·A PROBABJLl!['Y MODEL FOR RANDOM FIBER :BREAKAGE
by
Sung Won Lee
.
Institute of statistics
540
Mimeograph Series No.
1967
A PROBABILITY MODEL FOR RANDOM FIBER BREAKAGE
by
SUNG WON LEE
A thesis submitted to the Graduate Faculty of
North Carolina State University at Raleigh
in partial fulfillment of the
requirements for the Degree of
Doctor of Philosophy
DEPARTMENT OF EXPERIMENTAL STATISTICS
RALEIGH
196 7
APPROVED BY:
Chairman of Advisory Committee
ABSTRACT
LEE, SUNG WON.
A Probability Model for Random Fiber Breakage.
(Under the direction of JOHN WILLIAM BISHlR).
Two fundamental questions in connection with the random
breakage of fibers in a breakage system are set forth:
Given a fiber
group of any particular length, (1) What is the probability that a fiber
of the length groups breaks at all?
fiber breaks into s segments (s
:2:
(2) What is the probability that a
2)?
Since the existing models so
far are based on the restrictions of linear probability of fiber breakage
with respect to fiber length and two-s egment fiber breakage per
breaking fiber, the models are not suitable for the elucidation of the
above basic questions.
To clarify the questions above and to provide a more realistic
interpretation to fiber breakage phenomena, a new fiber breakage
model is derived which allows multi-segment fiber breakage and a
more flexible relation of the probability of fiber break to fiber length,
linear or nonlinear, in the breakage system.
First, a probability model for the uniformly random breakage of
fibers of uniform length is derived with multi-segment fiber breakages
taken into account.
The model expresses the output fiber weight
distribution in terms of the input fiber weight distribution and a parameter m, the average number of break points randomly occurring on an
initial fiber due to the breakage system.
The parameter m is a
quantitative characterization of the severity of fiber breakage in the
system, and simply estimated by the method of moments.
Then, the first model above is extended to the cas e where the
input fiber distribution is a nondegenerate arbitrary distribution.
In
this model it is assumed that the average number m. of break points
J
occurring on an initial fiber of length .t. is a function of the fiber
J
length .t. such that m.
J
J
= a.t~,
J
where a and 13 are parameters, which
permits flexibility of the probability of fiber breakage with respect to
length.
the form
This model is further put into a compact matrix expression of
.& = p .
!..:
where
.& and 1. are (rxl) column vectors for output
and input fiber weight distributions in discrete densities, respectively,
and P is an (rxr) stochastic matrix.
The applicability of the matrix model above is demonstrated on
two different lots of cotton fibers which had different pre-fiber proces s
histories in cotton ginning process.
The parameters a and 13 are
estimated by nonlinear least squares iterations using the GaussNewton method.
The estimated and observed output distributions are
in good agreement, which verifies the validity of the model.
Through
'" and '"13, some new criteria of practical importance and
the estimates a
usefulness are computed which provides a quantitative characterization of the different modes of fiber breakage.
The set-out basic
questions are completely elucidated for both cottons.
Thus a prob-
ability model which provides a realistic interpretation of fiber
breakage phenomena, in terms of probabilities of fiber breakage and
average number of breakage points, is established.
Through the applied examples it is found that the probability of
fiber breakage with respect to fiber length is nonlinear and some
three-segment fiber breakages occur, especially in longer fiber
groups.
Four or more segment breakages occurred relatively
infrequently.
breakage.
Of course, this will depend on the severity of fiber
Therefore, it can be concluded that the consideration of a
flexible relation between the probability of fiber breakage and length
and multi-segment fiber breakage in a fiber breakage model is
pertinent for realistic interpretation of fiber breakage phenomena.
iv
TABLE OF CONTENTS
Page
LIST OF TABLES
vi
LIST OF FIGURES
vii
INTRODUCTION .
1
Review of Literature
Scope of This Study
THEORY
.
Probability Distribution and Weight Distribution
of Fiber Length
.
Definition of Weight Distribution Function
of Fiber Length . . . . . . . . . . . . . . . . . . . . . ..
Duality of Weight Distribution Function and
Probability Distribution Function of Fiber Length.
Derivation of Probability Models for Random Fiber
Breakages
.
Input Fibers of Uniform Length
Poisson Law for the Number of Breaking
Points
.
Conditional Fiber Length Distribution,
Given n Breaks
.
Random Breakage Model for the Uniform
Input Fibers
.
•
Arbitrary Distribution for Input Fibers
The Model in a Matrix Expression . . .
Partitioned Probabilities and Interpretations.
The Matrix Model . . . . . . . . . . . . . . . . . .
Discussion of the Models.
2
8
10
10
10
12
13
13
14
16
22
25
30
30
34
37
v
T ABLE OF CONTENTS (continued)
Page
ESTIMATION AND PRACTICAL USES OF PARAMETERS
Method of Moments
Nonlinear Least Squares Estimation by Gauss-Newton
Method. . . . . . . . . . . . . . . . .
Practical Uses of the Parameters
Average Number of Breakage Points
Probability of Fiber Breakage
44
47
52
52
54
56
APPLICATION AND DISCUSSION
56
Experimental Data
Iteration of Parameters
57
Initial Values for a and 13
Convergence in Iterations
Discussion of Results
44
.
Estimated Output Distributions
Interpretation on Probabilities of Fiber Breakage
and Multi- segment Breakage
Remarks on the Model . . . . . . . . . . . . . . . . . .
61
62
67
67
67
77
SUMMAR Y AND CONCLUSIONS
81
LIST OF REFERENCES . . . . .
87
vi
LIST OF TAB LES
Page
1.
2.
3.
4.
5.
6.
Obs erved data for the changes in fiber weight distributions
due to the fiber breakage (Cotton H and Cotton L) . . . .
58
Convergence in the nonlinear iteration of the parameters
(Cotton H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
Convergence in the nonlinear iteration of the parameters
(Cotton L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
Obs erved and estimated. output fiber weight distributions
(Cotton H and Cotton L) . . . . . . . . . . . . . . . . . . .
68
Probabilities of fiber breakage and average number of
break points in each length group and over-all
(Cotton H) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
Probabilities of fiber breakage and average number of
break points in each length group and over-all
(Cotton L) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
vii
LIST OF FIGURES
Page
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Broken fiber segments weight distribution functions for
various number of breakage points when all input fibers
of uniform length are subject to breakages (Eq. 14)
38
Output fiber weight distribution functions for various
degrees of breakage for input fibers of uniform
length (Eq. 18)
.
