one
of simplicity,
yet it
generation in the
be introduced for the sake
that would naturally
proves
a satisfactory
representation for a
tions.
Without prior justification,
range of
practical condi-
the normal current density
in the
liquid at the liquid-tube interface that accounts for a rate of increase
of surface charge is regarded as due entirely to ion migration.
insulating tube is
limit the
surface,
in that no surface
no more than a
In this
passive charge collecting
charge accumulates unless an upstream flow
element supplies a net charge to the flow. It is when this limit applies
that flow-induced electrical stresses will
be reduced by insertion of a
the flow immediately upstream of
the insulating tube;
the conditions under which an
expansion volume will
"charge trap" in
Chapter 4 outlines
be effective as a trap. The normal diffusion current that generally competes with ion migration is accounted for in Chapter 5, leading to clarification of the conditions under which the latter is actually dominant.
Section 3.2 addresses the question: given the total axial current in the
the entrance to the insulating
liquid at
evolution of the electrical stress
tube,
what is the space-time
within the tube wall.
Conditions on
the liquid conductivity (the Debye length) and the hydrodynamic Reynolds
number (the diffusion sublayer thickness) are such that most of the convection current is carried in the core of the turbulent flow. The condition on the conductivity also renders the electrical transient initiated
by the flow
long enough (see Sec. 2.1
Thus, to
perimental observations.
must retain time rates
above) that it dominates the ex-
help interpret experiments the model
of change of electrical quantities.
the temporal transient also highlights
Analysis of
the essential role of the exter-
66
nal conductor
configuration, which supports image
bute to the generated field.
tion along the tube (but
be determined
less, the
proves
While the steady state potential distribu-
not the radial electric field in the wall) can
independently of
specified at the
that configuration, the latter
outset to determine any transient quantity.
steady state
to be
charges that contri-
stress distribution is
a convenient
basis
in Sec.
of interest
7.1.1 for
must be
Neverthebecause it
deducing general
guidelines from the model.
In Sec. 3.3 two versions of the experiment are described,
by the external conductor configurations.
tured in
Fig. 3.1,
nating the axial
wall.
electric field component in the liquid by elimi-
aspect ratio
the axial
establishment of a
of the insulating tube
is the
surface
processes of Sec. 1.2,
unopposed by leakage;
of the wall
as well.
In
charge accumulation proceeds
steady state is delayed well
The
justification for
in terms of the characteristic times of Sec. 2.1,
role of
inner surface of the wall
the small but
beyond the duration of
finite axial field
at the
is not recognized until this configuration is
taken up again in Chapter 5.
3.2, relaxes the
of the tube
field at the inner
terms of the basic
the experiment.
sleeve that accentuates
component at the outer surface
The extreme
neglecting
The first configuration, pic-
is a close fitting coaxial
the role of the radial
distinguished
The second configuration, pictured in Fig.
constraint on the axial field
and renders the leakage
process competitive with the accumulation process on a short enough time
scale that the approach of
the steady state can be discerned experimen-
tally.
There
is no
idealizes,
pretense that either
any practical
Moreover, while the
configuration represents,
configuration, nor
model of Sec. 3.2 could
would that
let alone
be desirable.
in principle be applied to
other configurations, including those identifiable as practical ones, to
do so would contribute nothing
underlie the
to a clarification of the phenomena that
flow-induced field generation.
here that these
Rather, the
phenomena are most profitably explored
view is-taken
in the simplest
67
metal
I
I
metal
i . .L
ioe
conducting
I
I
TU
U
U
I (o ,t)
'ing
i
I UL:
Fig. 3.1
Experimental arrangement with close fitting coaxial conducting
sleeve. For clarity the length of insulating tube is exaggerated relative to that of the sleeve.
be
:1
tube
U
1%
nal
_+t^
LUI ILUUULIUI
Fig. 3.2
Experimental arrangement with capped cylinder.
68
possible context,
and that once clarified, their implications for other
configurations will be clear.
3.2
Migration Model
What makes
the specific external conductor configuration
is the relatively simple
context for presenting the model
3.2 an ideal
mathematical description
shown in Fig.
of the potential distribution
in the external
region. The insulating tube lies on the axis of a cylindrical conducting
annulus, and extends between
Upstream of the
conducting caps that truncate the annulus.
insulating tube,
tion generates the convection
a potential-constrained metallic sec-
current that drives the charging process.
A similar section downstream collects
the liquid as well as part of the
leakage current driven by the rising potential of the insulating tube.
The objective is
to predict the temporal and
spatial evolution of both
the charge distribution along the insulating tube and the electric field
within the tube wall.
Fields
at the inner surface of the external con-
ductor are more conveniently measured than those that stress the insulation, so expressions
variables are
for the former are sought
needed to
specify the charge
choices are the volume and
ient, however,
effective
length
of tube,
density representing the
from both
and natural
It proves convennew variable, an
latter in favor of a
with contributions
Two dependent
distribution,
surface charge densities.
to eliminate the
line charge
as well.
total charge
the volume
per unit
and surface
charge densities. After the assumptions are clarified, a pair of coupled
partial differential equations expressing conservation of the volume and
line charge
densities is
identified (Sec. 3.2.2).
field component within the tube
axial electric
(Ez in Fig. 3.3) enters these equations
as a third dependent variable.
from solving
The
Laplace's equation
The additional relation called for comes
for the potential distribution
in the
external region andsexpresses the field in terms of the line charge density (Sec. 3.2.3).
When the influent convection current can be regarded
69
I·
external conductor /
insulating wall
IO t
E,
Er t
Ez
X 1',
r
Fig. 3.3
Definition of electric field components at the insulating
wall. Cylindrical coordinate system is defined, and typical
velocity and volume charge density profiles are shown.
vn rr&V%
r
1
0 .
Fig. 3.4
L
Cross sectional view of capped cylinder and insulating tube
with control surfaces S 1 and S 2 defined. For clarity the tube
radius (a) is exaggerated relative to the radius (R) of the
external conductor.
70
of these equations is available for
an analytical solution
as imposed,
the transient electrical quantities (Sec. 3.2.5). Numerical treatment is
required if
that current varies
the charging transient,
on a time scale comparable
to that of
perhaps through its dependence on the generated
field.
3.2.1 Assumptions
[a]
flow is
The
turbulent and fully
Fully
developed.
turbulent flow
occurs for hydrodynamic Reynolds numbers in the approximate range
2aU > 4000
Ry
(3.1)
Here a, U and v are respectively the tube inside radius, the mean liquid
velocity, and
the kinematic viscosity.
independent of position along the
of the
order of
100 tube radii.
and of
conditions become
tube axis beyond a development length
Thus, either
under study greatly exceeds this length,
the same inside diameter,
Hydrodynamic
the length of
the tube
or a metal entrance section of
sufficient length to complete the de-
velopment, lies just upstream of the insulating tube.
[bJ The Debye length exceeds the diffusion sublayer thickness,
X4
>>
(3.2)
8
For the high
Schmidt numbers typical of liquids
of interest (see Table
2.1) there is a thin layer near the phase boundary of thickness (22)
8
(118 RY-7/8 Sc-1/3 )a
(3.3)
within which the turbulent fluctuations are diminished so that molecular
diffusion
outweighs the turbulent
contribution to
mass transfer. :The
71
Debye length
(Eq. 2.9) characterizes
wall within which,
fusion,
the net
throughout.
not confined
where
by
virtue of a balance between
volume charge
to the Debye layer
It is perhaps
is
weak enough
eject the
weak
influenced
the flow
were laminar
that the
the liquid
also strengthens
by the
of the flow
eddy diffusivity
relax the inequality
of Eq.
(Fig.
3.2 by
~/Xm) internal to the "laminar" double layer
E
generated fields will
diffuse charge into
condition on
conduction and dif-
but extends into the core
tempting to
arguing that the field (E
is
would remain if
region near the
As a consequence of Eq. 3.2, the entrained volume charge is
its distribution
2.3b).
the width of the
disrupt the
the core of the flow.
However, the same
conductivity that renders the
the inequality
described by the model includes
layer and
internal fields
of Eq. 3.2. Moreover,
the transient
the initial portion where the generated
fields are not yet significant, and thus the inequality in question must
be retained.
[c] Most of
the convection current
is carried in the core
of the flow
where the mean liquid velocity is essentially the superficial velocity:
na2 p(z,t) >> 2naSp(a,z)
(3.4)
Here,
p is the volume charge density averaged over the core of the flow
(0 < r
<
a-8) and
p(a,z) is
the volume charge
density at
the liquid-
solid interface that typifies the density within the diffusion sublayer.
As discussed
in (22)
this inequality is
consistent with
Eq. 3.2, and
facilitates an integral approach in the description of the volume charge
distribution along the axis of the tube.
[d] The
migration component
of the radial
current density
within the
diffusion sublayer completely dominates the diffusion component.
tions that justify this assumption
are outlined in Sec. 5.5.2.
CondiIn view
of the large Peclet number (LU/De) axial diffusion is neglected as well.
72
[e] Departures of
ionic concentrations from their
respective values in
an undisturbed solution
remain small compared with those values.
the liquid conductivity
o1 is essentially uniform and
its value
%O where the
Thus,
remains close to
net volume charge density vanishes.
This condi-
tion appears to breakdown at the liquid-solid interface where if Jr(a,z)
Eqs. 2.10 and 2.12
Pw) is
of the order
suggest that the net charge density ((a,z)
<< J,
discussed below
c/b). However, as
of the individual densities (+
in Sec. 6.3.2
chemical equilibrium can be
expected to
diffusion sublayer, and now Eqs. A.11 and 2.12 yield
prevail within the
an approximate value for the bulk conductivity at the interface
o1
0, (a,z) =
+ (Pwb/%) 2
= 1.4%
(3.5)
that is still close to the assumed value.
[f] The bulk
time (w
charge relaxation
with the
long compared
(Tl of Sec. 2.1).
stored at the
s w/ow) of
time that characterizes the
As a consequence
charge
precipitated
be reduced only
liquid-solid interface as a surface
wall is
the tube
charging transient
from the flow is
density ()
that can
and perhaps along the
by conduction through the liquid
interface. This condition is relaxed in Sec. 5.3.
[g] The gas
that surrounds the insulating tube
on the time scale of interest (1
[f],
liquid
this implies that there
is
of Sec. 2.1).
is perfectly insulating
Coupled with assumption
no net charge anywhere other than in the
bulk, at the liquid-solid
interface,
and on the external
conduc-
tors.
[h] The axial electric field is
section.
essentially uniform over the tube cross
This presumes that the local axial field arises primarily from
distant sources (image charges on external conductors)
length scales large compared to the tube radius,
distributed over
while the local volume
and surface charge densities make no signficant net contribution.
73
[i]
on the scale
convection current Is(0,t) varies slowly
The influent
of the liquid residence time
(L/U) in the tube.
While probably of gen-
eral validity, this assumption is certainly appropriate if variations in
Is(0,t) relate to
the rising potential of the
to the grounded upstream
insulating tube relative
section that generates the convection current.
The time scale for variation of the potential is
1
which,
by the defi-
nition in Sec. 2.1 of an insulating system, is much greater than L/U.
3.2.2 Conservation Laws
Conservation of
volume charge
is expressed by
Eq. A.9
specialized to
reflect the assumptions of the previous section. Assumptions
justify neglect of
the term containing the
both the diffusion term and
remainder of Eq.
Integrating the
the liquid conductivity.
gradient of
d] and [e]
A.9 over the tube cross section yields
~a_ ·,
Uaa
;
[c] allows
where assumption
complete the specification
effective line charge
simple
form of the convection term. To
of the charge distribution
along the tube, an
density
instead of the
surface of the
tube.
(3.7)
surface charge density ()
This choice leads most
the axial electric field,
at the
the
(3.6)
a2 p(z,t) + 2na;(z,t)
X(z,t) -
is introduced
Q _B
tube entrance when
is
unknown
Conservation of line charge is expressed by
a=x~
1_a
_(3.8a)
8t
az
directly to evaluation of
and to application of
(0,t)
at the inside
the boundary condition
(see Sec. 3.2.5, case C).
74
where the net axial current
I(z,t) -
a2 (U~
+
0eEz(a,z,t)]
;
e
(6
+ (2s/a))'
includes conduction and convection in the liquid volume,
along the inside surface of
Cs ,
but
Assumption
definition of
allows the
within
the tube
simple form of
an effective axial
recognizes that no
and conduction
the tube via a uniform surface conductivity
excludes conduction
hi
-(3.8b)
wall
(assumption
the conduction term
conductivity
f]).
and the
e.
Equation 3.8 simply
current flows across the curved
part of the surface
S 2 in Fig. 3.4 which lies within the perfectly insulating gas.
With p(0,t)
slowly varying
(assumption [i]),
on the scale
the temporal derivative
of the liquid
residence time
in Eq. 3.6 is small compared
with the convection term and Eq. 3.6 can be integrated immediately:
P(z,t) =
(0,t)exp(-z/taU)
(3.9)
This is substituted into Eq. 3.8 to yield
8_x
ax + (na2e)
na2P(B
8gzE,
afz(azt)
na P(0.t] exp(-z/
at+ (a 2 e)-z(a 'zt)
U)
(3.10)
=
That Eq. 3.10 is written with the "excitation" on the right and the "response" terms
on the left is
local axial field is shown
made clear in the next
section where the
to be a linear integral function of the line
charge density. This inhomogenous equation is the starting point for the
analyses in
ation
the
remaining sections.
is treated
by setting
p(9,t) = 0.
applied to describe the charging
figuration (Fig. 3.2), and
In Sec. 3.2.5,
Eq. 3.10 is
transient for the capped cylinder con-
then specialized to the sleeve configuration
(Fig. 3.1) by setting Ez(a,z,t) = 0.
by Eq. 3.10 with ax/at =
In Sec. 3.2.4 surface charge relax-
Finally, the steady state governed
is discussed in Sec. 3.2.6.
75
3.2.3 External Fields
To complete the description of the model,
Eq. 3.10 must be supplemented
by a relation between X(z,t) and Ez(a,z,t), and this comes from solution
of Laplace's equation,
v2
=
;
=
(r,z,t)
(3.11)
in the region outside
for the potential
the insulating tube.
Gauss'
law applied to a cylindrical volume of differential axial extent and coincident with the outer surface of the tube (S2 in Fig. 3.4) yields
2na(Az)EE r + na2 c (Ez)
where c o
and
= XAz
(3.12)
9, are the permittivities
rounding the tube
of the free space
and of the liquid, respectively.
the axial field term is allowed by assumption
nor the permittivity of tube
The
simple form of
h]. Neither the thickness
wall appear in Eq. 3.12 because the former
is regarded as small compared with the inner diameter (a).
field satifies the
region sur-
The electric
electro-quasistatic limit of Faraday's law
if it is
the negative gradient of the potential:
zi-xial
W
rth =at0=
E (rzt)
With the
axial field
components in
;
r
tube-gas interface,
just those of Eq. 3.13
two equations,
the boundary condition on the
(3.13a,b)
ar
continuous at the
Eq. 3.12 are
Combining these
E°(r,z,t)
the field
and taking the limit
evaluated at r = a.
of
z 4 0, yields
potential at the outer surface of the in-
sulating tube
(Wa)
+
2[iQ z 2 _ _2
;
r = a
(3.14)
76
the external
on the potential being
boundary conditions
with additional
For the specific configuration
conductors.
imposed by
shown in Fig.
3.4, the additional boundary conditions on the potential are
= 0
O(r,,t)
;
O(r,L,t)
The complete set of solutions
=
0
;
O(R,z,t)
=
0
(3.15)
to Laplace's equation satisfying Eq. 3.15
is the series
(r,z,t) = E
n(t)Bn(R,r)sin(knz)
k
n
(3.16)
= n.
L
n
where
Bn( ,8)
(I0 (kn)K 0(knB) - I0 (kn)K(kna)
(3.17)
)
and In and
Kn are the nth order
modified Bessel functions of the first
and second
kinds, respectively.
Application of the
Eq. 3.14
is facilitated
by expressing
the
boundary condition
line charge
density as
a
Fourier series:
X(z,t) =
(3.18a)
Xkn(t)sin(knz)
n
where
Xn(t)
(3.18b)
(2/L)jX(z,t)sin(knz)dz
Now, substituting
Eqs. 3.16 and
3.18a into Eq. 3.14
yields a relation
among Fourier coefficients
Xn(,)
On(t)
where
= 2akn(a/2)knBn(R,a)
+
oCn(R,a)1
(3.19)
77
Cn(,8) _(Ig0 (knc)Kl(knS)
+ I(knB)K(kn)
Field components in the external
)]
(3.20)
region are expressed in terms of these
coefficients by substituting Eq. 3.16 into Eqs. 3.13:
Ez(r,z,t) =-
E
n(t)knBn(R,r)cos(knz)
(3.21)
n(t)knCn(Rr)sin(knz)
(3.22)
n
and
= +
E (rzt)
n
With no
at the outer surface
surface charge density
of the insulating
wall (assumption [g]), an expression for the radial electric field within the wall (Pig. 3.3) follows from Gauss' law as
E (z,t) =
r
S.
E 0 (a,z,t)
The desired relation
(3.23)
r
between X(z,t) and Ez(a,z,t) is
given by Eq. 3.21
(evaluated at r = a) with Eqs. 3.19 and 3.18b substituted.
3.2.4 Surface Charqe Relaxation
Unlike the
bulk conductivity,
determined with
the surface conductivity is
confidence in advance.
However, this
deduced from
observation of the surface charge
interruption
of the
current at
the tube
flow.
When
the flow,
inlet, is interrupted
and
not readily
parameter can be
relaxation that follows
hence the
convection
the residual volume charge
relaxes to the liquid-solid interface within a few bulk relaxation times
(c/aQ),
leaving only surface charge within the tube.
surface charge relaxation
at X8~
(naie)
+ az= =i;
(a2%
Thus, the ensuing
is governed by Eq. 3.10 with P(0,t)
0:
(3.24)
78
_
__
__
An explicit statement of surface charge conservation attributes a rate
of change of surface charge to a divergence of the surface current and a
bulk conduction current normal to the interface. To represent the latter
contribution
would
require
= E
0i(r,z,t)
the introduction of a Laplacian potential
(t)Ig(knr)sin(knz)
In terms of this potential,
for the charge-free region inside the tube.
and that given by Eq. 3.16 for the external region,
surface charge con-
servation is expressed by
at
-(Co)a+ (
To deduce the relaxation times
between the
+
=
(
i
of the spatial modes,
Fourier coefficients
n
and 0n would be
requirement that the potential be continuous,
a second relation
obtained from the
(a,z,t) =
i(a,z,t).
The purpose of this footnote is to show that Eq. 3.24 is consistent with
the explicit treatment.
equivalent to
that renders
The left
(1/2na)(a/at),
side of the conservation statement is
and
the axial field
it is in
the same
uniform over the tube
limit (kna <
1)
cross section (as-
sumption [h]) that the right side reduces to
2
(
-(V
BE
r + (°s) t
-
+ s)
3E
=-
z
taking the
as can be seen by substituting the series expression for 0i,
indicated
limit and
condition on
ka
making use
fails
of Faraday's law
for large enough
Eq. 3.13a.
n is unimportant
because the
amplitudes of these higher order modes are strongly attenuated.
_
_
_
_
That the
79
The generality of
Eq. 3.24 could be emphasized
3.14 to eliminate the axial
boundary condition on the
regardless of the
by using Eqs. 3.13a and
field and the line charge density, giving a
potential in the external region that applies
external conductor configuration.
dition could then be applied
This boundary con-
to the specific form Eq. 3.16 to determine
the surface charge relaxation times (38).
However, with a specific con-
figuration in mind at the outset it is more direct to insert Eqs. 3.18a,
3.19, and
3.21 in
Eq. 3.24, which yields the
homogeneous differential
equation for each Fourier coefficient:
dXn
nxn
+
dt
;
n
kn(t) =
(3.25a,b)
n(O)exp(t/Tn)
where
Zn
=
e
+ CFn(R,a)
o
+ (2Os/a)
(3.26)
aknBn(,8)
(3.27)
and
Pn(,S)2=
The
external configuration
trolling the charge
evidently plays an
relaxation process,
essential role
in con-
reflecting the fact that while
dissipative processes are confined to within the tube, energy storage is
primarily in
the free-space region outside the
result given by Eq. 2.1
of the tube wall
tube.
The more general
is obtained if axial conduction within the bulk
is included
in Eq. 3.8b.
All field quantities are expressible in terms of the same set of Fourier
coefficients,
Tn proves
and hence they decay with same relaxation times.
to be a decreasing
function
of n,
the relaxation
fundamental spatial mode, x 1, can be determined from
the initial portion (
mined from fields
< t <
2T 2 )
is excluded.
measured at the surface of
and as follows from Eq. 3.26 as
Because
time for the
relaxation data if
Thus,
1 can be deter-
the conducting enclosure,
80
c
+ eoFl(R,a)
a
(3.28)
V- I
(2Tl/a)
Us
The surface conductivity appears only in Eqs. 3.8 and 3.26,
which indi-
cate that it must be of the order of (aod/2) to be significant.
3.2.5 Charging Transient
The system of
equations to be solved is
ted by Eqs. 3.18 and 3.19).
can be
If the influent
regarded as imposed,
though not in closed form.
perimental external
Eqs. 3.10 and 3.21 (supplemenconvection current Q(0,t)
is available,
then an analytical solution
To illustrate
this solution for the two ex-
conductor configurations,
both cases A and B below
regard the charge density at the inlet p(0,t) as slowly varying over the
duration
ex of
generated
fields on
Case C
the experiment.
the
influent
arbitrary temporal variation
allows for
convection current
of p(0,t),
an effect
of the
by allowing
an
which necessitates a numerical
treatment.
A. Capped Cylinder and Imposed Influent Convection Current: While p(0,t)
varies slowly on
the scale of Tex,
the latter
is presumed long enough
1 that the entire transient is of interest. The homogene-
compared with
ous solution to Eq. 3.10 is given by Eqs. 3.18a and 3.25b.
The particu-
lar solution can be expressed as an exponential function, but to facilitate application of the initial condition, the right side of Eq. 3.10 is
represented
Fourier series
as a
of the same
form as
the homogeneous
solution:
exp(-z/d) = E ensin(knz)
(3.29)
n
where
en
=
(]
(
1
(d
2
1 + (L/nid)2
and en = 0 for n even.
3L;
n odd
The characteristic length d -
(3.30)
U.
Substituting
81
Eqs. 3.18a, 3.19, 3.21 and 3.29 into Eq. 3.10 yields
n...nn=a2 P.t)n
dXn ++
dt
where the
(3.31)
n
en
rn are given
by Eq. 3.26.
With
the right side of
Eq. 3.31
essentially constant on the scale of the longest time constant (
1
), and
subject to the initial condition of X(z,0) = 0, the complete solution to
Eq. 3.31 is
Xn(t) =
aP(t)
( - exp(-t/¶n))nen
(3.32)
Finally, the field components are obtained from Eqs. 3.19, 3.21 and 3.22
with Eq. 3.32 substituted.
face
of the
evaluated at
external
These can be evaluated at points on the
conductor
(a,z) to apply
with appropriate
Eq. 3.23 for the
values
of
(r,z),
uror
electrical stress within
the insulation. In retrospect, the assertion in Sec. 2.1 that the charging (Eq. 3.32)
and relaxation (Eq. 3.25b)
transients are characterized
by the same time constants holds only as long as the influent convection
current
is stationary
on the scale of
1.
B. Close Fittina Sleeve and Imposed Influent Convection Current!
case A,
the influent convection
current varies slowly on
As in
the scale of
Wex' However, the extreme aspect ratio of the tube and the proximity of
the equipotential surface (Fig. 3.1) render the time that now plays the
role of x1 (Sec. 5.3.3) long enough compared with
ex that axial conduc-
tion remains negligible. Letting E z * 0 in Eq. 3.10 yields
ax-=
a t -exp(
z/(3.33)
The assumption
that
perhaps more plausible
(s,t)
is
independent of
here than in case A
the generated
fields is
because the experiment ends
82
the inlet becomes appreciable.
before the axial field at
In fact, this
assumption receives strong support from experiments with the sleeve configuration reported in Secs. 3.3.2 and 6.2.
With the
(,t),
condition on
Eq. 3.33 is immediately
integrated with
respect to time to yield
X(z t) = t
a2p(,t)exp(-z/i
(3.34)
U)
'r9
quantity in this case
The measured
charge accumulated within the
ting Eq. 3.33 over the length
is the rate of change
tube (Qt)'
of the total
which is obtained by integra-
of the tube
dt
With Ez
0 the electrical stress within the insulating wall
E(z,t) ,r
=
X(z.t)
2nacw
=
t a(t)
2c.r
2
(3.36)
jexp(-z/tU)
follows immediately from Gauss' law and Eq. 3.34. The essentially linear
increase in
charge
electrical stress with
accumulation process
time reflects the dominance
over the leakage
compared with (the counterpart of)
process for
of the
times short
1.
C. Capped Cylinder and Arbitrary Influent Convection Current: In Abedian
and Sonin's theory (Sec. 2.2) for the convection current in a conducting
tube, steady electrical conditions are assumed to prevail well away from
the ends of the tube.
However, where the upstream conducting tube joins
to the entrance of the insulating tube, the rising potential of the latter produces an unsteady electric field that penetrates into the conducting tube a distance of the order of the tube diameter. It is for now an
83
open question whether the unsteady field in the neighborhood of the outlet of the conducting tube
municated to
such an
affects-the effluent convection current com-
the insulating tube.
effect if
tests of the
Thus,
it is essential to
model for processes in
allow for
the insulating
tube are to be unambiguous.
As
in case A the full transient
bitrary time dependence of
the influent convection
the upstream boundary
ured influent
rather
that
(0,t)
does
1
ex) but now an ar-
<
current of the generated electric field.
condition is now expressed in
on the time
as well,
scale of
and
1
hence the
it
is
3.8b must be
latter still
Eq. 3.9 so that Eq. 3.10 still holds.
used to eliminate the unknown
generated field
reasonable to expect
relative to the liquid residence time (assumption [i]).
still given by
and conduction)
Since the
(0,t).
(0,t)
Thus,
terms of the meas-
reflects both convection
initially unknown
varies essentially
(
is allowed to account for an effect on
(0,t)
current (which
than the
is of interest
varies slowly
Thus,
(z,t)
is
Now, however, Eq.
in Eq. 3.31 in favor
of the experimentally accessible I(0,t):
dxn
I I(0,t) - na2aeEz(a,O,t)
Xn
dx~ .r,~-r
2~U
en
(3.37)
where
co
Ez(a,,t)
~z
(a0,t)
=
=m
Z
-'X(t)Bm(R'a)
+ oCm(Ra)j
{2a((eta/2)kmBm(R,a)
(3.38)
follows from Eqs. 3.19 and 3.21. As in case A only the odd spatial modes
are excited.
To apply this result, the series Eq. 3.38 can be truncated
after k terms
so that
dinary differential
Eq. 3.37 represents
a set of k coupled
equations for the coefficients Xn
numerically with the given I(0,t).
linear or-
to be integrated
84
3.2.6 Steady State Solution
The steady state
crease of
results when axial conduction offsets
surface charge due
Thus, there is no basis
figuration because
to the space charge field
the rate of inin the liquid.
for a steady state solution for the sleeve con-
the corresponding
axial electric field has
not yet
been determined. That solution emerges from analysis of the same configuration in Sec. 5.3
recognized.
where the small but finite
Here, attention is confined to the capped cylinder configu-
ration, and the steady state
ax/at
axial electric field is
solution that is governed by Eq. 3.10 with
0
a2e)-z=I
a2 (
(na2 ae)
)]exp(-z/xU)
Substituting Eqs. 3.21 and 3.29
(3.39)
into Eq. 3.39 yields the relation among
Fourier coefficients
enP(0,c)
n(0)
=
en2
where en is given by
3.22
(3.40)
dneT)Bn(Ra)kn
the steady
Eq. 3.30.
state fields
With Eq. 3.40 inserted in Eqs. 3.21 and
are
expressed in
terms of
the average
volume charge density at the inlet. The latter can be expressed in terms
of the steady
state influent current I(0,w) by
recognizing that in the
steady state Eq. 3.8a requires the total axial current
I(z,.)
I na 2 (U (z,.) +
eEz(a,z,))
to be independent of axial position.
resolved by
substituting Eq. 3.9
line integral of the axial
=
(3.41)
I(0,")
The integration constant I(,oo) is
into Eq. 3.41 and
requiring that the
field over the length of the tube vanish (in
view of the grounded terminations):
85
p(
(1 - exp(-L/%U)
=
The quantity in
(3.42)
the large parentheses is evidently
the fraction of the
net influent current carried by convection.
It is notable that if only the steady state potential distribution along
the tube is of interest,
out specifying
3.42 are
a closed form expression can be obtained with-
the external conductor configuration.
combined and the
3.41, which
then gives
Equations 3.9 and
result is used to eliminate
the explicit dependence
p(z,w) from Eq.
of Ez(a,z,c) on
z and
I(,m):
L/r U
EZ(a,z,)
With respect to
point along
=
1
- exp(-L/rU)
(3.43)
na2ce
the grounded terminations, the potential
= -
(a,z,w) at any
Ez(a,z,)dz
If now the radial electrical
the potential
subject to
_______
) I(g.3)
exp(-zL//U)
..e tube follows immediately from a line integral of Ez:
O(a,z,)
tion for
1
-
1
stress E(z,c)
(r,z,o)
boundary conditions
ductor configuration
(3.44)
is desired, Laplace's equa-
in the external region
dictated by the specific
and the known potential along
stress follows from Eqs. 3.13b and 3.23.
must be solved
external con-
the tube.
