Effects of Temperature and Pressure on Particle Collection Mechanisms : Theoretical Review

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Office of Research
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EPA-600/7-77-002

January 1977

EFFECTS OF

TEMPERATURE AND PRESSURE

ON PARTICLE COLLECTION MECHANISMS:

THEORETICAL REVIEW
by

Seymour Calvert and Richard Parker

A.P.T. , Inc.
4901 Morena Boulevard, Suite 402
San Diego, California 92117

Contract No. 68-02-2137
Program Element No. EHE623A
EPA Project Officer: Dennis C. Drehmel

Industrial Environmental Research Laboratory
Office of Energy, Minerals, and Industry
Research Triangle Park, NC 27711
Prepared for

U.S. ENVIRONMENTAL PROTECTION AGENCY
Office of Research and Development
Washington, DC 20460
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ABSTRACT

A critical review and evaluation of the mechanics of aero-
sols at high temperatures and pressures are presented. The equa-
tions and models used to predict particle behavior at normal
conditions are discussed with regard to their applicability at
high temperatures and pressures. The available experimental data
are discussed and found to be inadequate to confirm the projec-
tions of aerosol mechanics at high temperatures and pressures.
A few examples of the effects of high temperature and pressure on
the collection efficiency and power requirement of typical col-
lection devices are presented. In general, particle collection
at high temperatures and pressures will be much more difficult
and expensive than the collection of similar particles at low
temperatures and pressures.
This report was submitted in partial fulfillment of con-
tract number 68-02-2137 by Air Pollution Technology, Inc. under
the sponsorship of the U.S. Environmental Protection Agency.
This report covers the period December 5, 1975 to September 10,
1976, and work was completed as of September 30, 1976.
111
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CONTENTS
Abstract iii

List of Figures v

List of Tables. „ vii

List of Symbols viii

Acknowledgement xii

Sections

Summary and Conclusions 1

Introduction 6

Fundamental Considerations 13

Collection Mechanisms 26

Properties 55

Example Applications 59

Bibliography 79
IV
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LIST OF FIGURES

No. Page
1

15
Collection efficiency of a nucleopore filter
as a function of absolute pressure
The effect of temperature and pressure on the
kinematic viscosity of air
Validity of Stokes' law at high temperature
and pressure
Validity of Stokes' law at high temperature
and pressure
The effect of temperature and pressure on the
ratio of the Cunningham slip correction factor
to the dynamic viscosity of air
The effects of temperature and pressure on the
particle inertial relaxation time
The effects of temperature and pressure on the
particle inertial relaxation time
The effect of interception on the particle
collection efficiency by a circular cylinder. .
The effects of temperature and pressure on
particle diffusivity
Thermal deposition velocity as a function of
temperature and pressure
The effect of pressure on the dif fusiophoretic
deposition velocity
Thermal coagulation of particles at high
temperature and pressure
Turbulent agglomeration tendency of particles
at high temperature and pressure
Sonic agglomeration of particles at high
temperature and pressure
Experimental and theoretical viscosity versus
temperature curves for air
Gas viscosity versus temperature
ii_
9

. . 28

. . 31

. , 33

. . 45

. . 50

. , 55

58
59
V
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No. Page

16 Mean free path of air molecules as a function
of temperature and pressure 61

17 Specific heat as a function of temperature
for air at 1 atm 64

18 The effect of temperature on the thermal
conductivity of air at 1 atm pressure 65

19 Molecular diffusivity of water vapor in air
as a function of temperature 67

20 The thermal expansion of silica brick at
high temperatures 68

21 Flow work (specific power) for impaction from
a round jet as a function of temperature and
pressure 70

22 The effects of high temperature and pressure
on the collection efficiency of a high
efficiency cyclone 73

23 The specific power requirements for a cyclone
as a function of temperature and pressure 75

24 The effects of high temperature and pressure
on the collection efficiency of a fiber bed .... 76

25 The effects of high temperature and pressure
on the inertial collection efficiency of a
packed bed 78
VI
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LIST OF TABLES
No. Page

1 Brownian Diffusion Reynolds Numbers for a
0.002 ym Diameter Particle at Various
Temperatures and Pressures 34

2 Numbers of Charges Acquired by a Particle:
a) Ion Bombardment; b) Ion Diffusion, 27°C;
c) Ion Diffusion, 1,100°C 39

3 The Effects of High Pressure and Drop
Diameter on the Growth of Drops by
Condensation 47

4 Effect of Temperature on the Agglomeration
of Charged Particles 51
vn
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LIST OF SYMBOLS

A - dimensionless constant in equation (14)

Aj - deposition area, m2

B - dimensionless constant in equation (14)

B- - ion mobility, s/kg

B - particle mobility, s/kg

c" - root mean square molecular velocity, m/s

C - isothermal slip coefficient, dimensionless
m r

Ct - temperature jump coefficient, dimensionless

C1 - Cunningham slip correction factor, dimensionless

Cn - drag coefficient, dimensionless

Cp - velocity of sound in gas, m/s

d - collector diameter, m
L-

di - drop diameter, m

d - initial drop diameter, m
o t- *

d - particle diameter, m

D - cyclone diameter, m

D - particle diffusivity, m2/s

D - turbulent diffusivity, m2/s

VG - molecular diffusivity, m2/s

e - unit electronic charge, e.s.u.

e - energy dissipation rate, m2/s3

E - collection efficiency, fraction

E - charging electric field strength, V/m

E - actual applied electric field strength, V/m

E - effective precipitating electric field strength, V/m

viii
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Fr " 8as resistance force, kg-m/s2

g - acceleration of gravity, m/s2

H - magnetic field strength, A/m

k - Boltzmann's constant, J/°K

kp - gas thermal conductivity, J/m-s-°K

k - particle thermal conductivity, J/m-s-°K

k. - thermal conductivity, J/m-s-°K

K - sonic agglomeration constant, m3/s
3.

K - coagulation constant, m3/s

K - inertial impaction parameter, dimensionless

Mp - molecular weight of gas, g/gmole

M - molecular weight of surface molecules, g/gmole

M - molecular weight of vapor, g/gmole

n - vortex exponent for cyclone, dimensionless

N. - ion number concentration, m~3

N,, - Knudsen number, dimensionless
K.n

N - particle number concentration, m~3

NT, - Peclet number, dimensionless
Pe

Nn - Reynolds number, dimensionless
Ke

NR r - fiber Reynolds Number, dimensionless

N~ - Schmidt number, dimensionless

p^ - vapor pressure in gas stream, Pa

- mean partial pressure of non-transferring gas, Pa
p - vapor pressure at drop surface, Pa

PP - gas partial pressure, Pa

p - vapor partial pressure, Pa

P - absolute pressure, Pa
IX
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Pt - penetration, fraction

q - particle charge, C

q - saturation charge, C

QP - gas volumetric flow rate, m3/s

Q - heat flux in "x" direction, J/m2-s
Jv

R - universal gas constant, J/gmole-°K

t - time, s

T, T , - absolute temperature, °K

T - temperature of incident gas molecules, °K
o

T - temperature of reflected molecules, °K

T - surface temperature, °K

u - centrifugal deposition velocity, m/s
L-

u, - deposition velocity, m/s

Ugp - average Brownian diffusion velocity, m/s

UD - diffusiophoretic deposition velocity, m/s

u - electrical migration velocity, m/s
C

Ur - superficial gas velocity, m/s

Up - gas velocity, m/s

u - magnetic deposition velocity, m/s

u - relative particle-gas velocity, m/s

u - gravitational settling velocity, m/s

u. - thermophoretic deposition velocity, m/s

Z - depth of granular bed, m
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Greek

a - fraction of diffusely reflected molecules, dimensionless
m

at - thermal accommodation coefficient, dimensionless

3 - proportionality constant in equation (49), dimensionless

Y - ratio of specific heats, dimensionless

6 - boundary layer thickness, m

An- - interception collection efficiency, fraction

e - dielectric constant, dimensionless

2, 2.
e - permittivity constant, C /N-m

n - single fiber collection efficiency, fraction
o

X - mean free path of gas molecules, m

Ur - gas absolute viscosity, kg/m-s

y - permeability constant, V-s/A-m

vr - gas kinematic viscosity, m2/s

PG - gas density, kg/m3

p. - liquid drop density, kg/m3

p - particle density, kg/m3

V - modified impaction parameter in equation (54), dimensionless

T - relaxation time, s

t - frictional shear stress in a gas, N/m2
XI
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ACKNOWLEDGEMENT

A.P.T., Inc. wishes to express its appreciation for excellent
technical coordination and for very helpful assistance in support
of our technical effort to Dr. Leslie Sparks of the E.P.A. and
Dr. Dennis Drehmel, E.P.A. Project Officer.
XII
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SUMMARY AND CONCLUSIONS