40
Continuous distribution part of Eq. (18) for Figure 2
pres ented in probability density functions
.
41
Obs erved data for the changes in fiber weight distributions
due to fiber breakage in the proces s (Cotton H)
.
59
Observed data for the changes in fiber weight distributions
due to fiber breakage in the process (Cotton L) . . . . .
60
Obs erved and estimated output fiber weight distributions
(Cotton H)
.
69
Obs erved and estimated output fiber weight distributions
(Cotton L)
.
70
Comparison of probabilities of fiber breakage with
respect to fiber length between cottons Hand L .
74
Probabilities of no fiber breakage, Uoj' and multi-point
breakages, u
: n = 1, 2, 3 (Cotton H)
.
nj
75
Probabilities of no fiber breakage, u ., and multi-point
oJ
breakages, u .: n = 1, 2, 3 (Cotton L)
.
nJ
76
INTRODUCTION
In science and technology, dimensional reduction of substances
due to certain causes, assigned or unassigned, is a widely observable
phenomenon.
In the field of textiles, in particular, breakage of fibers
in a mechanical operating system is typical of such a phenomenon,
which can be regarded as essentially a one-dimensional reduction.
The reduction of fiber lengths caused by mechanical actions in the
system is in most cases undesirable.
Normally, breakage of fibers
is an unavoidable consequence of processing the fibers through textile
machines which are designed for other us eful functions.
Ginning of
seed cotton, and opening, picking and carding of textile fibers are
typical examples of such processes.
Analytical characterization of the fiber breakage process is of
great value because it allows comparative study between different
modes of breakage phenomena and can indicate in what fashion or how
fibers undergo the breakage process.
Obviously, fiber breakages
occurring in a process will be reflected by any changes in fiber length
distributions of input and output fiber aggregates.
There arise such
simple but fundamental questions concerning a given input fiber aggregate as:
What proportion of fibers of any particular length is broken
due to the proces s?
What proportion of fibers undergoes breakage
into two segments, three segments, or ... , s segments .... ?
In
2
probabilistic terms, the latter is the probability that a fiber breaks
into s segments (s
~
2) and the former is the probability that a fiber of
a given length breaks.
These basic questions can be clarified through
a probabilistic formulation of the fiber breakage process.
•
•
tical usefulness of such a model may be manifold.
The prac-
Comparison of
different processes for a given type of fiber with respect to severity of
fiber degradation, comparison of different fibers for a given type of
process with respect to susceptibility of the fibers to breakages, and
characterization of different controls on a given process as to influence
on fiber length reduction, etc., are some meaningful applications of
practical value.
Review of Literature
Relatively few studies have been done on the problem of mathematical formulation of the fiber breakage process, and those works
merely span the past decade or so.
In the textile field, Byatt and
Elting (2), Shapiro, Sparer, Gaffney, Armitage and Tallant (8), and
recently Tallant, Pittman, and Schultz (9) can be cited.
Meyer,
Almin and Steenberg (7) also studied the fiber size reduction due to
beating in the paper-making process and mentioned the analogy in
textile operations.
Byatt and Elting (2) and Meyer, Almin and Steenberg (7)
approach the problem in essentially the same way.
The axiomatic
3
phenomena of the fiber breakage proces s can be simply described.
For a given fiber length L, the aggregate of the fibers of that length
after a breakage proces s is the sum of the original fibers of length L
which remained unbroken during the process and of newly generated
fiber segments of length L due to the breakages of fibers which were
initially longer than the specified length L.
This basic description of
the fiber breakage proces s is formulated into a proces s equation, considering process changes with respect to time.
Byatt and Elting
handle the changes of weight of fibers under random fiber breakages
while Meyer et al. consider the changes of number of fibers arising
from a more general expression for the breakage function.
Though
they differ in elaboration on the respective model, both groups of
workers begin with an integro-differential equation governing changes
of fiber aggregates with time.
In either cas e, due to the complexity in
the solution of such process equations, a set of assumptions is set
forth to ease the situation.
The common assumptions are:
(1) the
probability of fiber breakage is directly proportional to fiber length
(Meyer et al. term this the "selection function"), and (2) in an
arbitrary time interval, a fiber breaks, if at all, only at one point into
two segments.
Both assumptions seem quite restrictive, especially
the first one, in textile processing operations.
Emphasis in the treatment of the problem differs because
interest and availability of experimental data in the respective research
4
applications are different.
Byatt et aI., under uniformly random
breakage of fiber, stress solving the set-up process equation and
estimating the parameter of the model using an approximation based
on the method of moments.
Meyer et aI., with a general expression
for the breakage function, express the solution in terms of the
moments of the fiber length frequency function, and obtain an asymptotic value, independent of input fiber distribution, for the quotient
between the weight average and the number average of fiber lengths.
Due to the assumption that the fiber breaks only into two segments, an
expression for the breakage function can be obtained rather handily.
Some different modes of randomness for fiber breakages are
examined.
Meyer
~
al. state that the process equation presented by
Byatt and Elting (2) would seem to be incorrect, since the derivation
is bas ed on weight fractions and not on number fractions of the fiber
lengths.
However, this author feels that the incorrectness does not
stem from the fact that weight fractions are us ed in the study, but on
the fact that the expression for the breakage function (according to the
definition by Meyer et al. ) under the assumption of uniformly random
fiber breakage into two segments is not in accord with the consideration of fiber weight changes.
Shapiro, Sparer, Gaffney, Armitage and Tallant (8) investigated
mathematical models for fiber breakages caused by combing a fiber
sample tuft under two cases, (1) undamped and free sample model
5
and (2) clamped and held sample model.
Though the first case was
defined as the "fiber breakage model, " in the study the case is quite a
different situation from the previous studies by Byatt and Elting and
Meyer et al.
Although Shapiro et al. assume only two segments result
from fiber breakage when a fiber breaks, one of the segments is
regarded as being removed from the system.
Hence, the total number
of fibers remains constant whatever fiber breakages occur.
This cas e
is based on the particular mode of fiber breakage in which fiber
breakages occur by a combing action on a fiber sample aggregate and
one of the broken ends is carried away from the system by the comb.
It should be distinguished from ordinary fiber breakage phenomena
where all the broken segments are regarded being retained in the
system as in the previous cases of Byatt and Meyer.
The initial approach of Shapiro et al. to the problem differs
slightly from the previous two in that the axiomatic description of fiber
breakage phenomena is put into a mathematical model in terms of input
and output fiber length distribution functions on a fiber number base
irrespective of time, and probability functions are explicitly defined
and used.