Then the
86
3.3
Experiments
is
Because the model of Sec. 3.2
rameters
(with the
phrased in terms of known material pa-
exception of the
surface conductivity)
and simple
external configurations, support for the view of the charging process as
a competition between accumulation
and leakage tendencies can come from
simple controlled experiments. Following an overview of the experimental
arrangement
and materials,results
for the configurations
shown in Figs.
3.1 and 3.2 are summarized in Secs. 3.3.2 and 3.3.3, respectively.
3.3.1 Arrangement and Materials
The experimental facility
shown in Fig. 3.5 is
designed with three ob-
jectives in mind: to accommodate a range of experiments, including those
pictured in
ters,
Figs. 3.1 and 3.2 as well as those
to ensure
accounted
for,
that all
and to
tubes
of the same
latter by
significant net charge
maintain the
upstream and downstream of
diameter as
the insulating tube
Under nitrogen
gas pressure, liquid is forced
glass level indicator.
liquid
and joined
to the
the inlet to the insu-
velocity calculated from the measured
liquid height in the
Immediately
smooth hydrodynamic transition.
generate a convection current at
reservoir at a
is exhausted,
liquid conductivity.
can be
complete the development of the velocity
The upstream section serves to
lating tube.
and currents
the experimental section are stainless steel
Teflon unions that provide a
profile and to
reported in later chap-
is returned
out of the
rate of change of
After the liquid inventory
to the reservoir through
a separate
tube (not shown in Fig. 3.5) also under gas pressure.
The liquid comes
steel surfaces
into contact only with potential-constrained stainless
and the
insulating tube under test.
Observed
rates of
change of net charge are confined to the latter because the capacitances
of the metallic elements (with
minum screens) charge
respect to the pressure vessels and alu-
through the electrometer input impedances quickly
on the scale Tl of the transient in the insulating tube.
It is to allow
measurement of the influent and effluent currents in the insulating tube
87
level
indicator
experiment
I (IL,t)
essure
essel
Fig. 3.5
Experimental test facility. Flow path is completed by experiments shown in Figs. 3.1 and 3.2. All tubes and beakers are
stainless steel. Not shown are a dedicated three-terminal conductivity
cell on the floor
of the beaker
in the reservoir,
a
PTFE tube through which spent liquid is returned to the reservoir, and sections of aluminum screen "wrapped" around the
tubes and grounded to provide electrostatic shielding.
t
10- 8
ar
(S/m)
10
l-11
10- 4
10- 3
10 - 2
10- 1
1
10
s (g/ ) -- o
Fig. 3.6
C2 CI3P3 conductivity measured at the indicated frequencies as
a function of additive concentration. All conductivities reported in the text are measured at 200.0 cps.
88
that the liquid is not recirculated. The upstream section draws initially quiescent, and hence electrically
large enough
dimensions compared
neutral liquid from a reservoir of
with the tube
radii that
the liquid
presumably remains quiescent. The influent current I(0,t) is measured by
the electrometer
shown in
ground potential all of the
tion, including
both Figs. 3.1 and 3.2,
The effluent current
the electrometer shown in
to all of the metallic
maintains at
metallic sections upstream of the test sec-
the reservoir.
ground through
which
Fig. 3.5,
I(L,t) returns to
which is connected
sections downstream of the test section, includ-
ing the receiver.
The insulating
tubes under
ethylene copolymer
study are Tefzel,
an ethylene-tetrafluoro-
developed by Dupont (39, 40)
with slightly superior
mechanical properties than, but otherwise similar to, Teflon (polytetrafluoroethylene).
tively pure
The liquid is Dupont's Freon TF (41), which is a rela-
grade of Dupont's refrigerant
the properties indicated in Table 3.1.
through silica gel
is
exposed only
three
to its
terminal cell
own vapor and
immersed in the
a capacitance
ture, the liquid
materials, so
bridge.
the nitrogen
with
after which it
gas.
reservoir typically
A dedicated
indicates an
1.3 pS/m, as measured at 200.0 cps
In addition
to
its sensitivity
conductivity can be dominated by
once the liquid
2)
Liquid, as received, is filtered
into the closed experimental system,
initial liquid conductivity of about
with
Freon 113 (CC2F-CCF
to mois-
the effects of trace
has been filtered and
sealed, pumps are
avoided, as are materials containing plasticizers, antioxidants, etc. In
successive
addition of
experiments
the conductivity
is
gradually
an antistatic agent, Ethyl Corporation's
available from Mobil).
Very
(see Fig. 3.6)
additive is diluted
so the
raised by
DCA-48 (42)
the
(also
small additive concentrations are involved
with Freon TF
before being
injected by syringe through a small port in one of the pressure vessels.
The conductivity
as a
function of additive
concentration is
shown in
Fig. 3.6 where the linear dependence suggests a bimolecular dissociation
reaction (Sec. A.2)
that is
not uncommon in hydrocarbon
liquids (43).
89
TABLE 3.1
Room Temperature Data for Freon TF
formula
CC2 2F-CC2F 2
2.4
11/E
6.8 x 10
1565.0
4.4 x 10
x
10
- 8
b
3.
Dm*
8.0 x 1
Sc
550.0
Xm
1.3 x 10-10/
- 10
*f
calculated as in Table 2.1
90
This is
consistent with the
polymeric structure of the
additive mole-
28 amine - carboxylic acid functional groups,
cules which contain 24 to
and are thought to produce ions by a proton transfer reaction (44). This
structure
and ionization
negative ions have
process make it
similar mobilities,
plausible that
because their molecular weights
by that of the transferred
will typically differ
positive and
proton which is small
compared with the total molecular weight (6000).
Perhaps more likely is
a spectrum
which
of molecular weights and mobilities,
may still imply
similar effective mobilities for positive and negative ions.
3.3.2 Close Fittinq Sleeve
In the arrangement
shown
influent current I(0,t)
charge
on sleeve
in
Fig.
3.1, the electrometer
while the charge amplifier (C)
(= -Qt). As a check
formal experiments,
(E) measures
the
records the net
on the instrumentation
prior
to the
the second electrometer shown in Fig. 3.5 aids in a
demonstration that the relation
dQt
dt = I(0,t) - I(L,t)
(3.45,
is satisfied instantaneously, as required by the law of charge conservation. Subsequently, only two of the quantities in Eq. 3.45, the influent
current and net charge, are recorded.
Table 3.2
[a] and
in
summarizes the conditions
b] of Sec. 3.2.1
Fig. 3.7, where
current
tends
transient in
current
to
initial
constant
I(O,t) reflects
generated in
transient is
are satisfied.
after an
a
of two runs for
value.
transient the
Two factors
of run 36 is shown
measured influent
suggest
a temporal development of
the upstream
metallic section.
observed in experiments
sections, like those reported
The record
which assumptions
the
the convection
First,
in which there are
in Sec. 2.2.3.
that
the same
no insulating
Second, the fixed rate of
change of Qt following the transient indicates that conduction driven by
91
TABLE 3.2
Experiments with Close Fitting Sleeve
Parameter
Run Number:
Measured:
36
37
1.3
1.3
0.38
0.38
4.7
4.7
(m/s)
0.66
1.60
(nA)
-0.15
-0.7
Qt/t (nC/min)
-1.42
-4.0
p(0,t)b/a
-0.3
-0.5
3.46
Ry
3900.0
9450.0
3.1
a
(mm)
L
(m)
a9.
U
(pS/m)
I(0,t)
Calculated:
Xm
(in)
60.0
60.0
2.9
8
(wU)
14.0
6.2
3.3
6030-1.10
6030.0
5.30t
Qt/t (nC/min)
-1 .1
-2.1
3.35 & 3.46
Qt/t (nC/min)
-1.47
-3. 0
3.5 & 3.35
x1
Adjusted:
Equation
I
(min)
with data from Table 3.1
t
ith
w =
.3 mm and
-- 2.6co
92
time (min)
0
3
6
9
12
-.
15
18
21
24
T -o0.1
Qt
(nC)
(nA)
-0.2
-20
-0.3
-30
-0.4
Fig. 3.7
t
-I0
I (O,t)
-40()
Experimental results
of Fig.
3.1.
with close fitting
Conditions
are given
sleeve configuration
in Table
3.2 as run 36.
time (min) --
T
-0.4
-20
I (0,t)
E
(nA)
(kV/m)
-0.8
40
-1.2
60
-1.6
Fig. 3.8
Qt)
%JV
Experimental results with capped cylinder configuration
Fig. 3.2. Conditions are given in Table 3.3 as run 60.
of
93
the
generated
field
does
influent
current.
tion of
an imposed volume
case B of
Thus,
yet contribute
sults in
significantly
1 >>
the ordering of times
also consistent with
relatively fresh
tube
calculated from Eq. 5.30 with Os
which underlie
r1 given in
the assumed ordering of
times. Re-
samples
=
the
Calculated values of
Sec. 3.3.3 indicate that surface conduction
in
to
ex and the assump-
charge distribution, both of
Sec. 3.2.5, are confirmed.
Table 3.2 are
least
not
0 (and
is negligible, at
used here,
w
e
and
thus T
is
).
With axial conduction negligible, the volume charge density at the inlet
follows from Eq. 3.8b (with
(0,t)
Because
0) as
eEz
I(.t)
(3.46)
na2 U
I(0,t)
and
residence time L/U,
hence
(0,t)
vary
I(0,t) is constant.
agreement between the measured dQt/dt
improved by
enhanced
on
the scale
of
the
Eq. 3.35 applies even through the transient, but is
most easily applied where
is
slowly
adjusting the
conductivity at
integral approach used
Table 3.2 indicates that
and that calculated from Eq. 3.35
latter to take
the interface.
account of
This
to arrive at Eq. 3.9,
is consistent
which
the conductivity at the interface (Eq. 3.5) when
the slightly
with the
actually calls for
is evaluated for Eq.
3.35.
3.3.3 Capped Cylinder
The external conductor in Fig. 3.2
probes that
"view" the interior
and held at the same
Thus, the
is large enough to accommodate field
through flat tips mounted
flush with,
potential as, the inside surface of the enclosure.
probes do not
distort the local field, and
because they are
situated in regions of relatively low field intensity, discharges can be
permitted
to develop
in the
vicinity of
jeopardizing the instrumentation.
the insulating
tube without
Experiments consist of initiating the
94
current I(O,t) and the normal fields inci-
flow and recording the inlet
and -Ez(R/2,0,t). Just before the liquid
dent on the probes, E(R,L/2,t)
a downstream valve,
the flow is interrupted by
inventory is exhausted,
of liquid and allowing the accumulated
leaving the insulating tube full
surface charge to relax in a controlled way.
are found
Results
are initiated
runs
from
that successive
reproducible provided
to be fairly
allowing
conditions by
same electrical
the
surface charge to relax completely prior to commencing a new experiment.
of selected runs, and tabulates the
Table 3.3 summarizes the conditions
two lowest order (and hence longest) relaxation times as calculated from
Eq. 3.26 with
p(0,t)
= 0, making these
s
r 1, the latter characterizes
a longer time scale than
varies on
Provided that
values upper limits.
both the charging and relaxation transients (see Sec. 3.2.5).
exponential form
that the
enough
discernible in recordings
m(to )
the fundamental
of
from x1 = -A/m.
calculated and measured
mode should
be
a time
values of
to (
2
2)
has
Indicative of the agreement between
3.3 are the small values
1 in Table
of 2os/a (from Eq. 3.28) compared with
the bulk conductivities.
the materials and conditions investigated
least for
disparate
the fundamental mode relaxation
elapsed since interruption of the flow,
time is determined
are
the amplitude A(t o ) and slope
With
measured after
probe response
of either
and x2
of the probe responses following interruption
and this is the case.
of the flow,
T1
of
values
Relaxation Transient: Calculated
Thus,
at
here, surface con-
duction is unimportant.
Charqinq Transient: The solid
transient of
run 60, with
fundamental mode
calculated
duration
of the
discussed
in
represent the charging
flow conditions as given
time constant
natural transient in
connection with Pig.
interfere with the
analysis of
lines in Fig. 3.8
(
1)
is comparable
the influent
3.7.
Sec. 3.2.5 cannot be applied.
The
to the
convection current
Thus, the latter will
exponential form of the charging
case A in
in Table 3.3.
tend to
transient, and the
An alternative to
95
TABLE 3.3
Selected Experiments with Capped Cylinder
Run
Im
R (m)
arc
C
1
2
sI
-
12%c/aI
l
48
0.076
2.4
173.0
176.0
48.0
0.043
51
0.051
2.4
202.0
192.0
50.0
0.12
59
0.076
41.0
12.0
10.3
2.8
5.8
60
0.076
47.0
8.0
9.0
2.4
5.8
62
0.076
52.0
8.6
8.1
2.2
2.9
notes:
time constants are in minutes; conductivities are in pS/m
superscripts 'm' and 'c' denot& measured and calculated values,
= 0
the latter from Eq. 3.26 with
for all runs, tube
radius a = 1.3 (mm) and tube length
L = 0.3 (m)
Flow Conditions for Run 60
Ry =12000. 0
x1 = 20.0 (rn)
18 = 5.
(NWn
96
the
procedure given
as
case C
of
integration of Eqs. 3.6 and 3.8,
shown as broken lines in
between the
As in case C, the measured influent
While better
from more refined experimental or numerical
is strong encouragement from
experimental time
the favorable comparison
scale discernible in
Pig. 3.8
and that
for xi.
calculated
3.4
numerical
with the results
upstream boundary condition.
agreement might be obtained
procedure, there
a direct
as outlined in (38),
Fig. 3.8.
supplies the
current I(I,t)
Sec. 3.2.5 is
Discussion
The experiments tend to support
trical stress
competing
arises from
charge
a model in which the flow-induced elec-
a surface charge distribution
accumulation and
driven by the space charge
leakage
emerges
as the
time
The fundamental mode relaxation
scale on
electrical stress tend towards a
of the
is
factors and the conductivities of the
liquid bulk and liquid-solid interface.
1
The former
field of charge entrained in the liquid; the
latter is controlled by geometric
time
processes.
determined by
which the
surface
steady state where the axial component
a conduction current through
generated field drives
charge and
the liquid
that offsets the local rate of change of surface charge due to the space
charge field.
radial field
It is to
in the
enhance the axial
tube wall that
the external conductors
configured to support image charges
rather than along
tion of
in close proximity
of the
should be
primarily near the ends of the tube
the transverse boundary.
Fig. 3.1 illustrates to
conductor
field at the expense
Thus, the sleeve configura-
an extreme the "hotspot"
to the
tube as it
created by a
shorts out
the local
axial field.
The ultimate stress is that
basis
for
failure.
suggesting
Because
other
the bulk
numerator as well as
(0,t)
of the steady state, so Eq. 3.40 can be the
steps
to avoid
conductivity
helps
flow-induced
determines
insulation
en in
the
as discussed in Sec. 2.2, its appearance in
97
the
denominator in
deceptive,
Eq. 3.40
and further
discussion
Sec. 7.1 is a clarification
the ultimate
(through
reserved for Sec. 7.1. Also left to
radius (with the volume
However,
the
(through
e), and to the extent that it exceeds a/2
the
ultimate stress.
influent volume
conductivity) is
of the apparently complicated dependence of
stress on tube
surface
is
the effective
conductivity
appears
The proportionality
charge density
flow rate fixed).
only in
between
(0,c) is also
the
denominator
it tends to reduce
the stress
notable because
and the
of the
leverage over the latter afforded by the antistatic additives.
It is because
the
of side effects
additive from
implied
the liquid
bulk
by
the gradual
that an
depletion
alternative way
(44) of
to limit
p(0,o) would be desirable. Thus, in the next chapter, the subject is the
charge trapping
flow just
tendency of
upstream of
an expansion volume
the insulating tube.
characteristic time must
to be
be effective,
be short compared with that
transient in the insulating tube (1).
the expansion
To
when inserted
effective is an
charge originates primarily upstream
suggested by previous work (Sec.
in the
its own
which governs the
The rest of what is required for
indication that
the accumulating
of the insulating tube, as already
2.3.5).
Thus, in Chapter 5 the sleeve
configuration is taken up again, but now an expansion volume at the tube
entrance limits the influent
current to reveal convection currents gen-
erated within the insulating tube itself.
98
Chapter
4
CHARGE TRAPPIVC AND GENERATION IN FULLY MIXED FLOWS
4.1
Introduction
In the sense that the mean flow is obscured by essentially random turbulent motions, the flows in pumps and perhaps heat exchangers and certain
other
practical flow
elements are best
Much like the radial charge
turbulent core,
fully mixed.
density profile in the fully developed pipe
flow, species concentrations and the
form in a
characterized as
net charge density tend to be uni-
while suffering variation across a diffusion
sublayer at the solid boundary. This chapter explores the behavior of an
expansion region where
the resident liquid is presumed
to be entrained
and mixed by the influent liquid jet. When the diffusion sublayer thickness exceeds the Debye length
as a charge
the expansion region is shown to function
trap, in the sense that
less than that at the
inlet.
inserted in the flow just
the effluent convection current is
As discussed in Sec. 3.4, such an element
upstream of an insulating tube can help limit
the flow-induced electrical stress within the tube wall.
Limits to the
trapping tendency are anticipated when
influent liquid jet
increases to the point where
the energy of the
the turbulent motions
approach within a Debye length of the expansion wall.
Under this condi-
tion, double layer charge is swept into the turbulent core and withdrawn
by the effiuent liquid stream,
rendering the expansion
volume a charge
generator in that the effluent convection current can exceed that at the
inlet.
Thus, the ratio of the diffusion sublayer thickness to the Debye
length plays a role here that is analogous to that in the charge generation process in a conducting tube (Fig. 2.3).
Without specifying
ting, Sec. 4.2
rent in terms
whether the expansion wall is
develops an expression for the
conducting or insula-
effluent convection cur-
of the influent convection current
and the volumeicharge:
99
density at the liquid-solid interface.
and generating behavior with the appro-
for associating charge trapping
the ratio
limits of
priate
length.
liquid-filled expansions
in Sec. 4.3,
trapping behavior,
exhibit charge
to.Debye'
sublayer thickness
of diffusion
In experiments reported
uniformly
This general result is.the basis
while
those
filled with
liquid generally behave as charge genera-
porous media permeated by the
To show that these results are consistent with the model, empiri-
tors.
cal mass
transfer correlations
are invoked in
Sec. 4.4 to
(diffusion sublayer thickness) to
Sherwood number
relate the
the mechanical power
input and to a characteristic dimension of the expansion. The discussion
in Sec. 4.5
recognizes that the trapping tendency
expansion does
not rule out
of the liquid-filled
charge generation in a pump
where the me-
chanical power is supplied externally, and proposes modifications of the
interface intended
liquid-solid
charge generation
to inhibit
for any
power input.
4.2
Charge Distributions
Based on assumptions
summarized next,
an expression is derived in Sec.
4.2.2 for the resident volume charge in terms of the influent convection
current and material
In Sec. 4.2.3 the electrical
and flow parameters.
stress within the wall of an insulating expansion, pictured in Fig. 4.1,
is related to operating conditions,
but is shown to be controlled inde-
pendently of the volume charge by proper design.
4.2.1 Assumptions
[a] The net charge density p
essentially
fills the expansion
motions in the core are
turbulent
is uniform in a turbulent core region that
volume.
This presumes
intense enough that a Debye length based on the
diffusivity (Xt
vD t )
exceeds
the characteristic
dimension of the expansion (Le).' Thus, in the corO'
of time alone.
that turbulent
=
linear
(t), a function
100
8
.§ p
x
Fig. 4.1 Insulating
expansion
X '
. ,
. .
.
I :
with internalvolume V,
bounded
by the
surface S. At the outer surface of the insulating wall is a
grounded conducting layer. Two types of leakage channels are
shown: the dead-end type containing stationary liquid in contact with the conducting layer, and those carrying the influent and effluent liquid streams.
101
[b] The turbulent
core is
bounded by a
diffusion sublayer
that is small compared with t
thickness
of uniform
local radius of curvature of
the expansion wall. The molecular diffusivity is assumed to dominate the
turbulent one throughout the sublayer.
negligible within the sublayer so
[c] Convection is
that charge trans-
port there is entirely by diffusion and migration.
(in the sense of assumption [e] of
[d] The charge density remains small
Sec. 3.2.1)
so that the
liquid conductivity
lg
is
essentially uniform
throughout the expansion volume and close its value where the net charge
density vanishes.
[c] The influent,
convection
current
pi Q
varies
With the lowest conductivity
on the
scale of
This is consistent with obser-
the charge relaxation time xc (- e/or).
vations.
slowly
in the vicinity of 1.0 pS/m (see
Sec. 3.3.1) the longest relaxation time is of the order of 20.0 seconds.
Meanwhile, transients in
the flow-induced convection currents are char-
acterized by time constants of the order of minutes (see Sec. 3.3.2).
4.2.2 Volume Charge
With
assumption
[d],
the term in Eq. A.9
containing
the gradient
of the
conductivity can be neglected, and with Gauss' law, this equation can be
written in a form more convenient for present purposes as
=
at
-v.vp - V.J
(4.1a)
- DeVP
(4.lb)
where
=
is the
net current density,
in which De is the
local diffusivity. The
convection of charge by the turbulent velocity fluctuations is accounted
for by the diffusion term in Eq. 4.1, while the explicit convection term
102
involves the mean
velocity v which is presumed
to be negligibleevery-
where except at the ports.
Core Region:
With
assumption
charge of the uniform density
and carries
through out the expansion,
Q, where Q is the
4.1a over the volume V
contains volume
volume flow rate.
enclosed by the surface S in
4.1 yields
dt
dt
where
-(J.)da
(5·n)da
+
S-vf,
is the outward
Gauss' theorem was
bounding surface.
;
X
-~ r
r s Q
directed normal vector,
dence time in the expansion,
to
effluent liquid
(t) pevailing
a convection current
Integration of Eq.
Fig.
a] the
r is
the liquid resi-
iQ is the influent convection current and
used to convert the volume
It
(4.2)
integral to one over the
remains to relate the normal
current density J.-
by solving Eq. 4.1 for the volume charge distribution p(x,t) in the
diffusion sublayer.
Diffusion SublaYer:
With the surface S
edge of the diffusion
sublayer
in Fig. 4.1 coincident with the
at x = 8, the surface
integral
in Eq. 4.2
can be elaborated using Eq. 4.lb
(
(jI )da
S
S
+ Dm(S))da
max
=
J
IT 9.
A (8s)da
(4.3)
S ax
where x is defined within the sublayer as the distance measured from the
interface into the liquid (Fig. 4.1),
the second equality. With assumption
can be
4.1 for
and Gauss' law was
e], the time derivative in'Eq.-4;1
neglected compared with the conduction
the diffusion sublayer,
(assumption [c]) and De
used to write-
term.
To specialize'Eq.
the convection term is
Dm (assumption [b]):
also neglected.
103
-
2
where Xm
P
_
length given by Eq. 2.9.
is the Debye
Eq. 4.4
solution to
(4.4)
p =p(x,t)
;
2
is expressed in
terms of
With.P(S,t) = p(t) the
the values of-p' at the
boundaries of the sublayer as
p(x
(t)
-
(=,ticosh(8/Xm)
sinh(x/Xm)
sinh(8/m)
(4.5)
+ p(0,t)cosh(x/Xm)
With
Eq. 4.3 yields
Eq. 4.5 substituted,
(J.)da
=
+
T
(t)
Sm
coth($/xm)
- p(0,t)
where the uniformity of 8,
(0,t), and
Xm csch(8/
(4.6)
(t) on S allowed immediate eval-
uation of the surface integral.
Effluent Convection Current:
Substituting Eq. 4.6 into Eq. 4.2 yields a
differential equation for volume charge in the turbulent core
dp+P
dt +a
Pi
=
r
where
+
aand
and
Tb
X£L
+
+
P(O't)
b
coth(8/
csch(8/m)
(4.7a)
(4.7b)
(4.7b)
104
V/S is the linear dimension of the expansion. Although p(0,t)-
where Le
remains
to be specified,
Thus, the
it does
particular solution
not vary independently 'of p
of Eq. 4.7a
geneous solution is characterized by a time
the scale of
slowly on
exploiting the
reduces
(assumption [e]).
't
prevails because
or Pi.
the homo-
a < rn, while Pi(t) varies
limit Xm << 8, and
In the
inevitable inequality Xm << Le,
the particular solution
to
Pi(t)
pit)
5L 1+r/(4.8)
1 +
Because this limit implies a
short charge relaxation time ()
compared
with a diffusion time (82 /Dm) based on the sublayer thickness, it is not
diffusion is neglected
surprising that Eq. 4.8
is what would result if
at the outset.
migration limit the effluent convection current
In this
at the inlet (piQ ) and a trapping efficiency can
(pQ) is less than that
be defined:
E
Pi - p(t)1
-
(4.9)
Pi
which
increases as
increases. In
the
ratio
the opposite
of the
residence
to relaxation
limit (Xm >> 8) the particular
times
solution to
Eq. 4.7 reduces to
Pi(t) + (/8Le)P(0,t)
Unless the two
terms containing the Debye length
result is no different from
dition
>> 8 is evidently a
are significant, this
the migration limit Eq. 4.8. Thus, the connecessary but insufficient one.for charge.
generation. An additional condition emerges in Sec. 4.4 where a boundary
105
condition is suggested for
tions are invoked
p(0,t), and empirical mass transfer correla8 to the power
to relate
in the influent
liquid
jet.
4.2.3 Surface Charge in Insulating Expansions
As long
state influent and effluent
as the steady
convection currents
tendency for the inner surface of the insulating
differ there will be a
wall to collect a surface
charge.
electrical
the
stress within
through conduction paths in the
The surface charge generates both an
insulating
wall and
a leakage
current
liquid that tends to reduce the surface
charge density. If the dielectric strength of the wall permits, both the
electrical stress and the leakage current increase until the latter offThe objectives of this sec-
sets the difference in convection currents.
tion are
to relate the
ultimate electrical stress to
operating condi-
that stress can be rendered tolerable regardless
tions, and to show how
of the required trapping efficiency by creating additional leakage paths
through the insulating wall.
conducting layer applied di-
shown in Fig. 4.1, a
In the configuration
of the wall plays the role of the sleeve in
rectly to the outer surface
electric field in the wall E
Fig. 3.1 by constraining the
to be essen-
tially normal to the wall. The same layer seals and terminates the deadend liquid channels that
form leakage paths of predictable conductance.
The resistance encountered by the net leakage current (IN) is assumed to
be dominated
by that
of the channels,
over the inner
potential
pared with the
garded as
and
thus the variation
in the
surface of the insulating wall is small com-
potential drop across the wall.
essentially uniform over
With
the potential re-
the inner surface, the
net leakage
current is given by
N
IN
(FfA
®Gi =
E
NG =N-J
(4.11)
i
where the summation is over N channels, and G i is the conductance of:the
106
ith channel.
cross sectional
with
each
all channels are
For simplicity,
thickness (Fig. 4.1).
area
A
Of course,
taken to
length w
and
be identical,
to the
equal
the channels that carry
wall
the influent
and effluent liquid contribute to the total conductance as well.
obtained by equating the-net
the steady state stress is
An estimate of
leakage current IN to the difference between the steady state convection
currents (pi - P)Q. Then from Eqs. 4.2 and 4.11
E
NA1 ne
where the
(Le)
(f)da
(4.12)
typical radius
implicit that the
condition is
greatly exceeds the wall thickness (w).
and in
the migration
limit Xm
<
<
where
of curvature
With Eq. 4.6 substituted,
Eq. 4.8 applies,
the steady
state stress is given by
(piQ)E
NA
_piV
w ~ NAY(
Here, the inequality
The
result
efficiency and influent convection
by increasing either
be reduced
(13
the efficiency.E was
Xm << Le was used again, and
Eq. 4.9.
using
introduced
r/1Q)
+
indicates
that, for
a
given
current (PiQ), the stress can always
the number (N) or
the cross sectional
areas (A) of the dead-end channels. It is this leverage over the leakage
process that
makes it possible
to regard the efficiency
(Eq. 4.9) and
the steady state
stress (Eq. 4.13) as independent
design issues. If it
is of interest,
the steady state stress generated
in the thin sublayer
limit (Xm
>
) can also be obtained from Eqs. 4.6 ahd 4.12.
While the transient
in the electrical stress is
of no practical conse-
quence, it provides additional basis for testing the model.
change of the
average stress is related to
charge density ()
The rate of
that of the average surface
by Gauss' law applied to the closed surface r in Fig.
107
4.1:
1JdF1
dE
dt
e
N
eS
The first equality
negligible
(I(J.)da
I(4.14)
dt
(4.14
recognizes that the volume charge
contribution
to a stress
typically makes a
that is significant
on the scale
of
the dielectric strength. The time that plays the role here that
1
for
4.14 by
the insulating
tube
in Chapter
using Eqs. 4.11 and 4.12
to express
3 is identified
IN and E
from Eq.
in terms
plays
of the potential
0. The result
1
is written
t.NAagJ
so as
M(4.15)
to make
clear
its origin
in the
"lumped"
resistance
associated with the N chaninels,and capacitance formed by the liquid and
the outer conducting layer as electrodes and the wall as the dielectric.
This time,
which characterizes the transient in
the electrical stress,
is longer than that characterizing the transient in the effluent convection current (a
that
E
T1
S
w 1
Ta
NA
9 E
<
) essentially by a factor of
is, the ratio of the surface
area of the expansion
to the net cross
sectional area of the channels, multiplied by two factors that are close
to unity.
108
4.3
Experiments
There
are two ways
in which
the experiments can
model of the preceding section.
conducting expansions,
currents conform to
to the
First, at issue for both insulating and
is whether the influent
and effluent convection
or depart from the migration
consistent with those identified
the conditions
lend support
limit (Eq. 4.8) under
by the model.