The objective of this study was to conduct a critical review
and evaluation of all available information concerned with the
mechanics of aerosols at high temperature and/or high pressure.
The specific goal was to determine the availability, adequacy,
and useful ranges of the models, equations, and experimental data
which may be used to predict the effects of temperature and pres-
sure on particle behavior.
The literature review turned up a number of reports of stud-
ies dealing with the development of high temperature and pressure
particle collection equipment. Most of these studies were con-
cerned with operational problems and overall pressure drops and
collection efficiencies of the equipment when operating at high
temperatures and pressures. They were not concerned with the
effects of temperature and pressure on the fundamental particle
collection mechanisms.
There have been a few studies concerned with the effects of
temperature and pressure on the basic collection mechanisms. In
general, these studies have used accepted theory for the mecha-
nics of aerosols at low temperature and pressure and extrapolated
it to high temperatures and pressures by adjusting the gas prop-
erties. They did not discuss the validity of this extrapolation,
other than to note that experimental verification is lacking.
Although experimental data were presented in some studies, they
were not sufficient to test the theory at high temperature and
pressure. Generally the experimental results have been in quali-
tative agreement with theoretical predictions.
The work presented by previous authors has been reviewed and
evaluated in the present study. Our primary intent has been to
identify the regions of temperature, pressure, and particle size
where existing theoretical and/or experimental information is
either adequate or inadequate.
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REVIEW
For most particle collection mechanisms, the collection
efficiency, E, is a function of the fluid resistance force, which
is proportional to the ratio of the Cunningham slip correction
factor, C' , to the gas viscosity, y,,. The Cunningham correction
factor is a function of the particle diameter, gas temperature,
pressure, and viscosity. It increases with temperature and de-
creases with pressure and particle diameter. The gas viscosity
increases with temperature and pressure, although the effect of
pressure is negligible below about 20 atm.
Therefore, for particles greater than a few tenths of a
micron in diameter, particle collection efficiency decreases as
temperature and pressure increase. For very small particles,
particle collection efficiency will increase with temperature
but will decrease with pressure.
The above conclusions are based on available theoretical
models. The data needed to confirm the extrapolation of the
theory of aerosol mechanics to high temperatures and pressures
are not presently available.
EVALUATION
The effects of high temperature and pressure on the proper-
ties of gases and particles have been studied theoretically and
experimentally. Gas densities, viscosities, molecular mean free
paths, thermal conductivities, and molecular diffusivities may be
predicted as a function of temperature and pressure with good
accuracy. The theories have been verified by experimental data.
There should be no difficulty in predicting the properties of
gases (or gas mixtures) at high temperature and pressure.
The effects of high temperature on particle properties gen-
erally are less important, and less well defined. The most im-
portant particle properties are: density, thermal conductivity,
coefficient of thermal expansion, and dielectric constant. The
density, thermal conductivity, and dielectric constant usually
are considered independent of temperature, although data to con-
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firm this for fly ash at extreme temperatures (over 1,000°C) have
not been found. The coefficient of thermal expansion of refrac-
tory material (such as silica) increases with temperature, but
is too small to be of much significance (approximately II in-
crease in length as temperature increases from 0°C to 1,000°C.)
The major difficulty in extending the theory of aerosol
mechanics to high temperature is to determine what form of the
Cunningham correction factor, C', is applicable. The Cunning-
ham correction factor is an empirical factor which was used to
match experimental data at room temperature and very low pres-
sure. From theoretical arguments, the empirical constants in
"C"' [as presented in equation (14)] are functions of the momen-
tum and energy accommodation of the gas molecules impinging on
the particle surface. That is, they depend on the efficiency
with which the molecular momentum and energy are transferred to
the particle.
Available theory and experiment indicate that the molecu-
lar momentum and energy accommodation coefficients decrease
with increasing temperature. This would mean that the gas
(molecule) resistance to the particle's motion would decrease
more with temperature than would be predicted using the low
temperature form for "C"' [equation (12)]. The magnitude of
the resulting decrease in the drag force could be as much as
50% for submicron particles. The drag force on large particles
would be affected much less.
The above discussion is based on theory and experiment for
very pure gases and surfaces. The effect of temperature on the
accommodation coefficient may be much different for actual ef-
fluent gases and fly ash particles.
Gas pressure has been found to have no effect on the ac-
commodation coefficient, and thus would not be expected to in-
fluence the applicable form of "C"'. High pressures, however,
decrease the molecular mean free path and increase the inter-
action between molecules. At pressures greater than about 20
atm (and at room temperature), the gas begins to depart from
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the perfect gas law. The pressure then has an influence on the
transport properties of the gas (that is, viscosity, thermal
conductivity, and molecular diffusivity). There should be no
problem determining the effects of high pressure on the particle
collection mechanisms, as long as the effects of pressure on the
gas are considered.
At high temperatures, the influence of pressure on the gas
properties is decreased greatly. For example, at 1,100°C, the
gas viscosity is effectively independent of pressures up to more
than 300 atm.
Another uncertainty in the collection of particles at high
temperature is in the magnitude of the thermal force. The ther-
mal force is a function of the energy accommodation of the mole-
cules at the particle surface. It is expected, therefore, that
the thermal force would be less than predicted at high tempera-
tures because of the less efficient energy and momentum transfer
from the molecule to the particle. Experimental verification of
this prediction is needed.
CONCLUSIONS
The theory and experimental data needed for predicting the
effects of temperature and pressure on the important gas prop-
erties are available and adequate. The effects of temperature
and pressure on particle properties are less important, and less
well documented. Data on the thermal conductivity and dielectric
constant for fly ash particles at extreme temperatures (greater
than 1,000°CJ could not be found.
The major uncertainty in applying the theory of aerosol
mechanics at high temperatures and pressures is in properly pre-
dicting the gas resistance force on the particle. High tempera-
tures are expected to decrease the accommodation coefficients
between the molecules and the particle, and thereby decrease the
gas resistance relative to low temperature theory. This effect
would be most important for small particles and low pressures,
and would have its largest influence on the collection of par-
ticles by Brownian diffusion.
4
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However, the collection of particles larger than a few
tenths of a micron in diameter is expected to be much more dif-
ficult at higher temperatures. High pressures further aggravate
the problem.
Experimental measurements of the drag force and particle
diffusivity at high temperature and pressure are needed to ver-
ify the above predictions, and to modify existing theory for ap-
plication to high temperature and pressure particle collection.
It is unlikely that a new particle collection device will
be devised which can collect particles at high temperature and
pressure more efficiently than at standard conditions. It is
more probable that high temperature and pressure particle clean-
up will have to be achieved by equipment (such as cyclones,
metal or ceramic filters, granular beds) operating at higher
costs (larger power consumption) than would be needed for oper-
ation at standard conditions.
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INTRODUCTION

In many industrial situations, the life of critical com-
ponents, such as turbine blades and heat exchanger tubes, is
limited by the extent of erosion and corrosion damage that re-
sults from the presence of particulate matter in the process
gas streams. It is often economically desirable, therefore,
to remove the particulate matter while the gas is still at a
high temperature and pressure, prior to passing the gas through
heat exchangers, gas turbines, or other critical equipment.
Although high temperature and pressure particle collection
has been under investigation for over thirty years, no fully
satisfactory solution to the problem has been achieved. This
is in contrast to the great advances that have been made in
the collection of particulate matter at low and intermediate
temperatures. To better understand, develop, and evaluate par-
ticle collection equipment for high temperature and pressure
gas cleaning, it is important to obtain a firm understanding
of the effects of high temperature and pressure on the basic
particle collection mechanisms.
The basic mechanisms by which particles can be removed
from gas streams have been discussed by many authors, inclu-
ding: Fuchs (1964), Green and Lane (1964), and Hidy and
Brock (1970). The theory presented by these authors has been
developed for application to moderate temperature and pressure
particle collection. In order to extrapolate this theory to
high temperatures and pressures, it is not sufficient merely
to insert values for the gas and particle properties at high
temperature and pressure. It also is necessary to look at the
fundamental laws and assumptions on which the theory is based,
and thereby to determine whether or not the theory is valid at
high temperatures and pressures.
REVIEW OF PREVIOUS WORK
Fundamental investigations of the mechanics of aerosols
at high temperatures and pressures are very scarce. There
6
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have been few theoretical studies, and there have been even fewer
experimental studies to evaluate the theory. Most of the high
temperature and pressure experimental studies reported in the
literature have dealt with the operational characteristics of
specific particle collection equipment. Only general, qualita-
tive comparisons between theory and experiment were made.
Silverman (1955) discussed the technical aspects of high
temperature gas cleaning for steel making processes, and pre-
sented some results of attempts to use inertial cleaners, wet
cleaners, filtration units, and electrostatic precipitators at
moderately high temperatures (250°C to 600°C). He presented
the general theory describing particle collection mechanisms
in fiber filtration, and he used it qualitatively to explain
some of the experimental results. In general, he found that
particle collection is more difficult at high temperature, and
that inertial impaction and diffusion are the most important
mechanisms for high temperature filtration.
Thring and Strauss (1963) predicted the effects of high
temperature on particle collection mechanisms for temperatures
up to 1,600°C. They also presented the results of some experi-
mental studies for fiber filters up to 980°C, and for pebble
bed filters and electrostatic precipitators up to 650°C. Their
theoretical study considered electrical migration, inertial
impaction, direct interception, diffusion, and thermal precipi-
tation. The available experimental data were for particle col-
lection devices involving complicated combinations of collection
mechanisms, and therefore only a qualitative comparison between
experiment and theory could be made. Some of their results are
reported by Strauss (1966) .
To determine the effects of ash deposition on turbine
blades at high temperature, Morley and Wisdom (1964) studied
the deposition of ash particles onto a cylinder placed in the
outlet gas stream of a brown coal combustor (700°C). They made
no attempt to compare the amount of deposition with theory.
They tried cooling the cylinder (400°C) and did notice an in-
crease in deposition. This was attributed to thermal deposition.
7
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Strauss and Lancaster (1968) calculated the effects of high
temperature (800°C) and high pressure (100 atm) on the fundamen-
tal particle collection mechanisms, and concluded that the col-
lection efficiencies of all cleaning processes will be reduced
under these conditions. They considered electrostatic precipi-
tation to be the most promising method for collecting particu-
late matter at high temperature and pressure. Their theoretical
study included the effects of temperature and pressure on gas
properties, particle properties, and the important collection
parameters. They discussed some of the earlier experimental
work but were unable to make any quantitative comparisons with
theory.
Gussman and Sacco (1970) investigated the hazards that
might arise from the presence of aerosols in a high pressure
environment. They were interested in the effects of pressure
on the formation and behavior of aerosols in the atmosphere of
diving vessels (to 1,000 ft). They were concerned with an oxy-
gen-helium atmosphere and pressures up to 34 atm. They consid-
ered various air cleaning mechanisms and did experimental work
with electrostatic precipitation at this pressure. Their work
is of limited value to the present study because they were only
interested in particles smaller than 0.1 pm.
Spurn>? et al. (1971) conducted a theoretical study of the
effect of temperature (to 1,500°C) and pressure (to 50 atm) on
the collection efficiency of a nucleopore type filter. They
considered the collection mechanisms of inertial impaction,
direct interception, and diffusion. They also conducted an
experimental study of the effect of pressure on the collection
efficiency. Their results are shown in Figure 1. They were
concerned primarily with low (vacuum) pressure, but their high
pressure data do confirm their prediction that the efficiency
is not affected greatly by high pressures.
Calvert et al. (1972) investigated the effects of pressure
and temperature on the collection of particles in wet scrubbers.
Their study was limited to scrubber operating conditions of 1 to

8
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100
•I I
u
x
w
(H
U
X
o
t-^
H
U
in
. i
. !

U
20
0

0.01
1.0

PRESSURE, atm
10.0
100.0
Figure 1. Collection efficiency of a nucleopore filter as a function of
absolute pressure. From Spurn^, et al.
-------
10 atm pressure and temperatures from 0°C to 100°C. The collec-
tion mechanisms that were considered included: inertial impac-
tion, direct interception, diffusion, thermophoresis, electrical
migration, centrifugal deposition, and diffusiophoresis. No
experimental data were presented.
Kornberg (1973) experimentally investigated the high tem-
perature (to about 450°C) filtration of particles by diffusion,
and compared his results with theory. He concluded that current
diffusional filtration theory is adequate for predicting filter
efficiencies at high temperature. This work is discussed in more
detail in a later section.
A fairly recent study, reported by Rao et al. (1975), pre-
sents a review of the earlier theoretical work and also dis-
cusses some collection mechanisms and particle agglomeration
methods which were not considered in previous studies. In
general, particle collection efficiencies for all mechanisms
were predicted to decrease as temperature and pressure increase
for particles larger than a few tenths of a micron in diameter.
No experimental data were presented.
The high temperature and pressure particle collection
models described in the above studies were based on projections
of the available theory to high temperature and pressure condi-
tions. Basically, this involved adjusting the gas properties
to their high temperature and pressure values. There has been
no critical review aimed at determining the uncertainties and
applicability of this theory at high temperatures and pressures.
Also no experimental data have been found which could be used
to confirm the validity of extrapolating the theory of the mech-
anics of aerosols to high temperatures and pressures.
In addition to the above studies dealing with the mechanics
of aerosols at high temperature and pressure, there have been a
number of experimental investigations concerned with high tem-
perature and pressure particle collecting equipment. These
studies do not present sufficient data to evaluate the theory
of the basic particle collection mechanisms.

