AND EXPERIMENTAL APPLICATION
OF
MODEL OF DUST FILTRATION PROCESS
Prejget
Sc EfWIRONMENTAL PROTECTION AGENCY
Offie® ©f Research and Development
Washington. DC 2046n
mm
IPWMB
w fiPOio
-------
June 1984
PROGRESS AND EXPERIMENTAL APPLICATION
OF
PROBABILISTIC MODEL OF DUST FILTRATION PROCESS
Jan R. Koscianowski, Lidia Koscianowska
and Eugeniusz Szczepankiewicz
Institute of Industry of Cement Building Materials in Opole
45-6A1 Opole', 21 Osvi°cimska Str., Poland
Contract No. P-5-533-3
EPA Project Officer: Dr. James H. Turner
and Dr. Louis Hovis
Prepared for
U.S. ENVIRONf-TENTAL PROTECTION AGENCY
Office of Research and Development
Washington, DC 20^60
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ABSTRACT
The report summarizes 2 years of research works devoted to
the problem of progress of probabilistic model of the dust
filtration process. Both, the physical and probabilistic
models of the dust filtration process are presented. During
project realization, the oryginal method and methodology
of the pore size distribution in the structure of filtration
materials were elaborated. Using this method, the pore size
distribution for all American fabrics, which were investi-
gated in previous project /?-5-533-yV were performed.
The parameters of the pore size distribution depend on
technological parameters of fabrics and in this sense they
can be recognized as a universal magnitudes which describe
woven or non-woven filtration structures. On the basis of
theoretical considerations, it was found, that the geome-
trical pore radius distribution in the structure of dust
layer has a normal character and its parameters can be used
as a magnitudes describing the structure of dust layer from
the viewpoint of filtration properties. The results of
investigations of electrostatics effects in the ductive
filtration process, performed on the model layers of a cera-
mic and metal spheres, confirmed that they can increase
the mechanical effect of filtration.
ii
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TABLE OF CONTENTS
SECTION PAGE
Abstract . ......... ii
List of Figures vi
List of Tables x
Acknowledgments xi
I Conclusions 1
II Recommendations ............ A
III Introduction 5
Research Objectives ........ 8
General Program 8
Testing of Filtration Materials 8
Structural Parameters of Filtra-
tion Structures 9
Structural Parameters of Dust
Layers 9
Influence of Electrostatic Phe-
nomena On Dust Filtration Pro-
cess Performance ........ 9
Fabric and Dust Selection ..... 10
IV Problems of Dust Filtration Process
Modeling 11
Introduction ' 11
General Physical Model of Dry.
Filtration Process . 13
General Physical Model of Dust
Filtration Process . 16
Physical Model of Ductive Filtration
Process 23
Theory of the "Zero-layer" 30
Recapitulation and Conclusions ... 36
iii
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TABLE OF CONTENTS /con. /
SECTION PAGS
V Probabilistic Models of Dust Filtra-
tion Process 37
Introduction 37
The Model Based On The Isotropic
Random Fields And Stationary In Wide
Sense 41
The Model Based On The Isotropic
Random Fields and Quasi-stationary
In Wide Sense 48
Elements of Statistics 51
Dependences Between The Parameters . 53
The Correlation FunctionsAnd Their
Application 55
Applications 58
VI Determination of Fabric Pore Size Dis-
tribution 60
Introduction ..... 60
Theory 60
Equipment And Procedures 6l
Results And Discussion 70
Recapitulation 106
VII Determination of Pore Size Distribution
In Dust Layer 109
Introduction 109
Probabilistic Approach To The Pro-
blem of Dust Layer Formation .... 111
Final Remarks 119
VIII Study of Electrostatics Effects In
Dust Filtration Process 121
iv
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TABLE OF CONTENTS /con. /
SECTION PAGE
Introduction 121
Equipment And Procedures ..... 122
Results And Discussion of the Pre-
rninary Experiments ........ 128
Results And Discussion of the
Principal Experiments ....... 132
Final Remarks ..... 139
IX References 141
Appendix A List,of Nomenclature. ...
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LIST OF FIGURES
Number Page
1 General scheme of dry filtration process. . 15
2 General scheme of dust filtration process . 21
3 Four-element cofiguration of dust grains. . 25
4 Geometrical dependences on the surface of
dust layer 26
5 Filtration mechanisms in ductive filtration 28
6 Range of application of "zero-layer" and
ductive filtration theory .. 31
7a Comparison of non-woven and woven structu-
res 34
7b Comparison of non-woven and woven structu-
res 35
8 Illustration of the laboratory stand for
pore size distribution determination . . , 62
9. Diagram of the laboratory stand for pore
size distribution determination 63
10 Testing chamber
CD
11 The course of the curve P =. f/Q/ 67
12 The fabric surface with wetted liquid film
/first bubbles formation - bubbles formation
during the test/ 68
13 Superficial structure of clean Cotton fa-
bric style 960 73
I4a Superficial structure of clean Polvester
fabric style 862B and C8663 ."..... 74
I4b Sunerficial structure of clean Polyester
fabric stvle C868B "..... 75
15a Superficial structure of clean Polvester
fabric stvle 8553 and C8903 ...... 76
VI
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LIST OF FIGURES /continued/
Number
15b
l6a
l6b
17
18
19
20
21
22
23
24
25
26
27
28
Superficial structure of clean Polyester
fabric style C892B .....
Superficial structure of clean Nomex
fabric style 852 and 853
Superficial structure of clean Nomex
fabric style 19QR
Superficial structure of clean Nomex
fabric style 850
Superficial structure of clean Nylon
Polyamide fabric style 802B ......
Superficial structure of clean Glass
fabric style Q53-875
/
- Superficial structure of clean Glass
fabric style Q53-870 and 053- 878 . .
Pore radius distribution in the structu-
re of Cotton fabric style 960 ....
Pore radius distribution in the structu-
re of Polyester fabric style 862B . . .
Pore radius distribution in the structu-
re of Polyester fabric style C866B . .
Pore radius distribution in the structu-
re of Polyester fabric style , C868B . .
Pore radius distribution in the structu-
re of Polyester fabric style 865B . « .
Pore radius distribution in the structu-
re of Polyester fabric style C890B . . -
Pore radius distribution in the structu-
re of Polyester fabric style C892B . .
Pore radius distribution in the structu-
re of Noraex fabric style 852
Page
77
78
79
1 J
80
81
82
' 83
85
86
87
83
89
90
91
92
vii
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LIST OF FIGURES /continued/
Number
29
30
31 -
32
33
34
35
36
37
38
39
*
Pore radius distribution in the structure
of Nornex fabric style 853
Pore radius distribution in the structure
of Nornex fabric styl^ 190R
Pore radius distribution in the structure
of Nornex fabric style 850
Pore radius distribution in the structure
of Nylon Polvamide fabric style 8028 . .
Pore radius distribution in the structure
of G1ass fabric style Q53-875
Pore radius distribution in the structure
of Glass fabric style 053-870
Pore radius distribution in the structure
of G1 ass fabric style 053-878 .....
Dependence of air permeability utson the
Influence of warp thread count in 10 cm
on the mean pore radius .........
Influence of fill thread count in 10 cm
Influence of superficial filling with
warp and fill threads oh the mean pore
Pa^e
93
*/ ^
94
^ *
95
«/ ~r
95
97
^ t
98
•s ***
99
*j j
101
102
103
1 *^ ^
104
40 Variation of the pore size distribution
in used and unused Polish Polyester
fabric style ET-4 " 107
41 Three-elements dust grain system forming
the structure of the elementary dust
layers 114
42 Diagram of stand for dust efficiency of
the filtration layer of spheres measurement 123
viii
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LIST OF FIGURES /continued/
Number Page
43 Construction of measurement chamber , . 124
44 Particle size distribution of the silica
test dust 126
45 Dependence of the dust efficiency of the
filtration layer of spheres upon the
thickness of total layer 135
46 Dependence of the dust efficiency of the
filtration layer of a ceramic spheres
upon the thickness of total layer ... 138
ix
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LIST OF TABLES
Number Pago
1 Characteristic of Fabrics 72
2 Parameters of Pore Radius Distribution . . 8-'4
3 Dust Efficiency of the Filtration Layers
of a Metal and" Ceramic Spheres Tested. With
Silica Dust - preliminary tests 129
4 Electrical Properties of Test Dusts and Cera-
mic Spheres_ . 130
5 Dust Efficiency of the Filtration Layers
of Spheres Tested With Silica Dust
/principal experiments/ 133
6 Dust Efficiency of the Filtration Layers
of the Ceramic Spheres Tested With Silica
Dust /principal experiments/ 136
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ACKNOWLEDGMENTS
The authors would like to thank employees of the U. S.
Environmental Protection Agency who participated in this
endeavor. The authors extend special thanks to the Project
Officers Dr. James H. Turner, Dr. James H. Abbott and
Dr. Louis Hovis whose interest made possible realization
of this Project. We acknowledge with thanks the coopera-
tion with Dr. Jozef Malcher and his staff at the Institute
of Physics of Polytechnical University of Wroclaw; and
Dr. K. Skudlarski, the Head of the Electronic Microscopy
Laboratory and his staff, who performed microscopic
examinations.
xi
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SECTION I
CONCLUSIONS
In the result of the research program realization, the
complete general probabilistic model of the dust filtration
process as well as-its two specific applications for
description of the ductive filtration and "zero-layer"
processes, are presented. The physical model of the dust
filtration process and the theory of the random fields,
are a base of the elaborated probabilistic model, which
progress depends on the possibility to present all process
parameters in the form a certain random fields.
Coming from the requirements of the mathematical modeling,
the experiments were first of all concentrated on the pro-
blems of elaboration of a certain methods of the filtra-
tion material and dust layer structures distribution pa-
rameters estimation.
The oryginal method and methodology of the fabric pore si-
ze distribution estimation were elaborated and presented.
Results obtained from the laboratory experiments and also
from theoretical considerations lead to the following
general conclusions:
- Only on the way of the probabilistic modeling,
it is possible to describe fully the multi-
parameters dust filtration process, in the sen-
se of description all dependences between the
main filtration process parameters, as well as
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"to describe of their stability,
The progress of the probabilistic model of the
dust filtration process should be developed
in the order to description all main filtration
parameters by means of the certain random fields,
The introduction of the random fields of some
filtration parameters, as for examole: the ;~as
+ orT^OT*P"t~T1?'r*n "i "(~ ^ ~T~ 1"*^" ^ 1 T-\'»-*r»Q
^* '^, i.» k> V i C* ^ U, ^ ^— * _iw *— O L- W i- ^J, J_ y — \_*T>OLA.^. v- f JL s- O O U \-* w i_ *u
pressure, velocity of flow etc., is easy by
application of the continuous measurements of
these magnitudes and statistical analisis of
results,
The introduction of the random fields of the
other parameters is a more difficult, because
of a lack a certain methods of their continuous
measurements /dust concentration, particle
size distribution and shape estimation before
and after filtration structure/,
The introduction of the random fields of the
filtration material and dust layer structure
parameters, is possible only on the base of the
theoretical promises resulting from the espe-
cially developed theories /ductive filtration
and "zero-layer" theories/-,
From the above results, that the experimental
verification of the complete probabilistic
model of the dust filtration process recuire
to defeat of a many, mainly instrumentations,
difficulties. Experimental verification of
the model under a limited Quantity of the pro-
cess parameters, seems to be not suitable
because of a big costs.
-------
In the context of the general conclusions, the conducted
research works in the range ofthe fabric and dust layer
structural parameters distributions are very important.
On the basis of the obtained results, the following deta-
iled conclusions can be precised:
- Elaborated method of the fabric structure pore
size distribution can be utilized for the arbi-
trary kinds of the filtration materials,
- The pore size distribution of the woven and
non-woven filtration materials is normal,
- The mean value of the pore size and its stan-
dard deviation describe the structure of the
filtration materials from the viewpoint of
the filtration properties and air permeability,
- A certain dependences between the parameters of
the pore size distribution and technological
parameters of the filtration materials were
confirmed,
- The pore size distribution parameters depend
on the kind of raw material, kind of fiber and
kind of weave of the woven structures,
The existence of the discavered in previous
works, the basket free area was confirmed,
The parameters of the dust layer pore size dis-
tribution are a characteristics of the dust la-
yer structure, and they depend on the particle
size distribution after filtration structure,
Because of a specific electrical properties of
the most of industrial dusts /high specific re-
sistivity/ , the local electric fields inside
the dust layer structure can sigificance increa-
se the dust collection efficiency of process.
3
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SECTION II
RECOMMENDATIONS
The results of this project point to the need for further
investigation in the following areas:
- Determination of the best methods to conduct
the continuous measurements of the dust con-
centration and particle size distribution du-
ring filtration process performance, after and
before the filtration material structure,
- Development of the elaborated method of the
filtration material pore size distribution
by use the different kinds of the wetted li-
quids ,
- Empirical verification of the probabilistic
model in the laboratory scale with a special
attention to the influence of the concentra-
tion and particle size distribution on the
dust collection efficiency,at the different
phases of the dust filtration process,
- Searching the method to description of the
shape of a certain kinds of the industrial
dusts grain.
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SECTION III
INTRODUCTION
This report includes the results of research works which
were done during two years realization of the Project No.
P-5-533-3, between Environmental Protection Agency and
Institute of Industry of Cement Building Materials in
Opole in the range of Maria Skiodowska-Curie Foundation,
established by shares corning from United States of America
and Polish Government.
Because of lack of direct contacts between American and
Polish specialists, which were foreseen in the program
of the Project realization, it v;as not possible to discuss
a certain problems connected with research program of
realization as >-8ll as with its modification in the light
of obtained results. It is worth to notice that this
situation caused some troubles with realization of Project.
In this situation all decissions concerning the modifica-
tions of research program as well as a manner of a certain
problems solution, were taken up by Principal Investigator
Mr, Jan R. Koscianowski, personally.
The problems deliberated in this report are tightly connec-
ted with research vorks developed in the Projects P-5-533-3
and P-5-533-4 /realized in the range of FL-480 during
1974-1981/ and which were devoted to the investigations
of dust filtration process performance through different
textile filtration structures.
The conclussions resulting from these works become a base
-------
for the current considerations on the probabilistic model
of dust filtration process.
Because of short tine of this Project realization the main
stress was put on the problem of physico-matherr.at.ical
description of substantial process parameters deciding about
its performance. The test works were rather limited because
the utilization of previously otained results was foreseen.
According to the research program, the works were mainly
concentrated on the following problems:
1° Description of structural parameter of textile
filtration materials9
2° Description of structural parameter characte-
rizing the dust layer in the dust filtration
process,
3° Determination of the influence of electrostatic
phenomena on the dust filtration process perfor-
mance.
Presented above problems were already the subject of our
previous considerations which were not finally solved with
success because of lack a certain theoretical base explain-
ing some peculiarities of real dust filtration process.