Second,
for the insulating expansion, an indication that the ultimate electrical
Eq. 4.13 can come from comparison of
stress is correctly represented by
the observed time constants with those predicted by Eq. 4.15.
4.3.1 Arrangement and Procedure
Experiments were
conducted with
five cylindrical expansions,
insulating, and three with conducting walls,
two with
as described in Table 4.1.
In the 2-port expansions, liquid is injected and withdrawn through ports
at opposite ends of the cylinder axis.
Mixing is likely to be more com-
plete in the 3-port expansions, like that illustrated in Fig. 4.2, where
liquid is injected through ports
withdrawn through a
tween the ends.
at both ends of the cylinder axis, and
port on the curved part
of the cylinder midway be-
For the insulating expansions,
a close fitting conduc-
ting enclosure at the outer surface of the insulating wall is maintained
virtually at ground potential by a charge amplifier.
There are no dead-
end channels like those illustrated in Fig. 4.1, so the controlled leakage paths
include only those
exits the
expansion.
channels through which liquid
The expansions are
packed with a biphasic material
ated by the liquid.
with samples of
either filled with
enters and
liquid or
such as steel wool or fiberglass perme-
In the latter case, the packed expansion:is flushed
the liquid under study before
installation in the flow
loop.
The experimental arrangement is essentially that described in Sec. 33.1
but with the
in
Fig. 4.3.
insulating tube replaced by an
The tubes that
expansion volume, as shown
communicate liquid
to the
expansion are'
electrically .isolated from the latter (conducting expansion) or itscon-
109
TABLE 4.1
Descriptions of Experimental Expansions
expansion
J
material
volume (m3 )
__
II
I
x 10 - 5
A
Teflon
3.3
B
Plexiglas
1.0 x 10-3
C
Brass
3.3 x 1-5
D
Brass
3.3
E
Brass
1.0 x 10
x 10
-2.0
1-8
1
1
-1.2 x 1-7
2
1
1
1
2
1
2
1
x
5
notes:
Tlo9 is calculated from Eq. 4.15
tubes connected to the expansions have radii a = 1.3 (mm)
-
outlet
ports
inlet
ports
110
tU
insulating wall
Fig. 4.2
Three-port insulating
expansion corresponding to
expansion B
in Table 4.1.
U
Fig. 4.3
Experimental arrangement showing three-port expansion, electrometers E and charge amplifier C. The stainless steel tubes
are electrically isolated from the expansion or its conducting
enclosure.
111
ducting enclosure
(insulating expansion).
are the net influent and
Three
measurable quantities
effluent currents and either the current drawn
by a conducting expansion or
the net charge on the conducting enclosure'
of
As a check
an insulating expansion.
three quantities are
on the
instrumentation, these
shown to be consistent with
a statement of charge
conservation like Eq. 3.45, after which only two (typically the influent
current and the expansion charge
dead-end channels
or current) need be recorded.
in the insulating expansions,
and
With no
provided that the
wall is perfectly insulating on the time scale of the experiment, all of
the leakage
the
current (IN )
reaches ground through the
stainless steel tubes.
Thus, all of the
enclosure is "counted" by the
(negative of the)
electrometers via
charge on
the conducting
charge amplifier which then indicates the
net charge within the insulating
expansion and, with
Gauss' law, the electrical stress in the wall. Because charge convection
and leakage proceed through common channels, the convection currents can
be isolated oaly for times
much shorter than the time constant x1 iden-
tified in Sec. 4.2.3. Meanwhile,
for both the conducting and insulating
expansions, the quasistationary charge densities in Eqs. 4.8 to 4.10.can
only
be measured
after
a
time
a (Eq. 4.7b) has elapsed
since
commence-
ment of the flow.
4.3.2 Results
Results for the expansions described
in Table 4.1 are arranged in[Table
4.2 according to their letter designations.
-effluent currents, Ii and Io,
stainless
steel tubes
residence
times, as
the Reynolds number Ry of the flow.in the
is tabulated
well as
Along with the influent and
to
give an
for reference in
idea of
Sec. 4.4
the relative
Measured and
calculated efficiencies, Em and Ec, are determined respectively from the
first and second expressions for E in Eq. .4.9 using
Pi = Ii/Q
;
P= I/Q
112
TABLE 4.2
Selected Experiments with Expansions
Runt
1°Q
--
.
-I0
-I i
.
.
10- 8
Em
.
Ec
.
0.35
.0.15
0.57
0.54
0.65
0.35
0.46
0.44
1.4 x 10 - 8
0.90
0.60
0.33
0 .38
16000. 0
0.5 x 10
1.1
0.75
0.32
0.36
31 A
4.7
7300. 0
32 A
4.7
11000.0
1.0
33 A
4.7
14000. 0
34 A-
4.7
1.0 x
x 1S
8
69
B
47.0
16000. 0
4.1 x 1g - 7
2.1
0.i
0.95
0 .99
81
B
130.0
16000. 0
3.1 x 1-7
0.9
0.1
0.90
0.998
21
C
4.7
6500.0
0.6
0.23
0.62
0.56
22 C
4.7
11000.
1.1
0.48
0.56
0.45
23 C
4.7
14000. 0
1.5
0.70
0.53
0.40
D
7.6
12000.0
0.62
0.3
0.52
0.54
152 D
7.6
18000.0
1.1
0.68
0.38
0.44
153 D
3.3
12000.0
0.44
0.32
0.27
0.34
154 D
3.3
18000.0
0.85
0.65
0.24
0.26
155 D
7.6
12000.0
0.55
2.4
steel
wool
156 D
7.6
18000.0
0. 88
4.0
steel
wool
134 E
1.4
12000.0
0.015
138 E
7.0
12000.0
0. 002
142 E
17.0
12000.0
0.004
151
*
subscripts ''
oU in pS/m ; .la
'
letters
and 'c' denote measured and calculated efficiencies
in s-pS/m ; I i and Io in nA
identify corresponding entry in Table 4.1
113
For the insulating expansions these currents are measured at a time such
that
a << t <<
tion t >>
tube
a
l,
1l is calculated
the insulating
experiments with
Much as in
is required.
(Sec. 3.3.3)
the conducting expansions only the condi-
while for
from the
amplitude
and slope
of the
of the flow by a valve
charge amplifier response following interruption
downstream of the expansion. Three features of the results are notable:
the runs with
First, for
liquid-filled expansions
entries in Table 4.2) the
the last five
Em and Ec
is evidence that the migration
because
<< S or because the
m
4.10 are sufficiently small.
(that is, excluding
satisfactory agreement between
limit Eq. 4.8 applies, either
terms containing the Debye length in Eq.
Agreement is found for both conducting and
insulating expansions, consistent with the absence of any wall parameter
in Eq. 4.8. For one of the runs with
an insulating
expansion,
the record
shown in Fig. 4.4 illustrates (1) the expected transient in the effluent
current
characterized
by a time of the order
of
rTa (2) the transient
in
the influent current generated upstream that is familiar from Sec. 3.3.2
and (3) the relaxation of
the accumulated surface charge (following in-
terruption of the flow) from which
the runs in
Second, for
l
1 is calculated.
which steel wool fills the
expansion (155 and
156) the effluent current exceeds that at the inlet, indicating that Eq.
of these-runs,
As illustrated in Fig. 4.5 for one
4.8 does not
apply.
the effluent
current is not
characterized by a single
value, so those
Results
for expansions packed
with fiberglass were too erratic to characterize,
but for at least some
reported in
Table 4.2 are final values.
of these runs the effluent current was the larger.
Third, the
measured values of .'ldr are in fair agreement
proximate calculated values in Table
picture in Sec. 4.2.3
with the ap-
4.1, lending support to the simple
of the surface charge evolution.
The consistency
between runs 69 and 81 with significantly different conductivities gives
some confirmation that leakage through the insulating wall does not compete with that through the liquid.
114
time (min) 1
t
2
3
5
4
6
I56
--
__
9
10
-0.5
-40.0 t
I
Qe
(nC)
(nA)
Fig. 4.4
-1.0
-80.0
-1.5
-120.0
Experimental results with
liquid-filled insulating expansion.
Qe is the net charge within
en in Table 4.2 as run 34.
laxation time T = 4.5 s.
the expansion. Conditions are givResidence time
r = 2.5 s and re-
time (min)
1
2
3
4
5
6
7
8
9
10
-1.0
I
(nA)
-2.0
-3.0
Fig. 4.5
Experimental results with conducting expansion packed with
steel wool. Conditions are given in Table 4.2 as run 155. Res= 2.8 s.
idence time r = 3.2 s and relaxation time
115
With a view towardsexperiments
discussed
negligible current at the entrance
in the runs
the liquid-filled
with
fluent current.
in Chapter 5, which call
to an insulating tube, the
expansion
E
for a
objective
was to minimize
the ef-
Accordingly, an effort was made to minimize the current
generated upstream of the expansion, much as for the test section in the
experiment of
Sec. 2.2.3.
Because that influent current
was not meas-
ured, there is no comparison of efficiencies in Table 4.2. Nevertheless,
the effluent currents
are given for comparison with
larger currents encountered in
the experiments with insulating tubes in
Sec.
3.3.
4.4
Criteria for Trapping and Generation
The particular solution of Eq.
is compromised by
the typically much
4.7 is a general result whose usefulness
uncertainty about the wall charge
density p(,t)
by the tenuous
assumption of a well-defined diffusion
form thickness
.
Nevertheless,
ultimately accounts for
sublayer of uni-
when $ is large enough
migration limit (Eq. 4.8), neither
(,t)
and
to justify the
nor 8 enters explicitly, which
the satisfactory agreement between measured and
calculated efficiencies in experiments with liquid-filled expansions. In
the next two subsections, values are suggested for
limited
objectives of
reduces to Eq. 4.8,
clarifying
the condition
(0,t) and
with the
under which
Eq. 4.10
and of showing that this condition is satisfied for
the liquid-filled expansions but not for those packed with steel wool.
4.4.1 Conditions on the Sherwood Number
To exploit mass transfer correlations reported in the literature, 8 must
be eliminated from
Eq. 4.10 in favor of
definitions of the mass transfer
the Sherwood number.
From the
coefficient km and the Sherwood number
Sh (45) the latter is related to the effective diffusion sublayer thickness
8 by
Sh
k
e
Dm
mL
_
rmLe
(ac)Dm
Le
(4.16)
116
Here, rm is
the flux density due to
diffusion,.ac is the concentration.
difference between the interface and the solution bulk, and.8.isassumed
to characterize the concentration gradient so that
m
(c)Dm/8.
Intro-
ducing this relation into Eq. 4.10 yields
~
P(t)
3tt
1
tr
1+
/r
a << X m
;
(4.17)
(Xm/Le)2Sh
this result,
conclusions from
To draw
2-]
+
it is postulated that
the wall
charge density remains close to its value for an undisturbed double laythat p(0,t) N oP where Pw
er, so
To be definite
is given by Eq. 2.12.
about the influent volume charge, which is supplied by a conducting tube
in the experiments,
with the
the associated convection current
<<
limit of highly turbulent flow (
P
iSQQ
Pi
'
1+Pw~l
and
flow conditions and
For the experimental
in the
Xm) in the tube:
2I2
The subscript 'u' identifies the
omitted from P
by Eq. 2.11
convection current given
fully developed
iQ is identified
(4.18)
variable with the upstream tube. It is
because their magnitudes are independent of both
m
the solid phase,
by assumption in the
conditions of Sec. 4.3, the
in Eq. 4.18 ranges from about 0.1 to unity,
case of Pw.
factor multiplying'.p
the upper limit correspond-
ing to the lowest conductivities. With these characterizations of-p(0,t)and Pi, Eq. 4.17 reduces
Sh
Thus, Eq. 4.19
or
to Eq. 4.8 when
<< 1
is one of two sufficient
(4.19)
conditions for.validity of.the.:.X
117
there are
ingly,
the other condition being
(Eq. 4.8),
migration limit
trapping behavior:
<< Xm and
( ]
p(t)
X
h
for departure
conditions
two necessary
(4.20)
current (w
fully developed convection current
from charge
PW
p(0,t)
for which the effluent convection
>> xm. Accord-
Q)
is of the order of the
predicted for a conducting tube, in-
dependent of conditions upstream of the expansion. To assess the practirelate the Sherwood
it remains only to
of these limits
cal importance
number to flow conditions.
4.4.2 Mass Transfer Correlations
number of empirical studies involving
In a large
agitated liquids, the
Sherwood number has been found to be proportional to the product of powers of other
dimensionless numbers that characterize the configuration,
The conditions which justify the
liquid properties and flow conditions.
Chilton-Colburn
mass transfer
are not
analogy (46, p. 645)
counterparts of the heat
so the
too restrictive,
transfer correlations developed
from related studies (47) can be cited as well. In the majority of these
by a driven impeller, so the correlations
studies the mixing is induced
as formulated are most directly applicable to pumps.
for local isotropic turbulence
the condition
expressed in terms
can be
these correlations
The mass transfer correlations
specifying
enough that
of the
mechanical power
proposed for a variety of configurations
the magnitude of
the configuration.
details of
transfer coefficient
(45, p. 20) is satisfied,
permitting application to a jet-mixed expansion.
input per unit volume,
are similar
Nevertheless, when
S
h can be
An
estimated without
expression for
that has successfully correlated data
mass transfer to fixed bodies
agitated liquids is (45,
p. 78)
the mass
on heat and
(e.g. heat transfer jackets) submerged in
118
0.13
km
(S )-2/3
(P/V)n 1/4
S
PM~
2
where P/V is the mechanical
are, respectively,
-CDM
(4.21)
power input per unit volume and n, v and Pm
the absolute and kinematic viscosities
and the mass
density. The Sherwood number follows from Eq. 4.16 as
Sh = 0.13
where
(Sc)
the relation V
power input
Le
L3
(4.22)
1/4
was used.
is expressible in terms
For the jet-mixed
expansion, the
the pressure drop from the inlet (i)
of the the volume flow
to the outlet ()
rate Q and
channel (48, p.
176):
P = (ap)Q
;
p
(4.23)
pi P
-o
With the expansion dimension (Le) large compared with the channel radius
(a), the contraction
at the outlet
accounts
for
practically
the entire
pressure drop, which is approximately (49, p. 67)
0.75PmU2
hp
(4.24)
With Eqs. 4.23 and 4.24 substituted, Eq. 4.22 becomes
Sh
For the
X
0.1 (Sc)l/3
(Ry)3 / 4 (Le/a)1 / 4
expansions in Table 4.1
about 2.4 to 3.2, and
R = 2aU
y -v
the factor involving Le/a
(4.25)
ranges from
thus Eq. 4.25 is essentially the same as:the-cor-
relations proposed for heat transfer to jacketed walls in impeller-mixed
119
vessels
(47, p. 284),
if the Schmidt
is replaced
by the Prandtl
is replaced by one based on the impeller
number and the Reynolds number
dimension and angular
number
velocity.
As noted, use of Eq. 4.21 is justified
only under conditions of local isotropy,
that is, when the scale of the.
primary eddies (Le ) vastly exceeds the Kolmogoroff scale (Lk):
(a) 1 / 4 (Le)3/4
(n)3/4
Lk
12
4
3/
/ P/V)(Ry)/3
where Eqs. 4.23 and 4.24 have been used,
( Le
(4.26)
4
and the final inequality holds
in view of the high Reynolds numbers involved (Table 4.2).
With the Schmidt number of Table 3.1, Eq. 4.25 specializes to
/1
(Ry)3 /4 (Le/a)
n
Sh
For the experimental
0.1, a
-
10g 3
obtained from
Eq. 4.16,
from less than
(4.27)
conditions summarized in Tables 4.1 and 4.2
< 2 x 1)
Ry
4
Sh is
andX
at best
from
m
of the order
Table 3.1, the ratio
to greater than unity. However,
is easily satisfied which is
of 104.
(Le
With
8/Xm ranges
the inequality Eq. 4.19
sufficient to validate Eq. 4.8, consistent
with the experimental results for the liquid-filled expansions.
The charge, trapping tendency
of the liquid-filled expansions
does not
preclude charge generation in a pump for which the Sherwood number given.
by Eq. 4.22
must be re-evaluated
chanical power.
It is also tempting
4.2 and 4.4 to the
tical
based on the externally
supplied me-
to apply the developments of Secs.
expansions packed with steel wool (and to some prac-
filter elements)
by recognizing
that the
characteristic length
(Le) emerges in Eq. 4.10 from the ratio of the liquid volume to the area
of the liquid-solid interface.
th&t volume is
Relative to the liquid-filled expansion,
decreased while the surface area
is vastly increased by.
120
the inserted material,
charge generation are
likelihood that the conditions for
and a greater
satisfied.
to a much smaller characteristic length
leading
ere again, the
power input must be
re-evaluated, but more
as long as the liquid "ports" are lo-
important is the recognition that
the mass transfer process are now likely to
calized, the flow and hence
be highly nonuniform.
4.5
Discussion
Two aspects of
expansion render it a
the behavior of the liquid-filled
First, both of the
candidate for use in the coolant circulation system.
sufficient conditions for
the condition for high
charge trapping (Xm << 8
trapping efficiency (r
>
T)
are consistent in
Second, Eq. 4.7
satisfied with modest experimental volumes.
indicates that the
Le) and
and one of the sufficient conditions
their dependence on dC, L e and Ry,
was easily
>
Xm2 <<
or
within a.time
expansion operates at full efficiency
(-a <
) short compared with that which characterizes the
transient in the insulating tube ( 1 >> a, Sec. 3.2.5).
charging
As noted in Sec. 4.4.2, charge generation is more likely in both filters
(due to
length) and in pumps (due
the small characteristic
ternally supplied mechanical power).
to the ex-
In the case of the pump, an alter-
native to trapping the generated charge by means of a downstream liquidfilled expansion, is to circumvent a necessary condition for charge generation
the interface from
relative to
surface could
"tangle" of
ions
by displacing
(8 < Xm)
the
which
be "extended" into the
the solid material
but dense
to permeate,
edge
of
the diffusion
m is measured.
tenuous enough to allow
enough
to exclude
the solid
Thus,
liquid in the form of
sublayer
a "web" or
the liquid and
the turbulent
liquid
motions. To prevent re-entrainment of the "trapped" ions and allow their
escape to ground by conduction along the interface, the extended surface
layer must
be thick
polymer layer as
compared with X,.
The suitability .of
the extension of the solid
an adsorbed
surface is demonstrated by.
121
its ability
to reduce
the electrophoretic mobility of
indicating displacement of the plane of shear (50,51).
can extend the region at
dominates
is
also
host particles,
That such layers
the interface within which molecular diffusion
demonstrated
by
the attendant
reduction
in
Sherwood number measured for a rotating disc in turbulent flow (52).
is to
preserve flexibility in
the polymer layer
the choice of the
by evaluation of the generated
It
solid "substrate" for
that insulating walls are accommodated
opment of Sec. 4.2
the
in the devel-
electrical stress in
Sec. 4.2.3.
Apart
from anticipating
the behavior of
practical elements
where the
flow is well mixed, there are two ways in which the developments in this
chapter relate to phenomena in the insulating tube. First, the migration
model of Chapter 3,
whose range of
validity has yet to be defined, has
its parallel in the migration limit represented by Eq. 4.8;
lel is formalized in
Sec. 5.5.2.
of the migration model,
density
tube.
must be
Second, outside the range of validity
a boundary condition involving
specified for
To the extent that flow
this paral-
both the
expansion and
the wall charge
the insulating
conditions are uniform over a section of
the expansion wall serving as an electrode, and to the extent that those
conditions can be controlled by an external energy source, the expansion
offers the possibility of a relatively direct empirical investigation of
that boundary condition.
122
Chapter
5
FLOW-INDUCED CHARGING OF THIN INSULATING TUBES:
DIFFUSION EFFECTS
5.1
It is
Introduction
the premise of
the model of Chapter 3
exist for which diffusion makes
current density in the
that practical situations
a negligible contribution to the normal
liquid at the liquid-insulating solid interface.
In these situations, where space charge entrained in the influent liquid
drives the
charging process in
expansion is usefully inserted
importance of
the diffusion
the insulating tube, a
at the tube inlet.
influent convection
model which,
In this chapter, the
current is demonstrated in
which an insulating tube acquires
nificant
charge trapping
experiments in
a net charge in the absence of a sig-
current.
The results motivate
despite its more tentative character,
serves as the basis
for defining the range of validity of the migration model.
argument that
this range
a revised
includes practical conditions
A consistent
must reconcile
with the charge generation process in a conducting tube (Sec. 2.2) where
the normal diffusion and
migration current densities are generally com-
parable.
Like the
experiments of
incorporate
a close
Sec. 3.3.2, those
fitting sleeve to
reported below in
attenuate the
current so
that measurements will reflect primarily
density in
the liquid and
Sec. 5.2
axial conduction
the radial current
bear more direct comparison
to processes in
conducting tubes. Now, however, a large expansion volume at the entrance
to the insulating tube limits
split into electrically
of the phenomena.
insulating tube
the influent current, while the sleeve is
isolated segments to reveal distributed aspects
The observed effluent current is generated within the
itself, and
differs from that generated
by conducting
tubes in at least two respects. First, whereas Abedian and Sonin's theory appears to overestimate
the spatial development length for streaming
currents generated by the Freon TF/DCA-48 combination in stainless steel
123
tubes (Sec. 2.2.3),
tubes.
it appears to underestimate that
response to the
is a history dependent transient
Second, there
flow that suggests
length for Tefzel
an uncontrolled leakage process driven
by the small
axial electric field within the tube.
The model
developed in
Sec. 5.3 incorporates
diffusion in a
way that
closely parallels the model of Sec. 4.2.2 for the expansion region.
The
long spatial development length is accounted for with a revised boundary
condition
on the
that no longer
as the
volume charge density
diffusion sublayer
regards radial charge transport through
rate limiting
characterizing
within the
step of the
the observed
the liquid bulk
spatial development.
temporal transients
are
Time constants
too short
to be
accounted for by bulk conduction driven by a quasi-one-dimensional axial
electric field. Thus, the model allows for a significant surface conductivity at
the liquid-insulating solid
fixed and
uniform for
the duration of
interface,
which is regarded as
a single experiment,
but which
evolves over a much longer time scale as indicated by a gradual shortening of the observed time constants. While the role of surface conduction
in the experiments remains a matter of contention, an important implication of the model is that a controlled surface conductivity tends to reduce the ultimate' ectrical
stress regardless of whether the accumulat-
ing charge originates upstream or within the insulating tube itself.
5.2
Experiments
The main purpose of the
experiments described below is to qualitatively
characterize the charging process internal to the insulating tube within
the framework developed for conducting tubes (Sec. 2.2). Viewed in.their
entirety, the experimental
mation that
experiments are
and degree of repetition.
certain features of
Sec. 5.3,
observations are sufficiently rich in infQrmost helpful the greater
their simplicity
A preliminary interpretation i
the results guides the development
Sec. 5.2.3 of
of the model in-
-- -- II-
I II 1
124
5.2.1 Arrangement and Procedure
With details as given in Sec. 3.3.1,
the experimental arrangement is as
shown in Fig. 5.1.
connected to the expansion (or its
conducting
The electrometer
enclosure) also
constrains to
ground potential
the liquid
reservoir (not shown) and the stainless steel tube that feeds the expansion. Thus, the indicated current is identified as the net axial current
I(0,t) at
the entrance
virtually
grounds the
to the insulating
receiver and
tube.
A second electrometer
the communicating
stainless steel
tube to indicate the net axial current I(L,t) at the outlet of the insulating tube.
In most of the experiments the close fitting sleeve illus-
trated in Fig. 3.1 is split into electrically isolated segments; in Fig.
5.1 seven segments are
through a
shown.
third electrometer
These are grounded directly or virtually
which monitors, on a
time-sharing basis,
the currents flowing to ground from the individual segments.
The current
Ii flowing
quantities of
from the ith
interest by recognizing
sleeve segment can be
that to zero order
over the entirety of surface
(zero order) field
the electric
Sec. 5.3.2 below). Thus,
tube is entirely radial (see
field within the
related to
S1 in Fig. 5.2 the normal component of the
vanishes, and by Gauss' law,
so must the net charge
enclosed and its rate of change:
AiI(z,t) + Ii(t)
=
(5.1)
where
AiI(z,t)
and I(z,t) is
I(i,t) - I({i-}Q,t)
the local axial current and
ment. To check the instrumentation
of uncontrolled current paths
ments are preceded by a
.is the length of each seg-
and verify that the impedance levels
are sufficiently high, the formal experi-
demonstration that at any instant the summation
over all N sleeve segments
31
-
-
125
expansion
metal
tube
sleeve
segment
\
insulating
tube
metal
tube
L
I(o,t)A
I(L,t)4
ii~
receiver
Experimental arrangement for studying the convection current
generated within an insulating tube. Electrometers E monitor
influent and effluent currents and currents from individual
sleeve segments.
Fig. 5.1
insulating
I
_
-
^'
1
I
-
wall
I
"I\
/
sleeve
4
Sl
segment
Fig. 5.2
Section of the insulating tube showing generators of the
closed cylindrical surfaces'S1 and S..Not shown is a thin air
gap between the outer surface of the insulating wall and the...
.
conducting sleeve.
126
N
N
E (AiI + Ii )
0
I(,t) - I(Lt)
I
gIi(t)
i=l
i=l
as required
by Eq. 5.1.
flow and monitoring
of time,
Experiments
consist of initiating
the currents I(0,t), I(L,t) and
the liquid
Ii(t) as functions
with liquid conductivity and mean velocity as parameters.
,axial current
in
N-l) from the
the insulating
tube at a distance
tube inlet is deduced from
and Ii and repeated application
from 1 to k.
Finally,
z = k
The
(k = 1,2,...,
the measured-values of I(0,t)
of Eq. 5.1 with i ranging in succession
a smooth axial current profile
is drawn through
the discrete data points along the z-axis.
5.2.2 Results
Experimental results are divided into three groups. With the first group
the objective
mode that
is simply a
is independent
clear demonstration of an
of the original
internal charging
double layer;
here a single
sleeve segment suffices. The objective with the second group is to characterize the spatial development of the convection current; at least two
segments are needed to reveal this distributed process.
spans the
largest number of runs
The third group
with a single tube sample,
and it is
here that axial conduction first comes into evidence.
Evidence of Internal Charging Mode: With expansion B of Table 4.1 at the
inlet to the insulating tube (sample D; used in runs 90 through 109) the
measured influent currents, summarized
of
with those typical
in Table 5.1, are small compared
experiments discussed in Sec. 3.3.
Nevertheless,
the characteristic form of the response, illustrated in Fig. 5.3 for run
106, is essentially
same interpretation:
tion.
the same as that in Fig.
the observed currents are due
The influent and
effluent currents
which requires,
given run indicates a departure
according to
Eq. 3.9,
is subject
the steady
A comparison of these curfrom the migration limit
that the convection
crease in magnitude in the flow direction.
to the
entirely to convec-
in Table 5.1 are
state values measured at the end of the run.
rents for any
3.7 and
current de-
127
TABLE 5.1
Evidence of Internal Charging Mode
Runt
U
I(0,t)
I(L,t)
dc/L
102 D
13.6
1.2
-0.008
+0.057
0.23
2.2
0.31
103 D
13.6
1.2
-0.013
+0. 045
0.23
2.2
0.26
104 D
13.6
1.7
-0.004
+08.805
0.17
3.0
0.18
105 D
81.0
1.15
-8.8037
+0.89
0.58
0.43
8.17
106 D
82.0
1.67
-0.B5
+0.09
0.42
0.61
0.10
187 D
82.0
1.15
-0.028
+0.09
0.58
0.42
0.17
108 D
99.0
1.2
0(-8.801)
+0.20
0.62
0.37
0.32
109 D
99.0
1.7
-0.015
+0.15
0.45
0.52
0.14
*
oI
in pS/m
; U in m/s
;
I in nA
tube radius a = 1.3 (mm) and length L = 0.67 (m)
subscript 'c' denotes a calculated value
t
letter Identifies tube sample
128
0.15
I(L,t)
.t
0.10
0.05
t
3
2
I
I
(nA)
I
!
I
*
!
I
,
,
4
,
5
6
I
I
7
8
I
I
§
,
,
-
9
I
.
I
.
.
I
t.
time (min)
n)-…
-'I
-0.05
I(o,t)
-0.10
-0.15
Fig.
5.3
the arrangement
Experimental. results with
tions are given in Table 5.1 as run 106.
insulating
tube
L
-
k
of Fig 5.1.
sleeve
lent
----
I (L,t)
I(O,t)
(t)
I.
I (., t)
AId - I(L,t) - I(,t)
AI
Fig. 5.4
Condi-
= -Ii(t)
I(L,t) - I(0,t)
Definitions of current differences whose ratio is tabulated in
Table 5.2. Flow is from left to right.
129
A calculation of the charge
nal double
layer within
contained in the diffuse part of the origi-
the insulating tube
now serves
two purposes.
First, for the conditions of run 106, Eq. 2.16 gives
n
Qdl
2.0 x 10
while Eq. 2.18
time t
10
(C)
gives for the amount by
which the charge accumulated at
540.0 (s) exceeds that supplied at the tube inlet
t
t
f
I(,t)
- I(L,t)ldt -
indicating that departures of
the migration limit
Second, a
II(0,t)ldt
a
5.0 x
the steady state convection currents from
are not attributable to the
comparison of
the result for
transient portion of either curve
tion extrapolated back to the
10- 8 (C)
original double layer.
Qdl with the area
between the
in Fig. 5.3 and the steady state por-
time origin supports an interpretation of
the transient as a sweeping out of the original diffuse layer.
ent
with this
interpretation is
liquid transit time
of
evaluation from
based on the length of
velocity at the edge of
velocity
an
Eq. 6.11
Consistof the
the insulating tube and the
the diffusion sublayer (which characterizes the
the original
diffuse
layer).