Under the assumptions that the probability of breakage is
linear with respect to fiber length and that only two segments result
from a fiber breakage in the system (one of which is lost from the
system), the output distribution is expressed in terms of the input
distribution and the constant of proportionality for the linear breakage
6
probability.
Further, taking the breakage probability as being a
constant irrespective to fiber length, a numerical relation involving
the first and second moments of the input and output fiber length
distributions is deduced.
Most recently, Tallant, Pittman and Schultz (9) defined the above
study as the "lost ends case" and presented a further development
together with a new study on the ordinary fiber breakage process which
is defined as the "saved ends case ll with the implication that all the
broken fibers are retained in the system.
For the lost ends cas e, again under the assumptions of probability of breakage (defined as breakage function) being linearly
related to fiber length and two-segment breakage, the constant of the
proportionality is expr es s ed in terms of the moments of the input and
output fiber distributions.
Also, an interesting consideration is given
to this case; that is, the entire breakage process is regarded as an
n-stage Markov process of a single stage breakage process.
Under
linear breakage probability and two-segments breakage per breaking
•
fiber, the single stage breakage process is formulated according to the
previous work of Shapiro et al.
By superimposing the single stage n
times, the nth stage output distribution is pres ented in terms of the
input distribution, the proportionality constant in each stage, and the
number of stages n.
It must be noted that the fiber distributions are
handled on fiber number base.
7
The saved ends case, which corresponds to Byatt and Meyer
studies, is newly treated.
A basic equation is set forth relating the
input and output distributions and the breakage probability function
under the two-segments breakage assumption.
It differs from the
starting process equations of Byatt and Meyer in that the relation is
expressed in distribution functions and without time consideration,
and in that a normalizing factor for the output distribution appears,
due to the increase in fiber numbers when breakages occur.
general solution for the breakage function is derived.
A
Under the
restriction of linear breakage probability, the constant of proportionality is expressed in terms of the first moments of the input and output
distribution functions.
With the one stage model, the useful character-
ization of degree of fiber degradation due to any change in ginning
treatment controls is demonstrated by finding the proportionality
constant (inch -1) of the linear breakage probability as sumption.
It is
noted that the excellent agreement between the observed and computed
curves lends support to the acceptance of the hypothesis that breakage
was, indeed, proportional to length.
Also, it is mentioned that explicit compact expressions for the
n-stage Markov process on the saved ends breakage phenomena have
not been found due to the complication arising from the fact that the
number of fibers monotonically increases as the single stage process
is repeated.
However, a numerical iteration of an arbitrary n-staged
8
process, given the constant of proportionality for the breakage function
to be used for each stage and an input distribution, is presented
through a computer program.
Transition of fiber distributions due to
fiber breakages in successive stages is demonstrated for a given
number of stages and a given proportionality constant for each
breakage stage.
Scope of This Study
Earlier two basic questions were raised in connection with the
concept of fiber breakages.
questions are:
In probabilistic terminology, the
Given a breakage process and a fiber group of any
particular length from an input fiber population, (l) What is the
probability that a fiber breaks at all?
(2) What are the probabilities
that a fiber breaks into two segments, three segments, four segments,
.... , generally s segments (s
:<!:
2)?
In view of the approaches to the
problem and assumptions adopted by the investigators cited above, it is
not possible to obtain any clarification for thes e questions from their
works.
Therefore, it is of particular interest and the emphasis of
this study to derive a probabilistic model for the fiber breakage
process in which multi-segment fiber breakages and more flexible
fiber breakage probability with respect to fiber length are allowed.
then becomes possible to elucidate such questions and to give a
realistic interpretation to the random fiber breakage phenomena.
It
9
First, a fiber breakage model will be derived for fiber input of
uniform length.
Then, this model will be extended to the cas e of an
arbitrary fiber input distribution.
The model will be established in
such a way that any sample fiber distribution determined by existing
fiber testing techniques can be directly utilized.
This study is confined to uniformly random fiber breakages,
which is in general termed as random fiber breakages.
Application of
the theory to obs erved sample distributions and estimation of parameters are demonstrated.
In particular, some new measures directly
meaningful in practice are deduced from the parameters and also
applied to the data.
In view of the fact that all sample fiber distribu-
tions are obs erved in terms of weight proportion and fiber weight
distributions are obtained in practice, fiber length distributions on a
fiber weight basis are used.
The fiber weight distribution will be
explicitly defined and called specifically the "weight distribution, "
according to conventional practice in the textile field.
10
THEORY
Probability Distribution and Weight Distribution
of Fib er Length
In the study of fiber distribution in the textile field, us e of terms
such as "fiber distribution" and "fiber weight distribution" is apt to
present confusion.
The expression "fiber number distribution" is also
in frequent use, which in fact corresponds to the probability distribution function of fiber length.
fiber length is denoted F X (x)
If the probability distribution function of
= Pr
(X
~
x), X being a positive random
variable representing fiber length, then [FX (b) - F X (a)] is the probability that a length of fiber from the population belongs to length interval
(a, bJ.
A sample distribution for F X (x) will approximate the
relative frequencies of observing fibers of length in certain intervals.
However, in practice, the sample distribution of fibers is almost
always determined by relative weight proportions of fibers instead of
numbers, due to the impracticality of handling fibers by counting. This
distribution based on the weight proportion of fibers is termed the
"fiber weight distribution ll in contrast to the probability distribution of
fiber length which is conventionally called "fiber number distribution.
II
Definition of Weight Distribution Function of Fiber Length
Under the assumption of homogeneity of fiber diameter and
density, irrespective to length, the weight distribution function of fiber
11
length (in short, the fiber weight distribution) can be defined as
follows:
Definition:
Let fiber length be X, a positive random variable
with probability distribution function
= Pr
F X (x)
~
(X
x), 0 < x
~
maximum fiber length.
Then, the weight distribution function of the random variable X, F ~ (x),
is defined as
x
w
F X (x)
=S
w
d F X (x)
= Sx
: dF X (t)
StdFX(t)
o
o
(1 )
o
w
Note that F X (0)
=0
w
and F X (00)
= 1,
w
and it is clear that F X (x) is
indeed a well defined probability distribution function since all the
properties which F X (x) satisfies to be a probability distribution
function are pres erved in F ~ (x) due to the fact that X is a positive
random variable.
An approximate physical meaning may be attached
such that
That is,
Pr (x
~
tional to x [Pr (x
X
~
X
+ t::. x:
on weight base) is approximately propor-
~ X ~ x + t::. x: on numbe r bas e) ] under the as sumption
of homogeneity of fibers.