Since, the general physico-mathematical rr.cdel of dust fil-
tration process was elaborated it was possible to develop
sotne investigations to improve its applicable. It was also
possible to set sorse hipothesis in the range of structural
parameters of textile filtration materials as well as
dust layers,
The model of ductive filtration and also fundamentals of
zero-layer theory enable to describe the physical sense
of these parameters and their significance for shaping
-------
the values of external parameters of different kind of
dust filtration processes.
As it will be shown in the next sections, the main structu-
ral parameters describing the structure of textile filtra-
tion materials and also dust layers are parameters of
core size distribution, i.e. mean pore diameter and its
standard deviation.
The problem of experimental determination of pore size dis-
tribution for different kind of filtration materials was
successfully solved, A very simple instrument has been
constructed and also methodology of statistical results
interpretation was elaborated. Using this method, the pore
size distribution for the filtration materials which were
investigated in the range of previous Projects /P-5-533-3
and P-5-533-4/ were performed.
In results of carried out works, it was possible to extend
the probabilistic model of dust filtration process by one
of the most important parameter of dust filtration process
giving its more developed forms,
Determination of pore size distribution in dust layer was
considered in the aspect of ductive filtration model. In
this range, the problem of some differeces between the
values of theoretically and experimentally determined re-
sistances of flow ware discussed on the ground of a spatial
*
physical models.
Investigations of electrostatic effects on the dust filtra-
tion process performance, have been devoted to the estima-
tion of electrostatic forces magnitudes and to the kinetics
of charge increase during filtration and regeneration phase,
Majority of the problems discussed in this report, have
been presented to many conferences.
7
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The basic objectives of this program v;ere:
- Elaboration of the developed probabilistic mo-
del of dust filtration process basing on empi-
rical data obtained during realization of this
Project and research works conducted previous-
ly,
- Experimental application of developed probabi-
listic model of dust filtration process for
estimation and programing, from the view point
of its efficiency.
Total program research includes the following:
- Laboratory testing of selected filtration struc-
tures in certain conditions of dust filtration
process realization,
- Determination of pore size distribution of
selected filtration structures,
- Auxiliary studies; and
- Application of mathematical methods, including
modeling.
GENERAL PROGRAM
Testing of Filtration Materials
Laboratory testing of filtration structures was limited only
to a certain cases because of utilization of previously
obtained results. The conditions of nev/ experiments and auxi-
liary tests conducted in this Project, were given in the
8
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respective sections.
Structural Parameters of Filtration Structures
Work concentrated on the problem of pore size distribution
deteraination for different structures of filtration ma-
terials. Examination of filtration structures included:
- Elaboration of the method of the pore size distri-
bution deteraination,
- Determination of the pore size distribution para-
meters for 16th kind of American fabrics, which
were tested during realization of the. Project
P-5-533-^,
- Determination of the pore size distribution para-
meters for selected structures of Polish fabrics,
The analysis of obtained results from the view
point of clean air flow as well as dusty air flow,
- Investigations of the influence of technological
parameters on the pore size distribution of fil-
tration structures.
Structural Parameters of Dust Layers
/
The object of this research was to analize the growth of
filtration resistance during dust filtration process perfor-
mance on the basis of ductive filtration concept.
Some statistical methods were used in the range of results
analysis.
influence of Electrostatic Phenomena On Dust Filtration
Process Performance
-------
The problems of electrostatic phenomena in dust filtration
process was studied on the basis of laboratory tests carried
out using the special-constructed stands. These stands ena-
bled to measure of electrostatic effects during filtration
and regeneration phase.
Some of works were devoted to the determination of kind of
electrostatic effects by simulation of process performance.
Presented in this report works were carried out at the Insti-
tute of Physics of Wroclaw Technical University under direc-
tion of Dr. J. Malchor.
FABRIC AND DUST SELECTION
Selection of fabrics and dusts resulted from the problems
to be soluted. In the most cases the kind of fabrics was
similar to fabrics which were used in our previous works.
However in some cases the specisl structures were analysed.
The type of fabrics used in the individual experiments and
also their essential properties are given in respective
sections.
The typical dust which was used in our works was separated
/or unseparated/ fly ash with KMD =5,5
10
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SECTION IV
PROBLEM'S OF DUST FILTRATION PROCESS MODELING-
INTRODUCTION
The modeling is a procedure which enable to formulate some
dependences describing the certain phenomenon or process.
Dependly on a manner of the problem solution, the physical
and mathematical modeling can be distinguished. However,
in the both cases the problem drives to the formulation
of some mathematical relationships between parameters
describing a certain phenomenon or process, the character
of these relationships is different.
The physical modeling derives from a certain theoretical
premises resulting from consolidated and known physical
laws and also from a certain geometrical models or science
hipothesis. From its nature, physical modeling has a theore-
tical character, which degree of abstraction depends on the
complication of problem to be soluted. In the physical mo-
deling all parameters describing a given phenomenon or
f
process should have a certain physical sense from the quo-
litative view point and precisely definited. All simplifi-
cations should have a definite physical sense.
Physical models can be verified experimentally but with
preserve a certain conditions which are conformable with
assumptions taken into consideration during their creation.
The preciseness of such models depend on degree of compli-
cation of examined phenomenon or process, its peculiarities
11
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and possibilities of description of all its parameters.
The determined character of physical models unenable to use
of their to the modeling of multipararneters and random
phenomenon and processes. This property of physical models
is a big disadventage in their application.
The mathematical modeling has empirical character and des-
cribe the manner of accumulation, classification and estima-
tion of parameter values in the order to obtain some rela-
tions between them.
These relations can .have functional character of type :
y = f / x^f...,x /, v:here : f is a determined continuous
function if y, x^,...,x are a parameters of a given phe-
nomenon or process, or have stochastic dependences charac-
ter of type : y = f / x*,... ,x /, v;here : f* is a function
with limited oscillation and y, x..,...,x are a random
fields.
The relations vrhich have functional character are determi-
ned dependences, so they are determined models. The rela-
tions which have stcchastical character are the probabi-
listic models of a given phenomenon or process.
As it results from above considerations, only probabilistic
models should be used to description of multiparameters and
random phenomena or processes. But application of probabi-
listic models need to accumulate of an adoauate wide emi>i-
* *• *
rical base which enable using a certain statistical methods,
The problem of description from physical view point of all
parameters which create a probabilistic model is in manv
cases very complicated and it require to performs of the
physical analysis of a certain phenomenon.
In the light of mentioned above comments, it is clear that
only on the way of probabilistic modeling it is possible
to describe all phenomena, of the dust filtration process
12
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as well as a complete process.
However the lack of a certain theoretical base of this pro-
cess was a main abstacle in the range of develop of such
orobabilistic models, because of their later physical
interpretation.
In this situation it was necessary to create a certain
physical base of dust filtration process performance and
describe some relationships.
On the base of physical analysis of process, the general
determined model of dust filtration process was elabora-
1 2
ted ' . The general model of dust filtration process
enabled the creation of some more detailed physical models
explaining a certain peculiarities of process by means of
classical mechanics. Moreover it was possible to apply
some premises of classical filtration theory.
In results of these research works, the uniform theory of
dry dust filtration processes was elaborated, . This theo-
ry is still develop and is a base of a further considera-
tions on the field of dry dust filtration through porous
media.
Description of the relations between physical magnitudes of
process using the elaborated probabilistic models, seems
to be the main attainment of the previous research works.
In the next parts of this section a short presentation of
f
obtained results will be presented.
GENERAL PHYSICAL MODEL OF DRY'1' FILTRATION PROCESS
In each dry filtration process it is possible to distin-
guish two sets of parameters : internal and external pa-
rameters.
External parameters are the measurable manifestation of a
13
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continuous process that-can be determined Independently
of the physical phenomena that determine their absolute
values, External parameters of the dry filtration process
are : average and momentary efficiency of dust collection,
average and momentary fractional efficiency and filtration
resistance* The external process parameters can all be
measured in a real filtration process.
Internal parameters are factors determined by process phy-
sical phenomena and do not depend on filtration tine. Their
absolute value determines the measured values of the exter-
nal parameters. The internal parameters of the dry filtra-
tion process are : the state parameters /S?/, the filtra-
tion parameters /F?/ and structural parameters of the
filtration medium /SPFM/ . State parameters characterize the
aerosol before and after the filtration structure and are
the thermodynamic and physicechemical properties of the
dispersion and dispersed mediums. Filtration parameters are
the independent process variables that allov; comparison
of filtration effects. Structural parameters of filtration
medium describe, from physical point of view, the filtra-
tion structure as characterized by the technological parame-
ters of its production and also by ; the physicochemicai
properties that depend on the raw materials from which it
was manufactured,
Assuming the constant value of filtration efficiency, the
effect of groups of parameters on the filtration process
can be discussed. With this assuption, Figure 1 schemati-
cally illustrates the theory of dry filtration.
The transition from the initial aerosol state /1/ to the
final state /2/ docs not depend on the path but only on
the initial and final parameters of the aerosol state.
Many processes can yield the sane efficiency but with di-
-------
FILTRATION PARAMETERS;
ir2
FP,
n
FINAL STATS PARAMETERS
0? AEROSOL
STATE 2
INITIAL STATE PARAMETERS
OF AEROSOL
FILTRATION MEDIUM
PARAMETERS:
TTNTP
•"M
MP2 '
FMP.
n
SP1 4 SP2
FP1 and FMP1 ^
Efficiency of process
and FMP
constant
FPM and FMP..
n n
Figure 1. General scheae of filtration process.
L
15
-------
fferent filtration structures and filtration parameters.
However, it is worth to notice that frooi the energetical
ooint of vie1./, these processes are not comparable.
Basing on our previous deliberations, three types of dry
filtration processes can be delineated:
- High-efficiency air filtration, with initial
particle concentration in aerosol below 1 mg/n
and particle size distribution which is charac-
teristic for indoor aerosols,
- Air filtration, with initial particle concctra-
tion bolov? 50 rng/a and particle size distri-
bution which is characteristic for atmospheric
aerosols, and
- Dust filtration, with initial particle concentra-
tion above 50 ng/m^ and particle size distri-
bution which is characteristic for industrial
aerosols.
GENERAL PHYSICAL MODEL 0? DUST FILTRATION PROCESS
The primary property distinguishing the dust filtration
process frca other dry filtration processes is its cyclic
tine behavior. On the contrary to the high-efficiency air
filtration and air filtration processes, filtration median
after reaching a certain predetermined level of filtration
drag, is not replaced but is regenerated.
In this situation, the influence of the clean filtration
structure on process efficiency can be neglected because
the time during which the aerosol . contacts the clean
filtration structure is short compared to the total filtra-
tion tirse. However, interactions between the clean structu-
16
-------
re and the solid particles of the dispersed phase play
an important role in the dust filtration process because
of their influence on the structural properties of the
composite, partially dust-filled filtration medium.
From above considerations result, that the complete dust
filtration process consist with many elementary filtra-
tion cycles which are repeated continuously till filtra-
tion structure is destroyed.
To describe the complete dust filtration process it was
necessary to distinguish the characteristic filtration
processes. Three main types of dust filtration process
can be distinguished as a characteristic :
- Dust Filtration Type I that is characteristic
process for the first filtration cycle as the
filtration structure first time is in the con-
tact with aerosol,
- Dust Filtration Type II which includes those
filtration processes between the first regene-
ration cycle and the equilibrium state of the
dust filling the filtration structure, and
- Dust Filtration Type III which is the filtration
process characteristic of the equilibrium state.
r
As it is easy to see, the state of filling with dust of
the filtration structure after regeneration cycle is a
main criterion of complete dust filtration process parti-
tion.
To describe the state of the filtration structure in the
complete dust filtration process, the coefficient of
structure filling can be introduced:
17
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/i, - -fi- / 1 /
•where:
A. = coefficient of structure filling after rege-
neration cycle i ,
L. = structure dust load after "i" regeneration
cycle, g/ra , and
L»TV. = equilibrium structure dust load, g/n11".
The "structure filling" can also be described by the rela-
tionship:
flr = —^- I 2 /
I A i o
" NX
where:
/3 = relative coefficient of structure filling
1 after regeneration cycle i ,
?.T. = pressure drop of partly filled structure,
after "ifl regeneration cycle, N/m , and
P.,T, = pressure drop of fully filled structure
/after an equilibrium state is achieved/.
N/ra2.
/
In the conditions of stable values of the gas and dust loa-
ding of the filtration area /a = constant and a = cons-
g • "P
tant/, the following states of filtration structure can be
distinguished in the complete dust filtration process:
- Pure filtration structure /virgin fabric/ characte-
rized by the pressure drop, A P ,
- Structure partly filled with dust /virgin fabric
18
-------
plus dust not removed by cleaning and before the
steady state operation is reached/, characterized by
successive values of A ?„. and L»,. where i =» 1,...
. ..,k are the cycle numbers, and L... is the ereal
dust load after the i-th cycle. This fabric state
exists immediately after the regeneration cycle but
before the equilibrium state is achieved.
- Structure fully filled with dust /virgin fabric plus
dust not removed by cleaning and after the steady
state operation is reached/, characterized byAP.,,,
the pressure drop associated with an areal dust load
of LML,. This state represents the equilibrium state
fiK
for which Lvv =» constant independent of the cycle
Nn. ~ J
numbers.
- Structure covered with dust /fabric fully filled
with dust plus dust layer just before cleaning/,
characterized by the pressure drop A Pv , and
&
L a LNK + Lp , where Lp is the areal density of
the dust layer on the fabric surface.
From the above results, that the physical structural para-
meters of a pure structure change during the successive
filtration cycles of the complete dust filtration process,
until the equilibrium state is achieved. Differences in the
state of the filtration structure in the complete process
lead to the conclusion, that only the first filtration
cycle v/ith LN =» 0, and all cycles until the equilibrium
state is achieved /LNK =» constant/, can be treated as a
stationary in physical sense.
In this manner, the necessity of complete dust filtration
process partition on the three types of dust filtration,
is proved.
19
-------
On the Figure 2, the general scheme of the dust filtration
processes is shown. The transition from the initial aero-
sol state /1/ to the final state /2/ is for each of the
individual types of the dust filtration process different
because of differences in structure of filtration medium
caused by a degree of structure filling with dust.
Defining the rr.ean value of the total dust collection effi-
ciency of the filtration cycle as follows:
f(t) dt / 3 /
o
where :
ET = mean value of the total dust collection
efficiency,
t.,- = time of filtration cycle,
*. o
f (t) = the collection efficiency function.
it is possible to introduce the notion of initial dust collec-
tion efficiency, E , which is a hypothetical value of the
collection efficiency function at the time t_,, = 0.
Because the structural parameters of filtration raediun
change during a complete dust filtration process perfor-
mance, excluding first and also all cycles until equili-
brium state is achieved, the external parameters of respec-
tive types of dust filtration processes also differ:
E0fI(SPMF) < EQ>II (SPFK) < EQfIII (SPMF)
,
T,II
20
-------
FILTRATION PARAMETERS:
FPDFT I * FPDFT II * F?DFT III
FINAL STATE PARAMETERS
OF AEROSOL
FILTRATION MEDIUM PARAMETERS:
FMPDFT I ^ FMPDFT II ^ FMPDFT III
INITIAL STATS PARAMETERS
OF AEROSOL
SP
SP,
N
1,2
»• • •
»• • •
_Figure 2. General scheme of dust filtration process.