The result
for run 106,
8
k
4.4 (s), is of the order of the time constant of the observed transient.
A more
successful attempt
to account for
the steady
state convection
currents in Table 5.1 is based on Eqs. 2.11 and 2.15, which were applied
to
stainless steel
identifying
the
tubes in Sec. 2.2.3.
convection current
In the
Is(z)
with
limit of
the measured
I(z,t), elimination of Is(m) between these equations yields
Pw
Q
2Xm2J
1 - exp(-L/d)
8 << Xm and
current
130
where the
development length
density Pwb/o
wall charge
lated in Table
is
d is given
The normalized
calculated from this equation
5.1 for each run.
the tabulated values
by Eq. 2.15b.
and tabu-
Three observations are pertinent: (1)
are consistent with Eq. 2.12,
(2) they are fairly
consistent with each other, and (3) they are of opposite sign to, and an
order of magnitude
smaller than, the experimental value
for the stain-
less steel tube (-3.0; Fig. 2.5).
If this account
of the steady state convection
results given in
neglected, and
Table 3.2 must be reinterpreted.
an adjustment
charge within the tube was
Pwb/o
currents has merit, the
of the calculated
There, diffusion was
rate of change
of net
based on a normalized wall charge density of
m 1.0 and an enhanced liquid conductivity near the wall. However,
with the
values of
the normalized wall
charge density given
5.1, the conductivity
enhancement,
according
Here, a
of
and
further
test
Eqs. 2.11
in Table
to Eq. 3.5, is negligible.
2.15 (in the limit
8 << Xm ) is
applied by combining them with Eq. 3.45 to yield
dQt
dt
(I(0,t)
-
I(0,t)}(l
-
exp(-L/d)3
where
I(0,t)
Now,
+>
PwQ
using pwb/r, m
rameters from
0.1, the
mobility from Table
run 36 and Qt/t
Table 3.2 this equation
3.1, and
other pa-
yields Qt/tlm -1.65 (nC/min) for
-4.36 (nC/min) for run 37, which agree better with the
measured values in Table 3.2 than do the originally calculated values.
Evidence of Long Development Length:
2.11 and 2.15 are based
As discussed
in Sec. 2.2.2, 'Eqs.
on a limiting form (p(az)
P.) of the general
boundary condition Eq. 2.10. That limit is
applicable either'where.elec-
trical conditions are fully developed (Sec. 2.2.1) or where the exchange ...
131
J
current density
is much
current density at
greater than the normal
the wall Jr(a,z). An indication of whether Eq. 2.15b correctly estimates
the development length is based on the experimental arrangement shown in
The form
the results summarized in Table 5.2.
Fig. 5.4 and
of the re-
sponse is still typified by that shown in Fig. 5.3, so the measured currents are regarded as due entirely to convection. The ratio tabulated in
the last column is calculated from
Id
_
exp(-%/d) - ex2(-L/d)
tEIJc
which
ratio
- exp(-L/d)
t1
is just
an application
are tabulated
in
values.
indicate that if
of the runs that the
to the length
Eq. 2.15b is close
of the
An implication of these results is that if the
for the combination
condition Eq. 2.10 is appropriate
general boundary
is a
What makes this a fairly
the condition satisfied in most
upstream sleeve segment.
under study, then the condition
of materials
consistently
convection current exists, the associated development
d calculated from
value of
and are
these results
length is longer than that given by Eq. 2.15b.
sensitive test is
of the same
Because the calculated ratio
monotonically increasinq function of d,
a fully developed
values
Measured
the penultimate column,
the calculated
larger than
of Eq. 2.15.
Jr( a ,z) << Jw implicit in
Eq. 2.15b is not generally satisfied. A revised development length based
on the general boundary condition is obtained in Sec. 5.3.1 below.
Evidence of Evolving Surface Conductivity!
groups of experiments
their
evolution is
single short transient attributable either
insulating tube (Sec. 3.3.2) or
the insulating
5.2 are regarded as
summarized in Tables 5.1 and
convective because
entirely
Currents measured in the two
characterized by
the
to a process upstream of the
perhaps to the original double layer in
tube itself (see above).
It was
this circumstance that
permitted a direct comparison with expressions for the streaming current
developed for
conducting tubes.
In the group of
experiments described
next, there is evidence that conduction contributes significantly to the
,
\
X
-~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
132
TABLE 5.2
Evidence of Lona
~~ ~ Develooment
~ ~
I
Rune
U
aI
J .......
Lenath
L
l'aId]Ic
L
w
-
-
-
I
!
_
_
111 E
120.0
1.40
0.74
0.56
0.24
0.13
0.05
113 E
120. 0
2.13
0.74
0.56
0.36
0.14
0.09
114 E
120. 0
2.15
0.74
0.37
0.36
0.36
0.26
115 E
120. 0
2.08
0.74
0.37
0.35
0.35
0.26
116
E
120. 0
2.47
0.74
0.37
0.41
0.38
0.29
117 F
120.0
1.30
3.35
3.20
0.22
0.036
10- 7
118 F
120.0
1.65
3.35
3.20
0.28
t0.
Ia
41
in pS/m ; U in m/s ; L. I and dc in m
tube radius
subscripts
a = 1.3 (mam)
'm and 'c'
denote measured and calculated
t letter identifies tube sample
values
133
axial
illustrated
ig. 5.5, and
illustrated
in
in Pig.
the
distribution
state current
steady
5.6.
recirculated (Sec. 3.3.1) it must be
Because liquid is not continuously
periodically
temporal response
forms: the
takes two
This evidence
current.
recycled
temporal response of
the size
of the
segments
correspond
back
to
the
reservoir.
Each segment
Fig. 5.5 corresponds to the
the
to
time
required
to
gaps between
recycle
the
liquid.
the influent current
flow initiates a transient in
Commencement of the
the
flow time dictated by
volume flow rate; the
reservoir and the
of
to be completed before the liquid inventory
I(0,t) that is short enough
this transient accounts for the "hook" shaped segments in
is exhausted;
Fig. 5.5 and may be associated with a process upstream of the insulating
tube.
If
transient
the flow
much longer
on a
interpretation
is
the
the effluent current I(L,t)
than in
to
such as
in the terminal
convergence to steady
that axial
strongly suggests
that
the slow
reduce their
support of
In
convection.
with
observation
currents is
terminal
is dis-
and 5.3, and
now competitive
is
60.0 minutes,
scale, about
time
Figs. 3.7
a
continuous sequence
dramatically with their rapid
values in
conduction
a
This slow exponential variation
current.
currents contrasts
state
viewed as
more clearly in
cernible, perhaps
the influent
are
segments
variation
this
the
in
difference, and
hence
reduce the rate of change of net charge given by Eq. 3.45. Moreover, the
evident in Fig. 5.5 is observed only if
continuity of the flow segments
short on the scale of the long.transient;
interruptions of the flow are
if the
(liquid-filled) tube
is left idle
for longer periods
the long
transient is reinitiated when the flow commences again.
The sequence of runs shown in Fig. 5.5 is designated run 141,
sequences designated 133, 137, 145,
comparison to similar
in
Table 5.3.
described
in
Anticipating
successive
parallel
with the
heading
sequences, it
1.
170, and.187
charging
the time constants of these
Chapter 3,
ulated under the
a
and bears.
transient
sequences are tab-
:In view of the decreasing value
is pertinent
to note that
a single
of sl-in
tubeiis
134
0.15
t
0.015
0.10
t
0.010
I
I
(nA)
(nA)
0.05
0.005
40
80
120
160
time (min)
200
-
Temporal evolution of measured influent and effluent currents.
Fig. 5.5
Conditions
are given
in Table
z
0
__
I.
l
I
5.3 as run 141.
-
L
l
__
__
JI
11
t -0.8
Ii(c)
-0.02 t
I
I
(nA)
(nA)
-1.0
-0.04
I(zoO)
-1.2
Fig. 5.6
-0.06
Measured steady state currents from sleeve segments (discontinuous bars) and corresponding net axial current (continuous
curve). Conditions are given in Table 5.3 as run 187.
135
TABLE 5.3
Evidence of Evolving Surface Conductivity*
i
a
RunT
.
.
-1
U
..
--
.
133
H
1.4
1.7
134
H
1.4
135
H
136
H
1.4
1.4
137 H
7.0
138
H
7.0
139
H
7.0
140
H
141
H
142
H
143
H
144
H
7.0
17.0
17.0
17.0
17.0
2.1
2.4
1.1
1.6
2.1
2.5
1.0
1.7
2.1
145
H
110.0
1.7
146
H
110.,
2.1
147
H
110. 0
2.4
148
H
110. 0
170 H
7.6
172 B
7.6
1.1
1.4
1.6
173
H
7.6
1.9
187
H
57.0
1.0
70.0
70.8
60.0
2.4
1.1
25.0
14.0
2.0
I(L,m)
I ( ,o)
dc/L
Pwb/a
.-
.
0.55
+0. 008
7.0
7.4
7.7
-0.002
+0.015
5.9
0.57
+0.002
-0.001
-0.001
+0. 029
3.4
4.0
4.5
0.14
+90.005
+0. 022
0.16
+0.008
+0. 036
+0. 004
+0. 037
0.000
+0.010
+0. 037
2.4
1.9
2.2
2.5
1.3
+90.120
+0.250
+0.125
+0.130
+0.110
-0.265
-0.345
+0. 280
-0 .009
+0. 016
-8.015
+0.012
-0.023
+0. 030
+0.031
+0. 030
+0.185
0.35
0.43
0.49
0.23
-0. 180
2.9
-0.250
3.2
-0. 44
-0.330
3.6
-1.150
-0.875
0.40
+0. 300
9.48
0.48
0.12
0.11
0.97
0.06
0.96
0.09
0.20
0.16
0.14
0.32
0.05
0.01
0.01
-1.8
*n
a I in pS/m
; U in m/s
;
1 in min.
; I
tube radius a = 1.3 (mm) and length L =
in nA
.89 (m)
subscript 'c' denotes a calculated value ; 8/Xm <.0.7 in all runs
letter identifies tube sample
in runs 133-136 & 170-173 pure (undoped) Freon is used
runs 133-148 are with a liquid-filled expansion
in runs 170-173 expansion is packed with 3 mm diameter TFE spheres
in run 187 expansion is packed with fiberglass
136
involved:
sample B which was
Table 5.3 for which no
runs 122 to 192.
Runs listed in.
time constant is given represent segments at.the'
tail end of the preceding
been established.
used in
sequence,
where the steady state has already
Thus, for example, runs 142, 143, and 144 immediately
follow the last segment in Fig. 5.5, with the liquid velocity set to the
given values.
designated
The form
In all of
these runs the
expansion in Pig. 5.1
is that
E in Table 4.1.
of the axial
current distribution illustrated in
Fig. 5.6 is
typical of the steady state distributions deduced from the currents from
the seven sleeve segments shown in Fig. 5.1 by the procedure outlined in
Sec. 5.2.1.
The essential
absent if the
feature is
the inflection
point,
current is entirely convective, regardless
which is
of whether it
given by Eq. 2.15a or Eq. 3.9. A demonstration that the inflection point
is consistent
with an axial
conduction process is given
in Sec. 5.4.1
based on the model developed in Sec. 5.3.
Finally, as in Table 5.1, the normalized wall charge density is calculated from the
measured currents in Table 5.3.
However, that calculation
can be meaningful only if the measured currents are entirely convective.
Thus, a
further indication that
convection
alone while
slightly better
those of
currents given in Table
Table 5.3
consistency among Pb/0
5.1 represent
include conduction
in the former table,
is the
even if
runs with undoped liquid are excluded from Table 5.3.
5.2.3 Preliminary Interpretation of the Temporal Transient
If axial conduction is an essential feature of the results summarized by
Table 5.3 and
Figs. 5.5 and 5.6, it
currents were isolated
in Tables
Sec 5.4.1,
5.1 and 5.2.
in the experiments summarized in
As suggested in Sec. 3.3.2
bulk conduction is
the relaxation
time '1
is necessary to ask why convection
negligible for
based on bulk
Sec. 3.3.2 and
and demonstrated in
the sleeve configuration;.
conduction alone'is found .tobe
much longer than the times identified in Table 5.3. Thus, the importanca.
137
by the
results
of Sec.
appropriate conditions is indicated
Table
5.3.
Finally,
the
established (Fig. 5.5)
fact
3.3.3,
that
a
steady
it evolves
state -is eventually
of an experiment.
A specific
numerical evaluation of the
surface conductivities implied by the observed range of
l.
required of a model can be obtained by re-
Further insight into what is
writing
under
surface conductivity evolves
indicates that the
model developed next is a
objective of the
but that
by the decreasing time constants in
time scale than the course
over a longer
significant surface
That surface conduction is not an intrinsic feature is
conductivity os.
indicated
of a
from the presence
conduction stems
of axial
Eq. 5.1 as
(-)
dz
(5.2)
Ez
(5.3)
(i-1)%az
where in view of Eq. 3.8b
2
atIa
az
aUp
az
It is already
e azj
(5-3)
clear from Eq. 5.3 that
The steady state sleeve currents Iit)
inflection point in Fig. 5.6.
3.2.5,
in
so Eq. 5.2 implies that the two terms on the right
Fig. 5.6 are finite,
in Eq. 5.3
axial conduction may explain the
sum to a constant in the steady state.
the convection
term is
As in case B of Sec.
regarded as imposed,
terms are individually constant.
and thus
the two
In Sec. 5.3.2 the axial electric field
is shown to be proportional to the local axial derivative of the surfacecharge
density
distribution
influent
(z,t),
transfer
currents in
tendency of the
increase the surface
across
remains
to reconcile
the steady
state
finite difference between the steady state
(z,w) with the
and effluent
next, the
so it
Table 5.3.
In the
model developed
finite steady state net
influent current to
charge density is balanced by
a process of charge
the liquid-insulating interface.
138
5.3
Model for the Temporal Transient
The model of this section is developed with two objectives. The first is
to lend support to the
is important,
contention in Sec. 5.2.3 that surface conduction
by showing that
axial conduction is consistent
with the
inflection point in Fig. 5.6 and that bulk conduction is insufficient to
The second is to identify
constants in Table 5.3.
account for the time
be expected to dominate the normal
conditions under which migration can
current density at the liquid-insulating solid interface.
For
the external
conductor configuration the
retained to facilitate comparison with
three features
interface.
the
important on the time scale
axial
conductivity.
Third is
discussed in Sec.
the finite
conductivity of
5.2.3, it is necessary to
information,
rendered
the tube
wall. As
admit a current across the
to reconcile the steady state surface
finite difference between the steady state
at the inlet and outlet
specific
that is
of the experiments by a significant surface
charge distribution with the
of
the liquid-insulating solid
conduction process
liquid-insulating solid interface
currents
are omitted
First is the contribution of ion
the normal current density at
Second is
sleeve is
the experiments of Sec. 5.2, but
the present model that
are introduced into
from the model of Sec. 3.2.5 (case B).
diffusion to
close fitting
only
of the insulating
the
In the absence
description
of
charge
Thus, the wall is assigned an ohmic
transport in the wall is justified.
conductivity which is low enough
simplest
tube.
that (1) the wall carries a negligible
fraction of the axial current, and (2) the wall can still be regarded as
insulating on the time scale
the
exception
of
[d] and
of the experiments of Sec. 3.3.
[f],
the
assumptions
of Sec.
Then with
3.2.1
are
retained.
5.3.1 Volume Charge Distribution
With the normal diffusion current
included, the purpose of this section
is to develop the counterpart of Eq. 3.9 for the volume charge distribution
in the
liquid
along
the axis
of
the tube.
When the
boundary
139
condition p(a,z)
Eq. 2.15.
Pw
is appropriate, the
result is already
given by
However, as discussed in Sec. 5.2.2, Eq. 2.15b underestimates
length for the materials under
the spatial development
study, and this
motivates use of the more general boundary condition Eq. 2.10.
Conservation of net charge is expressed by Eq. 4.1:
a= -vVp - VJ
at
(5.4a)
where v is the mean liquid velocity and the net current density
E
J(r,z) =
in terms of
is expressed
(e] of Sec.
4.1
(0
3.2.1)
for the
< r < a-S)
5.4 is
(5.4b)
- DeVP
the uniform bulk conductivity
and the local
expansion region,
Fig. 5.7
by a diffusion
bounded
specialized to these
diffusion
diffusivity
renders the
De.
As illustrated
illustrates a
sublayer
essentially
by Fig.
turbulent core
> r > a).
(a-
two regions by recognizing
volume charge
6c (assumption
Equation
that turbulent
uniform in
the core
while molecular diffusion dominates in the sublayer.
Core Region:
With the differential form of Gauss' law inserted, and neof the large Peclet number LU/De) Eq.
glecting axial diffusion (in view
5.4 becomes
BP_
v _aa
at
zaz
T
+
1
a[ rD
r rDes
Following the development in Sec.
is neglected
c-X
]
(5.5)
3.2.2, the time derivative in Eq. 5.5
in comparison with the
convection term.
Then integrating
Eq. 5.5 over the cross section of the core region, where the mean velocity is essentially the superficial velocity, yields
140
__ I 1 j 5.nsu.aLu.ng
_
A
_
sleeve
wall
a-
-a -
z
Fig. 5.7
Section of the insulating tube showing core region where p =
5(z) and diffusion sublayer where
= p(x,z). Turbulent
velocity profile is shown and coordinate systems are defined.
Diffusion sublayer is enlarged for clarity.
insulating
wall
sleeve
r
"I
-a+w
-a
-p---~
Fig. 5.8
Z
Section of the insulating tube showing typical profiles of
quasi-one-dimensional electric fields. Radial and axial field
components are drawn to different scales. The small air gap
_-between the outer surface of the tube wall and the inner sur- ..
face of the sleeve is neglected..
. .
141".
-
U3zP E
where
+
r=a-
(5.6)
(z) is the net volume charge density averaged over the cross sec-
tion of the core, and
the diffusivity at the edge of the diffusion sub-
layer is approximated
by the molecular diffusivity Dm .
on the right in Eq. 5.6 can be expressed
characterizing
the interface
in terms
by solving Eq.
of
The second term
(z) and parameters
5.4 within
the diffusion
sublayer.
Diffusion Sublayer:
layer by
Equation 5.4
neglecting the time derivative,
law, and using the molecular
is
small compared with
terms
is specialized to
of
the
planar
the diffusion sub-
inserting Gauss' differential
diffusivity for De. The sublayer thickness
the tube
sublayer
radius, so the
coordinate
result is
x = a - r
written in
which
measures
distance from the interface into the liquid (Fig. 5.7):
P+ + D_
Vz(X)a
z
8Z
p = p(X,Z)
(5.7)
Dmx2
T
To justify neglect of the convection term, it is enough to show that the
liquid transit time,
based on the velocity at
(vs in Fig. 5.7) and
whatever length characterizes the axial derivative
in Eq. 5.7, is much
the edge of the sublayer
longer than the relaxation time ~T.
the development length d which,
That length is
according to Sec. 5.2.2, is at least as
large as that (dc ) calculated from Eq. 2.15b. Thus
s
>d
8
As given
only
by Eq.
=
!
v
applies
and L
exceeds
Lc
11.0(L/U
'
dc
L
)
6.10, the proportionality
weakly dependent
equality
U,
vs
on
Reynolds
number, so
to all of the experiments
given in
Table 5.3
factor between v
of Table
it is readily
. Thus, Eq. 5.7 becomes
the final
and
5.3.
U is
approximate
With
confirmed that
dc/L, ori,
. 8 easily
142
sDm3
00
-
(5.8)
With the turbulent diffusivity Dt given by Eq. 6.9 and a Reynolds number
typical of the experiments of Table 5.3 (Ry
a
the
)
turbulent
hundred
Debye
length
times greater
diffusivity,
than the
and that
Xt is
radius for these experiments.
essentially
(Xt
i7Dt
Debye
10 ), it may be shown that
is typically
comparable
length based
to or
about
on the
two
greater than
molecular
the tube
Thus, net charge in the turbulent core is
uniformly distributed
and its
value
at the
edge of
the
diffusion sublayer approximates the average value:
Po(z)
P(S,Z)
The solution
(5.9)
of Eq. 5.8
expressed
in terms
of
the values
of
p at the
boundaries of the sublayer is now
p(z) - p(,z)cosh(8/Xm)]
P(xZ) =
(
-
sinh(8/m)
)sinh(x/Xm)
(5.10)
+ P(0 ,z)cosh(x/Xm)
where Xm is the Debye length given by Eq. 2.9. Once P(0,z) is determined
from the boundary condition, the second term on the right in Eq. 5.6 can
be obtained from Eq. 5.10 and the relation between derivatives!
=a
arlr=a-8
-a
ap
axlx=8
(5.11)
consistent with the definitions of the coordinate systems in Fig. 5.7.
Boundary Condition: The general boundary condition Eq. 2.10 can be written in sublayer coordinates as
143
.o(O)As discussed in
Jw >> Jx,(,z )
Sec. 2.2.2, the limiting form
(0,z)
Pw leads to a satisfactory description of the streaming currents
generated by conducting tubes. There are two issues to address if Eq.
5.12 is to be applied to-the insulating tube:
First,
to account
for the results
(53,54) where it is suggested
the exchange
current density
should be
in
various redox systems
-5
10
to better than
than 108
J
A/m
2
2 SO4
lower for
insulator
(55).
solutions range
from less
While there is no
basis for
same range applies to the
assuming that the
on theoretical grounds
Experimental values of Jw for Teflon in
electrodes than for metal ones.
contact with
to re-
Some justification for this is available
length than the limiting form.
that
5.2, it is necessary
5.12 because it leads to a longer development
tain the general form Eq.
in the literature
of Table
present experiments, it is
encouraging to compare this with the observed average current density
<Jx(0,z)>
= II(0,t) _
'J,(B~z))
This expression
I(Lt)
2aL
is valid
only if surface
conduction is
excluded from
I(0,t) and I(L,t), so these currents must be taken from Table 5.1 rather
With the difference in currents
than Table 5.3.
current density
average
x 10- 8 (A/m2 ),
is 2.0
typically
.1 (nA) the
which lies
within the
given range of exchange current densities.
Second, steady state conditions prevail in the conducting tube, so it is
reasonable
contrast,
to
regard
pw
the accumulating
and
J
as
surface
fixed
material
charge and
parameters.
the evolving
In
electric
fields imply changing conditions at the interface between the liquid and
To make it plausible that
fixed
the accumulating
these parameters remain
the insulating tube.
in the face of
show that
only a
small fraction of
surface
charge
the available
it is necessary
to
interfacial surface
144
area may be occupied. If only one ionic species accounts for the surface
..--char-ge, the minimum area occupied per (monovalent) ion is
e
1.6 x
19
4
~
£w(E:rmax
where the
2
10
A
(2.3 x 10-11)(5.0 x 107)
maximum electric stress
is taken to
dielectric strength of the insulating
enough compared with
be of the order
of the
wall. This area is probably large
the actual area of the
surface in contact with an
ion to justify the use of Eq. 5.12 with fixed parameter values. However,
if the
same net
surface charge arises
surface concentrations of
from a small difference
two ionic species then the
area occupied per
ion is
smaller. Moreover,
parameters characterizing the
depend
on
electric
the
evolving
fields
or
the
in the
interface may
evolving
surface
conductivity, so Eq. 5.12 is used with less confidence in the insulating
tubes than in the conducting ones.
Solution for the Volume Charge Distribution:
To apply the boundary con-
dition Eq. 5.12, an expression for the normal current density within the
sublayer is needed. From Eq. 5.4b
Jx(xz)
= 1Ex-
(5.13)
Dma
Substituting Eq. 2.31 for -Ex yields
Jx(X,z) =
In Sec.
-
+
5.3.2 the axial
with the first term on
5.14 yields
z -
field term is shown to
(5.14)
be negligible compared
the right, so that with Eq. 5.10 substituted Eq.
145
jxfZ)
(5.15)
5, 2(-M + OP(OZ)
where
2
OL =
'CIi +
axDM
or in the limit 8/X m
8
<
1
iiI =1 el
+ 2X2
aj
M =
13 T 9 Ias
-
Eliminate J(0,z)
p(0,z)
=
a~2Dm
cothaS/Xm
X
(5.16)
J
between Eqs. 5.12 and 5.15 and obtain
3J(%
/!pl) + (a/2)(z)
(5.17)
Jw/iPw + a/2
Substitute Eqs. 5.10 and 5.17
into Eq. 5.11, and insert the result into
Eq. 5.6 to yield
uS = (-aL+ sp)
This has
+ 2JX
(5.18)
the solution
5(z) =
(0)exp(-z/d) +
(0)(1 - exp(-z/d))
(5.19)
where
8
MpW
Note
d =
U
that the development
aBI.
(5.20)
+ -2J
.r[la+lp.!
length d
in Eq. 5.20
is greater
than that
given by Eq. 2.15b by the factor in parentheses which tends to unity in
the limit of large Jw.
~~~~
I~~ ~ ·
Also
the fully developed charge densityp(M)
is
----
---
·----.
146
the same as that given by Eq. 2.11 in the limit 8 << Xm, consistent with
the fact that the boundary condition on P(x,z) (Eq. 5.12) is independent
of Jw in the fully developed region.
In case B
of Sec. 3.2.5,
the volume
imposed if the inlet value
at least
addition
for the close
to regarding
transverse
distribution
as
and in Chapter 6 indicate that this
fitting sleeve configuration.
(0) as
imposed, parameters
boundary condition
(Pw and
pendent of the surface charge
J)
Here, in
characterizing the
are also
considered inde-
transient, as discussed above. The volume
charge distribution of
Eqs. 5.19 and 5.20 plays
by Eq.
a surface charge transient that
3.9 by driving
is regarded
(0) is independent of the generated fields.
Experiments discussed in Sec. 3.3.2
is so,
charge
the role of that given
is described in
the following sections by an equation analogous to Eq. 3.10.
5.3.2 Axial Electric Field
As in
Sec. 3.2.3 for
the capped cylinder configuration,
here for the close fitting
the objective
sleeve is a relation between the axial elec-
tric field within the liquid
and the surface charge distribution at the
liquid-solid interface.
first application
justify neglect of
Sec.
5.3.3 where
The
the axial field term in
the axial
field enters
of this relation
Eq. 5.14.
into an
is to
The second is in
equation expressing
conservation of surface charge.
A
quasi-one-dimensional approximation
(14, Sec.
electric field in
the wall is motivated by
the tube
proximity of the
and the
form of Gauss' law applied
4.12) for
the radial
the extreme aspect ratio of
equipotential sleeve.
The integral
to the surface S2 in Fig. 5.2 yields for the
zero order field in the wall
Ew(z,
t)
p(z)
t) +
rN
SW _
)IM + a(z,
i
The first term represents the
.21
(5.21)
field of the fixed charge associated with
147
arise from
flow.
As
second and third terms
layer (Eq. 2.17), while the
the original double
the surface and
implied by
volume charge distributions induced
Eq.
5.21
and
illustrated in
Fig.
by the
5.8, E
is
essentially independent of radial position. This is a consequence of the
small wall thickness
(w) compared with the radius
of curvature (a) and
the absence of space charge within the wall.
The first order axial electric field component follows from the requirement that the field be curl-free within the tube wall:
w
3Ew
8~~Ez
%r
~
az
The integration
Ez
(r-
the outer surface of the
to
significantly
contribute
dielectric strength of
electrical
is the
the equi-
wall (Fig. 5.8). Neither
nor the space charge
(z)
approaching
the
stresses
the wall, so for most
surface charge density
the
consistent with
layer charge (see Eq. 2.17)
the fixed double
(5.22)
(a + w))
adjusted to be
constant is
potential surface at
2 88~
E(5.22)
~Er~~i
=_~
of the charging transient
dominant term on
in Eq.
the right
5.21. Then combining Eqs. 5.21 and 5.22 yields
Ez n
-w aj
·
r = a
for the axial field at
in Fig.
(5.23)
the interface.
interface
5.8 reflects both
and the
symmetry
the continuity of this
condition on
approximate uniformity of Ez in
field
must be
relation
curl-free in the
analogous to
The axial field profile sketched
Eq. 5.22
the
component across the
axis of
the tube.
The
the liquid is justified as follows. The
liquid as
applies
well as the
with E
in wall,
replaced by
so a
Si, the
radial field in the liquid. Then the fractional change in Ez between the
tube axis and the interface is approximately
148
a
Ez(a)
8Ez
a Er
ar
w Ew
r
where Eq. 5.22 was used with r = a. The ratio of lengths on the right is
about five,
while the ratio of
fields is small as soon
charge density dominates the right
side of Eq. 5.21.
as the surface
Thus, for most of
the transient, the axial field is essentially uniform in the liquid.
To justify neglect
of the axial field term in
Eq. 5.14 it is enough to
compare
P
el
,
I(L,-)
-
x 10 11
5.
>
a2Ue
(10-5)(2.1
=
2.0 x 10
11 )
X 1
with
2w
a Er
BEz(a,z,t)
az
-
A typical velocity
az
w
<
L
7
2(10 )
5.0 x 10
=
of U = 2.0 is used, and
to get a conservative test,
w
the minimum I(L,0) from Tatle 5.1 and a value for E
that approaches the
dielectric strength are used.
5.3.3 Conservation and Relaxation of Surface Charge
Exploiting the uniformity of the
axial field in the liquid, an equation
expressing conservation of surface charge
is obtained from Eqs. 3.7 and
3.8:
Bt
-Jw
r
1 aI
2na
z
(5.24a)
where
I - a 2 (UP + cE2)
,e -
+
s)
(5.24b)
I -1
-
-~
r
149
The additional term
Jw is the radial current
density in the insulating
wall evaluated at the liquid-solid interface. The time rate of change of
the volume charge density is neglected in Eq. 5.24a because that term is
small compared with the convection term retained as long as p(0) changes
slowly on the scale of the liquid residence time L/U.