12
Duality of Weight Distribution Function and Probability Distribution
Function of Fiber Length
It is interesting to show that the definition of Eq. (l) yields an
expression for the probability distribution function of fiber length in
terms of the fiber weight distribution function as follows:
xldFw(t)
F (x) =
t
X
X
co 1
w
-rdFX(t)
o
S
(2)
S
0
By the definition Eq. (1)
The proof is simple.
d F~ (t)
tdFX(t)
=
co
StdFX(t)
0
or by rewriting
d F X (t)
=
~dF~(t)
1
co
StdFX(t)
0
Noting that
co
1
co
StdFx(t)
0
=S
0
tdFX(t)
1
-t co
t dF X (t)
S
0
and using the definition,Eq. (2) follows.
co
= S ldFw(t)
t
X
0
13
Notice the dual relation between the expressions of Eqs. (1) and
(2).
The physical meaning of Eq. (2) is obvious:
on number bas e) is approximately proportional to
Pr (x s: X s: x
~[
+ t:.x:
Pr (x s: X s: x
+ t:.x:
on weight base) ] under the assumption of homogeneity of fibers.
Throughout the theoretical development which follows, the weight
distribution of fiber length will be called "fiber weight distribution"
and written with the superscript "w" as in Eq. (l), while the probability distribution will be denoted in the conventional way.
Derivation of Probability Models
for Random Fiber Breakages
Input Fibers of Uniform Length
A random fiber breakage model when input fibers are all of equal
length is to be dealt with in this section.
Suppose an aggregate of
fibers of length -t is put into a breakage system and a generated output
fiber distribution is Observed.
Let the input fiber weight distribution
be W~ (x), X being a positive random variable for fiber length. Since
the initial fibers coming into the system are uniform in length W ~ (x)
can be written as
w
WX (x)
=6
t
=0
(x)
for
0 < x < -t
(3)
=1
which is a degenerate distribution.
W~ (x) is particularly denoted as
6-t (x) to stress the fact that W~ (x) is the degenerated distribution
14
above.
For the input fibers, the probability distribution function
WX (x) also has the same expression Eq. (3).
Associated with the breakage system, a partition of initial fibers
according to respective groups of the initial fibers which undergo
breakage into two segments, three segments, ... , generally s segments (s
~
2) is conceivable.
An initial fiber breaks into s segments
if, and only if, it breaks (s - 1) points along the fiber (s
= 1,
2, 3, ... ).
Zero point breakage means no breakage of fiber.
Fibers will be assumed to be freely dispersed in the system and
subject to breaking independently of each other.
Random fiber break-
age implies random occurrence of a breaking point with uniform probability anywhere along the fiber.
Then, the random breakage system
can be modeled as a system where random occurrences of breaking
points on the initial fibers are achieved.
The degree of the occur-
rences of random breaking points on the initial fibers obviously
represents the degree of fiber breakages.
Poisson Law for the Number of Breaking Points.
Let the random
variable N be the number of breakage points occurring on an initial
fiber due to the breakage process.
Since every initial fiber is of same
dimension of length t and a breaking point can occur randomly on any
fiber and at any point along the fiber, it is postulated that the random
variable N follows a Poisson law, i. e. ,
15
Pr (N = n) =
e
-m
m
n
n!
n
= 0, 1, 2, 3, ...
(4)
m> 0
where m is the average number of breaking points occurring on an
initial fiber of the length
t. The expression Eq. (4) represents the
probabilities that initial fibers of length t undergo breakage at n points
for n = 0, 1, 2, 3, ... , respectively.
The Poisson assumption for the number of break points occurring
on an initial fiber is bas ed on the notion of spacial distributions for the
random occurrences of certain events in plane or space.
The postula-
tions for the distribution of random points in the plane or space (the
spacial distribution) to be Poisson distribution are:
that probability of
occurrences of a certain number of random events in any domain of
area or volume is solely depending on the dimensional size regardles s
of the position of the particular domain in the system (stationarity
condition of the system); that nonoverlapping domains are mutually
independent (independence condition); and that if a given domain is very
small the probability of two or more occurrences of events in the
domain is negligible relative to the probability of one or zero occurrences (nonclustering of occurrences) (3).
The physical assumptions
for the homogeneity of all fibers and independent fiber breaking
behavior, and the triviality of overlapped occurrences of two or more
breakage points lend support to the Poisson assumption for the number
of break points occurring on an initial fiber.
16
Conditional Fiber Length Distribution, Given n Breaks.
fiber of the initial length t results in (n
+ 1)
When a
broken segments at the
end of the breakage system, the broken segments present a newly
generated fiber length distribution which is, in fact, a conditional
fiber length distribution given n point fiber breakage.
Let the con-
ditiona1 probability distribution function of fiber length be denoted
'i'X/N (x/n).
It is simple to see that
x
'i'X/N (x/n = 1) =
J
d'i'X/N (x/n = 1)
o
x
=
J-
dt
t
=
x
t
if O<x<t
(5 )
0
=1
if
The expression Eq. (5) is used by all previous workers for the distribution of the length of a broken segment whenever the assumption of
two segments breakage (or one point breakage) per breaking fiber is
adopted.
The problem of multi-point breakage is much more complex
since the notion of sequential breakage or successive breakage comes
•
into the study.
points.
For example, consider a fiber which breaks at three
There are four distinct ways for the initial fiber to result in
four broken segments at the end of the process:
(l) Simultaneous
breakage at the three different breaking points into four segments,
(2) simultaneous breakage at two points into three segments first and
17
subsequent breakage of one of the three resulting segments at one
point, (3) breakage at one point into two segments first and subsequent
simultaneous breakage of one of the two resulting segments at two
points, and (4) sequential one point breakages.
As the number of
breaking points increase the problem becomes quite involved.
To simplify the problem we shall assume that a prior probability
is attached to each breaking point occurring on an initial fiber such
that each breakage point chooses a position at any point along the
initial fiber of length t with a uniform probability 1/t, irrespective
of the order of breakage occurrences.
Consider a realization involving
n break points on an initial fiber of length t, and take the r th breakage
~
point on the fiber in the breaking sequence (1
r
~
n).
Let the
distance of the r th breakage point from one end of the fiber be Z
r
Then the above assumption asserts that the positive random variable
Z
r
has a probability density function of the form
1
(z ) = -, 0< z <
r
r
t
t
(6)
r = 1, 2, 3, ... n
and joint probability density function
.