21
-------
And also
A?0 T (SPFM) < AP T, (SP?M)-J j -L -i. .*_
Dust Filtration Type I is a process characteristic of labora-
tory- scale experiments in v;hi. i the test material is dusted.
in- only a one-cycle operation without regeneration,, In the
conditions of settled, values of initial aerosol state para-
asters and filtration parameters, the external filtration
parameters differ due to filtration properties of filtration
materials what enable to investigate of structure influence
on process performance,. This property of process is utilized
in so-called comparative test investigations, when on the
basis of obtained experiment results, the quolitative analy-
sis can be performed.
Dust Filtration Type III is a process characteristic of dust
collection in industrial dust collection devices, cyclic
dusting, and regeneration of filtration elements composed of
certain filtration materials. Examination of Dust Filtration
Process Type III can be conducted directly, either by indus-
trial or "large-scale" testing. Because of a large costs of
industrial experiments, the large-scale testing stands are
usually used, and industrial conditions are simulated in
a wide range of parametric values.
The characteristic property of Dust Filtration Process Type
III is the equilibrium amount of dust regaining in the fabric
filter after regeneration because of the cyclic character
of the process.
22
-------
PHYSICAL MODEL OF DUCTIVE FILTRATION PROCESS
On the basis of statistic analysis of the results obtained
during laboratory as well as a large-scale experiments
with different kinds of filtration fabrics and dusts, the
physical concept of the Ductive Filtration Process, was
proposed .
The basic total assumptions for the proposed physical
model of the ductive filtration process are:
- The particle size distribution in the aerosol up-
stream of the filtration structure is independent
of time and position,
- The velocity distribution in the aerosol upstream
of the filtration structure is independent of
time and position,
The partcle size distribution in the aerosol is
normal,
- The spatial structure of the dust layer formed on
the surface of the filtration material *. is not
affected by dislocations or structural defects in
a geometrical sense,
- The dust concentration upstream of the filtration
structure is constant, and
- The dust particles are spherically shaped.
Microscopic inspection of the dust layers formed on the
surface of filtration materials as well as a theoretical
considerations lead us to the conclusion, that the nost
probable configuration of the individual dust grains is a
four-element configuration.
Assuming that the individual dust grains have diameter equal
23
-------
to the in can particle diameter in the aerosol upstream of the
filtration structure, the four-element configuration consists
of four individual dust grains whose centers for 21 a regular
oyranid-shaped cell. Figure 3 shows such a basic configura-
tion.
As a spatial geometrical fora, the cell /limited by the face
of regular pyramid with vertex in the centers of the indivi-
dual grains/ is characterized by superficial porosity £
- i_J
0,09 and spatial porosity £,, -. = 0953 which are independent
on the grains radius.
Froa the view point of gas flow through a such cell, three
characteristic dimensions of the pore size can be defined
/Figure 4/ :
The hydraulic radius of pore, R., , defined as a ra-
» * lv
tio of passageway cross sectional area, F, to the
passageway perimeter, U,
RKp - -- = 0,05rp
where: r = mean grain diameter
The equivalent diameter of core, D , defined
- cq.p
as a Egnitude equal to the four dimensions of
hydraulic radius :
Deq.p " O-r «
The gecnetrical diameter of pore, D-, , which is
u-p
a dianeter of a greatest sphere inscribed between
three grains forn the space of pore :
DQ? = .0,309rp / 6
24
-------
3* of
i 25
-------
Figure 4. Geometrical dependences on the surface of
dust layer.
-------
Because it is assumed that the particle size distribution
of the aerosol is normal, thus the hydraulic radius,
equivalent diameter and geometrical diameter of pores
distributions are normal, too.
Assuming additionally, that the collisions between indivi-
dual dust grains are non-elastic, the following mechanisms
of aerosol particle deposition can be delineated :
- Selective precipitation, which causes the dust
layer to grow,
Inertial deposition in the dust layer structure
because of acceleration of dust particle in pores
to velocity y »v /impaction effect/, where :
v is the gas velocity in the pore, and v is
the nean gas velocity upstream of the dust layer
structure,
- Difussion inside the dust layer structure, and
- Electrostatic precipitation inside the dust layer
structure because of local electric fields.
Selective precipitation is a major mechanisms of the ducti-
ve filtration process which includes the sieving effects
and also the mechanisms of simple inertial deposition and
interception in the velocity field, v.
The sieving effect in this case differs strongly in respect
to the classical conception because of change of the openings
dimensions with tine. It results from the fact that the
pore size distribution depends on the particle size distri-
bution in the aerosol.
Figure 5 explains filtration mechanisms in the ductive
filtration process performance.
As it is showed, all particles in a stream tube of diameter
27
-------
w » m
w » ra - 1
w » m - 2
Figure 5. Filtration mechanisms in ductive
ufiltration Tjrocess.
28
-------
D can penetrate the external elementary dust layer,
eq.p
if their diameter, d , is lov/er than geometrical diameter
of pore, D-, . All particles placed in stream tube with
d > Dp will be precipitated causing the dust layer to
grow. Dust particles with d
-------
TH20R7 0? THE "ZERO-LAYER"
The theory of the "zero-layer" was proposed by the author
of this report in results of analysis of research works
developed in the range of the ductive filtration performan-
ce.
It was found that the initial phase of each types of the
dust filtration processes during which the structure of the
filtration material is filled with dust /after the infle-
xion point of filtration resistance function is reached/
is not comparable with ductive filtration, deliberated pre-
viously. Because of a high initial dust concentration in
the aerosol, this phase cannot not also be considerated on
the ground of classical filtration in the range so-called
stationary filtration. In this situation , it was necessa-
ry to treat this process independently from the other ones.
The considerations related to the initial phase of the dust
filtration process was called the theory of the "zero-layer".
The theory of the "zero-layer" is tightly connected with
behaviours of the textile filtration materials.
Figure 6 shows the field of application of "zero-layer" and
ductive filtration theory.
Because of cyclic character of the dust filtration process
the problem of the first dust layer formation on the surfa-
ce of the filtration material is different in the range of
the Dust Filtration Process Type I and the Dust Filtration
Process Type III.
In the first case, the momentary initial dust efficiency in
the time t,_, = 0 -5- dt^ will be shaped by the similar rrecha-
M jrt - J
nisms as in the ductive filtration process. However, in
this case, the efficiency of selective precipitation is not
depends only on the particle size distribution but mainly
30
-------
FIELD OF
ZERO-LAYER
FILTRATION
FIELD OF
DUCTIVS FILTRATION
CHARACTERISTIC INFLEXION
A relate to DFP type I
B1 relate to DFP type III
CHARACTERISTIC INFLEXION
FUIWT
A relate to DFP type I
3 relate to DFP type III
'FA
FILTRATION TIME , t-
Figure 6.' Range of application of "zero-layer" and
ductive filtration theory.
31
-------
on the pore size distribution of the filtration material
structure.
In the second case, the momentary initial dust efficiency
depend on the pore size distribution of fully filled with
dust structure of the filtration material, /process charac-
terized by [b= 1/.
It is evident, that in the second case, the mean size of
riore diameter is lower than in the first case, so momenta-
ry initial dust efficiency is higher.
The pore size distribution of filtration materials depend
on the many technological parameters which describe spatial
as well as superficial structure of fabrics from the techno-
logical view-point.
Generally, the textile filtration materials used in dust
collection devices are distinguished as follows:
- Non-woven fabrics and felts,
- • V.' ov en fa bri c s,
- Knitted fabrics, and
Other structures.
Non-woven fabrics and felts have the raost uniform structure
comparing to the v/oven fabrics. Their structure is characte-
rized by definite order of elementary fibres composition,
enabling to assume isotropy in spatial'structure sense.
'•'oven fabrics have heterogenous and anisotropic structure
in which definite privileged directions can be found. The
characteristic property of these materials is their perio-
dic structure in the sense of superficial and spatial dis-
tribution of the structure elements.
Knitted fabrics have a medial structure properties between
non-v;oven and woven fabrics.
32
-------
On the Figure 7 the main structural differences of non-
woven and woven fabrics are schematically illustrated.
In opposition to the non-woven filtration structures
characterized only by inter-fibres spaces /not filled with
fibres/, the woven structures are characterized-by three
kind of spaces. There are inter-fibres spaces in threads
structure, inter-threads spaces and specific spaces /which
v?as found and described by us in our previous works / called
basket free area. Basket free area consist a free area
/not filled with fibres/ placed in a plane askew to the
fabric surface resulting from a threads stiffness.
Basket free area can influence the structure of the dust
layer formed on the fabric surface causing a dust layer
structure deffects, which decrease the total efficiency of
the filtration process . From this point of view the basket
free area are very undesirable in the filtration material
structure. Their influence can be limited by application of
the certain technological parameters.
Apart from the problem of technological parameters influen-
ce on spatial and superficial structure of the filtration
materials, it is possible to assume, that the each given
filtration material is characterized by values of statisti-
cal parameters of its pore size distribution. These para-
meters decide about the external filtration parameters
of the initial phase of the dust filtration process i.e.
momentary initial dust collection efficiency, total dust
collection efficiency and also filtration resistances after
the inflexion point is reached.
With assumptions accepted for the ductive filtration process
some general theoretical dependences were developed giving
the base for the further considerations.
To verify of these dependences and also to define of each
,33
-------
V
c
O
m
r/)
o <
M C.-.,
T7»f r»- »-^-»»f ~v^ ^'f
•o~~cr"cr~cr~
o o o o o
__Q__o__a__o.
O
M
CO
(O O O O O O
Figure 7a. Comparison of non-wovsn and v/oven structures.
-------
, ..
/
X
/ ' /
' ^ -v * X
k'v* / • ,
F ,•'
* . * •
^
,
• >
V
Figure 7b.
Comparison of non-woven and woven structures,
/Photo credit: SEM by K.Skudlarski, Electro-
nic Microscopy Laboratory of Polytechnical
University in Wroclaw/.
35
-------
filtration mechanism participation in the total filtration
aroccss after the first elementary dust layer is reached,
it is necessary to describe the real pore size distribution
of a given filtration materials. This problem was the
principal one during realization of this Project.
RECAPITULATION AMD CONCLUSIONS
Physical nsodels of the dust filtration process presented in
this section enable to develop so.7ie probabilistic raodels
on the way of application of some determined relations as
well as geometrical simplifications giving the main relation-
ships between parameters of filtration process.
It is worth to notice, that parameters used in the physical
models have a certain physical interpretation what enable
to utilize of all statistical conclusions which results
from testing of the probabilistic models.
-------
SECTION V
PROBABILISTIC MODELS
OF DUST FILTRATION PROCESS
INTRODUCTION
Because of lack of a consolidated theory of the dust filtra-
tion process performance, the conclusions concerning the
qualitative and quantitative parameters describing the dust
filtration process are mainly formulated on the basis of
test results. The process itself has a such degree of compli-
cation, that it is practically not possible to describe all
its physical parameters as well as its peculiarities.
In this case the probabilistic approach is necessary, which
enable to investigate and to infer about the process perfor-
mance by utilizing a certain mathematical methods.
The application of the elements of probability mathematics
to investigations of phenomenon and physical processes, and
also dust filtration process, is possible only if, they are
described in the form of probabilistic models. Then, it is
possible to apply of a certain statistic methods, mainly the
statistical inference.
The elements of the random field theory as well as the
different probabilistic models elaborated on the basis of
this theory will be presented, below.
ELEMENTS OF THE RANDOM FIELDS THEORY
Let we assume, that the dust filtration process can be
37
-------
described by noans of characteristics f,,,...,f , which
^ • A-*.
are understood as a determined physical parameters. Let D
be a planar set v>hich is filled by the filtration Tutorial.
The characteristics f15...,fn are tho functions defined
on the D , what Deans: f^f-,(?),... ,fn=fn(p) , peD .
So, they are a determined functions.
By determined physical nodol we consider the set of values
of f-jfp) »••• jfn(p) and all unions occur between f^,...,fn,
i.e. all functions be a unions between of these functions.
Because as it v:as mentioned previously, it is not possible
to investigate of f^>.,.?f and also all their unions, the
i n
elements of probabilistic should be introduced assuming that
f..,,..?f are the randcn fields on the set D .
By the probabilistic model of the dust filtration process
\,'Q consider the sot of random fields f 1 (p),... ,f ( p ) and
all unions betv/een these fields /regression curves/.
Let (E,A9P) bo a probabilistic space. The function f(p)
is described on. the filtration material. A certain valuer?
of f in the points p..,,..,p are investigated. A certain
functionals connected v/ith f e.g. total content, are esti-
mated
I
ffp)dp = Z
D
p
The coordinate cystea is randomly put on the R , what
means, that for each point eg. E of a certain probabilistic
space (E,A,P), the coordinate system C(c) is ascribed.
Then, for each of points p of coordinate system C(e) ,
a certain point p(e) in D or over D is adcquated.
Describing f(p(e)j by f(p9e) we obtain the random field
if f is as much regular, that of-* f(p,c) is a random
variable.
-------
Because of quasi-isotropy, which will be later described,
we will consider the randomness in the sense of parallel
displacements but not of rotations.
By f(p) we understand one of the parameters of the filtra-
tion material, which is investigated.
The random field f(p,e) we will describe by f(p) .
Let pfQeD^nD . The expression R [f (p) ,f (q)] is a corre-
lation coefficient of a random variables f(p) and f(q):
where: m » E[f(p)j, ma E[_f(q]Jare expected values of the
random variables f(p) and f(q) .
If the expression /?/ receive the forms:
R[f(p),f(q)] « R*(pq) /8/
for each p^eD^OD , that the expression /8/ is called
correlation function of the random field f(p) •
The random field f(p) is called stationary in wide sense
if:
* n - constant 4 +'oo /9/
D2[f(p)] » &2
-------
The random field stationary in wide sense is called isotro-
pic, if:
where: |pq| is a length of the vector pq .
Scuality /12/ !r.eans9 that for each jpcija constant, the
correlation function R(pq ) , is stable? on the circle
(pq | pq) « r , so if p =*(:<, y) and q ^(x^y^ 9 that it
in stable on the circle (x - r--\) ' * (>' ~ 7-0 3 r^ " cons.
The correlation function of the randcn field stationary in
v?ido sense 9 is estinated fron the follov/ing equality:
R(x) » J exp[- i(x|u)]dF(u)
rtn
where: x =»(x. ,., ,,xr), u a^u,.,...^. ) are the vectors
fron the space Rn , (T:[u) is their scalar product,
i is a in-aginary number „ F(u) is ^ distribution
function of sons n-dinensional random vector u ,
and x denote vector pq .