As discussed at the beginning
ohmic conductivity
of Sec. 5.3, the tube wall is assigned an
w so that using Eq. 5.21
w,r r =awv(az)
'w
;
xw
(5.25)
w
Substituting Eqs. 5.23 and 5.25 into Eq. 5.24 yields
aT
at
which
C+
y
W
8
2
=
2EwJaz2
is the counterpart
surface charge relaxation
the
Sec. 3.2.4
conductivity in
(5.26)
of Eq. 3.10.
Surface Charge Relaxation: The
in
aU ap
2 z
the
-
homogeneous form of Eq. 5.26 governs the
that ensues when the flow
objective
terms of the
is an
expression
time constant of the
is interrupted. As
for the
surface
fundamental spatial
mode. The series solution
d(z,t) = £
n(t)sin(knz)
;
k
=
5.27)
n
guarantees that the integral constraint on the axial field
Ez(z)dz
is
satisfied.
integral
=
The first
constraint
= 0
(Lt)-
is
equality
due
to
follows
the
(5.28)
from
grounded
Eq-5.23,.while
terminations
of
the
the
150
insulating
Eq. 5.27
tube. Substituting
into the
homogeneous form
of
Eq. 5.26 yields
0
dt+
n
;
°n(t)
=
(5.29a,b)
n(0)exp(-t/n)
where
(WM
1
Tn
[L
n
ew
aP_
The relaxation time
1 of
n = 1, and recalling the definition
Table 5.3. With
for the surface
Eq. 5.30 yields
as
='E
It is clear
2
1
1
w W IF1
II
to
get an
reciprocal
estimate
justifies the
of
of os
1
e (Eq. 5.24b),
(5.31)
w >
1.
Thus,
a lower
limit
on Tw is the
Table 5.3: zw > 70.0 minutes, corresponding to a
compared with
smaller values
of
conductivity
of o w < 5.5 x 10 -
conductivity
constants listed in
a
from Eq. 5.30 that
maximum value of r1 in
bulk
the fundamental spatial mode is evidently the
corresponds to the observed time
longest and hence
w
(5.30)
L2
from
without
(S/m). Now Eq. 5.31 can be applied
1/X 1.
This
specifying
by neglecting
estimate becomes
Table 5.3 are used.
assumption implicit in
w,
better as
This upper
Eq. 5.24 that the
its
limit on
the
w
wall carries a
negligible fraction of the axial conduction current.
5.3.4 Charging Transient and Electrical Stress
To facilitate application of the
of Eq. 5.26
and 5.19:
initial condition, the right hand side
is written in the form of
a Fourier series using Eqs. 3.29
151
-2U
=
2 3zn
Ansin(knz)
nn
k
kn
(5.32)
L
where
An-
-aUn
2d ()
-
with d (which also enters
5.27
and
5.32
(5.33)
(0)]en
into
Eq.
en ) now given by Eq. 5.20.
5.26
and
solving
the
Substituting Eqs.
resulting
ordinary
differential equation subject to the initial condition c(z,0) = 0 yields
a
=
F
[nAsin(kn (1
z )
- exp(-t/Tn))]
(5.34)
-
(5.35)
n odd
The electrical stress
Er =
[(TnAn/w)sin(knz)(1
exp(-t/n))]
n odd
follows from Eqs. 5.21 and 5.34.
5.3.5 Terminal Currents
Combining Eqs. 5.23 and 5.24b gives for the net axial current
I(z,t)
= a2 (
(z)U - We a )
-W
3
(5.36)
I
and with Eq. 5.34 inserted
I(z,t)
= ra2 {(z)U
_
e
w
n
nAnkncos(knz)
i
n(5.3
odd
(1 - exp(-t/An)
]
(5.37)
where
(z) is given by Eq. 5.19. The terminal currents follow as
152
I~~~
I(O) = na2 (a()U
- v - cWE
e
w
Irtak
n n- - exp(-t/ n))j
nAnkn(l
-
n odd
(5.38)
and
2
I(L) == .a
na2{PL)U
I(L)
~(5)U ++ -~w
£E
nodd
nAnkn(l - exp(-t/n)])
(5.39;
)
[
Finally, the currents
obtained from
drawn from the individual sleeve
Eqs. 5.1 and 5.37.
given by Eq.
5.37 in the steady state
unless Tw is finite. Thus,
state axial
It can be shown that
segments can be
the net current
is independent of axial position
the experimental observation that the steady
derivative aI/az(z,m)
is nonzero (Fig. 5.6)
justifies the
allowance for a nonzero wall conductivity in the model.
5.4
Comments on the Model
5.4.1 The Surface Conductivity
As discussed in Sec. 5.3.3,
the surface conductivities can be estimated
from
2
Os ~ wn
,
With
w
_V
2.6%,
2
w
3.0 x 10-4 and
other parameters as given in Table
5.3, the range of surface conductivity in (S) is
1.5 x 10-12
<
<
6
<
(av,/2)
5.1 x
1
1
while
9.1 x 1l
< 7.
x 10
14
I
.
.
, i ,I
153
indicating that
bulk conduction
does not compete with
surface conduc-
tion. Further evidence is available in Table 5.3 where the trend towards
shorter time
constants continues regardless of the
direction of change
in bulk conductivity.
In the steady state Eq. 5.37 yields
a I
a2j
"e
U
Ancosk
E
32
8Z2
nZ]
n odd
The convection term on the right has an exponential z-dependence, and by
itself cannot
account for
conduction term
the inflection point
included the
cannot be specified,
right hand side
in Fig. 5.6.
does have a
With the
zero, which
however, without information about the interfacial
parameters pw and J,
Depending on
one
may or
one's picture of
may not expect
the origin of the
it to
surface conductivity,
be uniform over
the entire
tube as
assumed in the analysis. It is not expected to be uniform if the rate of
evolution
is dependent
on
the
local normal
current
density at
the
interface. However, the fact that a steady state is reached at all (Fig.
5.5) suggests that the surface conductivity is fixed for the duration of
an experiment.
Additional evidence that
time scale is provided in Chapter 6.
of evolution
must be
much longer
If this is the case, then the rate
independent of the
vanishes while the liquid
it evolves over a
normal current
density which
is left stationary between experiments. Thus,
the surface
conductivity appears
exposure of
interface to
to originate simply in
the doped liquid,
the long-term
and the assumption
of its
uniformity is then reasonable.
5.4.2 The Continuum Approximation
In describing
terms of a
the charge distribution within the
diffusion sublayer in
smoothed volume density (Sec. 5.3.1),
there is the implicit
154
assumption
occur
that the
dimensions-over_ which
compared with
are large
the
variations in
mean separation
the density
between ions.
To
examine this assumption, the mean ion separation
i =
'e
1 3
(eb/j] l/
3
should be compared
with the smaller of the
Debye length and the diffu-
sion sublayer thickness. For simplicity the former length is used:
(EDml
1/2
m
l at
To justify
less than
i/Xm must be
treatment the ratio
the continuum
unity. Using the Einstein relation this ratio is
s = (e) /3(sEvt)-1/2(o /bj1/6
With
e = 1.6 x
(C), e
10
ion
mobility
of
mobility
Table
and
range
of
3.0 x
- 8
of
the
For
parameters.
Debye-Huckel
theory
criterion is usually expressed in
of
the
range
) and the conductivity
10
the approximation appears justified.
validity
atmosphere, the same
conductivity as
liquid
3.1 (b
interest (1-100 pS/m)
the
11 (F/m) and a room temperature
1
0.027 (V), this ratio is tabulated in Table 5.4
thermal voltage of v t
with
= 2.1
of
In defining
the
ionic
terms of solute
concentration rather than bulk conductivity (56).
5.5
Engineering Implications
There are
two issues
to disuss: first,
how would the analysis
of the
capped cylinder configuration in Sec. 3.2 be revised to reflect the normal diffusion current, and second, when can that current be neglected in
comparison with the migration current.
__
I___
155
TABLE 5.4
Ratio of Mean Ion Separation to Debye Lenqth
b (m 2/V-S)
ao
1.0
(S/m)
-
-
1.0 x 1
x 110
1.0 x 1
-9
1.0 x 10 - 8
1.0
x 1
-
- 12
0.33
0.23
0.15
0.11
3.2
x 1
12
0.40
0.27
0.19
0.13
1.0
x 10- 11
0.49
0.33
0.23
0.15
3.2 x 10-11
0.59
0.40
0.27
0.19
1.0 x 110
0.72
0.49
0.33
0.23.
3.2 xx 10
1.0
19
0.87
0.59
0.40
0.27.
3.2 xx l
1.9
1- - 91
1.1
0.72
0.49
0.33
3.2 x 1-9
1.3
0.87
0.59
0 .40
1.0
1.5
1.1
0.72
0.49
x 1-
-7
156
5.5.1 Charging Transient with Capped Cylinder
The counterpart of the master equation (Eq. 3.10) of the migration model
is obtained by substituting Eq. 5.19 into Eq. 3.8 with the result
where
=ia 2
Ut-
z
at + (na2 ce
a(a,z,t)
(a) and d are given
=
a
by Eq. 5.20.
3.2 can be generalized to include
p(,t)
(p(,t)
-
(w))
- P(0)Jexp(-z/d)
((0t)
Thus,
all of the results
(5.40)
of Sec.
diffusion by making the replacements
;
rU
-
d
(5.41)
In particular, the revised steady state result (Eq. 3.40)
~n=
where
enU
aedBn(R
,a)kn
2 (°(,x)
en is
3.2.5,
given by Eq. 3.30,
- p-
however,
generated fields
the
(5.42)
is called
assumption
is now more
that
for
in Sec. 7.1.1. As
p(z,t) is
independent
difficult to justify than in
in Sec.
of
the
the case of
the close fitting sleeve.
5.5.2 Range of Validity of the Migration Model
Conditions under which Eq. 5.40
reduces to Eq. 3.10 define the range of
validity of the migration model of Chapter 3. These conditions are
p(-)
<<
(O,t)
8 << xm
2Xi/a8
Dm1P,1/s
(5.43)
(5.44)
<
1
<(
(5.45)
Jw
(5.46)
157
ensures that most of the entrained charge
The first of these conditions
originates upstream
are just
of the insulating tube.
those for
U, where
which d
The
last three conditions
by Eq. 5.20.
The
the model of Chapter 3.
The
d is given
inequality of Eq.
5.44 is also assumed in
inequality of Eq.
5.45 implies that the diffusion
the charge relaxation time ().
compared with
time (aS/Dm) is long
Finally, the inequality
of Eq. 5.46 ensures that charge transport through the liquid bulk is the
rate
limiting
experiments
step in
spatial
reported
the
in
5.2,
Sec.
development.
Of course,
of
the inequality
Eq.
in
the
5.43
is
deliberately circumvented by means of the expansion element at the inlet
to the insulating
tube.
However,
Eq. 5.45
for all of these
is satisfied
experiments, except for those involving the lowest conductivities.
situations where the normal diffusion and migra-
There are at least two
tion current densities may be comparable in an upstream metallic section
while migration dominates in a dowstream insulating section!
Equal Tube Radii and Unequal Wall Charqe Densities:
dii, the left side of
With equal tube ra-
Eq. 5.45 has the same value in both the metal and
insulating sections, and this inequality is satisfied to the same degree
If diffusion and migration are in approximate balance near the
in both.
outlet of
the upstream
metal section, then
(0,t)
is
essentially the
fully developed volume charge density given by Eq. 5.16 and 5.20:
P(0,t)
1 + a8/2Xm2
where the subscript
2
(,)
'u' associates the variable with
tion. Now, Eq. 5.43 is satisfied if (Pw)u
the
(pw)u
results of Sec.
2.2.3 and
(5.47)
1 + a8/2Xm 2
>>
the upstream sec-
Pw. This is consistent with
those of Table
-3.0(a%/b) for stainless steel while Pw
5.1 which
suggest that
0.1l(6a/b)for Tefzel.
Unequal Tube Radii and Equal Wall Charge Densities: Even if (pw)u and Pw
are comparable,
Eq. 5.43 will
still be satisfied if the
radius of the
insulating tube exceeds that of the metal tube (see Sec. 7.1.1).
158
6
Chapter
STANDING-WAVE INTERACTION WITH IMPOSED FLOW
6.1
Introduction
There are both scientific and engineering incentives for the development
On the scientific side, the
of the technique described in this chapter.
of Chapter
model
5 remains
for lack
tentative
physicochemical
Thus, an independent experi-
description of the liquid-solid interface.
of the surface conduction and the boundary
mental approach to the study
condition on the volume charge
From the engineering
would be of value.
program for
a comprehensive
standpoint,
of a
controlling
the flow-induced
electrical stresses should include real-time monitoring of the processes
that are precursors of a hazardous condition.
The technique described in this chapter is one for probing the processes
of interest
near the liquid-insulating solid
interface through capaci-
tive interaction with the convecting part of the charge distribution. As
Sec. 5.3.1,
analyzed in
the density of
occuring volume
the naturally
charge in the liquid varies across the diffusion sublayer.
a
variation in
where
the liquid conductivity
equilibrium prevails,
chemical
applied
normal
Thus, a
sublayer.
surface
electric
of
field
standing
the tube
by
wave
means
within the sublayer where the
convection
shifts
distribution relative
confined to a
as
an
possible for
induce additional
of potential
of an
charge
in
at the
imposed
electrode
spatial
phase
an
the
outer
structure like
of
the
to the fixed electrode structure
that
induced
charge
is a reflection
the convecting charge was
free liquid surface, the spatial
image
The extent to
conduction and the velocity profile within
in previous work (57) where
imbalance in
it
makes
liquid velocity is finite.
the
of both the competition with
the sublayer. As
and
readily characterized
induce a standing wave of perturbation charge
pictured in Fig. 6.1 will
which
to
that is
This implies
charges
induced on
phase shift is detected
segments
of the
same
159
electrodes
p
Re{Voei wcos(kz)}
x
Fig. 6.1
An electrode structure on the outer surface of the insulating
wall imposes a potential distribution there that approximates
a standing-wave distribution. Typical velocity and conductivity profiles are as shown.
9
Pig. 6.2
Experimental configuration. Numbered circles identifywires of
the helical winding. L = 0.33 (m),;..X_ .01 m), a = 1.3 (mm),
w = .3 (mm) and R = 1
.
.
..
160
electrode structure that imposes the standing wave.
Section
6.2 describes
technique, and
the
experimental device
presents experimental
for implementing
results for the magnitude
this
of the
imbalance in image charges as a function of liquid velocity and standing
wave frequency
and amplitude.
model developed in Sec. 6.3
The frequency response predicted
exhibits the same trends discernible in the
observed response. This success encourages use
two issues confronted in Chapter
surface conductivity,
the
evolving
surface
by the
and the
of the results to address
5: the time scale for evolution of the
dependence of the convection
charge.
In
addition, the
current on
sensitivity
of
the
predicted response to the convection current suggests application of the
technique to noninvasively monitor conditions in operating equipment.
6.2
Experiments
6.2.1 Arrangement
The most convenient form of
insulating tube
consists of
(numbered
is the helical
eight individual
5-8)
with
virtually grounded
surface
of the
component as
an
AC
sets up
gradient is
6.2, which
wires. Exciting four
adjacent electrodes
potential while
other
the wavelength
indicated in the
induced, which
winding illustrated in Fig.
the
a standing-wave of
tube, with
the conductivity
charge is
the external electrodes to be used with the
four
potential at
of the
the outer
spatial fundamental
figure. In the diffusion
sublayer where
finite, a standing-wave component
in turn induces
remain
image charges on two
of net
of the
unexcited electrodes (2,3). These image charges are detected as charging
currents
drawn primarily
through bridge
potential drop remains small
limit
direct
electrodes,
capacitive
the
which the
compared with the excitation amplitude. To
coupling
intervening
resistances across
between
electrodes
the
(1,4)
excited
are
and
grounded,
sensing
and
a
grounded shield encloses the winding.
With the initially liquid stationary, the charging currents are balanced
161
by bridge adjustments. Then with the flow established, the standing wave
distribution of
induced charge is displaced downstream
relative to the
sensing electrodes, and the resulting imbalance in induced image charges
is detected
as a
difference in the
recorded experimental
frequency component of
as measured
the
response is
respective charging
the magnitude u
the fundamental
the differential voltage across the resistances,
by a lock-in amplifier. Because
tube would
of
currents. The
flow-induced distension of
modify interwinding capacities
and produce
a response
that is independent of processes in the liquid, the tube and winding are
clamped between
attached to
v-grooves cut
the grounded
into plexiglas blocks which
enclosure. No response
are rigidly
is measured
when the
pressure in the tube is raised while a downstream valve blocks the flow,
indicating that
the response measured subsequently with
the valve open
can be attributed to convection effects.
6.2.2 Experimental Results
Experiments with Freon TF in turbulent flow through Tefzel tubes exhibit
reproducible
behavior.
Figure 6.3
function of time. The observation
parameters
longer
that determine
than the
demonstrated by
important
an
evolve
can
be
obscured
shown in
increases, the frequency that
response as
over time
a
scales much
as is
readily
grounded shield, this
by temperature
interwinding capacities.) Next,
the results
typical
experiment. (However,
circulating warm air against the
amplitude Vo fixed, the response
with
the
of a steady signal indicates that the
the response
duration of
feature
slightly modify
shows
variations
which
with the standing-wave
is measured as a function of frequency
Fig. 6.4.
Note that
as the
mean velocity
produces the peak response shifts towards
higher values while the
resonance broadens. Finally, Fig. 6.5 indicates
a
the
linear
response
frequencyfixed.
as
standing-wave
amplitude
is
varied
with
162
_I
I
Io l
(arbitrary
units)
I
, i I·
I
I
I
I
I
flow
flow
on
off
I
I
I
1
I
II
2
Fig. 6.3
I
[
I
X
I
4
5
1
·I
8
9
I
7
time (min) -
3
6
I
Typical temporal response.
,
20
t
20
V)ol
((Lv)
10
0
1
2
3
4
5
6
7
8
f (cps)Fig. 6.4
Frequency response,. illustrating resonances that broaden and
shift towards higher frequencies with increasing flow rate U.
OL = .400.0pm..
.
163
1I
80
I
I
I
I
I
U= 1.0
L
f = 2.25
60
IUol
(NV)
40
U = 1.5
f = 4.0
20
i
50
I.
100
I
I..
150
200
250
300
Vo (volts)Fig. 6.5
Amplitude
response.
Frequencies (f)
correspond to
peak re-
sponses in Fig. 6.4.
= Re {
V ej(t-kz)
4BR
}
Io
%JI ~
A-
r =a
Fig. 6.6
-
-
IA
-Yw
x= 0
Cross sectional view showing insulating wall and typical ve-. ..
locity and charge--^density profiles. Coordinate systems and
surfaces (a - e) are defined.
164
6.2.3 Interpretation
The observed temporal (Fig. 6.3) and amplitude (Fig. 6.5) responses have
simple interpretations that form the
response.
exert
Once it is recognized that the unexcited helical winding must
the same
effect on
the
charging process
sleeve (Fig. 3.1), it becomes
shown in
Fig.
Fig. 3.7 is in
6.3 is
Yet
process driven
the reconciliation
u o attributed
tends
distribution of
reflects only a
to support
the
With the
entrained volume
rate of change
of the
the liquid-solid interface. With the response
to convection effects,
that electrical
response shown in
both of these experiments.
by a steady
observed transient
surface charge density at
close fitting
clear that a charging transient like that
underlie models for
charge, the
as the
progress even as the steady
observed.
pictures that
charging
basis of a model for the frequency
conditions are
the result in Fig.
steady in the
6.3 indicates
liquid volume
where the
velocity is finite, even if they are unsteady at the interface.
The linear
dependence of the response on
6.5) indicates
that, at least
perturbation theory
charge in
that is
for the amplitude range
is appropriate
the diffusion sublayer
independent of
be regarded
investigated, a
in which the applied
by virtue of a
that field.
as imposed by
standing-wave amplitude (Fig.
field induces
conductivity gradient
Thus, the conductivity gradient can
the steady distribution of
the flow-induced
volume charge.
6.3
Model
Consistent
charge in
with
the interpretation
the moving liquid
in
the
previous section,
is regarded as having
volume
two components: the
background charge induced by the flow (Sec. 5.3.1), and the perturbation
charge induced by
the standing
and turbulent flow model are
two
steps. First,
relate
the
charge within
wave of potential.
outlined, the response u o is determined in
the chemical
equilibrium
assumption is
invoked to
distribution of
background
sublayer (Sec. 6.3.3). Second,
the radial
conductivity gradient
the diffusion
Once the assumptions
to
the
165
component of
the perturbation electric displacement
tube by the applied
induced within the
potential is determined from the electroquasistatic
field equations, and evaluated at
the point of contact between the tube
and the helical winding (Sec. 6.3.4). Gauss' law converts this result to
expressions for
and,
in turn,
the perturbation
for
the
image charges induced on
charging currents
drawn
the winding
through the
sensing
resistors.
6.3.1 Assumptions
[a]
The
flow is
turbulent and fully
turbulent flow
developed. Fully
occurs for hydrodynamic Reynolds numbers in the approximate range
Ry
2aU
(6.1)
> 4000
while fully developed flow prevails beyond about 100 tube radii from the
inlet.
[b]
The Schmidt number is much greater than unity,
Sc
C
This is
D
V
(6.2)
>> 1
characteristic of the semi-insulating liquids
of interest (see
Tables 2.1 and 3.1).
[c]
The Debye length exceeds the diffusion sublayer thickness,
(6.3)
2 >> 82
The Debye
2.9
length and diffusion
and 3.3,
respectively. As
sublayer thickness are defined
a consequence
of this
by Eqs.
inequality, the
background volume charge distribution extends into the core region where.
166
the turbulent diffusivity is significant.
[d]
turbulent diffusivity
based on the
Debye length
The
exceeds the
tube radius,
> a
t
(6.4)
diffusion time based on the
That is, the
2
diffusivity (a /Dt) is exceeded by
thus
net charge
in the core region
tube radius and the turbulent
the charge relaxation time (),
(0 < r < a
-
and
) will be essentially
uniformly distributed.
Within the diffusion
sublayer the continuum approximation is valid
(Sec. 5.4.2) and chemical
equilibrium prevails for the unperturbed dis-
[el
tribution of
ions (Sec. A.2).
Thus, the local conductivity
within the
sublayer is given by Eq. A.11:
oC(x,z) =
1 + (P(x,z)b/)
(6.5)
the assumption of equal ion mobilities, which is
Implicit in Eq. 6.5 is
reasonable for
2
the antistatic
additive of interest (Sec.
3.3.1). Also
implicit is a condition on the strength of the applied fields (Sec. A.4,
Eq.
A.37c).-As
discussed in Sec.
6.2.3, experimental
results suggest
that the conductivity gradient is independent of the applied field. This
implies
a negligible
contribution to the
local conductivity
from the
perturbation charge, so there is no need to assume that perturbations of
the background charge are consistent with chemical equilibrium. Thus, in
the necessary condition
to be the
for chemical equilibrium (Eq. A.73)
liquid transit time based on the
diffusion sublayer (Sec.
wave excitation.
is taken
velocity at the edge of the
6.3.2) rather than the period
of the standing
167
6.3.2 Turbulent Plow Model
sublayer where the conductivity gradient, and
only within the diffusion
perturbation charge, is finite.
hence the
volume flow
the
profile is necessary
of the mean axial velocity
Accurate specification
rate,
the
While correctly representing
on the
based
simplified profile
Blasius
friction-factor correlation (18)
U
V=
(0.02 R
3
/4 x/aU
(6.6a)
0 < x <
(6.6b)
wall (x << a). The viscous sublayer thickness
is most accurate near the
by (18)
is given
A
0 < r < (a - a)
= (118 Ry
/ 8 )a
7
(6.7)
The thickness of the diffusion sublayer is given by (18)
=
Thus,
a
<<
a
<<
(SC-1/
S/ 3 )
high
sublayer
(6.8)
the high Schmidt number (assumption [b]), while
in view of
for the
diffusion
= (118 Ry- 7 /8 Sci/3)a
Reynolds
(x >
),
numbers (assumption
eddy
diffusion
[a]). Outside
dominates
the
the
thermal
process, the eddy diffusivity being given by (22)
Dt = (0 01 Sc Ry 7 /8 )Dm
As in
Sec. 5.3.1, this expression for
(6.9)
Dt can be used to
show that Eq.
6.4 is generally satisfied for the experimental conditions. The velocity
at the edge of the diffusion sublayer follows from Eqs. 6.6b and 6.8 as
168
Vz(8) = (2.36
The liquid transit
Ry - 1 / 8 Sc-1/
]
3
U
(6.10)
time based on this velocity
and the length L of the
helical winding
L
=
'tS
z(
easily exceeds the
iments of
(6.11)
charge relaxation time x
Sec. 6.2.2) and
application
Scl/3 )(L/U)
= (0.42 Ryl/B
(
0.05 s) for the exper-
thus the necessary condition
(Eq. A.73) for
of Eq. 6.5 is met.
6.3.3 Conductivity Gradient
By assumption
uniformly
d], charge
outside the diffusion sublayer is essentially
distributed. Within
the diffusion
charge density is distributed according
sublayer,
the background
to Eq. 5.10. Thus, in the limit
8 << X m
where
p(x)
,
Figure
6.7
0<r
-
(z)
illustrates
represented by
the
0
+ P(0,Z)
(0,z))
velocity
and
(6. 12a)
(6.12b)
x <
charge
and 6.12. Substituting Eq.
Eqs. 6.6
(a - )
density
6.12b
profiles
into Eq. 6.5
yields
cO(x,z) 2
(z) +
(6.13a)
(z)x
where
E(z)
and
I
_
=
o 1 +
P0 o
J
)3
(6.13b)
169
Y(Z)
-
-_oP(.
z)b]2
p(,[P(0z) - (z)
o
8
I
Co
to linear terms in the
fractional change in p(x,z) across the sublayer.
To apply the boundary condition
boundary layer coordinates to
P(a,z). Two cases are
is still developing,
(6.13c)
P (0, Z)
Eq. 5.15 it is necessary to revert from
cylindrical coordinates, so that po(,z)
considered. First, suppose the convection current
in which
case
(a,z)
is expressed
in terms
of p(z)
by Eq. 5.17:
p(a,z)
where
Jw(pw/lpwl) + (aa/2)p(z)
=
.
Jw/Ipwl +
and 8 are
(a/2(6.14)
a/2
given by Eq. 5.14. In the second case, the convection
current is fully developed so
that Jr(a,z) = 0 (see Sec. 2.2.1) and now
Eqs. 2.10 and 5.20 give
p(a,z) =
= PW
>(X)
(6.15)
In the fully developed case Eq. 6.13c becomes (with 8 << Xm)
...
Oo
=
Where
b
T-O
8
-h1
0
2
2
-1
+ j
(6.16a)
as-
electrical conditions
are still
ficiently small that p(a,z)
2
T= Y0or~wb
- (-PI
do (1
If the development
developing, and
if Jw
is suf-
pw, the conductivity gradient becomes
(6.16b)
-
length d (Eq. 5.18) is
extent of the interaction region
long compared with the axial
defined by the length L of the helical
170
winding, then
the conductivity gradient
is
essentially uniform along
the axis, in addition to being independent of time (Sec.6.2.3).
6.3.4 Perturbation Analysis
To determine the response uo
(defined in Fig. 6.2), it is convenient to
divide the tube cross section into three coaxial regions, within'each of
which the perturbation potentials
and normal components of the electric
displacements are governed by equations specialized to the given region.
The outermost
'a' and
region is the
'b' in Fig. 6.6.
surface 'e' which
tube wall, with bounding
Innermost
is the core region
lies at the radius r = (a
diffusion sublayer
with bounding surfaces
wall and the core regions, respectively.
variables are
evaluated at the
surfaces denoted
- 8).
bounded by the
In between lies the
'c' and 'd' adjacent
to the
In what follows, superscripted
surface (in Fig. 6.6)
corresponding to
the superscript.
The spatial
harmonics associated with the
actual square-wave potential
distribution imposed at surface 'a'
by the electrodes do not contribute
significantly
surface 'b'
to the
interaction region)
potential at
partly because their amplitudes at
reduced
relative
because
the amplitudes
electrodes.
electrodes per
potential
,a
to
With
(and hence
the
spatial
fall off
harmonics
neglected,
and
with
one
half
partly
from the
of
the
(Fig. 6.2), the
at 'a' is
2V°
- Re(t
and
with distance
wavelength remaining virtually grounded
imposed
surface 'a' are
fundamental component,
more rapidly
within the
(2V
-exp(jit)cos(kz) - Re---
xp(jwt)
(6.17)
Implicit in this axisymmetric form is a neglect of winding pitch that is
justified
as
long
as
wavelength
of the
represents
a uniform
the
circumference
excitation.
Because
distribution,
it
of
the
tube
exceeds
the
the second
term
in Eq.
6.17
makes no
contribution to
the
171
Thus, only the first term is retained, and
potential within the liquid.
this is represented as the sum of two traveling waves!
= Re( V(exp(jwt-jkz)
Because the potentials are
to these
+ exp(iwt+jkz))
(6.18)
described by linear equations, the responses
two traveling-wave
components can be-found
independently and
then superposed. The response to the forward traveling wave
0a = Re($aexp(jwt-jkz))
is
sought first.