1
=-
t
n
0< z < t
r
r = 1, 2, 3, ... n
where Z is a vector valued random variable, (Zl' Z2""
(7)
Zn)'
18
Now, if the distance of successive breaking points along the
fiber from one end of the fiber are Yl' Y2' . .. Yr' respectively, then
clearly 0
Y
+
n l
= Yo
< Y < Y 2 < ... < Yn- 1 < Y n < Y n +1
1
are dummy variables.
H1
broken segment.
function of (Y
HI
where Y
0
and
The Y.'s for i = 1, 2,3, ... n are the n
1
For i = 0, 1, 2, 3, .. n
order statistics from the distribution of Eq. (6).
the quantity (Y
= t,
- Y ) obviously represents the length of the (i
i
+ I )th
We shall show that the probability distribution
- Y ) is independent of i.
i
Let ~ Y Y
(y. , y. +1) be the joint probability distribution
i' H I l l
function of i th order statistic Yi and (i + 1)th order of statistic Y HI'
where 0 < Y < Y
<t
i
H1
and i = 1, 2, 3, ... n-l.
the joint probability density d I;X
It is shown (5) that
X
of any cons ecutive order
[HI]
statistics (X[i]' X [HI] ) from an absolutely continuous probability
[i]'
distribution F X (x) is
i-I
=
nl
(i - I)! (n-i-l)!
[
J
FX(x[i])
n-i-l
[1 -F X(X[H1])J
(8)
for
Therefore, by use of Eq. (6) in Eq. (8), the joint probability
density function of (Y , Y
),
i
H1
19
n!
Yi+l(Yi'Yi+l)=(i-l)!(n-i-l)!
1
. """2
t
(~ili-l (1
_ Yi,+1)n-i-1
'1.1
'1.1
dYi dYi+l
(9 )
for
By the transformation of variables such that
IJ I = 1, Eq. (9) becomes
for which Jacobian of transformation
(Xl' x ) =
n!
2
X
(i - 1) ! (n - i-I)!
2
. -
1
l
dX
l
dx
(~)i-l (1- Xl +x 2 )n-i-l
t
t
2
(l0)
for
Hence, the marginal probability density of X ' upon integration of
2
Eq. (l0) over Xl is
t- x2
d'¥X (x 2 ) =
2
J
0
=-
n
tn
(t - x)
2
n-l
dx
2
for 0 < x
2
<t
(11 )
20
It is noted that the probability density function of (Yi+1 - Y ) of
i
Eq. (11) is independent of i, indicating Eq. (11) holds true for any
i ::;: 1, 2, 3, ... ,n-l.
For i ::;: 0, (Yi+1 - Y ) ::;: Y1 which is merely the
i
first order statistic.
of Y
1
It is easy to find that probability density function
from the distribution of Eq. (6)
n-1
d'1'
Y
1
_ y.(,l )
(y)::;: ~ ( 1
1
.(,
which is identical to Eq. (11).
Y
n
is the nth order statistic.
°<
for
For i ::;: n, (Y'+
1
y 1 < .(,
- Y.) ::;: (.(, - Y ), where
l
n
1
Since the density function of Y
n
from
Eq. ( 6) iss impIy
d; Y
n
n
.(,
(y)::;: -
n
n
y
n-l
n
The probability density function of S
d'1'
S
n
°< y
for
n
n
<.(,
::;: (.(, - Y )
n
n
n-1
(s)::;: (.(, - s )
n
n
n
for
.(,
°< s n <.(,
which is also identical to Eq. (11).
Hence, Eq. (11) holds true for all i ::;: 0, 1, 2, 3, ... n, and
independent of i.
Now, simply letting (Y.
- Y.) ::;: X for all
1+1
1
i ::;: 0, 1, 2, 3, ... n, the probability distribution function of (Yi+l - Y ) .
i
can be written as
x
'1'X(x)::;:
J
o
n
(.(,-t)
n-l
dt
.(,n
for
°< x < .(,
( 12)
21
and 'l'X{O)
= 0,
'l'X{x)
=1
if x ~t.
However, distribution function
Eq. (l2) is in fact the conditional probability distribution function of
broken segments given the original fiber breaks at n points; i. e.,
'l'X/N (x/n).
It is readily seen that if n = 1 (two segments breakage per
fiber), the expression Eq. (l2) reduces to Eq. (5).
Finally, by Eq. (l2) and Eq. (l), the corresponding fiber weight
distribution function is found.
Letting the conditional fiber weight
distribution given n point breakage be 'l'~/N ' then
x
'l'~/N (x/n)
=
J
d
'l'~/N (x/n)
o
x
=
t d'l'X/N (tin)
-t--'----
J
o
J
t d'l'X/N (tin)
o
x
=
J
o
n (n + l) t {t _ t)n-l d t
t n +1
(l3)
for 0 < t <t
Therefore, upon integration of Eq. (l3)
(l4)
=1
if
Note that X is a positive random variable representing broken fiber
length such that 0 < x < t ,
'l'~/N (O/n) = 0
and
'l'~/N (tin) = 1.
Further, 'l'~/N (x/n) is a continuous distribution function.
probability density function
for 0 < x < t.
~ 'l'~/N (x/n) = ['l'~/N (x/n) ]
,
Hence the
exists
22
Random Breakage Model for the Uniform Input Fibers.
Having
derived the conditional fiber weight distribution given n point breakage
Eq. (14), we may now find the unconditional fiber weight distribution
of fiber length X.
As a conditional probability statement, Eq. (13)
can be written
~~/N (x/n)
= Pr w
(X s x/N
= n)
This is th.e probability that a broken fiber length is less than or equal
to x, given that an initial fiber undergoes n point breakage.
Together
with the Poisson assumption of Eq. (4) for the number of breaking
points occurring on an initial fiber, the joint fiber weight distribution
function of N and X is given by
Pr w (X s x, N s k) =
L Pr (N = n) Pr w (X s x I N = n)
nsk
(15)
Note that the Poisson law for the random variable N is invariant
regardless of whether probability is interpreted in terms of number of
fibers or weight of fibers.
Hence, if ~~ (x) is the marginal fiber weight distribution function of X
~~ (x)
(i. e., unconditional), by us e of Eq. (15)
= lim [ L Pr (N = n) . Prw (X
k- co
nsk
= Pr (N = 0) .
~~/N
S x/N
= n) ]
co
(x/O)
+ n~l
P (N = n) .