If, the random field stationary in vide sense is isotropic,
that its correlation function is estimated from the following
equality:
n-2 - *? n-2
Rfixll * 2
vrhero: (>:| is a length of the vector x = pq , A is parame-
ter which can be equal to A a jtT| ? Jn«^ is a
Bassel function of the first, kind d with index
n_2 , x
—=• , G(A) is a distribution function such th"t
-------
G(0) a 0 .
Utilizing the above equations, especially the reciprocal
Fourier's transformation of the equation /13/ having the
forms :
g(CI)
where: dx => dx1,...,dxn ,
the following correlation functions of the stationary
random fields in the wide sense and isotropic, was
obtained:
,,-k+l-,
R(Ix|) = exp[- A(|x!)^ J ,
*(!*>) - — - -k.1 /17/
-k-t-1
where: A>0 , 0
The model based on the isotropic random fields and statio-
nary in vide sense*
r ~" ---- *fc— ^^ - MB^MBMH L ~ ~ • ~ f
The random fields described bolow, can be used to investi-
gations of the problems of filtration materials homogeneity
and also other parameters.
Let we assume, that f .,.,., f are the parameters characte-
rizing the process.
Let we define, as previously, the planar random fields of
the parameters f1(p),...,f (p) . The following magnitudes
are estimated:
41
-------
M&(?U - HIT I f^)d-° /18/
D
which is a mean value of the parameter f in the area D ,
and
,
I — ' I
D
which we will call the variance of this paramo tor.
The equation /18/ we will consider as a stochastic limit
of the following suia:
/20/
for normal sequence of the partition |A] of the set D .
The equation /20/ we will call the estimator of the mean
value /18/« It was proved, that the best estimators of M
are: the weighted averages and for a sufficint renote points,
the arethmetic means. They have the following forms:
where: c^ + ...-»• c = 1 and c.> 0 (i =» 1,...,n) are
selected so that:
q = rain E[c1f1 * ... + cnfn - Mj^ /22/
cnfn]
.and
42
-------
M2(D) - c(f1 * ... + fj /23/
where: c a — ~ , and
[c(f1 + ... + fj] - E(M)
where: M » M^ is given by the relation /18/.
The errors of these estimators can be calculated from the
following formulas:.
s2 ,
a
, ^c.jCX^,^)* ... + cnol(f1,fn)-cL(f1fM)}+ ... +
* ca{c1^Cfn'fl)^— * cn R(PP) /26a/
2
" Rpda- /26b/
<=> rr r rr „/—^ i. /^ /
=* r < Rfcqldaldp /2oc/
(D!2 JJIJJ ^ | j
E(M) - -1-" ff E(f(p))dp . M /26d/
n ->J '
IDI
The problem of estimation of the equation /19/ is as
follows. If the random variables f(p1),...,f(p ) are
2
independent, and s is a standard deviation of a mean
value obtained from the sample, that estimation of
D [f(p)] has a form -—• s and is a good estimators.
-------
10
It was also proved , that for a sufficient remote of
samples points, the good estimator of the D [f(p)j is in the
form -^7^ s .
The other problems connected with the estimation of the mean
value is problem of sampling procedure. Let we assume, that
the mean value is estimated by arithmetic mean obtained from
the experiments. So
/27/
where: for i = 1 , the mean /27/ is estimated for the
values of f,.,... ?f obtained by means of random
sample; for i = 2 , the mean /27/ is estimated
for the values of f ..,.,.,£ obtained by means of
stratified sample; for i = 3 , the mean /27/ is
estimated for the values of f.,..., f obtained by
means of systematic sample,
The random, stratified and systematic samples are obtained
by the following manner.
Random sample. Let we assume, that the selection of points
peD is achieved with a monotonous probability. That moans
that selection of each point is the sase-probabie and that
selection is achieved independently frcn tha other points.
So, the probability of point selection is proporclonal to
the area D . In selected points the parameter f is measu-
red, and the set of the values f^?,..,f^ consist a random
sample*
Stratified sample. Let we divide the area D on the subsets
A*,...,An which we will call the stratifications. Let we
randomly select, the one point from each stratifications
A>j?...!,Ari • In selected points, the parameter f is measu-
surcd, and the sot of values f^9,..,fn consist a strati-
4/4
-------
fied sample.
Systematic sample. Let we divide the area D on the
congruence subsets A^,...,An • Point p^ we select random-
ly from the set A ^ . Point p^ (i« 2,...,n) we select
from the set A, in this manner, thet after displacement
of the set A by the vector P-jP.» > the sets A. and A>
were covering. In the points P^*...,? y the parameter f
is measured, and the set of values f^,...,f consist a
systematic sample.
Let we signify the variance of a mean /18/ as follows:
where: i » 1 for random saprale,
i = 2 for stratified saraple, and
i « 3 for systeaatic sample.
The following theorems were proved
Theorem 1, The equality
./29/
arrives in the optional area D .
Theorem 2. If the area D is a sum of the disconnected
and congruenced subsets A 1 , . . . ,An or if the random field
f(p) is isotropic and the sets A1,...,An are similar,
that
-------
Theorea 3» I£ "the area D is a sun of the disconnected
and congruenced subsets A^,...^^ 9 that
z:
i=»1 3-1
where: p- is a center of gravity of the set A . .
Theorem 4« If the area D is a sum of the disconnected
and congruenced subsets A_.5.,,,A or if the randoa field
f(p) is isotropic or if the sets A -,..., A are similar,
* n
that
s| < s^ /32/
Theorea 3. Let D be a sum of the disconnected and congru-
encad subsets A^,...,An . In order that
Sj < A /33/
it is sufficed that for each i and j the relation arrives
IJ [[ "(P5)dq
-------
Theorea6, Let f(p) be a random field with the correla-
tion function:
R(x,y) = exp[- A(x2 + y2)2 J /35/
where: A>0 ; k = 0, 1,...,ko<°o at the definite area
D . Let the area D be a sum of the disconnecte
circles A ,.«.,A of the same size.
-2k"1
If the diameter of the area D is lower than A ,
that the stratified sample in this set is better than the
systematic one. That means
s2
s2 /36/
Theorem 7. If f(p) is a random field with the correla-
tion function /35/ definited in the area D designing
with disconnected and congruenced subsets A ..,»..,A ,
that for a sufficient big number A , the systematic
sample is better than the stratified one, i.e. the equa-
tion /33/ is arrived,
Conclusion 1, Let A and A . be a such sets, that each
HB—WM-MBMIMmMMV^^^H^^BM • -*• jj
two points p^A and. q.6A. , i / J satisfy the
__k_ i ? v
condition: |pq| > A . The inequality /34/ is satis-
fied, that means that inequality /33/ is also satisfied.
If all sets A'<.,,.,,A* satisfy condition [pql>A"
for each two points p,q belong to the optional sets,
when p belongs to the one of the sets and q belongs
to the second ones, that the sample consists with measure-
ments f19.,.,f in the points p.,,.,,p , which are the
i S . IS
centers of the sets A„,...,A* , is better than the stra-
1y * s
-------
tified one,
At the end it is worth to notice that the random sample is
a worst from the all kind of the samples.
The model based on the isotropic random fields and quasi-
stationary in vide sense,
The random field f(p) depends on the additional paramete
t > 0 t that means that we have the random field f(p,t) .
Let we assume that:
D2[f(p,t)] = <32(t)>0 /38/
R[f(p,t) , f(q,t)] = R[A(t) p^l] /39/
where: A(t)^0 is a continuous function.
The random field f(p,t) satisfied the conditions /37/,
/38/ and /39/ v:e v/ill call the random field quasi-statio
nary in wide sanse and isotropic,
The correlation function /39/ we understood as a function
appointed by the correlation coefficients:
R[f(p,t),f(q,t)]
f(?,t)- n?(tf E[f (q,t)- mq(t)]2]1/2
where: mp(t) « E[f(prt)J ; m(t) »E[f(q,t)].
-------
The correlation functions of the quasi-stationary in wide
sense and isotropic random fields, can be the following
functions:
R[A(t)(x,y)] - exp[-A(t)(x2 +
R[A(t)(x,y)] . - L- - — /42/
1 + A(t)(x2 + y2f-
where: A(t) > 0 , 0<^<2 .
If the conditions /9/ and /10/ are satisfied and for
optional p,qeD1o D the following equality is arrived:
R(pq) -
where |[pqll is a norm of the vector pq , and such a
random field f(p) we will call stationary in wide sense
and quasi-isotropic. The quasi-isotropic random field is
a such stationary in wide sense random field for which the
correlation function /43/ has a stable value on the ed^o
of the curve appointed by the nana llpqll » constant.
The example of the stationary in wide sense field can be
the field, which correlation ^function /43/ is defined
on the elliptical norm llpqll » [a11(xTx1) + 2a12(x-x1)(y-y1)
+ a22(y"vi)2J2 • where P « (x»y)» q - (*<\9y<\)>
p.qeD.nD and ( a,^) is an unsingular syiaetric matrix
of the second order.
The correlation function in '.this case, has a stable value
p
on the edge of the ellipse in the form : a., x-x. *
n o I I I
" constant .
If the conditions /37/ and /38/ are satisfied and for
the optional p,q£D..nD and t6R1-f, the equality is
49
-------
arrived :
R[f(pft), f(q,t)] » RJA(t)!lp^!fj /4V
where: !l?ql! is a noria sontioned abov/o, that the random
field ffp) we call quasi-stationary and quasi-isotropic.
Sof it is a such random field,, which is quasi-stationary in
wide sense, and vhich correlation function /44/ is stable
on the elenent A(t)llp~qll = constant.
If the nora llpqll is elliptical, that A(t)||pql| »[A(tJ-
"
this,
we havGj that correlation function /44/ is stable on the
O O
ellipse ACtjQx^^-x^ -t- 2al2(x-x1)( y-y^ * a22(y-yi) ] =.
a constant.
The random fields have the following mean values:
M[f(p)] . ^-j f(p)dp /45/
D
M[f(p,t)] --j^j-
D
and variances
[f(p)] , -U J [f(p) -M(f(
D
D2[f(p,t)l - —JU- f[f(p,t) -,M(f(p,t))]2dp
D J
ID.
n
50
-------
ELEMENTS OF STATISTICS
The relations /45/ and /A6/ can be estimated by means:
1TJT
where: |A J (j » 1,...,m) are the fields of the sets A .
J J
on which the set D was divided, and p.eA. are the points
J 0
in which the value of the field f (p) is measured.
However, the expressions /49/ and /5O/ are too general
10
It was shown , that the following estimators of the expre-
and unuseful in practice.
10
It was shown , that the
saion /49/ are a best effective:
I + ... + cf(pm) /5V
where: d.(u,v) is covariance of random variable u and v,
o
and c^ is a variance of the field f(p).
if the sample points p..,...,p are on the sufficient
distance.
-------
Trie sinilar estimators can be given for /5Q/. It becomes
only to make f(?)- additionally depend in the points
The errors of the estimators /51/ and /52/ are given by
the following expressions:
•(p ) M* ) -ol fM* r*')> /C13/
• v.i'jj' »*'-i / ^ V'M » 1 / I I "-si
and
" I v« *J^ o ^v ^^ i-^* / v\ ^ ^ / \ i o
S ^ •"* ^^^™^?7^"* j i« \^? ^* t» *^ i—I '•p'^w 1 •* I *J •] #i( »J • y / ^ £»
_.—» I » ^ •£. ^ •/
n L i\ j ^ i= i
/5V
The expressions /53/ and /5V sre the estiaators of
/47/ and /48/.
The problem of the sample collection for the statistical
analysis is the next one. The sanple should be collected in
this nanner, that the error: of the statistical investiga-
tions was a least, and the inference concerning the sample
was a cost orobable,
10
As it was shown ff in the case of the discussed randon
fields the most effective sanples are: the stratified and
systematic samples.
The stratified sample is the best, if the sample points p
and q are in sense of distance jpqj or a norm ||pq|| in
a little distance.
52
-------
The systematic sample is the best, if the sample points
are in the sufficient long distance.
It is worth to notice that the these distances are appointed
from the correlation functions /16/ and /17/ or /41/
and /42/.
DEPENDENCES BETWEEN THE PARAMETERS
As it was previously mentioned, the dependences between a
characterises /parameters/ of the filtration process
are understood as a regression dependences.
The characteristics /parameters/ of the filtration pro-
cess f1(p),...f fn(p) a**e described by means of the
random fields f^pje),..., fn(p,e) . They will be denoted
as a f1(p),..., fn(P) -
These characteristics are characterized by M [f 1(p)]» • ••»
...., MffM(p)l given by means of relation /46/ and by
5 ~i 9-1
variance D [/.(p)],..., D [fn(p)J given by means of
relation /47/ .
The above magnitudes are estimated by means of estimators
/49/ or /52/ and /53/ and /54/.
In many cases, the information obtained by means of numbers
M[f .(p)J and D2[fj(p)] (j * 1,...,n) is not sufficed.
The dependences between the random fields f..(p) and ^(p)
j ^ k and also on a certain stable random field, on some
of others or on all remaining random fields, is often
necessary to be described.
Dependences between f^(p) and fi,(p) are appointed froa
3 "
the following conditions:
r\
Q = E[fj(P) -F(fk(p), c1 cr)J /55a/
53
-------
Q» =, Slf,/?) - F'(f,(p) , c' ..,c-')J /55b/
1^ "• \j • —•- —J
by minimizing with regard on a certain values of c ,...,c^
and c*,...,c-? . So, it is necessary to give a general form
of these functions from the conditions /55a and 5i»b/,
to appoint c^,..<,,c and c.,...,c, for which these func-
tions v;ill reach a least values. The conditions /55a and
55b/ are known in the least square methods.
From /55a and 55b/ v;e obtain:
f.(p) = F*(fk(p),c*,...fc*) /56a/
/56b/
which give the dependences between tv/o parameters of the
dust filtration process.
The dependence of one parameter frora the others, give a
regression surface, which is estinated fron the following
euation:
/57/
The regression surface /57/ is a locus of the points :
(5(f1 (p)],f2(pQ) t..«t*n(p0)) • Determination of the regre-
ssion surface /57/ in general forni is very hard. But
with some assumptions it is possible. This case will be
deliberated later.
-------
THE CORRELATION FUNCTIONS AND THEIR APPLICATION
The correlation functions of a simple random fields /called
autocorrelation functions/ , the correlation functions of
two random fields /called intercorrelation functions/
and the correlation functions of a many random fields
/more then two and called multi-correlation functions/
will be deliberated in this section.
According to the equality /13/ , if the distribution
function F(u) of_ the random field u is known, the co-
rrelation function R(x) is synonymously determined.
The intercorrelation function of two random fields u and
v is determined as follows:
/58/
if the distribution function F(u,v) of a Joint random
field (u,v) is known.
In similar manner, the multi-correlation function is
determined:
,...,
a™
J 0xp[i(u^7T777+tim| x)]
'
/59/
if the distribution function F(u1,...,um) of a Joint
random field (u,,...,u ) is known.