10
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Experimental measurements of the efficiency of cyclone
separators at high temperatures and pressures have been reported
by Parent (1946) and by Yellott and Broadley (1955) . Parent
considered temperatures up to 650°C. The collection efficiency
was found to drop off as the temperature was raised, for a con-
stant pressure drop through the cyclone. To maintain a constant
efficiency at a higher temperature (increasing from 25°C to
550°C), approximately 2% times the pressure drop was required.
Yellott and Broadley (1955) considered the efficiency and
pressure drop of cyclones operating at temperatures up to 700°C
and pressures to 5 atm. They also found that efficiency de-
creased with increasing temperature. To maintain a 95% effi-
ciency as the temperature increased from 25°C to 550°C, over
three times the pressure drop was required.
The filtration of hot gases has been studied by many au-
thors, including: Snyder and Pring (1955), Billings, et al.
(1955, 1958a, 1958b, 1960), First, et al. (1956), Kane, et al.
(1960), Spaite, et al. (1961),and Lundgren and Gunderson (1976).
Snyder and Pring were concerned with the operational and design
considerations for the filtration of hot gases. Billings, et
al. performed pilot-plant and full scale performance tests of
slag-wool filters operating at atmospheric pressure and temper-
atures over 500°C. They were concerned primarily with the
operational problems of the slag-wool filter. Their data did,
however, confirm that the filter efficiency decreases with in-
creasing temperature.
First, et al. (1956) measured the efficiency of a ceramic
fiber filter at temperatures up to 800°C, for various filter
depths, fiber diameters, and face velocities. In all cases the
collection efficiency decreased with increasing temperature as
predicted, qualitatively, from impaction theory. Kane, et al.
(1960) reported similar tests on a ceramic fiber filter at a
temperature of about 1,000°C. They were concerned with pres-
sure drop and dust loading, and did not measure the efficiency
as a function of temperature. Spaite, et al. (1961) discuss

11
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the operational problems and applications of high temperature
filtration.
Lundgren and Gunderson (1976) investigated a high-purity
"microquartz" fiber filter medium at temperatures up to 540°C
(1,000°F). Qualitative agreement between theory and experiment
was obtained, however the theoretical penetrations were only
within two orders of magnitude of the experimental penetrations.
More work is being done to improve the theory.
High temperature and pressure electrostatic precipitation
has been studied by many authors over the past thirty years,
including: Koller and Fremont (1950), Thomas and Wong (1958),
Shale, et al. (1963, 1964, 1965, 1967, 1969), Robinson (1967,
1969), and Brown and Walker (1971). These studies dealt pri-
marily with the problems of corona generation and the current-
voltage characteristics of electrostatic precipitators at high
temperature and pressure. Although Brown and Walker (1971)
demonstrated the feasibility of electrostatic precipitation at
a temperature of over 900°C, and at a pressure of 7 atm, the
particle migration velocity was much lower than normal for a
given applied voltage. This was attributed to the larger gas
viscosity at high temperature.
In the following sections, the equations and models for
predicting the behavior of particles at high temperature and
pressure are presented and critically reviewed. Emphasis has
been placed on identifying the regions of temperature, pressure,
and particle size where existing theoretical information is
inadequate.
In the final section, a few examples of the effects of
high temperature and pressure on particle collection devices
are presented. The available experimental data are not suffi-
cient to verify the theory for these examples.
12
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FUNDAMENTAL CONSIDERATIONS

The models and equations describing particle collection
mechanisms have some fundamental theoretical considerations in
common. The most important of these is the description of the
drag or resistance force which is exerted on a particle by the
gas, when the particle moves relative to the gas stream. In
particle collection theory, the motion of the particle relative
to the gas stream generally is characterized by a deposition
velocity, u_,.
In most particle collection equipment, the flow is turbu-
lent and it can be assumed that there is complete mixing of
the suspended particles. In this case, the deposition velocity
can be related to the particle collection efficiency, E, through
the particle penetration, Pt, as follows:
Pt = 1 - E = exp
where "A," is the deposition area, and "Q-" is the volumetric
flow rate of the gas. For any collection device, the penetra-
tion is defined as:

Outlet concentration ,--)
Inlet concentration

The deposition velocity for any collection mechanism de-
pends on the force balance between the driving force (deposition
force) and the resistance force of the gas. In this section,
the basic equations describing the gas resistance force will be
reviewed with regard to their application to high temperature
and pressure particle collection. The specific driving force
for each collection mechanism will be reviewed in a subsequent
section.
13
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STOKES' LAW
The major difference between the collection of particles at
normal conditions and at high temperature and pressure is in the
resistance force of the gas. For a rigid, spherical particle
moving through a continuous, viscous gas at constant relative
velocity, and for negligible inertial effects arising from the
gas being displaced by the particle, the resistance force, Fr,
is given by Stokes1 law:

Fr = - 3, yG dp ur (3)

where "Pg" is the gas viscosity, "d " is the particle diameter,
and "u " is the relative velocity between the particle and the
gas. The negative sign indicates that the drag force is oppo-
site to the direction of motion of the particle.
If the mean free path of the gas molecules (related to the
effective spacing between the gas molecules) is not negligible
with respect to the particle diameter, the particle no longer
sees the gas as a continuous fluid but rather as a finite num-
ber of discrete molecules. In this case, equation (3) usually
is modified by an empirical correction factor, C1, so that the
modified Stokes1 law becomes:
- 3iT y_, d u
F = - G_JL_
r f
The correction factor, C', is often referred to as the "Cunning-
ham slip correction factor" because of the pioneering work of
Cunningham (1910). The slip correction factor will be discussed
in more detail later in this section.
Equations (3) and (4) are strictly valid only for very
small Reynolds numbers, ND . That is,
Ke
d u pr
NRe ' P „! * ° (5)
14
-------
where MPG" is the gas density. According to Schlichting (1968) ,
equation (3) may be considered satisfactory for:
NRe

For larger Reynolds numbers, equation (4) can be written
in the form:
IT d 2 pr u 2
G r
F = - C p r m
r LD \ Tc^ / (7)

where MC~" is the drag coefficient. The drag coefficient for
various Reynolds number ranges can be obtained from equations
(8), (9), and (10) , as presented by Fuchs (1964) .

Stokes' Law /Nn < ~ I\ Cn = -^- (8)
\ Re / D N
Re

Oseen's solution /ND < ~ 3\ Cn = r,— (l + ^N^, | (9)
\ Ke / u NRe \ lo Ke/

Klyachko's solution (3
-------
temperatures, the gas is both less dense and more viscous so
that the inertia of the displaced gas is much less important
in comparison to the viscous forces. Figure 2 shows the kine-
matic viscosity of air as a function of temperature and pres-
sure.
The Reynolds number [equation (5)] also depends on the par-
ticle velocity relative to the gas. For ,a given particle velo-
city and diameter, Figure 3 may be used to determine the region
in which the drag coefficient is given by Stokes1 Law [equation
(8)]. For example, if the particle velocity is 100 cm/s, the
vertical axis reads directly as particle diameter in microns.
The area below each curve is the region in which equation (8)
is adequate. Therefore the drag coefficient for a 5 urn diameter
particle moving at 100 cm/s would be given by equation (8) for
all temperatures shown at atmospheric pressure. At a pressure
of 15 atm, equation (8) would only apply for temperatures lar-
ger than about 890°C. Figure 4 is an expansion of the vertical
axis in Figure 3. Where equation (8) is not sufficiently accu-
rate, equations (9) or (10), or the drag coefficients presented
by Lapple and Shepherd (1940) should be used.
If it is assumed that equation (4) is valid at high tem-
perature and pressure, the effects of temperature and pressure
will be contained in the terms "UG" and "C1". For most collec-
tion mechanisms, the particle deposition velocity is inversely
proportional to the gas resistance force, and therefore propor-
tional to the ratio "C'/V "• The effects of high temperature
b
and pressure on the ratio "C'/yG" for air, are plotted as a
function of particle diameter in Figure 5. "C1" was calculated
from equation (12) presented below.
At atmospheric pressure (solid lines), and for particles
larger than a few tenths of a micron in diameter, the ratio
"C'/y^" decreases with increasing temperature. This is because
G
the gas is more viscous at higher temperature and therefore is
more resistant to particle motion relative to the gas stream.
For large particles, "C"1 becomes very nearly unity and the

16
-------
e
u
CO
C
ro
C/3
U
w
200
400 600 800 1,000 1,200

TEMPERATURE, °C
Figure 2. The effect of temperature and pressure
on the kinematic viscosity of air.
17
-------
200
400 600 800

TEMPERATURE, °C
1,000
1,200
Figure 3. Validity of Stokes1 law at high temperature
and pressure.
18
-------
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^ 9
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i 8
i— i
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a 7
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O
£ 6
m
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^TTT
— — H
T: "~
~^*

•; ,;:;,•
= 10 c
—-7
i , i i
l
1,
, ' , ; , : ' t/rr -
itm ^T , . .-
, : : f . i — n —
i
-15 atm— ^
I • . ; i . t^H
rj\ , | ;i !i
"• p — | p-. — •
1 4--. 1 .!• ^\
-i — • ' f • • • • ; • • • — t-|-t-
- — T , j ; - i . . . i i ; -
1 -it ' • ' ' ' i ' 1
000 1,2
TEMPERATURE, °C
Figure 4. Validity of Stokes1 law at high temperature
and pressure.
19
-------
CONDI1'IONS
20°C, 1 atm
500°C, 1 atm
1,100°C, 1 atm
20°C, 15 atm
500°C, 15 atm
1,100°C, 15 atm
0.1
0.5 1.0

PARTICLE DIAMETER, ym
5.0
10.0
Figure 5. The effect of temperature and pressure on
the ratio of the Cunningham slip correction
factor to the dynamic viscosity of air.
20
-------
ratio "C'/VG" becomes independent of particle size. The fluid
resistance force is still a function of particle diameter [equa-
tion (4)] but the temperature and pressure functionality of the
resistance force becomes independent of particle size. For
small particles, the ratio "C'/VG" increases with increasing
temperature because the larger molecular mean free path at high
temperature causes a decrease in the gas resistance and hence
an increase in "C1".
At 15 atm pressure, the molecular mean free path is reduced
and "C"' is near unity, even for small particles and high tem-
peratures. The effect of high pressure, therefore, is to in-
crease the resistance force on very small particles, and hence
decrease the ratio "C'/y^"-

SLIP CORRECTION FACTOR
The major uncertainty in the use of equation (4) at high
temperatures and pressures is in the Cunningham slip correction
factor, C', used to modify Stokes' Law. It is an empirical
factor generally based on the data of Millikan (1923a, b) , and
is given by Davies (1945) as:
C1 = 1 + N
Kn
1.257 + 0.400 exp (-1.10/NKn) (12)
where "N" is the Knudsen number, defined by
NKn =
and where "A" is the mean free path of the gas molecules. The
temperature and pressure dependence of the mean free path is
discussed later, in the section concerned with gas properties.
The constants in equation (12) are based on experimental
data for Knudsen numbers from 0 to greater than 100 (Millikan,
1923b) . The experiments were at low pressure (about 0.003 atm)

21
-------
and room temperature (20°C to 25°C). Therefore some considera-
tion must be given to their applicability at high temperatures
and pressures.
Equation (12) may be written in the general form:

C' = 1 + NKn[A + B exp (-c/NKn)J (14)