~~/N
(x/n)
(16)
23
Since n = 0 m.eans no fiber breakages, 'i'~/N (x/O), the fiber weight
distribution given n = 0, is equivalent to the initial fiber weight distribution of Eq. (3), i. e. ,
'i'~/N (x/O) = 6t (x) = 1
if
=0
if
x
~t
(1 7)
0<
x
<t
Hence, using Eq. (4) and Eq. (14), Eq. (16) now becom.es explicitly,
After algebraic m.anipulation on the infinite series of the right hand
side above, it can be seen that
-m.
w
6t (x)
~X (x) = e
+1
1
-
+ m.x (~t
1 _ e-
or
= e -m.
5t (x) + (1 - e
-m.
x) ]
T{
1
)
1 - e
for
=1
w
Note ~X (0)
for
=0
w
and ~X (t)
= 1.
0 < x
~
+ mx
~~ - xl }
-m.
t
x~t
(18)
It is interesting to note that the fiber
weight distribution function Eq. (18) is a convex linear com.bination of
the discrete distribution function 6 t (x) of Eq. (17) and a continuous
distribution function
24
mx
1 - eH
where H
c
c
(x)
T [ I + =x (~;
Xl]
= --------~--m
for
=1
for
0 < x < t
(l9)
1 - e
(0)
=0
and H
distribution function.
c
(t)
= 1.
Thus, I W (x) of Eq. (18) is a mixed
x
It is the fiber weight distribution function for
the output fiber aggregates when the initial fibers are of uniform
length t, allowing multi-point breakage of the initial fibers.
H
c
(x) is a continuous distribution function d H
c
Since
(x) = HI (x) dx and the
c
probability function of X can be written
w
d IX (x) = e
-m
= (l - e-
or
=
?
x = t
for
m
) HI (x) dx
c
(20)
m
--x
x e t
[m (t
x)
+ 2t ] dx for 0 < x < t
It is clear from the above that the degenerate mass e
-m
represents an
unbroken portion of the initial fibers, while broken segments form a
new broken segments weight distribution for 0 < x < t
H
c
(x) of Eq. (l9) or HI (x) of Eq. (20).
c
according to
The parameter m
of the
model is the average number of breaking points occurring on a fiber
of the input fibers of uniform length
t.
Obviously a larger m
sents a greater degree of fiber breakage, and vice versa.
repre-
25
Arbitrary Distribution for Input Fibers
In textile practice, it is very rare to find a fiber population of
uniform length with a degenerate distribution function for the fiber
length.
Generally, fibers are of different lengths and present an
arbitrary distribution function.
extended to this cas e.
Hence, the model Eq. (18) will be
Let the initial fiber weight distribution be
F ~ (t), L being a positive random variable for initial fiber length.
The explicit expression for the F~ (t) for any fiber population is not
available in practice.
It is usually determined by experiments and
expressed in the form of discrete densities.
The range of fiber length
is das sified into arbitrary intervals and all fibers falling into any
particular interval are regarded having the length of the midpoint of
the interval.
Therefore, an expression for the initial fiber weight
distribution pertinent to actual practice is
F~ (t)
=
I: Pr
t.st
w
(L
= t.)
J
for
tst r
for
t~t
= t max
J
=1
r
or as a probability function
for
t
= t.
(21)
J
where
t.: the midpoint of /h length interval (cell), i. e., the
J
nominal fiber length of the cell, and j
= 1,
2, 3, ... r .
26
o<
t
l
< .t
2
< -- -- < .t . <-- -- < .t = t
max'
r
J
r
and
f. = 1
~
j= 1 J
Now, it must be noted that the fiber weight distribution function
for the output fibers after the random fiber breakage when the initial
fibers are of the same length t, w~ (x) of Eq. (18), is in fact a conditional distribution function given the initial fiber length t.
Putting
the w~ (x) into a conditional distribution function form as W~/L (xl t),
it can be written that
-m
j 6 t. (x)
J
= e
+ (1
(xl t.)
- e
for
=1
where H
c
(22)
J
O<x~t
for
(xl t.) is the same as Eq. (19) except that .t and mare
J
replaced by t. and m. respectively, and 6 . (x) is the same as Eq. (17)
tJ
J
J
with t. replacing t.
J
The joint distribution function of X and L is
Pr w (X
~
x, L
~
t) =
~
tj~t
Pr w (L = t.) Pr w (X
w
(x) = lim
X
t-co
Pr
= t.)
J
Hence, the marginal distribution of X, G
G
IL
~x
w
J
w
(x), can be found
X
(L=t.)
J
Pr w (X s xl L = t.)
J
27
Then by Eq. (21) and Eq. (22), the above expression in an explicit
form becomes
G
w
X
r
(x)
=
e
E f.
j= 1 J [
-mj
& 9 • (x)
""'J
+ (l
-m
- e
j) H
c
(xl t .)
]
(25)
J
Equation (25) is the fiber weight distribution function for the output
fibers when the initial input fibers pres ent an arbitrary fiber distribution.
The random fiber breakage model permits multi-segment fiber
breakages.
The parameter m. (j
J
= 1,
2, 3, ... r) represents the
average number of break points occurring on a fiber which belongs to
the /h length interval.
As to the relation between fiber breakage and fiber length, it is
reasonable to presume that the longer the fiber the more the breakage
points.
The average number of breakage points occurring on a fiber
in any length group can be regarded as a function of the fiber length.
Whether the function is linear or nonlinear is the very point of the
investigation.
Therefore, a flexible relation between the average
number of breaking points occurring on a fiber and the fiber length is
assumed such that
m.
J
= O:'t~,
J
j
= 1,
(26)
2, 3 ... r
which contains the linear relation as a particular case for
Since by Eq. (4) and Eq. (21),
= 1.
m. > 0 and t. > 0
J
Q'ts. > 0
J
which implies
~
J
for any j
Q' > 0 and -co <
~
< + co
(27)
28
From Eqs. (25) and (27), the random fiber breakage model, for an
arbitrary input fiber distribution F~ (t), can be finally expressed such
that the newly generated output fibe r weight distribution
G W (x)
x
= ;,
f. e
j= 1 J
-O't~
-O't~
r
J 6
+ ~
f. (l - e
j= 1 J
(x)
tj
J) H
for
=1
c
(xl t. : 0', S)
J
0 < x s: t
for
r
x~t
(28)
r
where
H
(xl t . :0',
c
J
S)
[
1
=
l-e
-O't.S
1 - e
_~t~-l
x{ 1+ ~t~J x(tJ - Xl} ]
J
2
t.
J
J
for
for
= 1
6t. (x) = 0
for
x < t.
J
for
x
J
=1
0< x < t.
J
x
~
(29)
t.
J
(30)
~t.