;
3a
are
Since they
= Vo/
(6.19)
solutions to
linear equations,
the
perturbation displacements and potentials assume the same traveling-wave
form. The present objective is
surface 'a'
to express the normal displacment at the
explicitly in terms
of that part of
the imposed potential
given by Eq. 6.19.
In
the insulating
tube wall,
the potential
is governed
by Laplace's
equation. With the wall thickness small compared to the tube radius, the
region
can be
amplitudes of
regarded
as
electrically planar
the perturbation potentials ($)
(Dx = -ca8/ax) evaluated at the boundaries
so
that the
complex
and normal displacements
of the region are related by
(14, Sec. 2.16)
[
j
coth(kw)
Dxb
sch(kw)
Within the liquid
-csch(kw)
a
-coth(kw)J
4b
core there is no conductivity
perturbation
charge.
governed
Laplace's equation,
by
Thus, the
(6.20)
gradient, and hence no
perturbation potential
and
perturbation
there
is also
amplitudes at
the
172
boundary of the core region are related by the cylindrical harmonic (14,
Sec. 2.16)
De = -De
D
where In
6
=
Iki(k
k a)
is the nth
=
(ka)
ekk I6.21)
e
(6.21)
order modified Bessel function of
the first kind,
and the prime denotes differentiation with respect to the argument.
Although Laplace's
those
obtained
equation no
for
diffusion sublayer.
spatial
the wall
longer applies, relations
and
core
regions
are sought
Axial diffusion of perturbation
variations on
the scale
of
analogous to
the wavelength
for
the
charge arises from
(x) and
hence is
characterized by the a diffusion time (X2/Dm) that is long enough on the
scale of
neglected.
the charge
It is
relaxation time ()
left to
the
that axial diffusion
Sec. 6.3.5
to show
that
can be
diffusion of
perturbation charge in the direction normal to the phase boundary can be
neglected as well.
Then using Poisson's equation to
convert Eq. A.9 to
an equation for the perturbation potential yields
Eat
which
+ (vz) z + 1]v2o + v .v
is linear
applied field
in 0
because
(Sec. 6.2.3).
Eq. 6.19), and
o, is
g
(6.22)
regarded as
independent
Substituting the traveling-wave
again regarding the layer as
of the
form (see
electrically planar yields
an ordinary differential equation for the complex amplitude:
(j¢Q(
where D
2
1o)D
$ + DDO 2 = 0
- kvz ) +
8/ax. Neglect
as long as the wavelength
of the term k 2
(6.23)
compared
with D2 $ is justified
X is much greater than the diffusion sublayer
thickness 8, and leads to a first order equation for the normal electric
173
displacement. Substituting Eqs. 6.6b
that D
and 6.13a into Eq. 6.23 and noting
= -Dx/¢E yields
+ DX(A + Bx- 1
DD
= 0
(6.24a)
where
+ j
A -
and
0.02
.
Z and
;
are
Y
B _ 1 - j
kUR
3 4
a
/
J
(6.24b)
defined by Eq. 6:.3. _Equation 6.24 is
separable and
readily integrated to yield
Dc(l + Sx/A)-'/ "
Dx(x)
(6.25)
Evaluated at x = 8, Eq. 6.25 yields the first of two relations that play
the role for the diffusion sublayer that Eqs. 6.20 play for the wall:
Dd = D(1
It
can be
+ B/A
-
/
B
shown that the
(6.26)
differentiation involving
complex constants
needed to verify that Eq. 6.25 satisfies Eq. 6.24a can proceed as though
the constants were real (58).
To complete the electrical description of
the diffusion sublayer, the relation
$d
is added
$c
(6.27)
which recognizes that
small fractional
change in
possible. Together
complete
which
(X >> 8) only a
potential across the diffusion
with Eqs.
specification of
must be
in the long wave limit
6.20 and 6.21,
the bulk
supplemented by
is
these last
two equations
for the
three regions,
conditions at
the interfaces
equations
boundary
aublay
174
between regions. At the d-e interface
Ad
e
DX
(6.28)
DX
of the potential as required by
The first condition expesses continuity
second reflects the absence of
Paraday's law, the
a charge singularity
at the d-e interface. Next, the boundary conditions at the b-c interface
= $b
(6.29a)
and
c
a =
-
Ab
(6.29b)
_3(OE
z ) - -_
express, respectively,
_k 22af
iwXb,
x
-
1
+
continuity of the potential
perturbation surface charge. Solving
j
and conservation of
Eqs. 6.20, 6.21, and 6.26-6.29 for
the complex amplitude of the normal displacement at surface 'a' in terms
of the potential at the same surface yields
D
=
(6.30)
T+$a
where
_(ewk ) 2 csch2(kw)
+
T+ = (wk)coth(kw) ++
k)coth(kw)
P+ + (wk)coth(kw)
and
I 1 + jp
P+
jwrtJ
'eskIl(ka),
f
J
I0 (ka)
vll/B
k2as
+ AJk
The response to the backward traveling-wave component in Eq. 6.18 can be
found
from Eq. 6.30
simply
amounts to replacing B by
by
making the
replacement k .- k,
which
its complex conjugate. Denoting the resulting
175
superposing the two responses, and recalling Eq. 6.19
coefficient by T,
to
express
a in
terms of
amplitude gives
the standing wave
for the
normal component of the electric displacement at surface 'a'
Dx = Re
between the wire electrodes is
Because the spacing
from the
their distance
intercept
+ T_exp(+jkz)]exP(iut)
-(T+exp(-jkz)
all
of
(6.31)
small compared with
grounded enclosure (Fig. 6.2),
flux
the normal
at
the electrodes
'a.' Thus,
surface
wire
the
electrodes can be regarded as flat, as pictured in Fig. 6.1, in applying
Gauss' law to express the charging currents defined in Fig. 6.2:
In(
aD
at d(kz)
i2 =
5n/4
i3 = j
(6.32a)
+ id(t)
aD
3
at d(kz) + id(t)(6.32b)
where because
current id(t) associated
of the balancing procedure, the
with any direct coupling between
the excited and the sensing electrodes
is identical for each of the latter. Finally, upon substituting Eq. 6.31
into Eq. 6.32, and carrying out the indicated integrations, the response
(in volts) follows as
Uo =
100
=
i3 1
i22 -_ i3
Rli
Once it
is recognized,
*
Eqs.
e.g.
6.20
quasistatic limit),
and
=
(2
IT - T_IV
IT+
- T_IV
-V2) wR
k
or by physical argument,
either by inspection
6.21
for
apply
including zero
(6.33)
any
frequency
frequency in which case
electric displacements arising from an
(within
the
the normal
applied potential of the:form of
Eq. 6.19 are clearly invariant under the replacement k b -k.
176
that the coefficients
k,
it is
uO = 0
shown that
easily
when U = 0,
flow. A
of
in Eqs. 6.20
and 6.21 are even functions of
P+ = P_ when
U = 0. It quickly
indicating that
computer program
the response
performed the complex
follows
that
is generated
by the
algebra called
for in
Eq. 6.33, and yielded the theoretical frequency response curves shown in
fixed and U as a parameter, and in Fig. 6.8 with U fixed
Fig. 6.7 with
and
as
a
parameter
(59). For
these
figures,
do
is
taken
to
be
negligible, while the conductivity gradient is essentially that given by
Eq. 6.16b.
6.3.5 Diffusion of Perturbation Charge
Neglect
of diffusion
showing that
than
in Eq. 6.22
the magnitude of
that of
the conduction
equates the time
is justified
here in
the omitted term is
term retained.
retrospect by
typically much less
The equation
in question
local net perturbation charge
rate of decrease of the
density to the divergence of the perturbation current density
= PVzj
z
with
v
-
- DmVp
(6.34)
representing the axial
4z
Eqs. 6.22
right do
where
and 6.23, the axial
unit vector. As noted in connection with
components of the last two
not contribute significantly
appropriate to focus on the
terms on the
to the divergence, and
so it is
normal component of J which has the complex
amplitude
i
C
- kx
Dr
Here, the potential
nd the
D]]X
F
Dm
DX
(6.35)
charge density have been expressed in terms
of the
displacement, the
first as in
Eq. 6.24a, and the
Gauss'
law.
approximate
equality recognizes
derivatives
excitation
The
second
must be
characterized by lengths
wavelength- in
order
to
be
second using
that
scales shorter
significant.
To
space
than.the
assess
the
177
-I
6U
20
t
I Uol
(MV)
10
0
5
20
15
10
f (cps)
-
Theoretical frequency response. Entrained volume charge density is fixed while flow rate varies.
Fig. 6.7
.,
.O
I
I
p =4
T
I
10-4
/.5 I0
20
'
4
U = 0.5
I Uol
(/.V)
AISxI
3
x 10-4
4
10
0
5
10
15
20,
f (cps)Fig. 6.8 Theoretical frequency response.
trained
volume charge,
and hence
Flow rate is fixed while.enconvectioncurrent, varies.
178
diffusion, Eq. 6.25 is substituted into
importance of
ratio
of
the
magnitude
of
the
Eq. 6.35 and the
to
diffusion current
that
of
the
conduction current is formed!
211
This
2A,
I
(6.36)
has a maximum
M =
ll
M at x = 0 given
by
(6.37)
4.
AR
JA21
Substituting Eqs. 6.13b and 6.24b yields
M
=2
II
where the approximation
is of the
i
l 12
+ j
(6.38)
j
doo used is justified as
Z
order of or less than
unity
(see Eq. 2.12).
4
Reynolds numbers of the order of 183 to 1
2.4Ry- 1/ 8 U
0.02Ry3 / 4 U
__
_8 _7_
118Ry7/ 8 a
a
Eq. 6.7 was
where
long as P(0,z)b/% o
For hydrodynamic
,
U
A(6.39)
_c
used. Furthermore,
since
an upper
M is
bound for
sought, w-r can be neglected
in the denominator in Eq. 6.38. Thus, with
Eq. 6.39 substituted and with
as given in Eq. 6.13c, Eq. 6.38 becomes
M
[](4
0.
52
j (+
[b/io)
°
,
,
'
:
(6.40)
179
Eq. 6.40
leading factor on the right in
Eqs. 2.12 and 6.3, the
Now-according to
is much
less
than unity,
and
thus to
determine whether
M
approaches unity, it is enough to examine the magnitude of M' where
Mb4
< ,
In
view of
2
91k
x2
(6.41)
for the
6.8) and
(Eq.
Schmidt number
high
the
(U
present
experimental conditions,
8 <
TkU
;
Then since
= 0(1)
the leading factor
order of or less than unity
the
correction
on the far
that would
(see for example
be
introduced
right in Eq. 6.41 is
Table
by
5.1),
of the
so is M'. Thus,
including diffusion
in
Eq. 6.22 would be small.
6.4
Discussion
The temporal
illustrated in Figs. 6.3
and amplitude responses
perhaps most valuable because they
respectively are
and 6.5
can be interpreted
without reference to the model, and because they tend to support some of
the assumptions invoked Chapters 3 and 5. Without a free liquid surface,
the observation of
clear indication
bulk. The
liquid
(see Sec. 6.2.1) is
a response to liquid convection
that the technique
steady
temporal
in the
is sensitive to conditions
response and
the linear
amplitude
response reveal three aspects of the phenomena initiated the flow:
(1) As long
diffusion
specific
steady
sublayer, the
distribution
assumed to prevail
equilibrium can be
as chemical
electrically-induced charge
of flow-induced
temporal-response
(Fig. 6.3)
net
reveals
volume
in the
corresponds
to a
charge. Thus,
the
this distribution
to be_.
180
imposed,
at least
for sleeve
configuration,
in
the sense
assumed
in
Secs. 3.2.5 and 5.3.1. The linear amplitude response (Fig. 6.5) confirms
that
at least
for the
charge distribution
modest applied voltages
is determined
by the flow
used here,
the volume
conditions and
not the
external field.
(2) Regardless of the electrical description of conditions in the liquid
bulk, the response (see Eq. 6.30) is attenuated by a significant surface
conductivity
at the
liquid-solid interface (surface
because it will tend to
Thus,
if a
'c' in
Fig. 6.6)
exclude the applied field from the liquid bulk.
significant
surface
conductivity is
present, the
steady
response of Fig. 6.3 indicates that it does not evolve on the time scale
of the experiment (consistent with assumption in Sec. 5.3).
(3) Though not
the
independent of (1),
steady temporal
illustrated
it is worth emphasizing
response overlaps a
in Fig. 3.7, the
latter
that since
charging transient
is attributable
entirely
like that
to surface
charge accumulation.
Two factors
prevent a quantitative comparison
(Fig. 6.4)
and theoretical (Fig. 6.7)
experimental Debye length
the same
order
flow
in conflict
rates
conductivity that
value. Second,
determine the
for
with a condition
First, the
indicated
in
is an order
of the theory
8 are of
(Eq. 6.3).
To
the model (Eq. 6.33) is applied with
Fig.
6.7
and
a
value
of magnitude less than
the influent convection
current I(0)
background charge density
of
the
bulk
the experimental
= Q(0)
was not recorded.
needed to
The value
indicated in Fig. 6.7 is based on measurements under corresponding
experimental
parameter
conditions (see
values to
and experimental (Fig.
exhibit
frequencies
6.33
and
5). Despite
that differ
the use
of
from experimental
correspondence between the theoretical (Fig. 6.7)
6.4) frequency responses is clearly discernible.
a resonance
with
Chapters 3
evaluate Eq.
values, a qualitative
Both
frequency responses.
Xm and diffusion sublayer thickness
be consistent with this condition,
the
between the experimental
which
increasing
broadens
superficial
and
shifts towards
velocity.
higher
Furthermore,
the
181
theoretical results
indicate resonant frequencies in the
those observed experimentally, and peak
same range as
amplitudes of the same order of
magnitude.
This qualitative correspondence
encouraged an examination of the depen-
dence on convection current at fixed flow velocity. Figure 6.8 indicates
that
while
unchanged,
square of
the
shape
the amplitude
the convection
quadratic relation is
it appears
the
and
of
the
on
the
frequency
response varies
current at a
interaction could
given flow rate.
a wide range of
provide
the
axis
remain
essentially as
not evident in the expressions
to hold for
convection current
position
the
Although this
for the response,
hypothetical conditions. Thus,
basis for
sensitive detection
levels downstream of current
of
generators or upstream
of insulating elements that tend to collect charge.
The
standing-wave technique
natural processes
only a
is
useful
as a
means
for studying
the
in the volume because the
induced charge constitutes
perturbing influence on the original
charge distribution within
the sublayer. Thus,
those that can
account
be generated by the flow
for
interface
the applied fields are typically
from
charge
one
injection
bulk
region
injection process, much higher
electrode structure
much smaller than
across
and which would be expected to
to the
the
liquid-insulating
other.
To
solid
investigate
the
potentials are applied when the external
is exploited
again in Appendix
B. There,
to help
distinguish injected charge from that induced by the flow, the liquid is
initially stationary, and then it is the mechanical response elicited by
an imposed traveling-wave of potential that is of interest.
182
Chapter
7
ENGINEERING IMPLICATIONS
By confirming
expected trends of
charging transient
3,
4 and 5
generation
1)
that
(see Sec. 2.1) the experimental
tend to
field generation
the time (
support the picture
in the
and accumulation
results of Chapters
of the
insulating tube wall
processes as
characterizes the
flow-induced electric
as the result
of charge
are opposed
by charge
they
leakage. While the clearest support for the basic ideas comes from study
of
the transient,
design issues
are
best addressed
in terms
of the
ultimate electrical stress
which results when these competing processes
reach a balance. Thus, the
objectives of Sec. 7.1.1 are to sort out the
dependencies of
stress,
on
these processes,
system parameters,
and hence of the
configurations other
and
to
clarify
than the capped cylinder.
ultimate electrical
the implications
for
Of particular interest,
is the dependence on parameters that can be controlled, namely, the bulk
and surface conductivities, the tube radius, and the influent convection
current. In
Sec. 7.1.2 the
conclusions of Sec. 7.1.1
are consolidated
and phrased in the form of design suggestions.
7.1.1 Dependence of Ultimate Electrical Stress on System Parameters
For the capped
cylinder configuration, the steady state
(see Fig. 3.3) is given
in the form of a superposition of spatial modes
by the combination of Eqs.
amplitudes
(see Eqs.
magnitude of
wall stress EW
3.22, 3.23 and 5.42. Examination of the mode
3.19, 3.26,
the stress
3.30
and 3.32)
is essentially that
indicates that
of the
the
fundamental mode.
Thus, for discussion purposes, only the first term of the series need be
retained,
and this is evaluated
EN "
Er(L/2,co)
t
r
at z = L/2 to yield
eLUC(R,a)j
SOeC(R*a)
_
SW edB1 (R,a)kl
(q(0,=)
the maximum
stress:
((7.1)
The usefulness of this result is hampered by the inclusion of phenomeno-
183
logical parameters (the wall charge
density
refers
J
enter p(m)
which
to a
radius R). In
specific
and d,
Eq. 5.20)
external conductor
to the stress
and a
parameter which
configuration (the
what follows these parameters are
upper limit
for an
density pw and the exchange current
cylinder
eliminated in exchange
useful provided
that will be remain
that limit is not too conservative. From Eq. 3.30
2
e
In the limit
(7.2)
1 + exp(-L/d)
1 + L/d)2
kla << 1 implicit
in the model
(see footnote
to Eq. 3.24)
and provided that klR is not too small, 'it can be shown that
Cl(R,a)
Bi(R,a)
-1/2
(kla/2)1n(kla/2)
(7.3)
given by Eqs. 3.17 and 3.20 respectively. While the
where B1 and C1 are
condition on klR excludes the impractical case of the sleeve (Fig. 3.1),
amounts to a << L) is fairly unrestrictive.
the condition on kla (which
The ratio
in Eq. 7.3
range 0 < kla/2 <
limit
<
is independent
l/e (
+
Jw
> xU(
are
readily
+
(7.4)
s-
into Eq. 7.1, the development length d appears
the denominator, so
corresponds to a
while
0.37). Finally, from Eqs. 5.16 and 5.20 in the
2Jw1
Once Eq. 7.2 is inserted
and
kla in the
Xm
d =t1
only in
with
of R, and decreases
a lower limit on d is appropriate. Thus, pw
eliminated
because
the
shortest
process limited by transport through
these parameters
enter only
spatial development is controlled by
when
Jw is
development
the liquid bulk,
small enough
that the
an interfacial process. In view of
the last three equations an upper limit for the stress is
184
r
+< 2/a6
2" 1so
Lw
E (L/2,w)
w.1
of
the definition
where
the coolant flow
as constrained,
tube radius (a).
the
radius (a),
from
e
liquid velocity (U)
so the
the total length of
is not independent
Hence, the design parameters remaining
and
which enters both the leading factor and kl
(m),
(O,c) and
in
and to
Eq. 7.5
order to apply
the former is sufficiently greater
determine whether
in Eq. 7.5 are
estimate the influent and fully developed volume
(- n/L). It remains to
densities,
of the
(as ) conductivities,
and surface
(c,)
the bulk
the
recalled. Because
Q is perhaps best regarded
path, the volume flow rate
perhaps the tube length (L)
charge
3.8b is
Eq.
only a small contribution to
insulating tube makes
(o
(7.5)
)
+ 2s/ac*jr(-kla/2)2n(kla/2)J
that an expansion
section (Chapter 4) will be effective.
To
Volume Charge Densities.
assumptions are made. First,
make
the
concrete,
discussion
two
the influent convection current Qp(0,*) is
fully developed near the exit of the upstream charge generating element,
and is governed by Abedian and Sonin's theory (Eq. 2.11). This envisions
an upstream element such as
that is long compared with
a winding channel
Second,
the channel
tube. This is
a heat sink with the liquid flowing through
radius is
the development length d.
that of
much less than
motivated by the expectation that
much greater in the upstream
the liquid. Then
the insulating
the liquid velocity is
element where heat is being transferred to
given that the inequality 8
<< Xm is satisfied in the
Sec. 3.2.1), it is also
satisfied in the upstream
insulating tube (see
section because at fixed flow rate
varies essentially as the square of
the tube radius (see Eq. 7.8 below). In this limit, Eq. 2.11 yields
(7.6)
P(0.a))
2
2xm J
where the subscript
'u' associates the variable with
the upstream sec-
185
tion.
For the fully developed convection current in the insulating tube
Eqs. 5.16 and 5.20 yield in the same limit
Pa.
s X(I
+
Regarding Pw and (Pw)u
(7.7)
as generally comparable and of the order given by
Eq. 2.12, it remains to determine the dependence of the quantity (as) on
tube radius at fixed flow rate. In view of Eq. 3.3
Then
,sRy7/8
1 2 ay a3v
a
as
with the
Ry 7 / 8
given
greater
typically
This result
a
'
-
assumptions
than
(7.8)
2Q
>
a
unity
Eqs.
because a/
(aS)u, and
7.6
and
7.7
Xm
indicate
of Sec. 5.5.2 by
completes the discussion
2
is
that
showing how Eq.
5.43 can be satisfied even when the metallic and insulating sections are
Thus, either of
characterized by comparable wall charge densities (pw),
the conditions (w)u
is,
>> P
au << a imply that p(0,o) >> p(s),
or
electrical stress
the generated
due to
the insulating tube.
upstream of
originates primarily
(Eq. 7.5) is
influent convection current (p(0,w)Q)
that
charge that
As long
as the
dominates in Eq. 7.5 an expansion
volume (Chaptar 4) will be effective in reducing the generated stress.
Now the upper limit Eq. 7.5 can be further clarified using Eq. 7.6:
P(0,w)
-
p()
Then taking (w)u
r
'
N
(w)u[l
+ (as)u]
(7.9)
to be of the order of od/b (Eq. 2.12) Eq. 7.5 becomes
E'w
Ew (L/2,a)
P(0.0)
E
o2Lsod
<
(1 + 2
9
6.wbl(
/aS)/(
2(7.10)
+ (a8)u/2Xm2)
1 + 2c5/acd
2 )(-kla/2)2n(kla/2).
186
l/e the right side of Eq.
With typical parameter values and with kla/2
7.10 is
0(1018d
(Cy
9
1
10
V/m), or 0(108 V/m) for the
largest bulk conductivity
0 S/m) consistent with the limit 8 << Xm under the present ex-
perimental conditions.
7.1.2 Control of Ultimate Electrical Stress
the pre-
by the development of
strategies are suggested
The following
vious section:
To the extent
Inhibit Charge Generation.
cally developed (Sec.
an
to the
supplied
current
2.2) in the flow element
tube, that
insulating
at the
tube.
element
immediately upstream of
determines
Thus, one approach
inlet to the insulating
are electri-
that conditions
the convection
to limiting
tube
would
be a modification
manner described
in Sec.
4.5.
the side
But now
of
perhaps in
the surface of the upstream element (typically a heat sink),
the
current
the convection
effects
of the
modified surface on heat transfer become a concern. An alternative is to
is to the right of the peak in the "bell"
point
is
not reflected
only the
Eq. 7.6 because
in
substance that may
be taken up by solid
(The
in Pig. 2.5.
limit
8 << Xm has
to avoid the addition of a
However, it is preferable
been considered.)
curves
increasing liquid conduc-
in convection current with
eventual decrease
tivity
that the operating
antistatic agent in sufficient concentration
use an
surfaces in the flow loop with
as yet unknown side effects.
There are two contributions to the charge that accumu-
Charge Trapping.
lates at the inner surface of the insulating wall: that which originates
upstream
of
accumulates in
the
the
Chapter 3, and
of
the absence of
first contribution
an
that
which
upstream sources (internal mode).
It is
tube
insulating
that is
(external
emphasized by
which can be diverted from
expansion
section
Conditions under which
.
.
operating
mode),
the migration
,
model of
the insulating tube by means
as a
trap
charge
the external mode is dominant
~~~~~~~~~
and
.
(Chapter
4).
are identified in
.
.
.
..
.
.~~~~~~~~~~~~~~~
187
Secs. 5.5.2
and are likely
and 7.1.1,
section
trapping expansion
charge
insulating tube, the expression
is
to be met
in practice.
inserted at
When a
inlet to
the
the
for the ultimate electrical stress (Eq.
a factor of (1 - E) where E is the trapping
7.10) must be multiplied by
efficiency given by Eq. 4.9.
Enhance Leakage. As discussed in Sec. 3.4 the external conductor configuration must be tailored to
proximity
field
to the
and
avoid "hotspots" where a conductor in close
insulating tube attenuates
the associated
controlled surface
a
conductivity,
leakage
the local
current.
axial electric
In contrast
conductivity
to the
bulk
(oC > ad,/2) at
the
inner surface of the insulating tube enhances the leakage proceess without influencing
the upstream element.
the charge generation process in
the single occurence of
This accounts for
in the denominator of Eq.
5
7.10.
Enlarge Insulating Tube Radius.
Eq. 7.10 decreases as the radius (a) of the
nificant, the right side of
insulating tube increases
are two
volume flow
rate fixed).
flow conditions,
the
increase in
in the range 0 < kla/2 < l/e (
bear in mind
issues to
Unless the surface conductivity is sig-
as the
First is the
due either to
radius is increased
possible transition
the reduction in Reynolds
diffusion sublayer
0.37).
thickness
There
(with the
to laminar
number or to
(relative to
the Debye
Second is the greater
energy stored by the accumulated charge
when the electric stress equals
the dielectric strength. That is, while
length).
the
ultimate field
tube, physical damage
is expected
to be smaller
is more likely in the
tric strength is exceeded.
in a
larger insulating
larger tube if the dielec-
188
Appendix A
CHEMICAL EQUILIBRIUM
A.1
Introduction
A theme of Secs. 1.3
and 2.2 is the existence of ionization equilibrium
in the liquid bulk where electrical conditions are fully developed (Sec.
2.2.1), and the departures from equilibrium that characterize the region
of spatial development (Sec. 2.2.2). There it is stressed that treatment
of the nonequilibrium process calls
one) number of partial differential equations
origin in the smaller (by
to
this distinction has its
constants. Section A.2 shows how
form of rate
needed
for additional specific data in the
the
describe
of
conservation
reacting
in
species
the
equilibrium case. It is this analytical simplification that provides the
incentive
strong
to
clarify
the
conditions
under
which
chemical
equilibrium may be assumed to prevail.
Classical studies of chemical
in experiments
rate processes have their empirical basis
of the
maintain homogeneous distributions
that ideally
chemical reactants. Following a disturbance, equilibrium sets in after a
a duration tcr
transient having
chemical
rate constants
convective
ionic
processes
imply
that may
in
the
laboratory frame.
fully
treatment
inhibit the local
liquid
bulk
electrical
of the
of the
rate
additional
time
electrical
processes
(diffusion
However,
analytical simplification outlined in Sec.
equilibrium
Sec. A.4
is to
avail
and
even if
viewed
argues that in the
conditions chemical
developmert
as
In
distributions of
chemical equilibration
remain stationary
scale Tcr-
terms of the
concentrations (60).
and dissimilar
Nevertheless, Sec. A.3
developed
regardless
the equilibrium
systems, the inhomogeneous
reactants
migration)
and
that can be expressed in
in
the
region of
prevails
shows that
itself of
if
the
A.2, this time scale must be
short relative to those that characterize diffusion and migration.
The condition on
time constants that justifies the
assumption of local
189
chemical equilibrium
is expressed in
ion recombination rate
that
attributes
kR. Section A.5 identifies an
when diffusion
is approached
recombination
process, and
of the
Thus, the price paid for
condition
for
is
which is
reactants.
dielectric media this upper
a
Sec. A.4 as a lower
It
the rate
limit
upper limit on kR
limiting step
independent of
is left
limit on the
to Sec.
specific
A.6
to
in the
chemical
compare
for
kl2 with the lower limit of Sec. A.4.
neglecting specific aspects of the reaction is
chemical
equilibrium
that
is
necessary
but
not
sufficient.
A.2
The Chemical Equilibrium Assumption
If only three
reactive species, two ionic and
one neutral, are present
in significant concentrations, the two simplest ionization reactions are
A + + B-
where
-
iC
;
i
=
1,2
(A.1)
i = 1 or 2 for a uni- or bimolecular
tively.
The conservation
equations that
dissociation
govern the
process,
respec-
associated charge
densities and neutral concentration attribute local time rates of change
to divergences of diffusion, migration and convection flux densities:
at -oe + G
ap
at
at=+J_
at7r
These equations
- =+p.+
R
;
+ G- R
n
;
(G - R) ee
+
J_
-P_v,+ b-_PIE + DQv_
r n='n.-
are coupled through
and recombination (R)
b+p+E- D+VP+
Dnvn
n
'
Gauss' law and the
terms which reflect the rates
(A.2)
(A.3)
(A 4)
generation (G)
of the homogeneous
reaction Eq. A.1. To ensure charge conservation, the same net generation
rate (G -
R) appears in both
of
the ion conservation statements. It is
190
to satisfy mass
conservation that the same net
rate, multiplied by i/e
(for monovalent ions), enters into Eq. A.4 for the neutral species.
Appropriate forms for the generation and recombination terms follow from
inspection of the homogeneous chemical rate equations
d rA+1 = d
=
D
(k[Ci
d[A]
= dt(-]
krddA+1rB1 ~
-
R=
- kR[A+]B-])'=
dij4
(A.5)
dt il
With the identifications (for monovalent ions)
n =
C]
;
+
P+ = eA
;
]
(A.6)
p_ =e[B-]
the generation and recombination terms must be
G = ek ni
R =
and kR are
where kD
(A.7)
e
e
the effective dissociation and
recombination rate
constants, respectively.
The chemical equilibrium
assumption regards the generation and recombi-
nation terms (Eq. A.7) as
large enough compared with other terms in the
conservation equations that they balance separately, with the result
P+P_
e 2 Kni
K
(A.8)
kD/kR
where K is the equilibrium constant. In Eq. A.8 the approximate equality
means the difference between the two terms is small compared with either
individual term. To
exploit this independent algebraic constraint among
the dependent variables, the conservation equations A.2-A.4 must be used
I
in
combinations for
appear.