Equation (14) may then be broken down into the two cases:

C' = 1 + A NKn (15)

for small NL; and

C' = 1 -«- (A + B)NKn (16)

for large NK . The constant "c" in equation (14) is used to
bridge equations (15) and (16) for intermediate Knudsen numbers.
The theoretical basis for equation (15) was discussed by
Millikan (1923a), and for air and oil drops at 23°C:

A = 0.998 /—_ _ i\ (17)
' m '

where "a " is the fraction of molecules that are diffusely re-
in
fleeted from the particle surface. The constant "0.998" was
obtained using the definition of the mean free path presented
by Davies (1945) and attributed to Chapman and Enskog [see equa-
tion (55)]. Millikan states that equation (17) is valid as long
as the term "2/otm" is not very large compared to unity.
At large Knudsen numbers, the gas molecules are independent
of each other and the particle is in the free molecular regime.
The drag force on the particle depends on the momentum transfer
between the gas molecules and the particle surface. Millikan
(1923b) showed that equation (16) is also a function of "am"
and determined bounds for the constant "A+B". That is,
22
-------
(A + B) = 2.455 for am = 0 (18)

(A + B) = 1.612 for am = 1 (19)

Therefore, the constants in equation (14) are a function of
the momentum transfer between the gas molecules and the particle
surface, and most likely are bounded by equations (17), (18) and
(19). The effects of temperature and pressure on the constants
in equation (14) will come in through the factor "a ", which may
be considered as a momentum accommodation coefficient.
A recent paper by Willeke (1976) discusses the temperature
dependence of particle slip in a gaseous medium. He considered
temperatures up to 727°C (1,000°K). He assumed that Millikan
(1923a) had recorded the maximum range of reflection coeffi-
cients (momentum accommodation), and estimated that the maximum
error in the drag force (and hence slip correction) resulting
from a lack of knowledge of the exact degree of accommodation
would be ±3$.
The maximum range of reflection coefficients found by Mil-
likan was from 0.79 (air-fresh shellac) to 1.00 (air-mercury).
These measurements were at room temperature. It is possible
that at extreme temperatures, the degree of accommodation could
be even lower than that measured by Millikan.
Wachman (1962) reviewed the theory and experimental data
for momentum transfer between free molecules and surfaces. He
showed that the drag coefficient is proportional to the square
root of the energy (thermal) accommodation coefficient, ou,
where "a." generally can be expressed by:
T - Tn
a = -^ i- (.20)
Ts - Tg

and where "T " is the absolute temperature of the incident gas
&
molecule, "T " is the temperature of the reflected gas molecule,
and "T " is the temperature at the particle surface.
23
-------
The experimental data reviewed by Wachman (1962) generally
showed "at" decreasing with increasing gas temperature. This
would imply that the drag coefficient, and hence "a " would
also decrease with increasing temperature and thereby influence
the constants in equation (14) . The accommodation coefficient
was found to be independent of gas pressure.
Other authors have also found the accommodation coeffi-
cients to decrease with increasing temperature, including:
Stickney (1962), Wachman (1966), Goodman and Wachman (1967),
Byers and Calvert (1967) , Goodman (1969) , and Ramesh and Mars-
den (1974).
Ramesh and Marsden (1974) obtained the following empirical
equation for the slope of the "at" versus surface temperature
curve, for nitrogen impinging on a surface.
d a.
= (-13.7 + 0.09 M ) x 10-14 (21)
d Ts

where "Ms" is the molecular weight of the surface material.
From equation (21), it can be predicted that "a." will decrease
with increasing temperature for any surface material whose mo-
lecular weight is less than about 150. If it is assumed that
fly ash is 50% Si02, 25% Fe203, and 25% A1203, then the average
molecular weight will be approximately 100.
From the above discussion, it can be inferred that high
temperature drag forces will be smaller than would be predicted
by using equations (4) and (12) . The constants in equation (12)
should increase with temperature to account for the less effi-
cient momentum transfer between the molecules and the surface
observed at high temperatures. These observations, however,
were based on data taken with clean, pure surfaces, and gases.
Wachman (1962) notes that the accommodation coefficient can
increase significantly with the buildup of an adsorbed gas
layer on the surface. Therefore it is not possible to infer
any quantitative information concerning the accommodation co-
efficient of effluent gases on fly ash particle surfaces.
24
-------
If the constants in equation (14) are bounded by equations
CIS) and (19), the maximum increase in "C"' for large Knudsen
numbers would be about \% times (that is, if a decreases from
m
1 to 0). For small Knudsen numbers, the increase in "C"' would
probably be smaller (because NK is smaller) but equation (17)
is not valid as ot^O , so the maximum allowable size of the con-
stant "A" in equation (14) cannot be determined.
In conclusion, therefore, it will be necessary to conduct
an experimental study to determine the value of "C"' at high
temperature. From the above discussion it is expected that at
high temperature, "C1" will be larger than predicted by equa-
tion (12) .
25
-------
COLLECTION MECHANISMS

In this section, the effects of temperature and pressure on
particle collection mechanisms and particle agglomeration mecha-
nisms are reviewed. In all cases, the resistance of the gas has
been assumed to be as predicted by equation (4). The validity
of equation (4) at high temperatures and pressures is the major
uncertainty in the theory presented below.

INERTIAL IMPACTION
One of the most important mechanisms for the collection of
particles greater than a few tenths of a micron in diameter is
inertial impaction. Inertial impaction is the collection of
moving particles by impinging them on some target. The relative
effect of inertial impaction for different particles and flow
conditions may be characterized by the inertial impaction para-
meter, "Kp", defined as:
C' p^ d * u
K = E-J2 § (22)
P 9 yGdc

where "p " is the particle density, "u " is the gas velocity,
and "d /2" is a characteristic length for the collector.
The inertial impaction parameter is equivalent to the ratio
of the particle stopping distance, "x ", to "d /2". The particle
o w
stopping distance is that distance the particle would travel be-
fore coming to rest if injected into a still gas at a velocity
"Up", when only the fluid resistance force acts on the particle.
By considering the particle stopping distance divided by "UG",
the particle's inertia can be characterized by a relaxation time,
"T", defined as:

t. !s. 5^ .c' *p V (2S)
u,, 2 ur 18 vr
(j b (j
A large relaxation time indicates that a particle will take
a relatively longer time to come to equilibrium with the gas

26
-------
stream when injected into the stream with some initial velocity.
Therefore a larger relaxation time implies that the particle en-
counters less viscous resistance relative to its momentum in the
gas stream. Figures 6a and 6b show the effect of high tempera-
ture and pressure on the particle relaxation time, plotted as a
function of the actual particle diameter (for unit density par-
ticles) .
In Figure 6a, curve 1 is for standard conditions while
curves 2 and 3 are for elevated temperatures and pressures. The
curvature in curve 2 reflects the effect of high temperature on
the molecular mean free path, and thus on the viscous drag of
the gas on small particles. Curve 3 shows that high pressure
effectively nullifies any beneficial effects of high temperature
at small particle diameters.
Below a few tenths of a micron, inertial forces are so small
that impaction is not an effective collection mechanism. There-
fore the reduced drag noticed at high temperatures for very small
particles may not have much significance.
Figure 6b shows the effect of high pressure at low temper-
ature. It is apparent that high pressure particle collection by
inertial forces will be somewhat more difficult than low pressure
collection. However, this effect is only significant for sub-
micron particles.
The inertial impaction parameter [equation (22)] is obtained
by non-dimensionalizing the particle's equations of motion. The
equations of motion are derived from Newton's second law of mo-
tion and the modified Stokes' law [equation (4)], assuming only
the drag force acts on the particle. The characterization of
inertial impaction by the impaction parameter of equation (22)
should be as valid at high temperature and pressure as the as-
sumption of the modified Stokes' law.

DIRECT INTERCEPTION
A particle is collected by interception when its surface
(not center of mass) touches the surface of the collector. This

27
-------
50
• >
«
10 -.
m
•
• •
. '
'
-
.-.
NO. CONDITIONS
1 20°C, 1 atm
2 1,100°C, 1 atm
3 1,100°C, 15 atm
10- 10

PARTICLE RELAXATION TIME, s
- 3
Figure 6a. The effects of temperature and pressure on the particle inertial
relaxation time.
-------
I
: .
NO. CONDI''IONS
1 20°C, 1 atm
2 20°C, 15 atm
0.1
10-' 10-
PARTICLE RELAXATION TIME, s
Figure 6b. The effects of temperature and pressure on the particle inertial
relaxation time.
-------
can happen without crossing any gas streamlines. Thus intercep-
tion occurs if the streamline the particle rides on passes within
one particle radius from the collector surface.
Interception is a function of the relative dimensions of
the particle and the collector, and of the flow field. The flow
field, or pattern of streamlines, is a function of the charac-
teristic Reynolds number. The effect of increased temperature
and pressure will be on the kinematic viscosity and therefore
on the Reynolds number. The kinematic viscosity of air was pre-
sented as a function of temperature in Figure 2.
For a circular cylinder, Fuchs (1964) predicts the increase
in impaction efficiency resulting from interception, An.:, will
be a function of the Reynolds number as follows:

An. « - 1 - (24)
1 (2 - *n 2
where "NR £M is the Reynolds number based on the cylinder dia-
meter and free stream velocity. Equation (24) is valid only for
small Reynolds numbers (NRef
-------

1,
o
oi n f\
PS u.b
N

rH
3

^—^ 0 A

t-r
g
l_l

fid
rn
::;;;;;;•
rv'
n ?
H U*Z

n i

0 JlauJi
0.05

' : '

0.1

+
•*

! iii I
_|_ .(!..
liil!

«•"-•
• «--
"* -- -

_l_4_

i
i
i. !..
T... I,
:
: : i
/
i' r

y*
. '
V
_|_

J f

i (
0.5 1
N
Re£
Figure 7. The effect of interception on the
particle collection efficiency by
a circular cylinder.
31
-------
and the boundary layer thickness will be valid. That is, from
Schlichting (1968):
^ X f o r *\
5
-------
lOr
.,,
B
o
w
ij
u
|H
[ I
irl
•
NO. CONDITIONS
1 20°C, 1 atm
2 20°C, 15 atm
3 1,100°C, 1 atm
4 1,100°C, 15 atm
10
10
10

10
PARTICLE DIFFUSIVITY, cm2/s
Figure 8. The effects of temperature and pressure on particle diffusivity.
-------
where "B " is the particle mobility. This equation assumes that
the drag force on a particle is proportional to the velocity of
the particle. This assumption is valid for very small Reynolds
numbers where the inertia of the fluid displaced by the particle
is negligible (NR <'vl) . Fuchs (1964) presents mean velocities
for particles undergoing Brownian motion at standard conditions.
The maximum Reynolds number occurs for the smallest particle
(d =0.002 ym) which moves at the largest velocity (4,965 cm/s).
Reynolds numbers at other temperatures and pressures can be ob-
tained using the kinematic viscosity curves presented in Figure
2, and noting that:
« m" (28)
where "u"BD" is the average velocity resulting from Brownian mo-
tion.
Particle diffusion Reynolds numbers for a variety of temper-
atures and pressures and for d =0.002 ym, are given in Table 1.