Because in the most cases, the distribution functions
F(u) , F(u,v) , F(u. , . .. »um) are unknown, they can be
determined from the empirical data by means of a certain
' 55
-------
criterion /% »A - Koiraogorov/ . However, this way of
proceeding is practically possible only in the case of one-
dinonsional random fields. In the another cases, the proce-
dure is following.
Let v;e estimate the autocorrelation, intercorrelation and
multi-correlation functions /what is ever possible/ .
Frorn equalities:
/&0/
v/e can determined the spectral density functions of the ran-
dom fields u , (u,v") - and (u..,.. ./u^) , respectivelly.
3y u , (u,v) , and (u.,,.».,u ) we have signified a random
fields, which was previously signified as a fj(?) t (^•<(?)»
fv(p)] and (f, (p) , ...,f E(p)) and we have assumed that these
fields 'are n-diaensional.
The functions R(xJ , R ,r(x) ,and R (x) are selec-
uv x u-i» •••»"—
ted from the class /16/ and /17/ for the random fields
stationary in wide sense and isotropic, or changing [3c j
on the norm ||x|| for the random fields quasi-isotropic,
or from the class /A1/ and /k2f for the randon fields
quasi-stationary in wide sense and isotropic, and changing
56
-------
o ? \ 1 ? ~*
(x + y J ' on the norm || x || for the random fields
quasi-stationary in wide sense and quasi-isotropic.
The empirical data from experiments are equalized by the
functions of the mentioned above class.
From empirical data, the correlation coefficients for the
distances t.,..., t , between the values of the random
field f(p) or random fields f,(p) , fk(p) (J / k) or
random fields f,(p),...,f (p) where t. « |p,q.| /orllp.qll/
' "* 0 «3 «J J
are calculated.
In the case of calculation of the autocorrelation function
the correlation coefficients P . are estimated from the
equation /?/. If the intercorrelation function is estima-
ted, the correlation coefficients are determined from the
following relation:
In the case of the multi-correlation function estimation,
the correlation coefficient is estimated from:
Using the least square methods, the criterion /the best
correlation function/ of the correlation function optimi
zation can be created. We have, respectively:
57
-------
s
S
1 k
Q
3 k
3=1
where: R is a function of class /16/ and /17/ or /41/
and /42/ .
Froa the conditions /65/ , /66/ and /67/ , the correlation
functions R_fx) , R~ - (x") and R~ o (3c) which arc
' s 1'***'rn
the best frosi the vie?? point of the empirical data, are de-
termined. The density functions can be determined on the
basis of relations /60/ , /61/ and /62/ .
Having the density functions, the criterion of compatibility
and differences can be constructed, which enable to settle
on a certain significance level . /usually cL » 0.05 or 0.01/,
APPLICATIONS
Generally, the random fields stationary in wide sense and
isotropic, can be used to description of such filtration
parameters, which expected value and variance have finite
values /so, they are stable/ and in the set in which the
random field is determined, the changeability is the same
in the all directions. That means, that the autocorrelation
function of this random field is stable on the circle of each
a settled circle.
58
-------
If, the filtration process parameter has an expected value
and variance and in the set of determinity of the random
field, the certain directions having the different varia-
tions of the field can be selected, that a such parameter
should be determined by means of the stationary in wide
sense and quasi-isotropic random field.
If, the filtration process parameter has a not stable of
exnected value and variance which depend on a certain
•\
t6R and in the set of determinity of the random field,
the directions having the different variations of the
field cannot be selected, that a auch parameter should be
determined by neans of the quasi-stationary in wide sense
and isotropic random field.
If, the filtration process parameter has a not stable of
expected value and variance which depend on a certain
teR and in the set of determinity of the random field,
the directions having the different variations of the
field can be selected, that such parameter should be deter-
mined by means of the quasi-stationary in ...wide sense and
quasi-isotropic random field.
In the case of woven filtration materials, two directions
of variation of the some parameters /along the fill and
warp/ can be selected. Sot for description of structural
parameters of woven filtration materials, the quasi-iso-
tropic random fields should be applied.
59
-------
SECTION VI
DETERMINATION 0? FABRIC PORF. SIZE
DISTRIBUTION
INTRODUCTION
Defining a structural parameter, by which the spatial ar,
well as superficial structure of textile filtration materials
can be characterized, is a r.ain research problem since a
many years.
In the light of elaborated theory of the ciust filtration
process, the msan pore size and its standard deviation, can
be ackov.'ledged as a parameters which satisfy of physical
and mathematical models.
On the basis of British Standard No. 3321 /19&0/11, concer-
ning the equivalent pore size of fabrics determination, by
neans of the bubble pressure test, the oryginal method of
extended bubble pressure test and manner of statistical
treatment of the empirical data, were elaborated.
/
THEORY
If, on the surface of soaked with wetted liquid specimen
of fabric, the increasing with time pressure of the air is
exercised, in a certain level of the air pressure dewatering
of the bigest pores is observed, in relation to the follow-
12
ing equation /Young and Laplace equation/:
60
-------
r~ PL. » 2T cos © / Rw /68/
' /V D
v;here: P. =* capillary pressure,
T a surface tension of the wetted liquid,
0 » contact angle,
R = capillary radius.
The equation /68/ can be presented in the other form:
RD = __2T cos 9 / Pk. /69/
Using a completely wetted liquid, cos ® » 1 , the equation
can be reduced to the following:
Rp - 2T / Pk /70/
From equation /70/ results, that for a certain levels
of the air pressure, the pores of a certain sizes will be
dewatered. Measuring, parallely, the air flow rate at the
definite level of the air pressure, the number of pores
of a certain sizes can easy be estimated.
Basing on this observation, the procedure of the pore size
distribution was elaborated.
EQUIPMENT AND PROCEDURES
The stand on which the measurements of the pore size dis-
tribution was conducted is presented on the Figure 8 and
9 . This stand includes the following:
- Testing chamber,
- Surge chambers, capacity 2 and 31,
61 i
-------
o>
*
P
, S
• -I
Figure 8. Illustration of laboratory stand for fabric pore size
distribution determination.
-------
r
FABRIC
TESTING CHAMBER
V
RH
B
SURGE CHAMBER,
V =• 3 1
PARAMETERS OF THE
AMBIENT AIR
PARAMETER'S''OF %IE
AIR IN ROTAMSTER
R
SURGE CHAMBER
ROTAMETER
PUMP
ESI
Figure 9. Diagram of the laboratory stand for pore size distribution
determination.
-------
- Rotameter,
- U-Xanometers,
- Pump,
- Valves and conecting pipes.
The testing chamber, which is a main part of the stand, is
equipeci with a special arranged specimen seat, in which the
2
specimens of diameter 3 cm /of 19,625 cm test area/ are
placed and clamped. The special shaped metal 0-ring is
used to assure of the air tight.
Over the specimen seat, a special shield is used to limit
the losses of the wetted liquid. The detailes of the
testing chamber construction was shown on the Figure 10.
Before testing the round fabric specimens, should be con-
ditioned at least 24 hourcs in the standard conditions and
then soaked with wetted liquid in a glass dish /about 5
minutes/.
After soaking the fabric specimen is placed into the
testing chamber and clamped in the specimen seat. The va-
tted liquid film is formed on the surface of the soaked
specimen.
After the pump is started, by the rotameter valve, the air
flow to the testing chamber is regulated. Under increasing
air pressure, the first bubbles appear on the liquid film
at a certain level of the air pressure.
In this phase of test, the largest pores are dowatered.
Increasing the air flow rate, the air pressure in testing
chamber increase too, and successive pores vith lowor and
lower diameters, are dsvatered.
Measuring the air flow rate through a soaked specimen for
a.certain levels of the air pressure, the dependence of
P = f(Q) is obtained /where: P is an air pressure and Q
64
-------
r
Przekrdj A-A
Rzut pionouL)
Figure 10. Testing chamber.
65
-------
is an air flow rate/.
During testing the temperature, humidit^r and. the atmospheric
pressure of the ambient air and also the temperature and
static pressure of the air into the rotarncter should be
recorded /reduction of the rotameter indications is nece-
ssary/.
The dependence of the true air flow rate upon the air pre-
ssure can be presented in the forn of a curve in the coordi-
nate system P-Q , as it was shorn on the Figure 11.
The dependence ? = f(0) is described by the following
equation:
P(0) =
-a,Q2 + b.,0 + c1 for Qs[o,QT]
'2
for Q€[oIf0ljLJ
/7^/
The course of the curve is limited by the value of On ^ ,
which depends on the endurance of the apparatus. In our
2
experiments, the air pressure up to the k 000 N/ra was
applied.
STATISTICAL TREATMENT OF TEST RESULTS
The course of the curve P = £(Q) is geometrically analized
to obtain a set of a certain parameters, for statistical
treatment.
At first, the range of the air pressure from P to P, .
- o JLini
is divided on the equal classes: (P , P.) , (p^Pp) , . . .
...,fP, ,P. ,.) , then inside of each class, the mean values
of the air pressure: PQ ^, P^ 2»*-'» p^ k+1 are
For the mean values of the air pressure, the suitable va-
lues of the air flow rate: Q0 ,, , (L ,,..., Qv Vj^ are
estimated.
,, ,
, I I •
v
n. ,
-------
lira
M
w
£
2 _
01
°01
°0 Q1
Q
12
Q.
Q
AIR FLOW RATE THRIUGH SPECIMEN £ Q
Figuro 11. Tho courso of a curvo P * f(Q) .
-------
a.
b.
f~\
*
1
1
•
> r
<- r JX
* VA'^
\
-
r
ff
* f *~
'-
^
,
-
f
.."'
. - . I-
.
/ ^
*.
-
Figure 12. Fabric surface with wetted liquid film /a. first
bubbles formation, b.. during test/.
68
-------
Because, the mean pore radii correspond to the definite
values of the mean air pressure in a certain classes,
it is easy to estimate the number of pores in each of the
classes. The number of pores in the first class is estima-
ted as follows:
NR . 2*1- 1121
where: NR =» numbers of pores with mean radius RQ .
' corresponding to the mean air pressure
OQ . a mean flow rate corresponding to the mean
air pressure PQ . ,
Fn ^ » cross-section of the mean pore with radius
u, \
R0,1 '
v a mean pore velocity /can be estimated
"o 1
9 from the following equation:
v a 2,49 P. , where: P, is a mean
air pressure in a given class/.
Estimation of the pore numbers in the next classes of the
pore radius, require to take into consideration the quanti-
ty of the air passing under the increasing air pressure,
through out the pores previously dewatered at the low
level of the air pressure.
So, for the second class v;e have:
- (NF)n .
°»1
... -.
69
-------
More general form of the above equation for the second and
next classes is follows:
n-1
Q_ , „ - KF . v"
n- 1 ,n , , D 0 ,.
= - !_ - -n-2,n-1
n-1'n * * F-1 n
? "
where: n = number of class,
NF = a characteristic paraaeter for each of the
classes v/hich represent "the free flow area"
in a certain class of the pore radii.
Having the set of the NF values, the frequency can be
estimated for the all classes:
P± - " mi
1 2NF
The parameters of the pore size distribution can be estima-
ted frcs the following dependences:
£
RESULTS AND DISCUSSION
Using the apparatus and procedure of statistical treatment
of the empirical data, as it was presented in the previous
sections, the pore size distribution for all American fa-
70
-------
brics, which were a subject of research works during reali-
zation of our previous Project P-5-533-3, were performed.
Laboratory experiments were performed for a five randomly
selected samples of a given fabric, then the mean value
of pore radius and its standard deviation were estimated.
As a wetted liquid, the white spirit /T « 25,06 dyn/cm
at the temperature 22,5 °C/ was used.
Table 1 presents the fifteen kinds of investigated fabrics
with a specisl attention to the kind of raw material and
kind of fiber.
Because of a big structural differences of investigated
materials, the microscopic auxiliary researches of the
surface structure were also performed. Figure 13 through
20 present the pictures of the fabric surfaces, obtained
during scanning microscopy investigations.
The results of pore size distribution of the pure fabrics
structures are contained in Table 2; This table includes
except the values of mean pore radius and its standard
deviation, the mean pore diameter and its standard devia-
tion.
Figure 21 through 35 present the theoretical distribution
of pore radius calculated for empirically determined
values of the means and standard deviations and also the
results of test data in the form of points.
There are some differences between the theoretical and
experimentally determined runs of the distribution
function. They result from the limits in application of
the high level of the air pressure /the empirically
determined functions are cut in the range of lowest
values of the pore radius/.
From the data presented in the Table 2, a certain regu-
larity can be found.
71
-------
Table 1. CHARACTERISTIC OF FABRICS
Kind of Raw Material
Cotton
Dacron Polyester
/staple fiber/
p
Dacron * Polyester
/continuous filament/
o
Noniex* Aromatic Nylon
/staple fiber/
Nornax" Aromatic Nylon
/continuous filament/
Nylon Polyaaide
/staple fiber/
Glass
/staple fiber/
Glass
/continuous filament/
Tvnc of
Filtration
Fabric
960
8623
C866B
C8683 .
8653
C8903
...C892B
852
853
190R
850
8023
053-875
053-870
053-878
Permea-
bility
in
.1^3 /£/„„-
Liu.** / .ii / *j c- W
Z^^j
382
240
163
166
107
70 _
457
187
97
148
'
140
226
58
219
Thread
10
V.'arp
384
1 "i0
164
164
302
292
254
122
154
-
380
140
210
210
176
Count
en?.
Fill
238
110
138
.158
178
262
232
100
144
-
288
136
204
204
96
72
-------
Figure 13.
Superficial structure of clean Cotton fabric
style 960 . /Photo credit: SEM by K.Skudlarski
Electronic Microscopy Laboratory of Polytechni-
cal University in Wroclaw/
73
-------
—-«-,
:
,
.
...
ft-.V'M "
'', I -f~~' ~ ' l~t
Figure l4a.
Superficial structure of clean Polyester fabrics
style 8623 and C866B . /Photo credit: SEM by
K.Skudlarski, Electronic Microscopy Laboratory
of Polytechnical University in Wroclaw/
74
-------
Figure I4b. -Superficial structure of clean Polyester fa-
bric style C868B . /Photo credit: SEM by
K.Skudlarski, Electronic Microscopy Laborato-
ry of Polytechnical University in Wroclaw/
75
-------
tj - - — — -- r—--- -r- p-r ——T- —.-~- "^ - -• — -
F
-
j
-
^
-
' • i-.w.
'
.- .::
•™\_r u •>>
-
-
n v
"
Figure 15a,
Superficial structure of clean Polyester fabrics
style 865B and C890B . /Photo credit: SEM by
X.Skudlarski, Electronic Microscopy Laboratory
of Polytechnical University in Wroclaw/
76
-------
-
)
1
f-
* * ; -
^.. -
_^— •
rn
_. : .