J
the initial fiber weight distribution function F~ (t) being expressed as
in Eq. (21). The generated output fiber weight distribution G~ (x) of
Eq. (28) is a convex linear combination of r discrete distributions of
the form Eq. (30) and r continuous distributions of the form Eq. (29).
Since the sum of the coefficients
r
~
-O't~
f. e
j=l J
J
+
r
~ £.(l-e
j=l J
it is a mixed distribution function.
Eq. (29) and Eq. (30) that
It is readily seen by the use of
29
G
w
(O)
x
= 0,
since 5 .(0)
t
J
=0
and H
c
(O / t. :a, ~)
J
=0
for all j = 1, 2, 3 . .. r
1, since for allj 0, (t )
r
-t-.
J
= 1 andHc (t r k.:a,~)
=1
J
as t.
J
~
t
r
=t
max
for all j.
The probability function for the length of the output fibers can be
written as
dG
w
(x)
X
-at~
= f
j
e
J
-Q't~
r
=
~
for x = t., j = 1, 2, ... r
J
£. {I - e
j=l J
J)dH
C
(x/t.:Q',~)dx
(3l)
J
or
for
0 < x < t
r
x ;. t., j = 1, 2, . .. r
J
The physical interpretation of the above model is similar to that for
the case of uniform length input fibers of Eq. (20), except the fact
Eq. (31) can be regarded as superimposition of breakage phenomena
of the uniform initial fiber length case for all different fiber length
groups.
As will be shown in the next section, a more realistic inter-
pretation of the breakage proces s naturally results when the model is
expressed in partitioned densities.
30
The Model in a Matrix Expression
To facilitate the practical us e of the models so far derived, it is
more convenient and realistic to partition the density of the output
fiber weight distribution into the same length intervals which were
described in connection with the expression for the initial distribution
in Eq. (21).
The sample distribution for the output fibers is also only
available in the form of discrete densities.
Therefore, the model will
be put into a discrete form compatible with the expression for the
input distribution.
The model Eq. (28) or Eq. (31) will be treated
first and a simpler analogy for the uniform initial fiber length case
will be noted.
Partitioned Probabilities and Interpretations.
Let the i
th
fiber
length interval be
= (c.,
1
c!)
1
where
~x:
width of an interval
c.
lower limit of the i
c!
Upper limit of the cell
1
1
th
cell
then clearly
=c
r-l
<.(,
r-l
<c'
=c <.(, =t
r-l
r
r
max
31
The partitioned probability for the i
th
cell Pr W (Xel.) = g. becomes
1
1
by Eq. (28),
[G
G
g. = Pr W (c. ~ x ~ c!) =
W (c ~) W (c. )]
1 1 1 1
x
1
x
or more explicitly from Eg. (28) and Eq. (30)
~
~
c'
-Ott. [
] i
g. = ~ f. e
J 6# (x/t.)
1 j= 1 J
'1.1 j
J
c.
r
r
[
+ ~
f.
j= 1 J
c'
-Ott'j [
1 - e
J
H
c
(x/t.:Ot,~)
i
(32)
J
C.
1
1
Denoting the first and the second component of the above as g..
IJ
g .. , simply
IJ
c
]
r
g. = ~ g ..
1 j=l IJ d
and
d
r
+
~
j=l
(33 )
g ..
IJ C
Since (t. < t.), (t. = t.), and (t. > t.) are disjoint events, the g. or
1
J
1
J
1
J
1
Pr
Pr
W
(Xel.) can be put as
1
W
(Xel.) = Pr
1
w
+ Pr
[(Xel.)
1
w
[(Xel )
i
r
~
=
n (t.1 <
(goo
j=i+l1Jd
n
+ goo
IJ C
t.)]
J
+ Pr
w
[(xeI.)
1
n (t.1 = t.)]
J
(t > t )]
i
)
j
+ (g.. + g..
lld
)
llC
+
i-I
~
j=l
(goo
IJ d
+ goo
)
( 3.4)
IJ C
by use of similar notation used for Eq. (33), which is a partition of the
summation of Eq. (32).
r
For
~
j=i+ 1
(goo
IJ d
It remains to evaluate each term of Eq. (34).
+ g ..
IJ C
): since c. < t. <
1
1
c~ ~
1
(c.') = 0, 6 #
5#
'1.1.
J
and
1
r
~
j=i+l
c. < t.
J
J
(c.) = 0
'1.1.
J
g.. = 0
IJ d
1
by Eq. (30)
32
Hence,
r
r
~
(g.. + g .. )
j=i+I lJ d
lJ C
=
~
g ..
j=i+I1J C
r
=
,
f.; 1
~
-Q'.t~)
e
J
j=i+I J \
[He (x/t j :C'.
~)]
c'
i
(35)
c.
1
For (g.. + g .. ): since c. <.t.
lld
llC
1
1
( c!)
t),
""'.
c
J
= I,
1
1
H
= .t.
< c!, (i
1
"",.
1
= 0
by Eq. ( 30)
1
1
(c!/.t.: Q', ~)
1
( c.)
t),
= j)
=I
by Eq. (29)
Hence
(g..
lld
+ g ..
) = f. e
llc
1
-Q'.t~
-Q'.t~
l+f.(I_e
1
l)[I_H
c
(c./.t.:Q',~)J
1
1
(36)
i-I
For
~
j=I
(g ..
lJ
+ g .. ): since.t. < c~ S c. <.t. < c!
J
J
1
1
1
lJ
d
c
t),
(c!) = 5, (c.) = I
1
""'.
1
J
J
by Eq. (30)
""'.
H
c
( c! / .t. : Q', ~)
1
J
= H C (c./.t.
: Q',
1
J
~)
=I
by Eq. (29)
Hence
i-I
~
j= I
(g..
lJ d
+ g ..
lJ C
)
=0
(37)
Combining the findings of Eqs. (35), (36) and (37), Eq. (34) now
becomes
33
r
Prw (Xe:I.) = g. = g..
1
1
lld
~
f
+
j=i+1
+ g.. +
11
C
~
g.
1J
C
0
j=i+l
.(1-
(38)
J
The above expression gives rise to a natural description of the
axiomatic phenomena of the fiber breakage process.
The partitioned
density of the i th length interval for the output fiber weight distribution
from a breakage system consists of three components as shown in
Eq. (38).
The first term for gii
is a portion of the density gi due to
d
the initial fibers of the i th length group unbroken and remaining in the
cell.
The second term for g..
11
C
is another fraction of g. due to some
1
broken segments from the initial fibers of the i th length group, which
are still long enough to belong to the length group.
The third
r
corresponding to
~
j=i+1
g.. is due to broken segments from fibers
1J C
which were initially in the longer length groups 1.