Table 1. BROWNIAN DIFFUSION REYNOLDS NUMBERS
FOR A 0.002 urn DIAMETER PARTICLE AT
VARIOUS TEMPERATURES AND PRESSURES
Conditions NR
20°C, 1 atm 0.007
20°C, 15 atm 0.110
1,100°C, 1 atm 0.001
1,100°C, 15 atm 0.016
Even in the worst case (20°C, 15 atm), the Reynolds number is
very small (NR =0.1) and therefore the application of equation
(27) should be valid at high temperature and pressure.

34
-------
The particle mobility, Bp, is the proportionality constant
between the drag force and the particle velocity, and is usually
given as:
C'
Bn = (29)
P 3, y, dp

Equation (29) comes directly from the modified Stokes law. There-
fore the use of equation (29) [and therefore equation (26)] at
high temperature and pressure depends on the applicability of
Stokes' law as discussed in the previous section.
Whenever a particle travels in a moving gas stream, it be-
comes necessary to consider the particle transport by convec-
tion as well as by Brownian diffusion. The Peclet number, Np ,
characterizes convective transport relative to Brownian diffu-
sion. That is:
yr d
Npe - -^ (30)

In general, the collection efficiency by diffusion is in-
versely proportional to some power of the Peclet number. The
principal temperature dependent term in equation (30) is the
particle diffusivity. However, the gas velocity will be a func-
tion of temperature and pressure if the mass flow rate is held
constant.
Kornberg (1973) experimentally investigated the high tem-
perature (to about 450°C) filtration of particles by diffusion,
and compared his results with the theory of diffusional deposi-
tion onto single, cylindrical fibers. He assumed the Cunningham
correction factor to be a function of temperature through the
mean free path only. If the effects of temperature on accommo-
dation coefficient (discussed in the previous section of this
report) are important, it would be expected that the theory used
by Kornberg would underestimate the experimental efficiencies.
This did not appear to be the case, however the experimental
uncertainty was large enough to have masked this effect.

35
-------
Kornberg's results indicated that the single fiber collec-
tion efficiency by diffusion, n , is proportional to absolute
s
temperature as follows:
n
-------
e = -. (32)
EP
Strauss (1966) , states that the dielectric constant is not a
significant function of temperature. Tassicker (1971) and Masek
(1973), however, have measured dielectric constants for fly ash
in AC electric fields and found that they are a function of tem-
perature at least up to 220°C. At low frequencies, the dielec-
tric constant increases with increasing temperature. At high
frequencies it is relatively independent of temperature.
The possible increase in the dielectric constant at high
temperature would not change the migration velocity very much.
At very large values of the dielectric constant, the term "e/
(e+2)" approaches unity. At normal temperature values of the
dielectric constant (for fly ash, e=3):

* 0.6 (33)
Therefore the maximum possible increase in the migration velocity
resulting from an increase in the dielectric constant would be
about 1.7 times.
Equation (31) is based on the assumption that the particle
is fully charged by field charging. A larger dielectric con-
stant means that the particle can maintain a larger maximum
charge. If "e" increases with temperature, therefore, the total
charge that a particle can hold will increase.
Equation (31) is also based on the assumption of Stokes'
law for the gas resistance force. Thus the temperature and
pressure dependence comes in through the ratio "C'/yG". White
(1963) calculated migration velocities for temperatures of 20°C
and 350°C, and predicted that they would decrease by a factor
of about 0.6 at the higher temperature. These predictions were
not verified theoretically.
High temperature and pressure also affect the formation
and operation of DC corona, and the current-voltage character-
37
-------
istics of electrical precipitators. A detailed survey of these
effects is outside the scope of the present study. In general,
however, the corona starting voltage and the sparkover voltage
decrease with increasing temperature. The sparkover voltage,
however, decreases more rapidly so that the voltage operating
range decreases with increasing temperature. Corona starting
voltages and sparkover voltages increase with gas density, and
therefore high pressure alleviates some of the problems asso-
ciated with high temperature precipitation.
Another effect of high temperatures is on the charging of
particles. Field charging is a result of the bombardment of
the particles by gas ions in the ionic current, and is the pri-
mary charging mechanism for particles larger than a few tenths
of a micron in diameter. The saturation charge, q , acquired
o
by a particle by ionic bombardment is :
For a given electric field, equation (34) will be a function of
temperature only through the dielectric constant, e. However,
the rate of particle charging, as pointed out by Strauss (1966),
is inversely proportional to the ion mobility, B-, and therefore
proportional to the absolute pressure, P, and to the inverse
square root of the absolute temperature. That is:

Th
Bi " — (35)

For very small particles, ion diffusion becomes an important
particle charging mechanism. Strauss (1966) presented a table,
attributed to Lowe and Lucas, comparing the numbers of charges
acquired by a particle by ion bombardment and by ion diffusion.
This table is reproduced as Tables 2a and 2b.
38
-------
Table 2. NUMBERS OF CHARGES ACQUIRED BY A PARTICLE
Tables 2a and 2b are from Strauss (1966).

a. Ion Bombardment
Particle
Diameter,
ym
0.2
2.0
20.0
Period of exposure, s
0.01
0.7
72
7,200
0.1
2
200
20,000
1
2.4 2.5
244 250
24,400 25,000
b. Ion Diffusion, 27°C
Particle
Diameter,
ym
0.2
2.0
20.0
0.01
3
70
1,100
Period of
0.1
7
110
1,500
exposure, s
1
11
150
1,900
10
15
190
2,300
c. Ion Diffusion, 1,100°C
Particle
Diameter,
ym
0.2
2.0
20.0
0.01
9
250
4,400
Period of
0.1
25
440
6,300
exposure, s
1
43
630
8,200
10
62
820
10,100
39
-------
The charge, q , acquired by ion diffusion may be calculated
from:
d k T F TT d c N e2 1
q^ = -E Jin 1 + i- i t (36)
P 2e I 2 k T
where "e" is the unit electronic charge (4.8xlO"10e.s.u.), "c"
is the root mean square molecular velocity, "N." is the number
of ions per unit volume, and "t" is time. The acquired charge
is proportional to absolute temperature.
Using the same numbers used by Lowe and Lucas, but a tem-
perature of 1,100°C, Table 2c was obtained for comparison to
Tables 2a and 2b. It is apparent that diffusion charging should
be much more important at high temperatures, and is the dominant
charging mechanism for particles up to a few microns in diameter,

GRAVITATIONAL SETTLING AND CENTRIFUGAL SEPARATION
The gravitational settling velocity, u , and the deposition
velocity of a particle in a centrifugal force field, u , may be
{**
approximated as:

1 C' dp2(pD " PG^ g
u = -i- E E S2 C37)
18
C' V<»p - P(
3> "t
where "p " and "PG" are the densities of the particle and gas,
"g" is the acceleration of gravity, "ut" is the tangential par-
ticle velocity at radius "R".
Equations (37) and (38) were derived using the modified
Stokes1 law for the gas resistance force. Therefore the ratio
"C'/Ug" contains the major temperature and pressure dependence.
The gas density also depends on temperature and pressure, how-
ever even at high gas density (pG=0.018 g/cm3 at 15 atm and 20°C)
40
-------
the density is less than one percent of typical particle densi-
ties (p = 2 to 3 g/cm3).
The tangential velocity in equation (38) is a function of
the inlet velocity to the cyclone. Therefore, for the same mass
flow rate, increases in temperature and pressure will affect the
volumetric flow rate and hence the gas velocity. High pressure
will decrease the gas velocity while high temperature will in-
crease it. The application of cyclones at high temperature and
pressure is discussed in a later section.

THERMOPHORESIS
Temperature gradients can give rise to deposition forces
which can improve the collection efficiency of particulate con-
trol devices.
Thermophoresis is the result of gas molecules impinging on
the particle surface from opposite sides with different mean
velocities. The particle receives a net impulse opposite to
the temperature gradient in the gas. The magnitude of the ther-
mophoretic force was first devised by Epstein (1929), and may be
used with equation (4) for the resistance force, to obtain the
thermophoretic deposition velocity, u^, as:

3 C' y
u »_ ±— VT (39)
d

where "k" is the thermal conductivity of the gas, "k " is the
thermal conductivity of the particle, and "VT" is the tempera-
ture gradient. The derivation of equation (39) assumes the
modified Stokes' law for predicting the gas resistance force.
The temperature and pressure functionality is fairly complicated
because all the terms in equation (39) are temperature dependent,
The individual temperature and pressure dependence of the gas
and particle properties is discussed in a later section.
Figure 9 shows the effects of temperature and pressure on
the thermophoretic deposition velocity per unit temperature gra-
dient, for a 5 ym diameter silica particle, as calculated from
41
-------
0.5
I-
I >
I'.
I '
<

E-HT
t—I O
w
ex
E- 6

I-H U
W Z

> W
LO W

O OS

PH 3

W H
hJ P4

< a,

Pi W

W E->
10-
10
i
10
>l
Particle diameter = 5x10 cm

Particle thermal conductivity^

= 1.0 J/s-m-°K

(0.6 BTU/min-ft-°F)
200
400 600 800

AIR TEMPERATURE, °C
1,000
1,200
Figure 9. Thermal deposition velocity as a function of temperature

and pressure.
-------
equation (39). The deposition velocity increases slightly with
increasing temperature but decreases significantly with pressure.
More elaborate equations for predicting the thermophoretic
deposition velocity have been presented by Hidy and Brock (1970),
and by Derjaguin and Yalamov (1972). For example, using the full
set of first order slip flow boundary conditions, Brock (1962)
derived the following expression:

3 IL^G CVkp + Ct NKn) VT (40)
Ud " " 2 pG T (1 + 3Cm NKn)(1 + 2kG/kp + 2Ct NKn)

where "C " is the isothermal slip coefficient, and "C " is the
m i
temperature jump coefficient. Both "Cm" and "Ct" are functions
of the energy and momentum accommodation between the molecules
and the particle surface.
The temperature and pressure dependence of the accommoda-
tion coefficient was discussed in the previous section in con-
nection with the Cunningham slip correction factor. In general,
the accommodation coefficient is independent of pressure and de-
creases with increasing temperature (this is discussed in detail
by Wachman, 1962). A decrease in accommodation coefficient means
that the thermal energy and momentum exchange between the bom-
barding molecules and the particle is less effective. Therefore
the thermal force on the particle will be weaker than it would
be with perfect accommodation at a given temperature, and hence
the thermal deposition velocity would be smaller.
As stated in the earlier discussion of accommodation coef-
ficients, the gas and surface purities are very significant in
predicting, quantitatively, the effects of temperature on the
accommodation coefficient. Therefore, experimental measurements
of the thermophoretic deposition velocity would be necessary in
order to determine the significance of the temperature dependence
of the accommodation coefficient.
43
-------
DIFFUSIOPHORESIS
Diffusiophoresis is the transmission of particulate matter
from one place to another by molecular diffusion forces. Gene-
rally, diffusiophoresis is important in situations where there
are large concentration gradients, or where vapor condensation
is occurring. It is unlikely that such conditions will exist in
high temperature and pressure particulate removal systems. Nev-
ertheless, diffusiophoresis may be important in situations where
high pressure and low or moderate temperature particle collection
is required.
Diffusiophoresis at high pressure and relatively low tem-
perature was discussed by Calvert, et al. (1972). Following a
similar theoretical approach, the diffusiophoretic deposition
velocity, u^., is given as:
/M" P D
PV/MV + PG/MG PG
VPV C4i)
where "M " and "Mp" are the molecular weights of the vapor and
gas, "p " and "pr" are the partial pressures of the vapor and
gas, "P" is the absolute pressure, "Vp " is the vapor pressure
gradient, and "D G" is the intermolecular diffusivity between
the vapor and gas.
The partial pressure of the vapor is a function of tempera-
ture but not of pressure. The diffusivity is also a function
of temperature. The diffusivity for a water vapor - air mixture
is presented later, in the section on gas properties.
Diffusiophoretic deposition velocities are illustrated as a
function of vapor pressure gradient, for various temperature and
pressure conditions, in Figure 10. The deposition velocity is
seen to increase with increasing temperature and to decrease with
increasing pressure.
44
-------
I/I
50°C, 1 atm
0°C, 1 atm
50°C, 10 atm
50°C, 15 atm
102 103