1M*K
'•
~*-^—
—I
--'* '|'1'"1 ~i"«V_-L!^ ~-
.^ ,
L- ^ _•>* M^L^L^ ^
r*-~ -----«^ -
. 73
'
- -
D • '-'1
Figure I5b,
L
Superficial structure of clean Polyester fabric
style C892B . /Photo credit: SEM by K.Skudlarski,
Electronic Microscopy Laboratory of Poly-
technical University in Wroclaw/
77
-------
^•s;x-?/'/.'/ -.vs^ivtf °» :VN:
'^•n '& x > ,
^v//^:^ 1
•;'..•**- -.
r — -- .'
^
- p:
/;- ?ll
filii i
Figure I6a. Superficial structure of clean Nomex fabrics
style 852 and 853 . /Photo credit: SEM by
K.Skudlarski, Electronic Microscopy Labora-
ory of Polytechnical University in Wroclaw/
t
78
-------
^~~- ^ V**
-, - ,.£3
^aSBTf
"H f -^ • >^; // .•<• ^"x*
\£*J £.
\£i=^Z2* ££J£i#-*?
'' ''s^^tk
% '^/^^VNS :£^^e^
t ^|f
-------
...... T
Figure 17. Superficial structure of clean Nomex fabric
style 850 . /Photo credit: SEM by K.Skudlarski,
Electronic Microscopy Laboratory of Polytechni-
cal University in Wroclaw/
L
80_
-------
&zs?&y*y
B^S^Il • -.
E^Ms
1 iV 2 *l'Ks^?S»^^^^«»w/jl \z^;v^i5
f - -*•» -^ J J -^.^L. ^ji-——- * V •~-^^_.J. __ t m __
Figure 18.
L
Superficial structure of clean Nylon fabric
style 802B . /Photo credit: SEM by K.Skudlarski,
Electronic Microscopy Laboratory of Polytechnic
cal University in Wroclaw/
81
-------
.
rrr~ -:
-
n
Figure 19. Superficial structure of clean Glass fabric
style Q53-875 . /Photo credit: SEM by K.Skudlarski,
Electronic Microscopy Laboratory of Polytechni-
cal University in Wroclaw/
82
-------
..-. •-
. __
—~"'
i • . „ r
S * " ' ft • "^ ""
• ;•
•
- •
;
• I ;
ivfr m^
1 \ i
:''A \ij'.
l-'\- III'
• 1 - u • • 41 illi'
11
' ; ,-*
• j ' 1- . • i , , .
1 * If
i
Figure 20,
Superficial structure of clean Glass fabrics
style Q53-870 and 053-878 . /Photo credit: SEM
by K.Skudlarski, Electronic Microscopy Labora-
tory of Polytechnical University in Wroclaw/
83
-------
Table 2. PARAMETERS OF PORE RADIUS DISTRIBUTION'
Kind of Raw Material
Cotton
D
Dacrcn" Polyester
/staple fiber/
t>
Dacron Polyester
/continuous fiiaeent/
D
Nomex Aromatic Nylon
/staple fiber/
t>
Nomex Aromatic Nylon
/continuous filament/
Nylon Pol yami do
/staple fiber/
Glass
/staple fiber/
Glass
/continuous filament/
Tvr»e of
Filtration
. Fabric
960
862B
C866B
C86S3'
865B
C8903
C892B
852
853
190R*
850
802B
Q53-875
Q53-870
053-878"'"'"
Distribution
yx'r>?
18,82
56,67
31,62
34,69
26,94
27,33
54,37
34,82
24,10
31,78
29,52
52,10
30,28
56,18
Parameter,
.
»l
4,34
15,14
11,78
3,81
7,52
4,54
4,60
12,65
8r58
3,70
4,68
8,82
9,71
10,48
15,75
+ Non-woven filtration material
•M- Texturized thread in the fill.
84
-------
0,25-
0,20-
0,15-
0,10_
0,05-
0
'i^uro 21. Pore radiun.distribution in ihn struetur-
of Cotton faerie style 9cO /r,tr.plo fiber/,
-------
CO
0,1
0,05-
0
0
10
—r
20
30
40
50
60
—r~
70
80
Figure 22. Pore radius distribution in the structure of Polyester
fabric style 8623 /staple fiber/ .
90
-------
0,1
0,05
0
0
10
20
30
50
60
70 80
RP
Figure 23. Poro radius distribution in the structure of Polyester
fabric style CC56B /staple fiber/ .
-------
0,1
0,05 -
CO
03
0
Figure 24,
Fore radius distribution in the structure of Polyester
fabric style CGoOB /staple fiber/ .
-------
0,05 -
10
Figure 25. Pore radius distribution in the structure
of Polyester fabric style 8553
/continuous filament/.
-------
0,1
0,05 -
0
10
Figure 26. Poro radius distribution in the structure of Polyester
fabric stylo C890B /continuous filament/ .
-------
0,1
0,05
0
0
10
20
Figure 27. Pore radius distribution in the structure of Polyester
fabric style G092I3 /continuous filament/ .
-------
r\>
Figure 28. Pore radius distribution in the structure of Homex
fabric style 832 /staple fiber/. .
-------
0,1
0,05 -
0
10
Figure 29. Poro radius distribution in tho structure of Ncriox
fabric stylo 053 /staplo fiber/.
-------
0,15
0,1
0,05 -
50
100
Figure 30. Pore radius distribution in the structure of
Noraex fabric style 1903. /staple fiber/ .
-------
0,1 -I
0,05 -
10
20
30
'40
50
60
R.
Figure 31.
Poro radius distribution in the structure of
Nonox fabric style 850 /continuous fil./.
' 95
-------
VO
en
0
10
Figure 32. Pore radius distribution in the structure of Nylon
fabric stylo 802B /staple fiber/.
-------
0,1
0,05
O
0
33. Pore radius distribution in structure of Glaaa fabric
style 033-075 /staple fiber/ .
-------
CO
0,1
0,05 ~
0
0
10
Poro radius distribution in the structure of Gains fabric
style Q53-870 /continuous filament/ .
-------
vO
.0
Figure 35. Poro radius distribution in the structure of Glass fabric stylo
053-8/3 /cont. filament in warp and tcxturizcd thread in fill/
-------
The fabrics performed with continuous filament are charcte-
rized by.a low values of mean radius and standard deviation
in comparison to the fabrics sade on the base of staple
fibers. That means, that in statistical sense, their struc-
tures are core homogeneous.
The fabrics performed with staple fibers have nearly two
times biger the pore radii and nearly four times bic;er the
standard deviations. So, their structures seems to be a low
homogeneous.
As it was in our previous works pointed out many tines,
the physical parameter of filtration structure, should be
adequated for a conditions of physical and mathematical
modeling and more over it should be functionally connected
with technological parameters of fabrics.
One of the most important technological parameter of filtra-
tion naterials is their air permeability. On Figure 56 ,
the dependence between cean pore radius upon a permeability
of investigated fabrics, is shown. This dependence is
evident and conformable to the expectations. The permeabi-
lity of fabrics increases with increse of the mean pore
size.
Because, the perrceabilty of filtration materials depends
strongly on the other technological parameters deciding
about filling with threads of their spatial structure,
the influence of the thread count in 10 cm of warp and fill
and also the number of crossing threads on the area of 100
cm , due to relation /n + n^.//2 /where: n.f is a thread
count in 10 cm of warp, and n^ is a thread count in 10 en
of fill/ , were investigated. The results are presented
on Figure 37 through 39.
For the fabrics which structure is pernorned with continuous
filament, the decrease of the warp thread numbers in 10 era
100
-------
o
CM
n
•t
(H
M
S
400
300
200
100
0
a staple liber
O cont.fila.7icn
0
100
P
Figures 3^. Dopondcnco of pornoability upon the i^oan pore x-a
-------
o
o
o
M
O
U
(X
300
330
100
0
\
\
/
/
D
\
\
\
X?
A?'-'
O0/
/
^ /
x'
•^ /ON
•
c
O
/^
'
<^/
^
L xj
n
a
.4
•
/
\
^
DV
\
.___ ... . _
»
a
a
w
p
\
\
\
\
\
\
a staple fibc
O cont.filanic
\
r
nt
^
50
100
R.
Figure 37. Influence of warp thread count in 10 era on the mean poro radius.
-------
o
4oo
*.— 4
u
300
o
j-j.
c-«
3 200
o
rij
r.>-<
"• *
^4
KH 100
r\
\
\
\
'%%
\^'
a
«
>
^
\
\>,
0N
(
•x.
*»
c
0
^
1 '"'^
\
x° *'
a
i
|\
*
x *X
D «**^«
\
n
»5^>,
^^^fi
• i-i^^
N^Up
\
^
^^^-T^-
^<;
•
\
\
I
^^
p 5;taplc fibci'
o cont.f ilanont
•^
b
x
•s
0
50
100
P
Fifuro 30. Influence of fill thread count in 10 en on tho ;nonn pore
-------
400
300
200
O
C\J
100
0
X
— . TV^-.
D
. .1 ' - . -:
X
o
_ . _._
o
^xC
c
o
COMrI
— _ _ . — ,
O
J
p
D
]
inuous
- . . .
^
°X
FILA:
_ . — .
p
•Qs
a
ENT
> « — — . .
p
X ^
a
staple
fiber
O cont.f ilarnei
-^^ • - 1 HL
?
\
• J_H • •»
X
.„ . , .
X
'
t
) 50 100 _• _
R~
P
p
on the moan pore radius.
-------
causes the decrease of the mean pore radius. This effect
results from deformation of the warp thread structures
/elementary fibers displacements in the structure of thread/
and in consequence of the interyarns spaces decrease.
In the case, when the fill threads numbers in 10 cm are
decreased, the increase of the mean pore radius is obser-
ved. This observation is in accordance with the concept of
basket free area * in the structure of woven filtration
materials.
The basket free area /BFA/ effect is especislly visible
for fabric structures basing on the continuous filament,
Loosening^the woven continuous filament structure along of
the warp cause to increase of the basket free areas which
is characterized by a big plane sizes.
Moreover, the obtained results show, that the influence
of so-called "free area" , proposed by D,C. Draemel ,
on the dust filtration process performance, is lover than
the basket free area, what directly flows from the presented
above considerations.
For the fabrics which structure is performed with staple
fibers, the loosening of the structure along of a warp and
fill give a similar effects. Decreasing the number of
threads in one of the thread systems cause the increase
of the mean nore size.
t
It is possible to lead out a similar dependences for the
other technological parameters.
It was shown in presented above considerations, that such
dependences can easy be obtained from a certain empirical
base, and moreover that, the mean pore size is a strong
physical parameter of the filtration materials structure.
Satisfactory results of the experimental works, lead us
to the conclusion, that the pore size distribution of fully
105
-------
filled, with dust fabric structures can also be estimated,
using the elaborated method.
Auxiliary experiments wore performed, for the Polyester
Polish fabric ;styie ET-4 , which samples were taken from
the used bag of cement mill dust collector.
Figure ^0 shov:s the results of the auxiliary experiments
which are. corsparized with results obtained for a pure
unused structure of the Polyester fabric style ET-4.
As it was expected, the pore size distributions are not
comparable. The structure of filled up with cement dust
filtration material has very lover value of mean pore size
/mean pore radius/ as well as a value of standard deviation.
These results confirm the principal assumptions and concepts
of the "Zero-layer theory" and "ductive filtration pro-
cess".
However, the problem of experimental determination of pore
size distribution of the fully filled with dust filtration
structures by using the extended bubble test, should be
more detail!y investigated. It is possible that during the
test performance some grains of the dust can be extracted
from the filtration structure in the result of the air flow
through out the soaked fabric sample. In this case seme
difference between the real and measured values of the pore
size distribution parameters can be observed.
RECAPITULATION
Elaborated method of the pore size distribution in the flat
samples of pore structures enable to estimate the pore size
distribution parameters of any kinds of the dust filtration
materials.
This method can be extended by using another kinds of the
106
-------
f R.
o
--0
0,1
0,05 -
0
Figure
Variation of pore radius distribution in used and unused
Polish Polyester fabric style ET-'* /staple fiber/.
-------
wetted. liquid, characterized by a special properties
/surface tension and contact angle/. The author of this
method expects a certain results of its extension especia-
ly from the viewpoint of structural investigations of the
pore structures.
Because of, limited range of this program and also a lack
of very specialised instruments /^article counter etc./,
it was not possible to perform a certain research works
in the range of estimation of the pore size distribution
influence on the monentary dust filtration efficiency in
the different types of the dust filtration process.
By utilizing the theoretical considerations in the range
of the "zero-layer theory" , it is possible, for a certain
parameters of the particle size distribution after the
filtration structure and assumed, value of the momentary
dust filtration efficiency, to estimate the largest pore
radius of the pore structure /fabric/ that the efficien-
cy of filtration not drops below of the assumed level.
In this nanner it is possible to compare the results of
the experiments with a theoretical foresights.
The procedure is very simple. Assuming the particle size
distribution of dust N/n, &/ and the value of the momen-
tary dust filtration efficiency E , the maximum pore size
R of the filtration structure, can be estimated from the
*
following:
R - a
-------
SECTION VII
DETERMINATION OF PORE SIZE DISTRIBUTION
IN DUST LAYER
INTRODUCTION
Similarly, as in the initial phase of each dust filtration
process, the structural parameters of the filtration struc-
ture /or fabric-dust system characterized by a different
degree of fabric structure filling with dust/ decide
about the hydraulic and filtration properties of fabric,
in the second phase when the first dust layer is formed
on the surface of filtration structure, the parameters of
dust layer influence the filtration resistance and dust
collection efficiency.
However, in contrary to the fabric structure, the structu-
re of dust layer is more complicated for investigation
because of its non-stability. So, more sophisticated re-
search methods are needed.
Practically, despite the microscopic examinations which
range is rather limited, does not exist any methods which
enable to investigate of the dust layer structure. Some
trials of dust layer stabilization using the synthetic
resins or glues, are very controversial because of possible
dislocations of the individual grains during preparation
of the samples.
One of the most effectve methods of the dust layer structu-
re investigations is utilization of hypothetical physical
109
-------
models and. their experimental verification with applica-
tion of a. probabilistic modeling.
This v/ayjaf the problem solution was assumed in our works.
Assuming, that the dust layer structure consists of four-
element space forms /see Section IV/ , it is easy to esti-
mate all characteristic geometrical and physical magnitu-
des characterizing of a such fora.
The compression effects of the dust layer caused by shocking,
which were investigated during realization of our previous
program ' shown, that the decrease of the bulk porosity of
the loose dust layer during shocking depends on the kind of
dust, degree of the dust dispersion and shocking tine.
Under a suitable length of the shocking time /more than
2 000 s/ , the equilibrium of the spatial structure of dust
layer is observed, and a stable value of porosity is reached.
The interesting observation was made, that for fly ash dust
which grains satisfy the theoretical assumptions, the theo-
retical and experimentally determined values of porosity
were the same but only for the unseparated dust.