Thus,
which the
subtracting Eq.
net generation
rate (
- R)
A.3
A.2 and
inserting
from
Eq.
does not
the:
I,.
-
191
differential form of Gauss' law yields
ot
=
_,vov
R
a +
V(Dp)eV
--k
atDt V
v(A
(A.9)
where
P S P+ - p_
(A.10)
b(p+ + P_)
aI
and
are the net charge density and the local conductivity, and for simplicity equal
ion mobilities
assumed. Eliminating
and incompressible
(see Sec. 3.3.1)
the individual charge densities from
conductivity expressed in terms of
A.10 leaves the
flow are
Eqs. A.8 and
the net charge den-
sity
ai
Here
o
= 0o
1
is
the
+
generation term
(pb/ 0 ) 2
conductivity
simplification
2be K
where
the
A.4 can likewise
in Eq.
the new
density and
neutral
o
;
dependent
the
achieved:
three
(A.ll)
net
vanishes.
charge
be expressed in terms
and ai.
variables
original
partial
Ne
The
of the
the
differential
equations are replaced by the two partial differential equations A.4 and
A.9 supplemented by the algebraic equation A.11.
Further simplification is
possible for dielectric media where ionizable
solutes remain so weakly dissociated
that the neutral density suffers a
much smaller fractional departure from its equilibrium value than do the
ions.
Then n
can be
approximated
as fixed
and uniform
so
that the
neutral species is extraneous.
Now when
conservation equation
for the
Eq. A.8 is justified,
the original three partial differential equations
are
replaced
by
supplemented by a
the
single
equation
differential
single algebraic equation (A.11). Note
rate constants do not enter,
Eq. A.11.
partial
(A.9)
also that the
only the equilibrium constant does through
192
A.3
Fully Developed Electrical Conditions
The
demonstration that
fully
Eq. A.8 holds
developed regardless
of the
where electrical
conditions are
rates is
reaction
based on
three
assumptions:
1.
Total solute
fully developed. This
concentration is
is consistent
with the fully developed electrical condition.
2.
Ion
and recombination
generation
liquid bulk. That is, the
the interface, and
example, the
are
confined to
the
reactions of Eq. A.1 do not take place at
any transformation of one of
the interface results in a
or C at
processes
different chemical species. For
1.1 is
thought to
undergo a proton transfer reaction of the form of Eq. A.1
with i = 2
(44).
polymeric neutral
the species A+, B-
transfer
individually participate
resulting ions
If the
solute of Table
interface, the neutral
at the
entities so
in electron
produced will
have lower molecular weights than the original neutral form.
3.
of the
Rate constants
ionization reaction are uniform.
Because kD
and kR are temperature dependent this assumption presumes isothermal
conditions, certainly
Furthermore, if
reasonable in a
convecting turbulent liquid.
the recombination process
is diffusion controlled,
then the mean separation between reactants must be small relative to
the scale of the turbulence so that the uniform
molecular diffusivi-
ty is controlling regardless of the local turbulence intensity (61).
A corollary is the demonstration that radial ionic flux densities vanish
identically.
Equation 2.7 implies
the two equalities among
the first of
the radial
flux densities
r+(r,) = r(r,w)
= -i(r,
"
)
r(r,)
(A.12)
where i = 1,2
for uni- and bimolecular
dissociation, respectively (see
Eq. A.1). If
the net generation rate (C
- R) is eliminated between the
193
steady, fully developed forms of Eqs. A.3 and A.4 the result
1 d(rrnr
shows
that r
r
constant.
=)rr
and rn
can differ
r
However,
identically
(A.13)
zero,
only by
to avoid
a
and
second
the
the term A/r
singularity
on the
equality
in
where A
axis,
Eq.
is a
A must
A.12
be
follows
immediately.
In view of the first two assumptions above the individual flux densities
in Eq.
A.12 must vanish
at the interface: rr(a,0) =
finite value of rr(a, ) would,
in
form of
the
solute,
integrating the steady,
0. Otherwise, the
by the second assumption, imply a change
which the
first
assumption precludes.
fully developed form of any
Then
one of Eqs. A.2 to
A.4 over the tube cross section yields
f(G - R)2nrdr = 0
(A.14)
The stronger statement that
on manipulation
of the
the integrand vanishes identically is based
expressions in Eqs.
A.2 to A.4 for
the radial
flux densities:
dc
rr(r)
= ±bc+Er
d
- De dr
;
(A.1abc)
i
(A.15a,b,c)
where the definition in Eq.
A.12 has been used, and the same De appears
in
the
each
diffusivity
case
reflecting
over most
of the
dominance
cross
of
the
nonspecific
section. Multiply
eddy
Eqs. A.15a and
A.15b by c_ and c+ respectively and sum the results to obtain
(c++ c_)rr'= c+crE(b- b_)-Der(C.._),
(A.16)
194
by kD and kR respectively and sum the
Now, multiply Eqs. A.15c and A.16
results to yield
kR(c+ + c_ + iK)rr
=
De[kDn
(A.17)
+
- kRc+c_]
.kRc+cEr(b+ - b_)
where
K
kD/kR
is
the
equilibrium constant.
Note
that the
third
assumption above has been invoked to bring the rate constants inside the
derivative
to form the
net generation
rate in the
first term
on the
right.
To argue that r (r,x) and
it
is enough
to show
rate vanish
the net generation
A.17 is
that otherwise Eq.
for
< r < a,
generally violated.
First, suppose the ion mobilities are identical, to eliminate the second
term
on the
right
in
generation
the core
recombination
A.17.
in Eq.
near
of
derivative of the
the
the
To allow
this
satisfy
implies
a
is
Eq. A.14,
negative
net
net
radial
left hand side of Eq.
in ion mobilities
neglected term remains much less
of the
there
The same contradiction arises if rr < 0
for a difference
The ratio
product c+c_
But
to
net generation rate, while the
that the previously
retained.
(say) then
flow and,
wall.
A.17 is positive by assumption.
is assumed.
rr > 0
If
two
terms in
Eq. A.17
it must be shown
than one of those
that
contain the
is
aEr(b+ - b_)
(A.18)
VTb(Dt/Dm)
where
the
tube
radius
a
is
assumed
to
derivative, b is the average ion mobility, and
introduced by
the Einstein relation.
Reynolds
numbers
of
the
order
the
ratio of the
4
for Schmidt
1l3, :while the
fractional
of the order of 1
of
radial
T is.the thermal voltage
Prom Sec. 3.2.2 the
turbulent to molecular diffusivities is
and
characterize
195
difference in mobilities is at most of the order of unity. Finally, with
fields (ap/2e.) encountered in the experiments
the typical space charge
does not exceed
inner wall
axis and
much
typical potential drop between
previous chapters, the
of the
as Dt
that the
conclusions holds
original
nearly as
the thermal voltage by
the ratio
Dm . Thus
exceeds
Eq.
A.18 is
the tube
small, and
equality Eq.
approximate
the
A.8 is
satisfied in the sense that both sides vastl--exceed their difference.
A.4
Chemical Equilibrium in Inhomogeneous Systems
The
simplification
outlined in
any other
term. To investigate
the
section addresses
in the
implicit
to dielectric
specialized
without actually
into normalized form.
characteristic times
questions: what
equations, how
conservation
normalized
greater in magnitude
this assumption
requires that they be cast
(A.1-A.3)
the generation (G)
on the assumption that
(R) terms are individually much
solving the equations
This
equations
coupled conservation
Sec. A.2 hinges
and recombination
than
the
of
solvents, and what
ordering of
are
can they
be
these times
justifies the equilibrium assumption.
and time scale of interest
Define a characteristic length
r
=
It will
;
that
t =
;
be supposed that the
differences
so that
v = v(I/T)
(A.19)
net charge density is small
between instantaneous
in the sense
concentrations and
ion
their
respective equilibrium values remain small compared with those values, a
condition
usually
satisfied in
(defined with Eq. A.9) is
practical
bipolar
systems. Thus,
the characteristic conductivity,
o
o/2b is the
typical value for P+, and the typical neutral density is determined.from
the equilibrium condition Eq. A.8.
196
= ad o ;+
where
.
2
i = 1 or
Eq. A.l.
= P+
-2b
or bimolecular
to
of
of an
field make
the electric
specify a typical field strength.
difficult to to
reactions
and the contribution
potential,
distribution
(A.20)
4e
to the uni-
of an applied
surface charge
ka 2
ni = ni
2 corresponding
The absence
evolving
a
it
For purposes of nor-
malization, the characteristic field is taken to be that associated with
one species of ion when the counterions are discharged. From Gauss' law
.,So
E = E
(A.21)
- cS2b
However,
because
are in fact comparable,
cncentrations
the two ion
the
normalized field is likely to satisfy
E <<
(A.22)
1
In normalizing the conservation equations to these characteristic values
the rate constant
e(b+ + b)
kL
(A.23)
(m /s)A.23)
emerges, which is the
recombination rate constant predicted by Langevin
for gases at high pressure (62).
Given this normalization, what characteristic times emerge, and for what
ordering
through
of these
the
maintained
region
in
discerned from
the
an observer
times will
of
interest
approximate
a normalized
following a
chemical
find that
sense
of Eq.
A.8?
liquid element
equilibrium
The
equation for the convective
answers
is
are
derivative of.
197
the net generation rate (G - R).
Multiplying Eqs. A.2 and A.3 by p_ and
p+, respectively, substituting Eq. A.7, and adding the results yields
Dp+P
=
tDt
=
2
E
.t
+
where for
kR
i
ce 2 , + -p - +) + l kL
B,(P+VP
- P2+)
Vo(n
p
TD
+ P V2 p+)
simplicity incompressible
-
+p_)
;
i=1,2
(A.24)
flow and equal ion
mobilities are
assumed, and the underlined equation number indicates that all variables
in the equation
are normalized to the characteristic
values defined in
Eqs. A.19 to A.21. The relaxation and diffusion times are defined by
(A.25)
=
TD-
O
/e
Because of the exponent i in Eq. A.7, the uni- and bimolecular reactions
must be considered separately when normalizing the conservation equation
for the neutral species.
Unimolecular Dissociation: With i = 1, Eq. A.4 has the normalized form
Dt
-tkD(n
Et D
d=_ 2
'r
Dn
+p_)
+ zV2n
+
To~~~Ti
Subtracting Eq. A.24
from Eq. A.26 yields the
(A.26)
convective derivative of
the normalized net generation rate
Dc-
= -
)
cr~~~~~
- (P2P
where
-'
+ p V 2p+) + V 2 n
+
(A.27)
198
2
=
cr
2
be
kR
(be K+1)
T kL
o
K
kR
+ 1)
-R
t0kL
where Eq. A.23 and the definition of.t
approximated
introduced
by unity.
total solute
to dielectric
were used, and both n and
The equilibrium
by recognizing that
o
=
degree
2eb[A+]
concentration. To specialize the
media it proves
(A.28)
2fs
of dissociation
=
2ebs,
were
was
where s
is the
chemical relaxation time
convenient to eliminate K
and express the chemical relaxation
i
from Eq. A.28
time in terms of kR and
alone. To
that end note that
[A+ ] = [B-] =
s
;
[C] = (1 -
)s
(A.29)
and that in equilibrium Eq. A.5 (with i = 1) requires that
+ 1 =
y2 (s/K)
+
1
=
11 --
2-fs
where
the second
2
(A.30)
l-Y
relation follows
after
some algebra.
Now Eq.
A.28
becomes
T cr
=
=
R
The
approximate
0
< 1.
<
.
"
Tk
kL
kL.
(1 - )
(A.31)
9kL
equality
Bimolecular Dissociation:
recognizes
that
(1 -
After multiplication by
/2)
=
(1)
2n, and with i
since
=.2,
Eq. A.4 assumes the normalized form
Dn2
_
Dt
~
(2
i~-~kkD(n
Subtracting Eq. A.24 from Eq.
-
p_)
P)+
2nv2 n.A.32)
d
(A.32)
A.32 yields for the convective derivative
199
of the net generation rate
n2 - P+P-2-(
) = _
2
5tDn
'tcr
r_
-
_
P- <p.
- -9
I
+
2 p+)
2p_ + p
+ 'n2n 2
(A.33)
where
2- kR
~k2K
i+
'r 9kL
/=
c/~r
l
As before K is eliminated
by Eq. A.5 with
2
(A.34)
in favor of kR and
. The equilibrium implied
i = 2 is
_ K(1 -
)2 =
where the second
2ffi + 1 =
I
+
Y
- y
relation follows after manipulation of
(A.35)
the first. Now
Eq. A.34 becomes
't/TCC
t 1 _-Y
: 2
where the
'kR
2
2
rkR
(A.36)
that (1 + Y)
fact was used
=
(1). Note that
Eqs. A.31 and
A.36 are the same.
In
view of Eqs.
A.27 and
A.33 the same
criterion for
maintenance of
chemical equilibrium applies in both the uni- and bimolecular cases:
1/1D
0(1)
;/'
d
< (1)
;'
(-/r)_ i 0(1)
(A.37a,b,c)
while
t
Tcr
>>
1
(A.38)
200
Note that as
long as
range of T/T.
Thus,
in dielectric
media
q. A.22 holds,
the "universal"
for which
Eq. A.37c will be met
criterion
for a wide
for chemical
equilibrium
<< 1 is
2-kR
Given
A.5
1
k>
L
9
(A.39)
and -x, this
is a lower
limit on the recombination
rate kR.
Physical Significance of the Rate Constants
If the criterion
poorly
for chemical
equilibrium
characterized solutes,
about the
it is
component steps, one of
process, and
what can
to
be said
referring to specific
reaction of Eq. A.1 is resolved into two
which is identifiable as a diffusion-controlled
the other a chemical transformation.
more slowly
is to be applied
natural to ask
recombination rate constant kR without
reactants. In this section the
proceeds
Eq. A.39
than any
of
its component
The overall reaction
steps, and
thus the
apparent recombination rate kR is bounded from above by the forward rate
of the
diffusion step k1 2.
It is left
prediction for the limiting
to Sec. A.6 to
compare Debye's
recombination rate k1 2 with Langevin's rate
kL.
As the basis
imply
for a description of macroscopic
an ionization
process in
effects, Eqs. A.2 to A.4
which only two
ionic and
one neutral
species have significant concentrations. The identification of component
reaction
steps
introduces a
elucidate the mechanisms
also
serves
Equation
notice that
A.1 is elaborated
fourth
reaction
participant that
implicit in the overall reaction
Eqs.
A.2
as follows:
to A.4
are
helps
Eq. A.1, but
not always
adequate.
201
step
step
1
k2 3
k 12
+
A
+ B-
CE
iC
3
k2l
(A.40)
T
'k 32
free ions
where
2
encounter
complex
transformed
complex
i = 1 or 2 for uni- or bimolecular
dissociation,
respectively.
The
encounter (CE) and transformed (CT) complexes are neutral species in the
sense that they make no contribution to conduction.
With nothing
implied regarding its stability, the
defined here as what results
charged
ions and
from the mutual approach of two oppositely
contact between
change occurs in the solvation
The transformed
significant
encounter
product
complex.
solvation shells,
before any
pattern or any other structural feature.
as any neutral species,
that
results
Defined this
of Coulomb
energy, while
their
complex is defined
concentration,
encounter complex is
way,
the
from
production of the transformed
modification
encounter
interactions in competition
present in
of
the
complex is
the
only with
the thermal
complex generally involves
Coulomb and short-range interactions and a desolvation process.
Whether Eqs. A.2 to A.4
represent the
limiting
remain sufficient hinges on whether Eq. A.5 can
two-step reaction.
cases where
this is
constants (kD and kR) and
The present objective is
so,
and to
express the
to identify
effective rate
the concentration of the "observable" neutral
species [C] in terms of the rate constants and concentrations of neutral
complexes defined in Eq. A.40.
rate constant
(kR) and
interest because
The relation between the overall
that of the
first step (kl2 ) is
the former enters into the
forward
of particular
equilibrium criterion
(Eq.
A.39) while the latter is the subject of the theory of Debye.
For the homogeneous
chemical reaction Eq. A.4g, the
tions remain equal (A+]
two ion concentra-
=- B-] _ [I]), and the rate equations are
202
dt[I]
= -
= k2[I]
dt[C]
- (k2 1 + k 2 3)[CE] + k 3 2[CT]i
dt[CT] = i(k2 [CE]
3
This
(A.41)
k1 2[I] + k2l[E]
set of equations
(A.43)
- k32[CT]i
can be reduced
dtrI]= (k[C]i -
(A.42)
to the two represented
[Idt2) =_
by Eq. 5
(C/i)
(A.44)
in three of the four limiting cases considered below.
(A)
Negligible concentration of transformed complex: [CE] >> [CT]
The identifications
;
kD = k2
kR = k1 2
are the obvious ones. It
trations
of the
minimizing
(A.45)
is tempting to anticipate the relative concen-
encounter and transformed
free energy gained
by
[C] = [CE]
;
complexes by
the
upon desolvation to the loss
coulombic
potential
intervening solvent molecules. This would
as
comparing the
in free energy secured
a
result
of
removing
suggest that case A refers to
solvents of moderate to high polarity, but the possibility that specific
solvation
effects
precludes
generalization. Nevertheless,
statement
is
and
short-range
possible: bimolecular
interactions
in
may
one case
dissociation
be
important
a more
definite
is clearly
excluded
because by the definition of the encounter complex it is produced at the
rate of only one per encounter.
(B)
Negligible
In the case
ions
concentration
of encounter
complex:
[CE] << [CT]
of bimolecular dissociation, an encounter
produces
two
neutral
entities.
involved, and the neutral status
Thus charge
between two free
transfer
must
be
is maintained by covalent bonds rather
203
attraction. Here, the ion pair
than Coulomb
represents-an intermediate
form, present in small concentration, that is still produced at the rate
For unimolecular dissociation, this
per encounter.
of one
ordering of
of the encounter complex, and is
concentrations reflects an instability
perhaps more typical than case A when the solvent polarity is low.
successful approximation in chemical kinetics is
A common and generally
(64)
provided
has
of an
concentration
=
[CE]
=
In
justification.
the
this,
unstable intermediate
relatively
small
approximated as
species is
0, Eq. A.42 yields
constant. Thus with dCE]/dt
[C
which Hirschfelder
approximation (63) for
steady state
the Bodenstein
k12I]2 + k32CTi
k3
(A.46)
k21 + k2 3
from Eqs. A.41 and A.43 leads to equations
Using this to eliminate [CE]
of the form of Eq. A.44 if the identifications are made
k21 k32
23
k2 1
kD
k 12 k 23
kR =
in agreement with Eigen et al (65). Note that'kR
approached
only
of k 2
in the limit
transformation step
is fast
3
=
[C]
k
21
[CT]
(A.47)
k 12 with the equality
>> k2 1 when the
forward
rate of the
the transformed
enough that production of
complex is limited by the forward rate of the first step.
(C)
First step is rate limiting
Now
allowance is
complexes, and
can be
concentrations of
made for significant
it is striking
that a single effective
identified. Only for unimolecular dissociation
A.41 to A.43
be reduced to Eq. A.44.
both neutral
neutral species
(i = 1) can Eqs.
On the relatively slow time scale
of changes in the ion concentrations [I], the two neutral species remain
in instantaneous equilibrium:
204
k23
(A.48)
k2
K2
;
K2[CE]
[CT]
This relationship can be inserted in Eq. A.41, but not in Eq. A.42 where
the faster time scale is
also implicit. To obtain equations of the form
of Eq. A.44, Eqs. A.42 and A.43 are summed yielding (with i = 1)
+
dt(CE
CT)
which is -d(I]/dt
=
]2 - k21[C E]
k12
(A.49)
as required by mass conservation.
Next, advantage is
taken of Eq. A.48 to write
k21[CE] =
2(CT]
1
+
(A.50)
Substituting Eq. A.50 into Eq. A.49 shows that the desired form has been
obtained if the identifications are made
kD
1 + K2
kR
= k12
and
(A.51)
[C] = [CE] + [CT] = [CE](1 + K 2 )
In the analysis
of case B, the small
concentration of the intermediate
form was assumed to remain constant. Here, the intermediate participates
in the
fast second
Thus, at least
step, and its
rate of change cannot
as analyzed here, cases B
be neglected.
and C are mutually exclusive.
On the other hand, when K2 << 1 cases A and C are indistinguishable, and
tney remain
when
(D)
2
so as
concerns the equilibrium
criterion (Eq.
A.39) even
is not small.
Second step is rate limiting
Now the concentrations of the
two neutral forms are not instantaneously
205
coupled as
in case
required in
C, and two
addition to those
independent conservation
for the ionic species.
equations are
Now Eqs. A.2-A.5
are insufficient. This case may
have practical importance when the ions
have localized
or inaccessible
charge groups. In
transformation
step
may
involve
a
the former
reorientation
process
case the
that
is
inhibited by the confining effects of the solvent molecules.
In summary, Eqs.
A.2 to A.5 represent the
in three limiting cases, and in general kR
the equilibrium criterion
the rate of
two-step reaction (Eq. A.40)
<
k 12 . Because kR enters into
(Eq. A.39) and is bounded
formation of the encounter complex
Portunately, as discussed in the
from above by k1 2,
is of special interest.
next section, k1 2 represents a process
that is amenable to a simple model.
A.6
Limiting Recombination Rate
The
equilibrium criterion
overall recombination
Eq.
A.39 calls
rate constant
for
a lower
kR, while Sec. A.5
forward rate constant of the first step (Eq. A.40) k2
for kR, Thus,
a necessary
condition
for Eq. A.39
limit on
the
identifies the
as an upper limit
to hold
is
2rk 12
e
>> 1
(A.52)
This section reviews the classical
the
diffusion-controlled encounter
theory of Smoluchowski and Debye for
frequency,
with a
view towards
a
comparison of k12 and kL for dielectric solvent media.
For the purpose of modeling its rate of formation, the definition of the
encounter complex
given in
Sec. A.5 yields
two benefits, even
if the
concentration of that complex (so defined) is negligible. First, because
the
complex is
ions, the
the initial
constant (k1 2 )
product of
an encounter
that governs this
between solvated
rate reflects
a diffusion
process and not any specific chemical attributes of the reactants or the
206
solvent.
Thus, provided
equals many
the
average initial
solvent molecule diameters, a continuum
that an
encounter complex will
contact
between
of the
ions
encounter complex
phrased in
unity, and
is
is identical
approach of and
the rate
thus
to the
of
frequency of
the transformation step in Eq. A.40
encounters. In contrast, models for
cannot be
description of the
result from the mutual
two solvated
ions
Second, the probability
provisional, ought to suffice.
solvent, though
formation
separation between
terms of macroscopic parameters
because specific
short range forces and the discreteness of the solvent come into play.
Smoluchowski's model
for the frequency of
encounters between uncharged
particles in thermal
motion (66) was developed to
describe the rate of
coagulation in colloidal systems. As discussed by Overbeek, the original
model
does not
account
for
double layer
repulsion forces,
evolving
particle size, and the spectrum of particle sizes. The simplicity of the
original model proved a virtue, however, for the omission of refinements
pertinent
only to
the colloidal
system
encouraged its
other
physical contexts.
Not surprisingly,
model
to
rates
ionic reaction
long-range
interaction
issues. These issues
in
Debye's adaptation
solution
between reactants
adaptation to
by including
raises
a
a
of the
general
different set
are identified by the subheadings
of
in the overvieA
of Debye's theory given below.
Boundary Condition at
the Reaction Radius: The Smoluchowski-Debye arti-
fice designates one member of
a reference
its reactivity by replacing
ion, and represents
sink that consumes
the edge
one of the reactive species, A+ (say), as
of the
sink. The diameter
distance between
complex, while
members of the B- species
centers of the
its charge
of
it with a
when the latter encounter
the sink corresponds
reactants when they form
and diffusivity are
those of
to the
an encounter
the reference
ion.
In
the
frame of
spherically
the
symmetric
sink,
fashion
the
B- species
with
diffusion
diffuse
and migrate
coefficient
D_.
in
This
207
isotropy on the molecular scale reflects the continuum approximation. To
exploit this symmetry,
at the center
the sink is regarded as
stationary and situated
of a spherical coordinate system,
is accounted for
and its thermal motion
by replacing the diffusivity of
the B
species by the
sum of the diffusivities 2D = D+ + D_.
The frequency of encounters between the reference ion and the B- ions is
given by the net inward
flux -4nr 2 rb(r,t) of the latter across a closed
surface coincident with
the edge of the sink
disappearance of Bmultiplied
ions is the net flux
by the concentration
ddB-] = [A+]4,2rb(,t)
[A+
]
at r =
. The net rate of
associated with a single sink
of sinks:
(A.53)
-]
- k 2 [A+][B
The rate constant follows as
4nk2rb(,t)
k12(t )
-
(A.54)
CB-]
The radial flux
density rb(r,t) and the local
concentration cb(r,t)
of
B- ions are coupled through Fick's first and second laws:
ac
rb(rt)
=
-2D(
r
+
acb
.ta-
kT ar
-1.
-a2
r
-ara (r 22rbr b 3
(A.55a,b)
where U is the potential of the average force between ions. Solutions of
Eqs. A.55 are subject to the boundary-conditions
Cb(a,t) =
;
U(,t)
cb(",t)
= [B-]
;
U(m,t) = 0
cb(r,U)
= [B-]
(r > )
=
m
(A.56a,b)
(A.57a,b)
(A.58)
208
The
widely expressed
insistence
complex
called
that the
here
a
theory apply
transformed
complex as
transformed
r =
to
complex
that
condition Eq.
condition,
their
of what
(Eq.
They
regard. the
that is
present in
A.40).
reaction product
equates the inward flux of
flux" that
has been
here by case B
of Sec.
they require a
the reaction with one step,
a "reaction
and
the formation
a situation modeled
order to represent
boundary condition that
radius"
to
the only
significant concentration,
A.5. In
with
consistent
the boundary
of these authors to define an
the failure on the part
A.56a stems from
intermediate
criticism (67,68) of
B- at the "reaction
reflects a
net rate
in the
second step in Eq. A.40.
The intermediate consisting of a B- ion at the
reaction
identified
radius is
not
as a
distinct
species, and
thus
b(Q,t) is nonzero.
With the
formed by
the entity
two ions with
centers separated by a
of such a
the introduction
distance
is regarded as distinct.
encounter complex which
constitutes an
zero because
A.5), cb((,t) must be
present definitions (Sec.
complex that, despite its
It is
relatively small
concentration, permits application of the Smoluchowski-Debye method with
the
original
boundary
of the
separation
(Eq.
condition
reaction into
applies, the two approaches are
A.56a),
two steps.
necessitating
while
Whets case
B of
Sec. A.5
reconcilable, as they must be since the
negligible concentration of the intermediate complex implies that it has
no unique representation.
Conditions for Quasistationary Reaction Rate: When U =
, Eqs. A.55 have
the solution
cb(r,t) =
The rate constant
B-] 1 -
terfc(r
follows from Eqs. A.54, A.55a
(A.59)
(with U =
), and A.59
209
as
8
k 1 2 (t)=
i
2D a
[B-]
D
b
a t
Collins and Kimball (69) note
for t >>
2
-1 = <s >/6D,
the diffusing
v is
molecules and <s2>
=
that it
(given by Eq.
does not
the frequency
"jumps"
of
jump length. Thus
0 must be ignored. These authors calculate
on random walk diffusion equations, and
significantly differ
A.60 with t
of random
is the mean square
the initial reaction rate based
find
n2
that Fick's second law is applicable only
where
the singularity in k1 2 at t
(A.60)
+
L
) when the
from the
ratio a/<s>
stationary rate
1. As this ratio
increases, so does the ratio k1 2 (0)/k1 2(m), and k1 2 can be regarded as a
constant only on time scales long compared with the time
O2
= 2nD
that
(A.61)
characterizes
particular,
relaxation of
Tr must
be
short
the
concentration distribution.
compared with
the time
In
characterizing
changes in the average concentration [B-], which from Eqs. A.53 and A.60
is
·c = (kl2 [A+]-
1
= (8nDA+])
-1
(A.62)
Thus k1 2 is quasistationary if
OL
( <
e
c
<
dr
where dr - [A+]
1 /3
this condition
is consistent with the
for the solvent.
(A.63)
is the mean spacing between reactive ions. Note that
original continuum approximation
210
No
solution of
the
interaction energy
long time
time
U is
dependent Eqs. A.55
significant.
solution with U = 0
equation directly.
Thus,
is
available when
However, the usefullness
encourages solution of the
with rb regarded as a
the
of the
steady state
constant, Eq. A.55a is
readily integrated subject to Eqs. A.57:
[
cb(r) = exp(-U/kT) (B-
+
r2rb(r)
(UkT
exp(U/kT)
2D
dre
(A.64)
Finally, with Eqs. A.56 applied, Eqs. A.54 and A.64 yield
k12 = Bnaeff
which is
eff -t
exp(U/kT)
Debye's general result.
d
1
(A.65)
Definition of an effective
ion size
aeff points up the similarity between Eq. A.65 and Eq. A.60 when the
quasistationary condition Eq. A.63 holds, and suggests that the latter
be generalized to
%eff << d
(A.66)
When the interaction is attractive (U < 0),
eff )>
, and thus while Eq.
A.66 is more tentative than Eq. A.63, it is also more restrictive.
Application to Turbulent Flows:
A.4)
are applied
to
When the conservation equations (A.2 to
turbulent
flows, the
diffusivities that
appear
explicitly in those equations must include the eddy contribution. If the
recombination
whether
Eq. A.65
implicit
answer
rate (R)
still
in the latter.
hinges on
is
still
applies
As in
to be
since
modeled
the question
arises
a diffusion
process
is
the third assumption
in Sec.