VAPOR PRESSURE GRADIENT, mm Hg/cm
Figure 10. The effect of pressure on the diffusiophoretic
deposition velocity.
45
-------
PARTICLE GROWTH BY CONDENSATION
If high pressure and low temperature particle collection is
required, the collection of fine particles can be improved by
the condensation of vapor onto the particle, thereby increasing
its size and the ease with which it can be collected by inertial
methods.
After a drop of critical size has been nucleated it grows
at a rate determined by the vapor pressure in the gas phase and
the conditions at its surface. For a spherical, motionless drop,
assuming constant drop temperature and vapor pressure, and ideal
behavior of the vapors, the drop diameter can be calculated from
Maxwell's equation (1890) (presented by Calvert, et al., 1972):

g DyG M P
KL " * PBM
where "d^" is the drop diameter, "d0" is the drop diameter at
t=0, "PT" is the density of the drop, "R" is the universal gas
constant (8.317 J/gmole-°K), "?„," is the vapor pressure in the
gas stream "p" is the vapor pressure at the drop surface, "p "
is the mean partial pressure of the non-transferring gas, and
"t" is time. Equation (42) is independent of absolute pressure,
and is a function of temperature directly and indirectly through
the vapor pressure. The growth of particles by condensation,
however, is not likely to be very important in high temperature
particle removal systems.
For drops condensing while moving relative to the gas,
Fuchs (1959) presents the following expression:
°'276 N N (43)
where the Reynolds number, NR , and Schmidt number, Ng , are
defined as:
(44)
46
-------
N
Sc
PG DvG
(45)
and where "u " is the relative velocity between the drop and the
gas stream.
The Reynolds number will increase with increasing pressure
while the Schmidt number will decrease with increasing pressure.
\- k
The term "NR2 Ncc > therefore, will increase with the sixth root
of pressure, or by about 501 as pressure increases from 1 atm to
15 atm. Nevertheless, the pressure dependence of equation (41)
is relatively weak for normal Reynolds numbers and Schmidt num-
bers. The term "x=(l + 0.276 NR|f NgJ?)" is shown as a function
of pressure and drop diameter in Table 3. The velocity and gas
properties were assumed so that at atmospheric pressure and at
dd=lym, NRe=NSc=l.

Table 3. THE EFFECTS OF HIGH PRESSURE AND DROP DIAMETER ON THE
GROWTH OF DROPS BY CONDENSATION
Drop Diameter
ym
1
10
100
1,000
x= 1
P = 1
1
1
3
9
+ 0.276
atm
.00
.87
.76
.73
NRe NSc
15 atm
1.43
2.37
5.33
14.71
Therefore high pressure can increase the factor "x" and
thus increase the rate of particle growth by condensation.
The similar problem of liquid drop evaporation in a high
temperature and high pressure environment has been investigated
by Matlosz, et al. (1972). They found the effective mass diffu-
sion coefficient to be in good agreement with theory for temper-
atures up to about 250°C and pressures up to about 7 atm. At a
47
-------
pressure of 100 atm the mass diffusion was much greater than
predicted by theory.

MAGNETIC PRECIPITATION
When an electrically charged particle with no intrinsic
magnetic properties moves through a magnetic field of strength,
H, it will be acted on by a force at right angles to both the
direction of the particle motion, and to the magnetic field
lines. Therefore the particle will obtain a magnetic deposition
velocity, u , which may be calculated from:

Cf y H q ur
= % ^__G }
L J
and
where "q " is the particle charge, "u " is the gas velocity
"y " is the permeability constant (y =4irxlO"7V-s/A-m) . Here the
deposition velocity is a function of the gas velocity; the higher
the gas velocity, the higher the magnetic deposition velocity.
The effects of temperature and pressure on equation (46)
come in primarily through the ratio "C'/yp", which comes directly
from the assumption of equation (4) for the gas resistance force.
Also the rate at which a particle is charged is a function of
temperature and pressure [equation (35)], and the amount of
charge obtained by ion diffusion is a function of temperature
[equation (36)].

PARTICLE AGGLOMERATION MECHANISMS
One way to improve the collection efficiency for fine par-
ticles is to cause the fine particles to agglomerate into larger
aggregates which can be collected more easily.
Thermal Coagulation
Thermal coagulation is the agglomeration of particles under-
going random Brownian motion. The rate of agglomeration (or co-
48
-------
agulation) is generally considered to be proportional to the
square of the particle number concentration. That is:
d Np
__,£. = -K N2 (47)
dt o p <• )

where "N " is the particle number concentration, and "K " is the
proportionality constant, or coagulation constant. Assuming
equation (4) for the gas resistance force, Fuchs (1964) presents
the following equation for the thermal coagulation coefficient
of a particle undergoing Brownian motion in a still gas, assuming
particles stick upon touching:

.„., 4 C' k T r /\Q~\
Ko = 4* Dp dp = 3 -TT" (48)
b

The thermal coagulation constant is shown as a function of
temperature, pressure, and particle diameter in Figure 11. The
agglomeration of particles increases with temperature and de-
creases with pressure. The net effect of high temperature and
high pressure (20°C, 1 atm to 1,100°C, 15 atm) is to increase
the rate of thermal agglomeration for a 1 ym diameter particle
by a factor of 1.5 (K increases from 3.5 x 10~10cm3/sec to
5.3 x 10"10 cm3/sec). For a 0.5 ym diameter particle, the agglo-
meration rate increases by a factor of about 1.4 (KQ increases
from 3.9 x 10"10cm3/sec to 5.6 x 10~10 cm3/sec) . For a 0.1 ym
diameter particle, the rate of agglomeration remains relatively
constant (K = 8.5 x 10"10 cm3/sec) . Therefore it appears that
at high pressure and high temperature, there is a small increase
in the rate of agglomeration for particles larger than 0.1 ym.
At high temperature and atmospheric pressure, the rate of ag-
glomeration of fine particles would be greatly increased; how-
ever, high pressures effectively nullify the benefit of high
temperature.
49
-------
70

40
^ 20

•a
t-

.-
:
.

z 10
o

- 9

^ 8
=
r
_
_

=
PARTICLE

DIAMETER
6 ^
3 T^i
:
200
400 600 800

AIR TEMPERATURE, °C
1,000
1,200
Figure 11. Thermal coagulation of particles at high temperature

and pressure.
50
-------
Agglomeration of Charged Particles
Thermal agglomeration can be improved if the particles are
charged. Fuchs (1964) discusses this question and predicts that
the coagulation constant will be increased by a factor "B" which
depends on the magnitude and sign of the particle charge. That
is:
K = B KQ (49)

where "K" is the actual coagulation constant, and "K " is the
uncharged coagulation constant of equation (48) . For the case
where the particles have the same charge, the rate of agglomera-
tion decreases because of electrostatic repulsion (B<1). For
particles with opposite charge, agglomeration increases because
of electrostatic attraction (B>1). For a collection of charged
particles with zero net charge, the arithmetic mean "B" is still
slightly greater than unity. Thus even though there is no ne.t
charge, agglomeration is still greater with charged particles.
Table 4 shows the effect of temperature on the thermal agglom-
eration of charged particles. The agglomeration of particles
with like charges (3 ) increases towards unity while the ag-
glomeration of oppositely charged particles (3att) decreases
towards unity. The average agglomeration for a mixture of posi-
tive and negative particles with zero net charge (B ) decreases

Table 4. EFFECT OF TEMPERATURE ON THE AGGLOMERATION OF CHARGED
PARTICLES

1
T
20
100
500
,100

°C
°C
°C
°C
B
0
0
0
0
rep

.771
.81
.90
.94
8
8
8
B
1
1
1
1
att
.271
.208
.098
.054
B
1
1
1
1
avg
.021
.013
.003
.001
51
-------
towards unity. Therefore, the net effect of temperature on the
agglomeration of charged particles is to decrease the rate of
agglomeration relative to uncharged particles. That is, high
temperature nullifies the beneficial effect of particle charge.
For a given charge on the particles, pressure should not have
any effect that would not be present with uncharged particles.

Polarization Agglomeration
When a particle is under the influence of an electric field,
it is inductively charged and the rate of agglomeration is af-
fected. Fuchs (1964) discusses this problem and shows that "0"
is inversely proportional to the cube root of absolute tempera-
ture. Therefore, agglomeration of inductively charged particles
decreases slightly with increasing temperature.
Turbulent Agglomeration
Particles can also agglomerate as a result of turbulence in
the fluid. Turbulent agglomeration has been discussed by Beal
(1972) and is proportional to the turbulent diffusion coeffi-
cient. For the general case where the particle diameter is much
'smaller than the turbulent microscale, X , Beal presents the fol-

lowing equation for the turbulent diffusion coefficient, Dt:

(50)
where "e " is the energy dissipation rate for unit mass of fluid
and "v " is the gas kinematic viscosity. Therefore, "D " is
O i . L
proportional to "u,"5". Figure 12 is a plot of "v"'5" against
temperature for a range of pressures. Turbulent agglomeration
increases greatly with an increase in pressure, at low tempera-
tures. This beneficial effect of pressure is almost completely
nullified at high temperatures.
Sonic Agglomeration
Another way to cause particles to agglomerate is by the ap-
plication of sonic vibrations. Sonic agglomeration has been

52
-------
e
u
JT
i
-
--
•J-.
C
U
-'
•~
->
—
-
=
1!
-
:
200 400 600 800

AIR TEMPERATURE, °C
1,000 1,200
Figure 12.
Turbulent agglomeration tendency of
particles at high temperature and
pressure.
53
-------
studied by Mednikov (1965) and has been shown to be inversely
proportional to the square root of the product of gas density
and the speed of sound. That is:
K
a
where "K" is the sonic agglomeration coefficient, and "Cr" is
a \j
the speed of sound in the gas. The speed of sound of an ideal
gas is given by:
(52)

where "y" is the ratio of specific heats, and "/?" is the univer-
sal gas constant, and "M^" is the gas molecular weight.
Figure 13 shows the relative sonic agglomeration coefficient
for a variety of temperatures and pressures. Once again, high
temperatures can slightly improve agglomeration at atmospheric
pressure, but high temperature and high pressure together sig-
nificantly reduce sonic agglomeration relative to standard con-
ditions.
54
-------
200 400 600 800

AIR TEMPERATURE, °C
1,000 1,200
Figure 13. Sonic agglomeration of particles at
high temperature and pressure.
55
-------
PROPERTIES

In previous sections it has been shown that the removal of
suspended particulate matter from a gas is strongly dependent on
the resistance of the gas to the motion of a particle. The gas
resistance force is a function of a number of gas properties,
the most important of which are the absolute viscosity of the
gas and the mean free path of the gas molecules. Also, but to
a lesser extent, other properties such as density, thermal con-
ductivity, and molecular diffusivity influence particle collec-
tion.
In this section, the effects of high temperature and high
pressure on the important gas properties are reviewed.