For the separated dust /9G& of the dust grains below IQ&m,
MMD = 5,5/um/, the porosity obtained during testing was
higher than theoretically calculated, and reached value
£ =* 0,64.
iJ
The above phenomenon, can only be expla'ined by a low energy
of shocking due to the mass of the investigated dust samples,
so, the individual grains of the dust have not reached a
level of the final packing.
It can be assumed that the dust layer formed during dust
filtration process performance is very closely packed, becau-
se of a specific conditions of the individual grains preci-
pitation on the surface of fabric or on the surface of dust
layer previously formed.
110
-------
Basing on the geometrical model of the dust layer, which
was a subject of considerations in the range of the ductive
filtration process, it is possible to introduce some ele-
ments of the probabilistic modeling.
This approach is substantiated by random character of the
particle size distribution as well as the process of the
dust layer formation.
PROBABILISTIC APPROACH TO THE PROBLEM OF DUST LAYER
FORMATION
The complete dust layer consists with a many individual
dust layer, which can ce treated indepandly. Let we divide
the complete dust layer on the following elementary dust
layers w..,..,,w /reserving the notation w for a zero-
layer i.e. filtration material structure/. Let we intro-
duce the random field as it was shown in previous sections.
Let we define in aech elementary layer, the random field:
The correlation function for each layer can be estimated:
R[fis(p(e)),fis(q(e))] , (i * 1,...,k), (s ~ 1,...,
Let we assume, that the random fields fi/p(e)) are a same
type for the each layer. For each random field, the auto-
correlation function can be estimated:
111
-------
for D£Dxv
1
mi
for r>e Dxw
The correlation function for the complete dust layer con-
sisting the layers u^,...,\f can be written as follows:
R[f11(p(e)),...,fs^p(e))]
for o e Dxw,
/78/
for pe Dxw,
Let v:e assume that:
- The dust grains are a spheres of radius r.,,...
•«
- The particle radius distribution is noraal,
- The dust concentration in the aerosol after fil-
tration structure is uniform.
Under of these assumptions, in the set . D , the dust
grains toward;:the saxinum areal packing.
Because B is an area of an arbitrary shape /in parti-
cular circle - laboratory samples/, the tree-eler.ents flat
system is a most probable for the structure of the elemen-
tary dust layer.
112
-------
The tree-elements grains system is shown on Figure 4
The random field f-,(p (e)),...;fk(p(e)) is consider as a
zero state of these random fields and is denoted f.
The first state is defined in the layer w1 and etc.
As it results from deliberations performed in the Section
IV and V , the geometrical pore size is a most important
from the view point of the ductive filtration process effi-
ciency, so it is important for the dust layer formation, too,
From the geometrical relations of the grains system, shown
on the Figure 41 , result that the geometrical radius of
the pore, is a function of the dust grains radii.
R
Gp * f(r1'r2'r3)
Moreover, it is evident that the bigest radius of the pore
will be reached if all dust grains radii be equal.
Because of previously assumed conditions, the random varia-
ble (ri»r2»r-5) nas normal distribution in the form:
1
5s- x
S3
exp
P. _J ] 1 /79/
1—2 2 2
where:
-2 n2/ N
' • a u\!jj /. i o z^
i v i' , (i = 1,2,3;
113
-------
Figure 41. Threc-elesents dust grains systen forming the
structure of the elementary dust layer.
-------
Fron the above results that the geometrical radius of pore
has also normal distribution in each layer.
Let we assume, the distribution of random variable Rr
is normal in the set DxW, so we have:
/80/
where: raf » EJf(p(e))] , <£2 a D2[f(p(e))]
and they have the following properties:
W6W /81/
and they are non-negative functions and m^ going to zero.
It is worth to notice, that these random fields do not
satisfy the assumption about stationary in the set DxW ,
despite the fact that in each layer w. they are statio-
nary. These fields, however, can be investigated by using
the methods elaborated for the random ,f ields stationary in
wide sense.
So, the correlation function in this case has a form:
exp-
pq
] /82/
Then the function A = A(w) exist, such that:
•R(pq) * exp[- A(w)|p^)2J /83/
-------
in the set DxW , we w and A(w) > 0 .
Because of, the correlation function of the stationary' in
wide sense random field is a characteristic function of the
random variable f(p(e)) , the distribution /SO/ has a form
P / '
F[f(p(e}
LV
2A(w)
The variance has forra: D2[f(p(e))] = G §(>.') = A (w) for
wew , n^ = nf(w) and is regression curve.
If we assume, that each of layers >;. , w_ , . . . is represen-
ted by a mean pore radius r1,r2,... , the probability of
events that the dust grain will be precipitated on a certain
/previously precipitated/ elementary dust layer, can be
estimated as follows:
fr^r.) =• 1 -- =- — exp -- dr /85/
-
The above equation is a criterion of the dust layer growth.
Among the many problems connected with dust layer structure,
the problem of the hydraulic resistance during the gas flow
seems to be very important.
The hydraulic resistance to the gas flow depends on many pa-
rameters characterizing the porous structure of the dust la-
yer as well as the behaviour of the gas.
The external parameter , of the flow process equal to the
hydraulic resistance is a static pressure difference of the
flowing medium before and after the dust layer structure,
which can be experimentally measured.
116
-------
Introducing the random field as it was shown in the pre-
vious sections and denoting f(p) = f(p(e)) as a hydraulic
resistance of the i-th layer, the random field of the
hydraulic resistance is described.
This random field has the following properties:
E[f(p(e})] = m(w..) /85/
l) /87/
and R(pq) = R^Cw^ jpqf] /88/
From the previous considerations result, that the correla-
tion function in each elementary dust layers is a same
class, so it differs only on a constants: A 1tA p,...
•"'Awn- " Tiae cons"tan'i;s: \;i»Aw2' •* * *Awn can be ecluali~
zed by a continuous curve A = A(w) , which is a regre-
ssion curve estimated from the experimental data.
As it was experimentally verified, the course of the A(v)
curve is a parabolic shaped:
A(w) = at2 + bt + c > 0 /89/
\vhere: t = is a time of the dust layer growth, which is
equivalent to the thickness of the dust layer,
a,b,c a are a constants.
From the above results, that the hydraulic resistance of
the elementary dust layers is not the same in the spatial
of complete dust layer and that it is higher for each
next layer in relation to the previously formed.
117
-------
That means:
/90/
The above property of the dust layer results from the influ-
ence of selective precipitation mechanisms which was discu-
ssed in the Section IV.
The linear course of the regression curve A(w) = at ,
is possible only in the case if the dust layer is formed
with grains of the sane size.
Then:
(RGP)
/9V
wn
what is conformable with reality.
Described, properties of the dust layer are very important
from the viewpoint of the dust grains deposition in its
structure.
Inequality /90/ lead to the conclusion, that the influen-
ce of inertial deposition, which is a second major mecha-
nisms of the ductive filtration process, is higher in the
external elementary dust layers .
Because of a normal distribution of the, geometrical pore
radius in each elementary dust layers as well as in the
structure of complete dust layer, the distribution of the
hydraulic resistance is normal, too.
So, the correlation function is as follows:
R (pq) = exp [- A(w) JpaJ2] /92/
Utilizing the deliberations presented above, it 'is possible
118
-------
to estimate the hydraulic resistance in each of the eleraen-
tary layer and also in a complete dust layer.
The similar considerations can be developed for the others
parameters of the gas flow process through the porous
structure of the dust layer 1\
It is also possible to take under consideration all process
parameters by using the multiparameter probabilistic model.
FINAL REMARKS
The probabilistic approach to the problem of the dust layer
structure investigation is very important from the viewpoint
of understanding the partial mechanical processes in ran-
domly formed porous spatial structures.
Some conclusions resulting from the statistical analysis of
the physical phenomenon forme a base to the very interesting
considerations, which till now were not taken into conside-
ration, because of a certain troubles in the range of some
physical magnitudes definition.
In this manner, the develop of a certain problems on the
ground of the theoretical mechanics by using the occlusions
of the statistical analysis, was made possible.
Given in theoretical parts of this report: statistical pro-
cedures, random field estimators and also the sampling
methods, enable to utilize of the random fields theory
not only in the range of modeling of the dust filtration
process, but also they are a certain research instrument
which enable to solve many not decided problems in the field
of flows and particle movement.
From the r>ractical viewpoint, the most hard problem is samp-
ling.
To practical utilization of the presented theory, it is
119
-------
necessary to present of all process parameters in the form
of a certain random fields. It cause a certain conditions
of the experiment realization. The problem of the measure-
'rr.ent of all process parameters at the same time is a cardi-
nal one, and roust to be satisfied. Moreover, because of
change with time /filtration time/ nearly all process pa-
rameters, there is necessary to use of a continuous rejestra-
tion of the parameter values. Under of these conditions of
the experiment performance, it is possible to utilize of the
random field: estimators,,
From the experimental viewpoint, to satisfy of these condi-
tions is especially hard in relation to the some of aerosol
parameters, for example: continuous measurement of the dust
concentration and the particle size distribution before and
after the filtration structure.
In the case of some process parameters /e.g. estimation of
pore size., distribution of the filtration material structure
partly filled rith dust/ , the random fields can be estima-
ted on the base of the post-test measurements /after filtra-
tion cycle/.
This problem can be solved by developing the theory of the
filling process of the filtration material structure with
dust. Such theory would be accepted some simplifications
in relation to the complete dust filtration process.
The ways in order to resolve of the presented above problems
result from the considerations given in this and previous
section.
At the end, it is worth to notice, that the crogress in the
range of the dust filtration process modeling is closely tied
with resolving all measurement problems which enable to uti-
lize of the elaborated modeling methods.
120
-------
SECTION VIII
STUDY OF ELECTROSTATICS EFFECTS
IN DUST FILTRATION PROCESS
INTRODUCTION
The results of investigations concerning the electrosta-
tic effects,in the dust filtration process presented in
this Section, are a continuation of the research works,
which have begun in the Institute of Physics of Polytech-
nical University in Wroclaw in the range of the previous
2 5
projects * .
The earlier works, were first of all concentrated on the
problems of the electrical properties both the industrial
dusts and fabrics determination. Moreover, the problem
of kind of forces existing between the individual grains
and fibers and charging and discharging processes during
filtration and regeberation phase, were also investiga-
ted.
These works were utilized to interpretation of the experi-
mental results obtained in the field of the dust filtration
process investigations, and were taken into consideration
when the theoretical base of the dust filtration process
were created.
The subject of the works presented in this section is
estimation of the influnce of electrical forces on the
ductive filtration process performance with utilization
of the model layer formed with conductive and non-conducti-
121
-------
ve spheres. Additional experiments were also performed v/ith
application of the external electric field.
All of these works, were performed bv a worker's group of
the Institute of Physics of Poiytechnical University in
Wroclaw under direction of Dr. J. Malcher.
EQUIFT-ENT AND PROCEDURES
Laboratory testing- of the artificially forned layers of the
spherical elements, was conducted on a stand illustrated on
Figure 42 , and specially designed by Institute of Industry
of Cement Building Materials.
This stand includes the following:
- A vertical metal pipe with a metal net in its upper
part, a measurement chamber in its middle part,
and outlet air duct in its lover,
- A rotarseter for measuring air flow,
- A micro-manometer for measuring pressure difference,
- A power supplier with regulated voltage from
0 - 3 kV and different polarization, and
- - A sction air blower v/ith regulated rotation.
The measurement chamber, which is a main part of the stand,
consists of a metapiex /kind of teflon/ ring with a spe-
cially mounted nets to support a spherical elements layer
and to creation of the external electrical fields.
The detailes of the measurement chamber are shown on Figu-
re 43 •
The filtration layer was formed with ceramic spheres of dia-
meter range 2 - 2,2 mm and with metal spheres /lead/ of
diameter range 2 - 2.1 mm .
122
-------
EARTH
HIGH
VOLTAGE
SUPPLIER
MEASUREMENT CHAMBER
ABSOLUTE FILTER
TO ROTAM3TER
AND BLOWER
Figure 42. Diagram of stand for dust efficiency of the
filtration layer of spheres neasurcsient.
•-123
-------
a. Preliminary experiments
EARTH _
UPPER NET
b. Principal experiments
NET A
NET B
NET C
Figure 43. Construction of measurement chamber.
-------
The thickness of the filtration layer of spheres differs
in the preliminary and principal investigations. In the
preliminary experiments, the layer thickness was stable
and it was 5 mm . In the principal experiments, the
layer thickness changes from one to five numbers of the
elementary layers of the spheres.
After filtration layer of spheres, the absolute filter
was applied.
The dust efficiency of the filtration layer of spheres,
was estimated from the following equation:
where: K » dust efficiency of the filtration layer
of spheres,
m. = mass of the dust collected in the structu-
re of the filtration layer of spheres,
mass of the dust collected
re of the absolute filter.
m. a mass of the dust collected in the structu-
In all experiments, the separated silica dust /in the
three ranges of the particle sizes/ was used. The particle
size distributions of the test dusts, are shown on Figu-
re 44 .
The following general conditions of the tests performance,
were applied in all experiments:
- filtration velocity measured after .-the filtration
layer vf =» 25,2 cm/s ,
- initial dust concentration after the filtration
layer c1 = 2,5 g/cr ,,
- air temperature 19 - 22 °C ,
125
-------
o,/
CO
a
<
H
CO
EH
-r"
t-i
CO- :_; ,i I I i_L^
{^J I ^_, : : i ; :
I i_;_J_ :.._:J i_i_L_f.
E-
CO
£-•
ss
K
O
f-t
•~r-»
t— .
O
10 20 . 30 40 50 ICO
PARTICLE DIAMETER , MICROMETERS
Figure 44. Particle size distribution of silica test dusts,
126
-------
- air relative humidity 47 - 52# .
The program of the preliminary testing included:
- Determination of the electrical properties of
test dusts,
- Determination of the ceramic spheres electri-
cal properties,
- Determination of the dust efficiency of the
filtration layer of a ceramic and metal spheres
without application of the external electric
field and in the conditions when the external
electric field is applied /in tests with metal
spheres, the difference of voltage 1500 V and
3000 V were applied, while in the testa with
ceramic spheres, the difference of voltage
3000 V was used/.
The program of the principal experiments included:
- Determination of the dust efficiency of the
filtration layer of a ceramic and metal spheres
without of the external electric field,
- Determination of the dust efficiency of the
f
filtration layer of a ceramic and matal spheres
in the conditions of application of the exter-
nal electric field, /the different combinations
of the external electric field creation, were
applied. T see Figure 43/.
The upper net A was in all experiments ground-
ed while the nets B and G were grounded or
connected to the high voltage.
127
-------
RESULTS AND DISCUSSION OF THE PRELIMINARY EXPERIMENTS
During realization of the preliminary experiments, the
following observations v;cre made:
- In result of the dust deposition in the structu-
re of the filtration layer of spheres, the charge
— Q
of 0,5.10 ^ C is measured independly on the
kind of the sphere natarial,
In the experiments with application of the exter-
nal electric fields, the polarization of the
active net, not influence on the value of the
dust efficiency of the filtration layer of spheres,
- The efficiency of the filtration layer of spheres
depend strongly on the change of the air relative
humidity,
The results of the dust efficiency of the filtration layer •
of spheres measurements in the conditions of the narrow
range of the temperature and relative humidity changes and
for the different dust fractions and different kind of the
spheres material, are presented in the Table 3 .