A.3, the
whether the smallest
effects prevail, are still large
eddies, within
also
which molecular
enough to contain many reactants (61).
Thus the ordering of lengths
II
I
211
aeff
<<
<<
dr
turbulence scale
(A.67)
appears necessary.
Nature
of the
tioned the
Concentration Gradient: Collins
nature of
and Kimball
(67) ques-
the postulated concentration gradient
that helps
maintain a flux of reactants
because a
reaction has
towards the sink. If the gradient develops
depleted the local concentration
of reactants,
then the gradient is no longer of interest because the reference ion has
already been
by showing
discharged. These authors proceed to
that the knowledge
that a given
resolve this dilemma
ion has not yet reacted
at a
given time diminishes the probability of a reactive partner being in the
immediate neighborhood. The probable distributions are still governed by
Fick's laws,
and in
that sense the
postulated gradients are
real and
must be recognized by treatments of chemical kinetics.
Interaction Enerqy:
Equation A.65 expresses
the limiting recombination
rate k1 2 in terms of the general interaction energy (U) and the ion size
parameter ().
Now
U is specified, the rate
the result is compared to
Langevin's rate. Consistent with Eq. A.67 the
concentration of reactants is
randomly distributed,
constant is evaluated, and
assumed sufficiently dilute that they are
and thus
the time average electric
potential in
the vicinity of the reference ion is that of a point charge, so that
(r)i =z
U(r) = -
e 2
r(A.68)
Debye (70) carries out the integration called for in Eq. A.65 and finds
k12
8"D
exp(U()/kT
-
exp(U(%)/kT) - 1
(A.69)
Inserting Eq. A.68 and the Einstein relations into-Sq. A.69, and forming
212
a ratio with Eq. A.23 yields
k12
1
1 -
kL
where bij is
the
-zZe
exp(-biV)-
2
4jn/
2 kT
(A.70)
the separation between ions of
coulomb and
ziz j < 0, and
thermal energies
are
valence z i and zj at which
that bij > 0
equal. Note
that the predicted rate constant
when
exceeds Langevin's rate
to a degree that is significant when
(bij/M)
i
0(1)
(A.71)
that is, when the coulomb
energy of two oppositely charged reactants in
contact has a magnitude of the order of or less than the thermal energy.
To appreciate
the reason, it
Debye's rates account for
evidently represents
should be noted that
both Langevin's and
the coulomb interaction, but while Langevin's
a point
charge solution, Debye models
the finite
size of the reactants. That Debye's rate is the larger reflects the fact
that
Brownian motion
will ultimately
molecules even without
mechanism for
bring together
two finite-sized
the coulomb attraction. However, this additional
producing an
encounter can only
be noticeable
when the
coulomb interaction fails to dominate the thermal energy, as is the case
when Eq. A.71 holds.
For a monovalent electrolyte at room temperature, find that
= 5.6 x 10-8
bi
where er = E/%
the
order
unimportant
(A.72)
eLr
t
of
O
is
10
when s r
0
the relative permittivity of the solvent. With
A,
Eq.
is less
A.72
than
indicates that
about A.71,
the
finite
and thus
k12
size
X
of
is
kL for
213
dielectric
media.
screened by
For
less
dilute
the Debye-Buckel ionic
solutions,
the
atmosphere and k2
interaction
is
kL. Thus, the
necessary condition for chemical equilibrium Eq. A.52 reduces to
I 9
<<
In Chapters 5
(A.73)
and 6 the characteristic time
based on the liquid velocity
tube and the length
is a liquid transit time
at the edge of the diffusion sublayer in a
of the tube
(
8
= L/vS).
214
Appendix
B
TRAVELING-WAVE PUMPING OF LOW VISCOSITY LIQUIDS
B.1
Introduction
Like the standing-wave interaction of Chapter 6, the technique described
below for
probing the
liquid-insulating solid interface
avoids direct
liquid-electrode contact by capacitive coupling between excited external
electrodes and
frequency
an internal
()
and
potential is a
charge distribution. Again,
wavenumber
(k)
traveling wave rather than the
and backward waves that comprise
whereas
in the
combined
are imposed,
induction
effects
but
now the
the standing wave (Eq. 6.18). However,
sensor the
of convection
flow
and
is also
conduction
imposed, and
on the
distribution are sensed electrically, now
the initially
quiescent liquid
are induced by
a convenient way to constrain
the
the
described
in Sec.
of
B.2 the
insulating tube,
induced
motion is
difference developed across the ends
needed to induce
field. The
the potential at
in
experiment
detected as
a pressure
of the winding. The applied fields
by an imposed
is intended as
the motions of
the
observable motion are typically of
inadvertently generated
the technique
and
the
field-induced
the applied
helical winding remains
surface
applied
superposition of forward
volume charge
outer
the electrical
the order of those
flow (see Chapter 3),
one for studying natural
and hence
processes under
controlled conditions.
Like
the standing-wave
origins in a study (71)
accounted for
by a
conductivity and
hence
the
presence of
interaction,
the traveling-wave
pump has
its
of a free liquid surface. There, the pumping is
model that regards
the liquid as having
permittivity, and thereby confines the
electromechanical
coupling
thermal gradients in
to
the free
a uniform
net charge and
surface.
the liquid, which imply
In
the
gradients.in
electrical conductivity, an ohmic model (72) still predicts pumping even
if a free surface is
absent, because the applied field can induce a net
215
charge in the liquid bulk.
observed bulk electro-
model cannot account for the
However, the ohmic
mechanical coupling in a (presumably) isothermal liquid. In a model (73)
With the thickness of the coupling layer determined by the
charge-free.
small enough
an
effects on the induced charge.
order of 106 V/m
layer (Lm
2 x 10
b
is
For example, with
a
0.1,
of n
viscosity
dioctyl-phthalate
in
mobility
to
description of field quantities and the
temperature
room
absolute
excitation wavelength
with the
compared
justify a quasi-one-dimensional
neglect of convection
the requirement
relatively viscous liquids by
restricted to
Lm be
that
, applica-
= bEi) based on the .excitation period
migration length (Lm
bility is
liquid remains
of the
the bulk
while
the interface
layer at
in a thin
net charge is induced
predict that a
conservation equations
liquid
the fundamental species
pumping of relatively viscous liquids,
for the
10
With
E
ion
typical
of the
and
of the coupling
and 1.0 s respectively, the thickness
mu) is easily exceeded by the wavelength associated with
200
the wire electrodes used in practice.
typical of cooling applications the larger
In the low viscosity liquids
ion mobilities (see Table 3.1)
imply migration lengths that may be com-
parable to both the excitation
wavelength and the tube diameter.
Thus,
the electromechanical coupling is distributed throughout the liquid bulk
and
convection
the
component
double
interface.
hinges
or
layer
applied field.
they
namely, the passage
the
under
The
influence
of the
originate in
ions may
result from
physical process
same
in Sec.
outlined
model
transfer
charge
as pictured by the model,
At least
on the
may
the neglect
model nor
of
B.3
the liquid bulk as being due to ions ejected
liquid-solid interface
of the
The
is appropriate.
effects
pictures the net charge in
from
dimensional
the quasi-one
now neither
normal
a charge
across
the
the pumping interaction
that underlies
of ions between the solution
electrodialysis,
and the interface (or
even the solid phase).
However, while in electrodialysis the passage is
detected as
in an external
a current
circuit or
by an effect
on the
216
liquid
conductivity,
injection
exerted,
into the
and the
in the
traveling-wave
liquid bulk
is reflected
pump
motion induced,
by the
the
by the
extent of
ion
electrical force
tangential component
of the
applied field.
B.2
Experimental Arrangement and Preliminary Results
Figure
B.1 illustrates
section of insulating
the pump
experiment.
A helical
tube supports a traveling wave
winding on
a
of potential that
advances essentially along the tube length. The tube extends beyond both
ends of
the winding and
fashion
of a
rises vertically to contain the
manometer. Exciting
longitudinal electrical
the
liquid in the
winding induces
force density in the liquid
a time-average
bulk. The pressure
difference developed as a result between the ends of the winding is then
determined from liquid displacement in the vertical sections.
Figure B.2 illustrates the pattern
data acquired for
when the
of response typical of the extensive
Freon in Tefzel tubes. Pumping
average electrical force
is said to be forward
is in the same direction
which the
traveling wave advances. Backward pumping
force and
the phase
velocity
and
velocity being oppositely
traveling wave
amplitude
as that in
corresponds to the
are
directed. As
varied,
the phase
regimes of
both
forward and backward pumping are observed.
Apparatus for
the traveling wave
pumping of liquid through
a two-foot
glass column packed with insulating particles has also been constructed.
Preliminary
tests
with
Teflon
particles
typically 3mm
in
diameter
indicate that pressure differences can be developed which are comparable
to
those observed
pumping
through
for the
the column
thirty-foot length
appears
to
be
of open
tube. However,
significant over
a
much
by
the
narrower range of liquid conductivities.
There
are
two
questions
one
experiments. First, is the flow
would
like to
have
answered
the fully developed laminar one assumed
217
d
fl li d
r,
Fig. B.1
Experimental arrangement for traveling-wave pumping.
displacement
(arbitrary
units)
forward
pumping
ibackward
pumping
frequency (Hz)
Pig. B.2
Typical dependence of -displacement d (Fig. B.1) on travelingwave amplitude and frequency.
I
218
in
the
model
the
whether
the
indicate
of
Alternatively,
Doppler
an
next section.
laminar
experimental
anemometer to
second question
Theoretical
flow
i:s
approach
reveal the
is whether there
considerations
likely
is
to
be
may
unstable.
possible using- a. laser
details of
the liquid
motions. The
charge injection.
is any evidence of
This evidence might take two forms: a threshold field for pumping, and a
qualitative change
in behavior when
the migration length Lm
tube radius. An unambiguous threshold
order of the
is of the
proved difficult to
observe, partly because the liquid was slow to settle completely after a
disturbance.
However,
involved is
a strong
indication
the correlation between
that
charge injection
the cross-over points in
is
Fig. B.2
and the excitation periods obtained by setting the migration length L
bkVT
=
radius. Here, k is the wavenumber 2n/X, where X
equal to the tube
is the excitation wavelength.
B.3
Model Outline and Preliminary Numerical Results
model for the
The simple
when the
dominant source of
net charge in
the liquid bulk is
jection from the liquid-insulating solid interface.
ions travel
imposed by the applied potential.
along characteristic lines determined by
and an initially
ion in-
The net charge den-
that the electric field in the liquid is
sity is assumed tenuous enough
essentially that
intended to apply
pumping presented below is
unknown velocity profile that must
Thus, the injected
the imposed field
be consistent with
the average tangential electric force density. To specify the net charge
density
(and
hence
the force
density)
at
a
given point,
the
ion
trajectory through that point must be traced back to the interface where
the
normal field
component determines
charge through a (trial)
"replenished' as
the
"injection law."
the ions eventually
magnitude of
the injected
The interface is continually
return to the interface
where it
intercepts a characteristic line.
Because the
velocity profile
is needed to
evaluate the
force density
219
which then determines that profile, a trial profile must be specified to
initialize
the
analysis.
In
may
what
generally
not
be
a
becomes the
velocity profile then
the predicted
convergent procedure,
or may
A self-consistent result
new trial profile in the succeeding iteration.
is obtained when the trial and predicted velocity profiles agree.
The
penetration of
injected ions
of the
into the bulk
low viscosity
liquid makes it difficult to
develop a model that is "universal" in the
sense of being applicable to
a general configuration of the liquid con-
tainer.
For its simplicity, and because it represents the tube configucompared with the excita-
tube radius (a) is large
ration whenever the
tion wavelength, a planar half-space of liquid (Fig. B.3) is the context
for presenting the model. Four assumptions are made:
response is a fully developed
[a] The fluid mechanical
low enough Reynolds number that
Thus, in the
flow model.
laminar flow of
the velocity profile satisfies a creep-
configuration of Fig. B.3
the velocity field
takes the form v = vz(x)z and a stream function can be defined such that
vz(x)
d
;
volume charge
[b] The
(x) =
vdx
(0)
;
density remains
enough that
small
electric field dominates the space charge field.
limit the
potential in the liquid half-space
tion subject
(B.l)
0
the applied
In this weak injection
satisfies Laplace's equa-
that specify the potential
to boundary conditions
at the
interface and require the potential to vanish far from the interface!
$(x,z,t) = Vocos(ky)exp(-kx)
Without loss of generality, the
;
ky
(wt - kz)
(B.2)
thickness (w) of the insulating wall is
regarded as small compared with the excitation wavelength (2n/k) so that
the potential at
the liquid-solid interface is essentially
that at the
220
C = V cos (t
a
insulating
wall
- kz)
I
2a
4 = V0 cos (wt - kz)
Fig. B.3
Planar model with channel half-width (a) much greater than excitation wavelength (X).
221
electrodes. If kw is not small, the amplitude and phase of the potential
at the
interface relative to
that at the electrodes
is readily deter-
mined from "transfer relations" like those used in Sec. 6.3.4.
[c] Ion diffusion is negligible. This is justified if the diffusion time
(X2 /Dm) vastly
exceeds the transit time
that is equivalent
(X/bEo; Eo
kVo), a condition
to requiring the applied potential
to vastly exceed
the thermal voltage.
[d] The
transit
time
= sl/o).
relaxation time (
conduction
relaxation
(X/bEo )
processes.
time
(see Eq. A.31
short
compared
with
the
charge
Thus, injected ions are not neutralized by
When
cr is
is
recombination
of the order
with kR 4 kL and
is
Langevin
of the charge
4 0).
the
chemical
relaxation
time
T1
Thus, the transit time is also
exceeded by the time that characterizes ion recombination, and hence the
injected charge
density retains
its initial magnitude over
the entire
trajectory.
Bernoulli's equation relates
the measurable lateral displacement to the
pressure gradient induced by the traveling wave of potential:
ead= cotcot((e8)]
~t~e)
cdo
ad = cot(e)Ah
Here
p
beyond
is the
the ends
negligble.
(assumption
= ct(e)
Pmg
creep-flow
Pmg
) dz =cot
of the
winding (of
Consistent
with
[a]), the
final
points in the
length L)
fully
equality
where the
developed
in Eq.
B.3
the excited section of the
appropriate
limit of
the
the
to
the
(B.3)
dz
pressure difference between
pressure gradient along
coordinates
ap
developed
velocity is
velocity
profile
implies a
uniform
tube. In the planar
model
Navier-Stokes equation
liquid just
of
Fig.
relates the
B.3
the
pressure
gradient to the electrical force density:
d_ =
dz
.
4'
+ d2V=
d2
dx2
(B.4)
222
This is
to be solved subject
the condition
to the no-slip condition at
stress vanish at the plane
that the shear
the wall and
of symmetry a
distance from the interface equal to the tube radius (a)!
vz(0)
=
For the static
;
dv z
dx(a)
B.5)
=
head measurement, there is no
net flow across any plane
normal to the z-axis:
rvz(x)dx = 0
(B.6)
Subject to these conditions, Eq. B.4 is integrated twice to give
Z(X) =
Fz(x)dx dx'
-
(ax
- x2/2
]
(B.7)
and integrated third time to give
dz
I
-d 3
=J
(B.8)
Pz(X)dX
x dx' dx
The electric force density is the averaged one
FZ(x) =
With assumption
(B.9)
= 0
P(y)Ez(x,y)d(ky)
b], field components in the
half-space are consistent
with the Laplacian potential Eq. B.2
Ex = Ecos(ky)exp(-kx)
;
:Ez = -EOsin(ky)exp(-kx)
(B.lOa,b)
223
where
Eo
kVo. In
view of
assumption
c]
the
by migration and convection.
transported only
inertialess ions
are
Thus, their trajectories
or characteristic lines are described by
b(
dx
dt
p
dt =
where vp =
(B.11)
+bEx = ±bEocos(ky)exp(-kx)
-
d
=
(v - v-z(x)+
bE&oi(ky)exp(-kx))
(B.12)
/k is the phase velocity of the traveling wave and the upper
and lower signs correspond
to positive and negative ions, respectively.
If now dt is eliminated between Eqs. B.11 and B.12 the result
dx(vp - Vz(x)
bEosin(ky)exp(-kx))
(B.13)
+ dy(T bEOcos(ky)exp(-kx)) = 0
is a perfect differential because the time dependence is eliminated with
the definition
of
(ky) in Eq. B.2.
vector potentials (in the frame
In what
amounts
to a determination
of
of the traveling wave) for the solenoi-
dal flow and field, Eq. B.13 is integrated to give
xvp - O(x)
where
view
(bEo/k)sin(ky)exp(-kx) = C+
is defined in
towards
(B.14)
Eq. B.1 and C.. are integration constants.
application of
an
injection
law,
With a
these constants
are
expressed in terms of the position (x,y) = (g,y±) on the interface where
the characteristic lines are intercepted:
(kx)vp- k(x))
Jsx+ sin(ky)exp(-kx) = sin(ky±)
(kxv,
~[ ~bE°
(B.15)
224
desired
one
that
is
corresponding
case
unnecessary to
it proves
because
characteristic line both
the proviso
coincide, and
same
the
species
thus
However, in the present
magnitude
enters and leaves the liquid
that a charged
the
the normal field
two solutions
between the
distinguish
has
field
normal
the
bulk.
of
velocity is
the liquid
field lines
into (out of) the liquid
) the
<
entry
of
can be injected only where
positive (negative) charge
E:-is directed
the point
to
and
characteristic lines
negligible,
(-1 < y
y
for
the interface where
line. Near
characteristic
of Eq. B.15
solutions
of the two
In general,
can be injected
a
where
given
bulk. Thus, with
from only one end of
a characteristic line, the injected charge densities have the functional
form
P
=
f+(
IEx(O,yO)I
(B.16)
field intensity (Eq. B.1la) evaluated
where the normal
entry
)
at the point of
is
IEX(0,y)
=
Eocos(ky±)
Thus, given assumption [d]
and
=
E(l
-
sin2 (kyo)1/2
(B.17b)
a trial stream function
(x), the charge
density at (x,y) is the algebraic sum of the individual charge densities
by eliminating
Po determined
Eqs. B.15,
Ex(B,y±) and sin(ky±) between
B.16 and B.17.
As might be
expected intuitively, there are points
(different for the
tions.
two species) for which Eq.
For the planar
(x,y) in the liquid
B.15 has no (real) solu-
half-space, this implies that the characteristic
line through that point does not intercept the injecting wall, and it is
expedient to simply set the charge density (and hence the force density)
there to zero.
However,
the more complicated
with a view towards generalizing
case of a liquid channel
the model to
bounded by two injecting.
225
walls spaced closely enough that the characteristic line through a given
point may
intercept the
interpret
the absence
critical
more distant wall,
of solutions
characteristic lines.
in
it is more
terms of
instructive to
critical points
Critical points
,
are
and
points of
zero force, and hence satisfy
(Vz(x) - vp)Z ± bE(x±y±)
Thus while y± = n/2 and -/2,
= 0
is given implicitly by
- vp) i bEosin(ky±)exp(-x)
(vz(x)
Critical characteristic lines are
points. For
for
(B.18)
= 0
(B.19)
those which pass through the critical
the planar half-space the critical
the two
species)
set
off regions
of
lines (again, different
the liquid
which are
not
accessible to injected charge. For the liquid channel with two injecting
and interacting walls,
corresponding
to
combinations of
possible destinations
This situation
the critical lines define regions
of charge
is complicated by
the
two
possible
on the enclosed
of four types
origins and
two
characteristic lines.
the fact that the
critical points do
not lie on the channel plane of symmetry.
With all of the quantities of interest expressed as integrals over space
(Eqs. B.7 to
of the
B.9), acceptable accuracy can be
net charge density
the net charge
to B.17,
on a relatively coarse grid
density is determined at each
the pressure gradient
profile are obtained in succession
The predicted velocity
of points. Once
grid point from Eqs.'B.15
the average longitudinal force density,
and the velocity
and B.7.
expected from evaluation
from Eqs. B.9, B.8
profile generates the new trial stream
function through Eq. B.1, and the procedure is repeated until either the
trial and predicted profiles are in satisfactory agreement or it becomes
clear that the two profiles are not converging.
226
In
a preliminary
profiles
computer implementation
did converge.
The results
of
indicate
this procedure,
that at
the two
least for
some
excitation amplitudes and frequencies, both forward and backward pumping
are simultaneously
consistent with the model.
is
the
predicted
for
appears to be
one.
initially
quiescent liquid,
the only accessible final state,
There is,
that backward
however, some suggestion
pumping is
may result
even when
initial
state.
excitation
forward pumping
At
pumping
provided it is a stable
stable state, and
it is not
is actually recorded, an
thus forward
directly accessible
frequencies
and amplitudes
from the
for
which
initial backward displacement
observed before the forward pumping
when the excitation is applied abruptly.
backward
from experimental observation
not always a
pumping
is sometimes
Because backward pumping
sets in, particularly
227
NOTATION
(MKS units are used unless otherwise noted)
a
tube inside radius (m)
A
cross sectional area of dead-end channels (Sec. 4.2.3)
A
(Sec. 6.3.4)
A+
positive ionic species (Sec. A.5)
b
average ion mobility (b+ + b_)/2
b+
positive and negative ion mobilities (m2/V-s)
bij
effective range of coulombic interaction (Sec. A.6)
B
(Sec. 6.3.4)
Bn
(Eq. 3.17)
B-
negative ionic species (Sec. A.5)
cb
local concentration of B- (m- 3 ) (Sec. A.6)
Ci
concentration of inert ions (Sec. A.6)
cr
concentration of reactive ions (Sec. A.6)
c+
molar concentrations of ionic species (moles/m3 )
C
neutral solute species (Sec. A.5)
Cn
(Eq. 3.20)
CE
encounter complex (Sec. A.5)
CT
transformed complex (Sec. A.5)
d
development length (m)
dr
mean separation of reactive ions (m) (Sec. A.6)
D
a8/ax
228
De
local effective diffusivity (m2 /s)
Dm
average molecular diffusivity (m2/s)
Dn
diffusivity of neutral species (m2 /s)
Dt
eddy diffusivity (m2 /s)
D+
diffusivity of ionic species (m2/s)
Dr
complex amplitude of perturbation radial electric displacment
Dx
-Dr
e
electronic charge, 1.6 x 1
en
Fourier
E
expansion efficiency (Eq. 4.9)
E
electric field (V/m)
Ew
electric field within expansion wall (Sec. 4.2.3)
Er
radial electric field in liquid at tuie wall (V/m)
Er
radial electric field in gas at tube wall (V/m)
Er
radial electric field inside tube wall (V/m)
Ez
axial component of electric field (V/m)
F
Faraday's constant, 96,500 (C/mole)
Fn
(Eqs. 2.2 and 3.27)
Fz
time-averaged axial electric force density (Sec. B.3)
G
generation rate (C/s-m3 ) (Sec. A.2)
G,G i
conductance of dead-end channel (Sec. 4.2.3)
i2 , i3
coefficients
19
(C)
(Eq. 3.30)
charging currents (A)
I
total axial current (A)
Ii
current from the ith sleeve segment to ground (A)
19.
axial current carried by liquid bulk (A)
229
In
modified Bessel function of the first kind and nth order
Is
convection or streaming current (A)
Iw
axial current carried by the tube wall (A)
IN
leakage current carried by dead-end channels (A)
j
'
3
current density (A/m2 )
3+
2
current densities carried by positive and negative ions (A/m )
'.w
2
exchange current density at liquid-solid interface (A/m )
JW
radial current density inside wall (A/m2)
k
Boltzman's constant, 1.38 x 10- 23 (J/K° )
k
wave number (2n/X)
km
mass transfer coefficient (m/s)
kn
wave number (
Kn
modified Bessel function of the second kind and nth order
kD
effective dissociation constant (Sec. A.2)
kL
Langevin's recombination coefficient (m3 /s)
kR
effective recombination constant (m3 /s)'(Sec. A.2)
k12
recombination rate constant (m3 /s)
k21
dissociation rate constant (m3 /s)
k2 3
rate constant (s-1 )
k3 2
rate constant (5s- 1
K
equilibrium constant (K = kD/kR)
K1
equilibrium constant (K1 = kl2 /k21 )
K2
equilibrium constant (K2 =k
+
K
coefficient (m/s)
n/L)
; i=l) or (m3 /s ; i=2)
2 3/k 32 )
230
K-
coefficient (m/s)
I
length of a sleeve segment (Chapter 5)
9.
length of one wire electrode (Chapter 6)
.
characteristic length (Sec. A.4)
L
tube length (m)
L
axial length of helical winding (m)
Le
linear expansion dimension (m)
n
density of neutral species (m- 3) (Sec. A.2)
N
number of dead-end channels (Sec. 4.2.3)
p
liquid pressure (Kg/m-s2 ) (Sec. B.3)
Pi
liquid pressure in expansion, inlet channel (Kg/m-s2 )
pO
liquid pressure in expansion outlet channel (Kg/m-s2 )
P
power (W)
P+
coefficients (Sec. 6.3.4)
Q
volume flow rate (m3/s)
Qdl
total charge in diffuse part of double layer (C)
Qe
total charge in insulating expansion (C)
(Qt
total charge in insulating tube (C)
r
radial coordinate
R
radius of external conducting cylinder (m)
R
resistance (Fig. 6.2)
R
recombination rate (C/s-m3 ) (Sec. A.2)
Re(Z)
real part of the complex argument Z
Re
electric Reynolds number (
Ry
hydrodynamic Reynolds number':( 2aU/v)
sU/oL)
231
s
total solute concentration (m- 3)
S
surface area (m2 )
Sc
Schmidt number (- v/Dm )
Sh
Sherwood number (m kmLe/Dm)
T
absolute temperature (K° )
T+
coefficients (Sec. 6.3.4)
U
mean liquid velocity (m/s)
U
potential of the average force (Sec. A.6)
vp
phase velocity (m/s)
vx
local time-averaged velocity (m/s)
vz
local time-averaged velocity (m/s)
VT
thermal voltage (kT/e
vs
liquid velocity at edge of diffusion sublayer (m/s)
V
volume (m3 )
Vo
standing wave amplitude (V)
w
thickness of the insulating wall of a tube or expansion (m)
x
coordinate measuring distance from interface into liquid
z
axial coordinate
zi
unsigned ionic valence (Sec. 2.3.3)
zi,zj
signed ionic valence (Sec. A.6)
OL
(Sec. 5.3)
aL
ion size parameter (Sec. A.6)
%eff
effective ion size parameter (Sec. A.6)
8
(Sec. 5.3)
etl
·
(Sec. A.4)
0.02? V)
liquid permittivity (F/m)
232
permittivity of free space (F/m)
SW
permittivity of tube wall (F/m)
dle
effective conductivity (
c1 + (2/a)os)
local liquid conductivity (S/m)
(lo
liquid bulk conductivity
os
surface
conductivity
here p = 0 (S/m)
(S)
conductivity of tube wall (S/m)
c(z,t) surface charge density averaged over tube circumference (C/m2 )
M(t)
surface charge density averaged cver entire tube (C/m2 )
+
surface charge densities (C/m2
)
conductivity (Sec. 6.3.3)
y
conductivity gradient (Sec. 6.3.3)
y
degree of dissociation in equilibrium (Appendix A)
rb
radial
rn
flux density of neutral species (m-2s-
r±
radial
r
flux density
flux density
of B- ions
of ionic
(m- 2 -1
s
species
)
(Appendix
)
A)
(Appendix A)
(m-2 s-1 )
electric potential (V)
On
Fourier coefficients (V)
$O
complex amplitude of perturbation potential
V)
time characterizing evolution of surface charge in expansion (s)
characteristic time (Appendix A)
temporal period of traveling-wave excitation (Appendix B)
time characterizing charging
2.1, 3.2, 5.3)
transient in insulating tube (Secs.
233
x1
time
characterizing
evolution of
surface
charge in
expansion
(Sec. 4.2.3)
a
relaxation time of the ion distribution (Appendix A)
Tc
relaxation time of the average ion concentration (Appendix A)
tcr
chemical relaxation time (Appendix A)
cd
diffusion time for neutral species (Appendix A)
D
diffusion time for ionic species (Appendix A)
Tex
duration of an experiment
'T%
charge relaxation time in liquid '(- e/1
'Tm
migration time (Appendix A)
)
time characterizing the nth Fourier mode (s)
tr
liquid residence time in expansion (
V/Q)
shear stress at interface (Kg/m-s2)
charge relaxation time in wall (e¢/ao)
s
transit
time
(L/v 8 )
p(r,z) local volume charge density (C/m3 )
p(x,z) volume charge density in diffusion sublayer in tube (C/m3 )
p(x,t) volume charge density in diffusion sublayer in expansion (C/m3 )
p(z,t) volume charge density averaged over tube cross section (C/m3 )
p(t)
volume charge density in turbulent core of expansion (C/m3 )
Pi(t)
3
volume charge density at expansion inlet (C/m )
Pm
mass density (Kg/m3 )
tP
= ao/2eb (Appendix A)
volume charge density at liquid-solid interface (
P+
= e[A+], eB-]
(C/m3 ) (Appendix A)
(a,z))
234
effective line charge density (C/m)
X
wavelength of standing- or traveling-wave excitation (2n/k)
Xm
molecular Debye length (m)
Xn
Fourier coefficients (C/m)
Xt
turbulent Debye length (m) (-eDt/a)
8
diffusion sublayer thickness (m)
viscous sublayer thickness (m)
in
absolute viscosity (Kg/m-s)
v
kinematic viscosity (m2 /s)
u
mean axial ion velocity (m/s)
uO
response in volts (Chapter 6)
angular frequency (s-1 )
zeta potential (V)
235
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