ABSOLUTE VISCOSITY
The absolute viscosity of a gas is defined by Newton's law
of friction:

'G - »G 0

where "TG" is the frictional shearing stress in the fluid, and
"du/dy" is the velocity gradient perpendicular to the flow. The
viscosity is thus the proportionality constant relating "TG" and
"du/dy", and characterizes the resistance of the fluid to shear-
ing stresses.
The absolute viscosities of gases and gas mixtures at tem-
peratures up to 1,200°C have been studied by Saxena (1971), and
experimental values are presented for many gases by Weast (1968) .
Gas viscosities also can be predicted from a number of em-
pirical equations presented and discussed by Reid and Sherwood
(1958). They found the average error in these empirical equa-
tions to be about 3%. If experimental viscosity values are
available at two temperatures, Reid and Sherwood (1958) recommend
using the following approximation:
56
-------
1.5
yr = — (54)
G ml + b

where "T" is absolute temperature in t|0K","vi " is in micropoise,
and "m" and "b" are constants which must be obtained empirically
for each gas. Figure 14 compares the theoretical curve obtained
from equation (54) with experimental data from Weast (1968) , for
air.
Equation (54) and data from Weast (1968) were used to pre-
dict viscosity versus temperature curves for many common gases.
The results are presented in Figure 15. The viscosity of water
vapor increases most steeply with increasing temperature. As
the temperature increases from 100°C to 1,700°C, the viscosity
of water vapor increases by a factor of about 5.5. The other
gases are less sensitive to temperature, with a similar tempera-
ture increase only causing the viscosity to increase by a factor
of 2 to 2.5.
In practical situations, mixtures of various gases are
present. The viscosity of a gas mixture can be obtained by
using equations presented by Reid and Sherwood (1958) and attri-
buted to Bromley and Wilke (1951) . The calculations are quite
tedious and generally are not necessary in determining the ef-
fects of temperature on the gas viscosity. Usually it is satis-
factory to assume the gas viscosity will behave similar to that
of air.
From simple kinetic theory it would be expected that the
viscosity should not depend on pressure. In a dense gas, how-
ever, the forces of attraction and repulsion between molecules
causes the viscosity to increase rapidly with pressure. This
effect is especially noticeable at low temperatures. However,
for air at 0°C, the viscosity is not affected significantly by
pressures below about 20 atm. At 1,000°C the viscosity is in-
dependent of pressures up to about 300 atm. Therefore the ef-
fect of high pressure on the viscosity has not been considered
57
-------
8
W
-
-
-
700
600
500 i
5 400
c
u
_
-

—
c
L-
—
300 5
200
100
0.068Tabs * 7.8
500 1,000

TEMPERATURE, °C
1,500
Figure 14. Experimental and theoretical viscosity versus
temperature curves for air. Data from Weast
(1968).
58
-------
r.
• -
c
-
z
•~
L
'-
-
.
C
_
600
500 i
400
300
200
100
200 400 600 800

TEMPERATURE, °C
1,000 1,200
Figure 15.
Gas viscosity versus temperature.

The solid lines are interpolated

between data, and the dashed lines

are extrapolated from data. Data

are from Weast (1968).
59
-------
important in this study. Methods for predicting the effect of
pressure on gas viscosity are presented by Reid and Sherwood
(1958).

MEAN FREE PATH
The mean free path of gas molecules is defined as the aver-
age distance between molecular collisions. From the kinetic
theory of gases, the mean free path, X, is given by the Chapman-
Enskog equation:

X = £ (55)
0.499 pr c
b
where "c"" is the root mean square molecular velocity, and is
given by:

VT (56)

Making use of the ideal gas law and equation (56) , equation (55)
becomes:

X = £ fl 5_I\" (57)
0.499 P \8 M /

If the mean free path for a gas is known at one temperature
and pressure, its value at any other temperature and pressure
can be obtained using the formula:

t = ^ "M^J C58)

where the subscript "o" denotes values at the known conditions.
The mean free path of air molecules is presented as a function
of temperature and pressure in Figure 16.
The mean free paths for mixtures found in actual process ef-
fluent gases can be obtained from equations (57) and (58) if the
molecular weight and viscosity of the gas mixture are known.

60
-------
0.4

0.1
0.05
X
-
«a
—

—
_
pi
—

2
•*
—
0.01
0.005
0.003

&:^E3 1 atm .-

f_ _' \ u.j ^_ r_ ..i .L".-__HTU
f

-~^>^i 5 atm -

*"T~ „ L ~~ ^i

15 atm —^~
PRESSURE

200
400 600 800

AIR TEMPERATURE, °C
1,000
1,200
Figure 16.
Mean free path of air molecules as a function
of temperature and pressure.
61
-------
GAS DENSITY
For a perfect gas, the density of the gas is related to the
temperature and pressure through the equation of state:
M P
PG • ~ (59)

This expression is suitable at high temperatures and pressures
providing that the gas still acts like a perfect gas. If the
pressure is large enough, the compressibility of the gas becomes
important and equation (59) must be modified by a compressibility
factor which is itself a function of temperature and pressure.
Equation (59) is suitable up to about 20 atm pressure at 0°C and
up to over 300 atm at temperatures larger than 1,000°C. The use
of compressibility factors is discussed by Reid and Sherwood
(1958) and most general thermodynamics textbooks.

THERMAL CONDUCTIVITY
The collection of particles by the mechanism of thermophor-
esis depends on the ratio of the thermal conductivities of the
gas and the particle. The thermal conductivity, k , of a sub-
stance is defined by:

<5x - - kt I (6°)

where "Q " is the heat flux in the "x" direction, and "T" is the
Jv
temperature. Thus the thermal conductivity characterizes the
rate of heat transfer through a substance.
The thermal conductivity of gases is a function of the spe-
cific heat and the viscosity of the gas, and hence is a function
of temperature. It is not a significant function of pressure
for pressures below about 15 atm.
Experimental investigations of the thermal conductivity of
gases and gas mixtures have been conducted by Saxena (1971, 1972)
and by Chen and Saxena (1973). Also Reid and Sherwood (1958)
present empirical equations and some data for thermal conducti-
vity as a function of temperature and pressure.
62
-------
Both the viscosity and the specific heat are functions of
temperature. The constant pressure specific heat for air is
presented as a function of temperature in Figure 17. This was
taken from a nomograph presented by Liley (1963).
Gambill (1963) presents a relatively simple empirical ex-
pression relating the thermal conductivity, kG, to the gas vis-
cosity and specific heat, expressed in English units as shown:
kr(BTU/hr-ft-°F) = yr(lb/hr-ft)
C (BTU/lb-°F)
2.48
MG
(61)
where "M " is the molecular weight of the gas. Equation (61)
and Figure 17 were used to determine the thermal conductivity
of air as a function of temperature. The results are presented
in Figure 18 together with the experimental data obtained by
Saxena (1973) for nitrogen.
The thermal conductivities of other gases may be approxi-
mated in a similar manner using equation (61), or may be esti-
mated by more elaborate methods described by Reid and Sherwood
(1958), by Gambill (1963), and by Saxena (1971).
The thermal conductivity of solid particles, kp , may be
predicted from equations presented by Gambill (1963), however
the temperature dependence of the thermal conductivity of ma-
terials likely to be found in fly ash (FeaOs, Si02, A1203) is
quite small. Therefore the temperature dependence of the gas
to particle thermal conductivity ratio will depend primarily
on the thermal conductivity of the gas.
Figure 18 indicates that the ratio of "k^/k " will in-
crease by a factor of 3 if the temperature is increased from
0°C to 1,100°C.

MOLECULAR DIFFUSIVITY
Particle collection by diffusiophoresis involves the molec-
ular diffusivity of the vapor in the gas. The rate of molecular
transfer, N^, in molal units per unit area in unit time is given
by Pick's law:

63
-------
1.20
I
00

1-3

s
1.10
—
a
-
—
u
r
—
1.00
I
200 400 600 800 1,000 1,200

TEMPERATURE, °C
Figure 17. Specific heat as a function of

temperature for air at 1 atm.

From nomograph presented by Liley

(1963).
64
-------
0.10
200 400 600 800 1,000 1,200

TEMPERATURE, °C
Figure 18. The effect of temperature on the thermal
conductivity of air at 1 atm pressure.

(0.58 x J/m-s-°K = BTU/hr-ft-°F)
Data from Saxena (1973), for N2
65
-------
NA ' - °VG ^ C62)

where "dc./dx" is the concentration gradient in the direction of
diffusion, and "D „" is the molecular diffusivity or diffusion
coefficient of the vapor in the gas.
The molecular diffusivity can be calculated from a number
of empirical equations presented by Reid and Sherwood (1958),
and in general they predict:

T3*
DvG * T (63)

Reid and Sherwood (1958) state that the product "D *P" is con-
stant for a given temperature for pressures up to 20-25 atm.
They also present data for the diffusion coefficients of
water vapor in air, at 1 atm pressure, for temperatures up to
1,200°C. These data are plotted in Figure 19. The dashed line
is the prediction based on equation (63) and the experimental
diffusivity at T=100°C. Thus equation (63) underestimates the
temperature dependence of "DvG".

THERMAL EXPANSION
The thermal expansion of particles at high temperature will
increase their size and could thereby improve the collection ef-
ficiency of devices operating at high temperature.
Most fly ash is composed of refractory material such as sil-
ica and alumina. Monroe (1963) presents data for the thermal
expansion of refractory materials, and the data for silica brick
are reproduced in Figure 20. The thermal expansion increases
with temperature up to about 700°C, and then remains constant
at higher temperatures. The maximum expansion is only about
1.5%, and therefore the increase in size of a particle at high
temperature is expected to be negligible.
66
-------
200
400 600 800 1,000 1,200

TEMPERATURE, °C
Figure 19. Molecular diffusivity of water vapor
in air as a function of temperature.
Data from Reid and Sherwood (1958).
67
-------
0.02

200
400 600 800 1,000 1,200

TEMPERATURE, °C
Figure 20. The thermal expansion of silica brick
at high temperatures. From Monroe (1963)
68
-------
EXAMPLE APPLICATIONS

In most particle collection devices, many collection mecha-
nisms are acting simultaneously. Often one mechanism dominates,
as in electrostatic precipitation, and the deposition velocity
resulting from this mechanism can be used to characterize the
collection efficiency of the device. Sometimes, however, many
mechanisms are important and they must be combined, as in fiber
filtration, in order to characterize the device.
In this section, therefore, a few examples of the effects of
high temperature and pressure on the theoretical efficiencies of
particle collection devices (specifically: a cyclone separator,
a fiber filter, and a granular bed filter) are presented. We
have not attempted to critically review the theoretical models
used in this section. We have taken typical models from the li-
terature, which are based on the b-asic mechanisms discussed in
a previous section above, and used them to predict the efficien-
cies at high temperature and pressure. Our purpose in presenting
these examples is to provide an indication of the relationship
between the fundamental mechanisms and collection devices.
Because particle collection equipment usually is necessary
in order to meet the air pollution emissions standards, perhaps
a more important consideration is the amount of power required
to maintain a high collection efficiency for a given device. The
first example presented in this section, therefore, considers the
power requirement for one of the most important collection mecha-
nisms, inertial impaction.