Table 4 , presents the results of the electrical properties
of the dusts and ceramic spheres.
The measured dielectric constant & for the ceramic spheres
is an effective value which results from a certain porosity
of the filtration layer of spheres. The value of the di-
electric constant £•' for the material of the spheres, can
be estimated from the following equation:
1
e» « __ - - - /9V
128
-------
Table 3.
DUST EFFICIENCY OF THE FILTRATION LAYERS OF A METAL
AND CERAMIC SPHERES TESTED WITH SILICA DUST -
preliminary tests /Temperature 19-22°C, RH » 41-53/
Dust Kind
fraction of
sphere
/Uni
ceramic
5 „ 20 ceramic
metal
metal
ceramic
20 - 40 ceramic
metal
. metal
ceramic
ceramic
>6° metal
metal
Voltage
of
lov;er
net
V
0
+ 3000
0
+ 3000
0
+ 3000
0
• + 3000
0
+ 3000
0
-»• 3000
Number
of
tests
4
12
3
5
3
6
3
4
4
5
5
6
Dust
efficiency
/mean/
0,7904
0,7909
0,8025
0,8329
0,3877
0,5824
0,3789
0,4738
0,4781
0,5796
0,4813
0,5222
129
-------
Table 4. ELECTRICAL PROPERTIES OF TEST DUSTS AND CERAMIC
SPHERES.
Kind
of
material
Silica dust
Silica dust
Silica dust
Ceramic spheres
Grain
diameter
5 < dp < 20 ,am
20 < d^ < 40 ^um
dQ > 60 ^m
2 < d^ £^ 2 , 2 n:in
Specific
resistivity &
s-\
diem
2,8.108 4,0
2,3.108 3,8
3,4.108 6,4
2,9.:1010 2,13
130
-------
where: |i = is a reciprocal of the bulk porosity of the
filtration layer of spheres,
In the case of a ceramic spheres, the dielectric constant
£' = 11,2 and » 2,9.1010 & cm.
Because of a high values of the dielectric constant and
specific resistivity of the ceramic spheres, the external
electric field cause a high polarization in the structure
of the filtration layer of a ceramic spheres.
In this conditions^ the high non-homogeneous electric
fields in the areas between the ceramic spheres , can be
created.
For a spherical particle of radius r placed in non-
homogeneous electric field, the polarization force is
expressed by the equation:
Fr, - 4- Srt **3 Srad £2 /95/
P 2 ° £, + 2
where: £Q =» permittivity of vaccum / Q» 8,85.10 F/m/,
g. = dielectric constant of the particle ma-
terial,
£ = dielectric constant /for air,5= V
/ E s» local value of the electric field inten-
sity.
Under this force, the partcle is precipitated on the sur-
face of a ceramic spheres in the place where the highest
value of the surface density of the charge exist.
In the case of the metal spheres layer, the electric field
exist only on the external surfaces of the layer while
in inside structure the electric field not exist.
131
-------
So, in the conditions of application of the external electric
field, the dust efficiency of the layer consist with metal
spheres, should be lower than in case of the ceramic spheres.
The experiments confirmed of these expectations.
Basing on the results presented in the Table 4 , the follo-
wing preliminary conclusions are precised:
- The dust efficienccy of the filtration layer of
spheres in the case of absent of the electrical
field depends on the dust particle size but not
depends on the kind of the spheres material,
- The external electric field causes the increase
of the filtration layer of spheres efficiency
in order of 20 to 50% for the silica dust with
particle diameter > 20^^ ,
- The increase of the dust efficiency in the result
of the external electric field application, is
higher for the filtration layer with a ceramic
spheres then for the netal ones.
RESULTS AND DISCUSSION OF THE PRINCIPAL EXPERIMEiNTS
Comparison of the filtration effects of the ceranic and metal
spheres layers.
f
Firstly, the neasurenents of the dust efficiency of the
filtration layers of spheres without the external electric
field were performed. The results of these experiments are
presented in the Table 5 and on the Figure 45 .
The measurements were performed with grounded nets A,B and
C .
In this conditions of the experiments, the dust efficiency
of the filtration layer of a ceramic and netal spheres are
132
-------
Table 5. DUST EFFICIENCY OF THE FILTRATION LAYERS OF SPHERES TESTED WITH
SILICA DUST /principal experiments/.
Description
Ceramic spheres
Metal sphere
Variant of the
external electric
field formation
AB
EBC
EAB => 6000
E
BC
EAB " °
EBC - 0
EAB * 60°°
Efic = 0
Description of
efficiency
K
Number of the
elementary layer
of spheres:
1 layer
3 layers
5 layers
0,2579
0,3719
0,4169
0,3504
0,4503
0,5695
0,2688
0,3789
0,4371
0,3448
0,4738
0,5664
-------
nearly the sane and they increase with the numbers of the
elementary sphere layers.
The measurements with the external electric field under the
filtration structure of spheres, were ccducted with con-
nected nets B and C and with application of the high
voltage. The value ol the applied eelectric field intensity
was £„„ a 60QO V/ca.
Ab
The dust efficiency was higher then in the case of filtra-
tion without of the external electric field application.
The value of the dust filtration efficiency was also nearly
the same independly on the kind of the spheres material and
increased linearly with the numbers of the elementary sphere :
layers.
The influence of the external electric field on the dust
efficiency of the filtration layer of a ceramic spheres.
Because the electric field not exist inside the filtration
layer formed with metal spheres, the influence of the
external electric field on the dust efficiency of the filtra-
tion layer of spheres, can only be observed in the case when
the ceranic spheres are applied.
The dust efficiency of the filtration layers of spheres was
measured for the three thickness of the, sphere layers i.e.
for the one, three and five numbers of the elementary layers
of the spheres.
The following combinations of the electric field creation,
were applied:
- EAT> = 0 , £ar, a 0 /the nets: A,B and C grounded/,
An liU
- E»n = 6000 V/crn , Enr, = 0 /the nets 3 and C'
A£s iJv^
: connected to the high voltage, the net A grounded/,
15/+
-------
100
u
o
60 -
Q
20 -
I I ' I
01 3 5 7
NUMBER 0? ELEMENTARY LAYER OF SPHERES
Figure 45.
Dependence of dust effiency of filtration
iaiyar'..;of >opheres on thickness of total
layer.
135
-------
Table 6. DUST EFFICIENCY OF THE FILTRATION LAYERS OF THE CERAMIC SPHERES
TESTED WITH SILICA DUST /principal experiments/.
Description
Variant of the EAQ- » 0
external electric
. field formation EBC =0
Description of
efficiency K-.
Number of the
elementary layers
of spheres:
1 layer 0,2579
3 layers 0,3719
5 layers 0,4169
Ceramic spheres
EAB = 6000 E^B =o EAB . 6000
. EBC » 0 EgC =» 2000 Efic .» 2000
K2 . K3 K4
0,3504 0,3022 0,3875
0,4503 0,5982 0,6893
0,5695 0,7929 0,8243
-------
" EAB * ° ' EBC * 2°°° V/Cm /the nets A and B
connected and grounded/,
- EAQ = 6000 V/cm , EBC « 2000 V/cm /the net A
grounded/.
The results of these experiments are presented in the
Table 6 and also on Figure 46 .
The results of the experiments lead to the following
conclusions:
- In the all cases, the measured dust efficiency
of the sphere layers increase with the number of
elementary layers of spheres,
- The application of the external electric field
in the spatial of the filtration layer /field
Enr./ is a more effective than creation of the
iSL
external electric field after the structure of
spheres,
- The abdication of both EAn and EDr, electric
Ais nu
fields is more effective than application of one
of them.
It is possible to assume, that in the case of a ceramic
spheres, when the nets A,B and C are grounded, the
filtration process performs without of the electric field
/it was confirmed by nearly the same values of the dust
efficiency measured in the case of a ceramic and metal
spheres/ . Denoting the dust efficiency of a such process
as a K1 , it is possible to estimate the increase of the
dust efficiency , as a difference of a certain measured
dust efficiency /when the certain creation of the external
.electric field is applied/ and K^ , as follows:
137
-------
2
M
M
O
w
Q
100-
80-
60_
2CL
0 -4
.0^'
,.<<
o k
q K|
A ^4
o K*
013 5 7
NUMBER. OF ELEMENTARY LAYERS 0? SPHERES
Figure 46. Dependence of dust efficiency of filtration
layer of a ceraaic spheres on thickness of
total layer.
M38l
-------
AK1 . K2 - K1 /EAB=6000, EBC»O/ /96/
AK2 =» K3 - K1 /EAB»0, EBC=2000/ /97/
If the assumption concerning the physical sense of the ma-
gnitudes K1 , AK1 and AK2 is rightful, these magnitu-
des should be additive. So, the following equation should
be satisfied:
K* » K1 + AK1 + AK2 » K^ /98/
As it results from the data presented in the Table 6 ,
there are some differences between K* and K^ , especially
when the thickness of the sphere layer excess the three
elementary layers of spheres. It ia probably caused by the
particles with diameter d > 40,-uni in the test dust, which
are hardly precipitated in the electric field.
The magnitudes AX,, and AK2 can be treated as a quantita-
tive parameters of the filtration layer of spheres efficiency*
FINAL REMARKS
The results obtained during realization of the research
program, which was devoted to the problem of electrostatic
effects estimation in the ductive filtration process,
confirmed the concept of this process and also the influen-
ce of the electrostatic effects on the variation of the
process dust efficiency.
However, the applied model of the dust layer, in the form
of multi-elementary layers of a ceramic and metal spheres,
is a very simple approximation of the reality /especially
concerning the ratio of the dust particle to filtration
139
-------
structure element diameters/ , the obtained results are very
important from the viewpoint of understanding the process
precipitation of the dust particles due to the local electric
field inside the dust layer.
Application, in the experiments developed with a ceramic
spheres, the external electric field enables to increase of
the electrostatic forces interactions to the dust particles
in relation to the real dust layer.
Because of similar electrical properties of the material of
a ceramic spheres and dust particles, it is possible to
assume, that in the real conditions the electrostatic effects
will be similar, however instead of the external electric
field, the inside local electric field caused by the charge
accumulation p'rocess, will be created.
It is also worth to notice, that electrostatics effects in
dust filtration processes depend strongly on the relative
humidity of the dispersion medium, so, their influence can
changes from the one moment to the another during the conti-
nuous dust filtration nrocess.
140
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SECTION IX
REFERENCES
1. Koscianowski, J.R., and L. Koscianowska. Effect of
Filtration Parameters On Dust Cleaning Fabrics.
EPA-600/2-76-074, U.S. Environmental Protection Agency,
March 1976.
2. Koscianowski, J.R., L. Koscianowska, and E» Szczepankie-
v/icz. Filtration Parameters for Dust Cleaning Fabrics.
EPA-600/7-79-031, U.S. Environmental Protection Agency,
January 1979.
3. Koscianowski, J.R., et al. Effects of Filtration Para-
meters On Bust Cleaning Fabrics. EPA-600/7-81-028,
U.S. Environmental Protection Agency, March 1981.
4. Davies, C.N. Aerosol Science. Academic Press, London
and New York, 1966.
t
5. Friedlander, S.K., Smoke, Dust and Haze. A Wiley-
Interscience Publication, New York, London, Sydney,
Toronto, 1977.
6. Dennis, R., Handbook On Aerosols. Tchnical Informa-
tion Center, U.S. ERaDA, 1976.
7. Fuchs, N.A., The Mechanics of Aerosols. Pergaraon Press,
Oxford, London, Edinburgh, New York, Paris, Frankfurt,
1954. '
-------
8. Koscianovski, J.R., L. Koscianowska, and E. Szczepanr
kievd.cz. Physical Model of Dust Cake Filtration Pro-
cess. Paper 30-62.5. Presented to 73 APCA Meeting,
Montreal, Canada, June 1980.
9. Koscianowski, J.R, Prespectives of Filtration Techno-
logies Development for Submicron Particulates Collec-.
tion. Presented to U.S. - Japan Seminar, Kyoto,
Japan, 1980.
10. Szczepankiewicz, E. Pewne klasy pol losowych i ich
zastosowanie /A Certain Class of Random Fields And
Their Application/. PV/N, Y/arszawa-Wrociaw, 19S1.
11. British Standard No. 3321, 1960.
12. Adarason, A.Yf. Physical Chemistry of Surfaces. Inter-
science Publishers, INC., New York, 1960.
13. Draerael, B.C. Relationship Between Fabric Structure
and Filtration Performance In Dust Filtration.
EPA-R2-73-288, U.S. Environmental Protection Agency,
July 1973.
/
14. Koscianowski, J.R., and E. Szczepankiewicz. Application
of Random Field Theory To Description of Gas Flow
Through Porous Layer. Presented to 14 Conference of
Stochastic Processes And Their Application, Geteborg,
Sweden, 1984.
142
-------
APPENDIX A
LIST OF NOMENCLATURE
143
-------
BFA = Basket free. area
c.j = Initial dust concentration
Do = Equivalent diameter of pore
= Geometrical diameter of pore
d => Diaaeter of dust grain
E = . Electric field intensity
E = r-lomentary initial dust collection efficiency
ET = Mean value of total dust collection efficiency
F = Passageway cross-section area or cross-section
of pore
F" « Mean cross-section of oore
K = Dust efficiency of the filtration layer of
spheres
a Structure dust load after "i" regeneration
cycle
= E equilibrium structure dust load
m, = Mass of dust collected in the structure of
spheres layer
= Mass dust collected in an absolute filter
-------
N = Number of pores
NF =» Characteristic parameter of pore class
P a Capillary pressure
AP a Pressure drop of pure filtration structure
= Pressure drop of partly filled structure
after "i" regeneration cycle
Pressure drop of fully filled structure
AP,, =« Pressure drop of covered with dust structure
is.
a Filtration resistance of filtration process
Q =» Pore rate of flow
Q =r Mean value of pore rate of flow
R~ a Geometrical radius of pore
JT - Hydraulic radius of pore
R « Capillary radius
r » Dust oarticle radius
D
= Surface tension of the wetted liquid
a Time of filtration cycle
145
-------
= Passageway perimeter
= Gas velocity before filtration structure
=* Pore gas velocity
^
= Filtration velocitv
(b i =* Coefficient of structure filling after regene-
ration cvcle "i"
. - Relative coefficient of structure filling after
regeneraSTion cycle "i"
£, =« Superficial porosity
P = Soatial oorositv
\^* ^T •*• •*• •*
£ = Dielectric constant
£' =« Material dielectric constant
£ =» Permittivitv of vaccum
o
£/. =« Dielectric constant of particle material
= Specific electrical resistance
Contact angle
146
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