SINGLE STAGE IMPACTOR
Using equation (4) to represent the fluid resistance force,
it is possible to predict the collection efficiency and work re-
quirements for particle collection by inertial impaction. Fig-
ure 21 shows the particle cut diameter as a function of flow work
(or specific power) for a single stage impactor at various tem-
perature and pressure conditions. For an impactor, it may be

69
-------
0.01
SPECIFIC POWER, HP/MSCFM

0.1 l.Q
10
u
*

o
1—I

X
Q

E-

W
1.0 lie
0.5 ^
0.1
Particle density = 1.0 g/cm
CONDITIONS
20°C, 1 atm
800°C, 1 atm
800°C, 10 atm
1,100°C, 15 -atm
10
103

FLOW WORK, J/kg
10
10
Figure 21. Flow work (specific power) for impaction from a round jet as
a function of temperature and pressure.
-------
assumed that all particles larger than the cut diameter will be
collected. The flow work is equivalent to the energy require-
ment per mass of gas. The power requirement would be equal to
the flow work multiplied by the mass flow rate.
Figure 21 illustrates that the work required to collect
submicron particles by inertial impaction increases rapidly
with decreasing particle diameter. The effects of temperature
and pressure may be seen by comparing the four curves. Curve 1
represents standard temperature and pressure conditions. Curve 2
shows the effect of high temperature at atmospheric pressure.
The work required to collect submicron particles does not in-
crease as rapidly as that required to collect larger particles.
The reason for this is that, for a given particle diameter, the
fluid resistance force decreases as the mean free path of the
gas molecules increases. The mean free path increases with
temperature, for a constant pressure.
Curves 3 and 4 are for simultaneous high pressure and high
temperature conditions. The beneficial effect of high tempera-
ture on the mean free path is completely nullified by a decrease
in mean free path with increasing pressure. The work required
to remove submicron particles is greatly increased. For example,
the collection of all particles greater than 1 ym by inertial
impaction would require about 450 J/kg (0.31 HP/MSCFM) at stan-
dard conditions (Curve 1). It would require about 4,000 J/kg
(2.8 HP/MSCFM) at 1,100°C and 15 atm (Curve 4). This is equiva-
lent to approximately a 9:1 increase in the power requirement
to maintain a similar degree of particle removal.

CYCLONE SEPARATOR EFFICIENCY
A further illustration of the effect of high temperature and
pressure on particle collection may be obtained by predicting the
collection efficiency of a cyclone separator operating at various
temperatures and pressures, for the same inlet velocity.
Leith and Licht (1972) derived an equation for predicting
cyclone efficiency from theoretical considerations (this work

71
-------
is reported by Calvert et al. (1972). Their equation is:

E = 1 - exp - 2 (CY)1/(:2n+2) (64)

where "C" is a function of the cyclone dimension ratios only.
The temperature and pressure dependent terms are contained in
the modified inertial parameter, Y, and the vortex exponent, n,
defined by:
P d2 ur
± (65)
18
1(0.00394 D
1 -- c_ [_i~ (66)
2.5

where "D " is the cyclone diameter, "u~" is the gas velocity,
and "T" is absolute temperature.
Equations (64), (65), and (66) were used to calculate typi-
cal collection efficiency curves for a high efficiency cyclone.
The cyclone was assumed to be about 15 cm (6 inches) in diameter
with a volumetric flow rate of about 1.4 m3/min (50 ft3/min).
Figure 22 shows a typical efficiency curve for a high effi-
ciency cyclone (Curve 1) . The cyclone cut point (50% efficiency)
occurs at a particle diameter of about 1 ym, and the cyclone is
better than 99% efficient for particles larger than 15 ym.
Curve 2 shows the estimated cyclone efficiency for a gas at
1,100°C and atmospheric pressure, and for the same inlet gas
velocity as in Curve 1. The cyclone efficiency has dropped sig-
nificantly and now has a cut point occurring at 2.0 ym and is
only 96% efficient for 15 ym particles.
Curve 3 shows the estimated cyclone efficiency for a gas at
1,100°C and 15 atm, for the same inlet velocity. The cyclone
efficiency has decreased again slightly for small particles, but
is relatively unaffected for larger particles. The cut point
now occurs at 2.5 ym and the efficiency for 15 ym particles is
95%.
72
-------
100
CONSTANT INLET VELOCITY
CONDITIONS
20°C, 1 atm
1,100°C, 1 atm
1,100°C. 15 atm
8 10 12 14 16 18 20 22 24 26 28 30

PARTICLE DIAMETER, ym
Figure 22 . The effects of high temperature and pressure on
the collection efficiency of a high efficiency
cyclone.
-------
SPECIFIC POWER FOR A CYCLONE SEPARATOR
Figure 23 presents another example of high temperature and
pressure effects on the performance of a cyclone separator. The
specific power ratio is the ratio of the specific power (HP/MSCF)
at high temperature and pressure to that at standard conditions.
The curves in Figure 23 show the power requirement relative to
standard conditions to collect various particle sizes while main-
taining a constant collection efficiency for each particle size.

EFFICIENCY OF A FIBER BED
Figure 24 shows the effects of temperature and pressure on
the collection efficiency of a fiber bed. The curves were cal-
culated from the theory presented by Calvert et al. (1972) and
attributed to Torgeson (1958). The collection efficiency is the
combined efficiency resulting from inertial impaction, intercep-
tion, and Brownian diffusion.
From Figure 24 it is apparent that the filter collection
efficiency for particles larger than about 0.5 um in diameter is
reduced significantly at high temperature and pressure. This is
a result of the smaller inertial impaction parameter [equation
(2)] at high temperature and pressure. For particles smaller
than 0.5 ym, the collection efficiency is somewhat increased
because of the increased Brownian motion at high temperatures.
At high temperature and atmospheric pressure the collection
efficiency is greatly increased for particle diameters up to
about 0.9 ym. For particles larger than 1 ym the collection ef-
ficiency is reduced at high temperature and atmospheric pressure,
although not as severely as at high temperature and high pres-
sure .

COLLECTION EFFICIENCY OF A GRANULAR BED
No generally accepted theory is available for the detailed
design of granular bed filters, especially for high temperature
and pressure applications.
74
-------
10
9

7
w
u ^==r
u
-
—
-

300
500 1,000

AIR TEMPERATURE, °K
2,000
Figure 23.
The specific power requirements

for a cyclone as a function of

temperature and pressure.
-------
9n
'

'•
•

:• '
i
: .
; I

i '
I '
• .

1 •
100

10
1
'
t
CONSTANT l-'ACIj VELOCITY
IT

!
1
2
I
20°C, 1 atm
1,100°C, 1 atm
1,100°C, 15 atm
0.1
Figure 24.
0.5 1.0

PARTICLE DIAMETER, u
5.0
The effects of high temperature and pressure on
the collection efficiency of a fiber bed.
-------
Pt = exp - 21.4 -f- K
A simple model for particle penetration was developed by
Calvert, et al. (1972) and correlated with experimental data on
collection of particles in packed beds. This model may be used
to predict the efficiency of granular beds. Actual efficiencies
may be higher if a filter cake is formed, or if significant ag-
glomeration, diffusion, or thermal deposition is present.
Penetration through packed spherical collectors may be ap-
proximated by:
f n ~\
(67)

where "Z" is the bed depth, "d " is the collector diameter, and
"K " is the inertial impaction parameter. In this case, "K "
is defined as:
u. d2 C' p
K = JL-E B. (68)
P 9 UG dc

where "u^" is the gas velocity based on empty cross-sectional
area (superficial velocity).
The collection efficiency of a granular bed, as a function
of temperature, pressure, and particle diameter, is shown in Fig-
ure 25. It is assumed that the bed is 0.04 m deep, the collec-
tors are 0.002 m in diameter, the particle density is 2,500 kg/m3
(2.5 g/cm3), the superficial velocity is 1 m/s, and the gas is
air.
Because this model is based primarily on inertial impac-
tion, the increase in collection efficiency at high temperature
and small particle diameter is the result of the decreased drag
force at larger mean free paths. Diffusion would further in-
crease the collection efficiency for small particles, especially
at high temperature.
77
-------
100
I
oo
.
• )
i •
; •
O
80
60
PH
u.
i,.

I
40 -
20 -
20°C, 1 atm

1,100°C, 1 atm

1,100°C, 15 atm
0.1
0.5 1

PARTICLE
,0

DIAMETER, ym
5.0
10.0
Figure 25. The effects of high temperature and pressure on the inertial

collection efficiency of a packed bed.
-------
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA-600/7-77-002
2,
3. RECIPIENT'S ACCESSION-NO.
4. TITLE AND SUBTITLE
Effects of Temperature and Pressure on Particle
Collection Mechanisms: Theoretical Review
5. REPORT DATE
January 1977
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
Seymour Calvert and Richard Parker
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
A. P.T. , Inc.
4901 Morena Boulevard, Suite 402
San Diego, California 92117
10. PROGRAM ELEMENT NO.
EHE623A
11. CONTRACT/GRANT NO.

68-02-2137
12. SPONSORING AGENCY NAME AND ADDRESS
EPA, Office of Research and Development
Industrial Environmental Research Laboratory
Research Triangle Park, NC 27711
13. TYPE OF REPORT AND PERIOD COVERED
Final; 12/75-9/76
14. SPONSORING AGENCY CODE
EPA-ORD
is. SUPPLEMENTARY NOTES IERL-RTP Project Officer for this report is D C. Drehmel, 919/
549-8411 Ext 2925, Mail Drop 62.
16. ABSTRACT ,
The report is a critical review and evaluation of the mechanics of aerosols
at high temperatures and pressures. It discusses equations and models used to pre-
dict particle behavior at normal conditions, with regard to their applicability at high
temperatures and pressures. It discusses available experimental data, concluding
that the data are inadequate to confirm the projections of aerosol mechanics at high
temperatures and pressures. It presents a few examples of the effects of high tem-
perature and pressure on the collection efficiency and power requirement of typical
collection devices. It concludes, generally, that particle collection at high tempera-
tures and pressures will be much more difficult and expensive than collection of
similar particles at low temperatures and pressures.
17.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS C. COSATI Field/Group
Air Pollution
Dust
Aerosols
Collection
Temperature
Pressure
A.ir Pollution Control
Stationary Sources
Particulate
13. DISTRIBUTION STATEMENT

Unlimited
19. SECURITY CLASS (This Report)
Unclassified
21. NO. OF PAGES

96
2O. SECURITY CLASS (Thispage)
Unclassified
22. PRICE
EPA Form 2220-1 <9-73)
84
-------