, 1. 2' ... , I .
1+1
1+
r
It is noted that the second term g..
11
arises from the fact that an
c
arbitrary length classification is made and the output fiber weight
distribution is partitioned into the different length groups according to
the way in which the fiber weight distributions are determined in
i-1
practice.
It is natural to find
.~
J=1
(go
1Jd
0
+ g ..
1J C
) = 0 of Eq. (37), which
34
asserts that no broken segments from the shorter length groups
II' 1 , ... ,I _ can contribute to the output density gi of the i
2
i l
th
interval.
The Matrix Model.
i = 1, 2, 3, ... , r.
Clearly the expression Eq. (38) holds for
Rearranging the terms, Eq. (38) can be put into the
form
g.
1
= £.
1
P ..
11
+
r
~
j=i+ 1
J
(39)
for i= 1, 2, 3, ... r
£. P ..
lJ
where
P ..
11
=e
-Q't~1 + ( 1-
( 40)
and
Pij (1 -
(j > i)
=
(41 )
The set of r linear equations (39) can be expressed in the matrix form:
=
o
o
o
o
,
" , , ...
, ...
, ...
... ,
... ...
... ,
... ...
o
o
(rX1 )
... ,
(42)
"',
......
' ... ,
,
'
...
o
P
P
o
o
P
(rxr)
r-l r-1
r-l r
rr
f
r
(rX 1)
35
where
~
P ..
11
= -e
-ext i c i
~-2 2
(ext.1
c.
1
~-l
-ext.1
(43)
c. -1)
1
and
(44)
Equations (43) and (44) are obtained from Eqs. (40) and (41) by use of
Eq. (29).
The (rxr) matrix (P .. ) = P exhibits some interesting
IJ
following properties:
P
= 1, since the lower limit of the first length interval,
ll
c
r
~
P ..
i=l IJ
=
j
~
P ..
i=l IJ
l
=0
=e
in Eq. (43).
-ext~
J
j
+ ~
i=l
for j = 1, 2, 3, ... r
c'.
j
since
~
[He (x/t j "".
i=l
~)LJ
Hence, the matric P is a stochastic matrix.
densities, g., it is now easy to see that
1
0
1
by Eq. (29).
For the partitioned
36
~
; g.=
[f.P ..
i=l 1 i=l
1 11
r
~
f. P ..
i=l 1 11
=
=
+
+
~
j=i+l
J
1J
r
j-l
r
f. ~ Poo = ~ f. (P ..
j=l J i=l 1J
j=l J
JJ
~
j
r
i.P .. ]
+
j-l
~
i=l
P .. )
1J
=1
~ f. (~ Poo)
j=l J i=l 1J
by the property of P (stochastic matrix), the f. 's being the initial
J
fiber distribution express ed in a probability function as shown in
Eq. (21).
The expression Eq. (42) is the fiber breakage model
allowing multi-point breakage of initial fibers when an arbitrary initial
fiber weight distribution is assumed.
Both input and output fiber
distributions are expressed in probability functions.
parameters ex and
~
appearing in the stochastic matrix P have the
defined parameter space shown in Eq. (27).
have the uniform initial length t
f
k
= 1 and f = 0 for j
j
--- and, gk = f
k
P
gi = P
kk
ik
I
k.
k
H
c
(x/ t
We then have gl = f
k
P
lk
, g2 = f
k
P
2k
,
, or simply
, since f
k
The cas e when all fibers
is included in Eq. (42) by taking
k
= 1, for i = 1, 2, 3, ---, k
Here
and
The model
) is as shown in Eq. (19).
( 45)
37
It is clear that P
+ gkk
) of Eq. (38)
c
and possesses the analogous physical interpretation as before. P
is
ik
the newly generated density in the i th interval for i = 1, 2, 3, ---, k-l,
kk
above is equivalent to (gkk
d
respectively, due to the breakage of some initial fibers in the t
k
length interval.
Discussion of the Models
The breakage model Eq. (18) for input fibers of uniform length
primarily depends on the conditional fiber weight distribution function
'l.'~/N (x/n) of Eq. (14). It is, therefore, illuminating to observe the
behavior of this conditional distribution function for different values of
n.
For a fixed initial fiber length
t, the family of conditional weight
distribution functions given that all fibers undergo n point breakage is
shown in Figure 1 for n = 1, 2, 3, 5 and 10.
If n = 0, that is, no fiber
breakages at all, 'l.' ~/N (x/O) is the initial fiber weight distribution
function of the form Eq. (17).
For n
fibers of length less than t is clear.
~
1, the presence of broken
Needless to say, a larger n
represents a more severe fiber breakage.
As n increases, the
gradual increase of shorter fiber groups becomes apparent.
If n
•
becomes very large, the theoretical curve shifts close to the ordinate
axis, implying an accumulation of pulverized fibers.
The accumula-
tion of short fiber groups as n increases is a reasonable theoretical
interpretation pertinent to the fiber breakage phenomena.
38
Length
Figure 1.
Broken fiber segments weight distribution functions
for various number of breakage points when all
input fibers of uniform length are subject to
breakages (Eq. 14)
39
The :model for the unifor:m input fiber length case shown in
Eq. (18) is a :mixed distribution function and the continuous distribution part H
c
(x) of Eq. (19) repres ents the broken fiber length distri-
bution due to so:me breakages of initial fibers of the length
•
t. So:me
fiber weight distribution functions for output fiber aggregated are
shown in Figure 2 for various values of:m.
The para:meter :m is the
average nu:mber of break points occurring per fiber of length t, and it
clearly :measures the degree of severity of fiber breakage for the
particular length group fibers.
For a s:mall value of :m (:m = .1) a
s:mall portion of broken fibers of length between (0, t) are seen, and
the :majority of fibers still re:main intact as displayed by the ju:mp at t
.
-:m
of the Slze e
.
As:m beco:mes larger (:m = .5, 1, and 3), gradual
accu:mulation of short fiber groups and a di:minishing portion of
unbroken fibers beco:me apparent, which is an expected fiber breakage
interpretation for the case of unifor:m initial fiber length.
To give a direct presentation for the accu:mulation of short fibers
as is us ed in the textile field, sche:matic curves for the continuous
distribution part H
•
c
(x) of the :model are also presented, in Figure 3,
in ter:ms of probability density functions of the for:m expressed in
Eq. (20) for 0 < x <
t. It :must be noted that the area under a curve is
(1 - e-:m) for each case since it represents the distribution for
0.
< x < t.
The way in which the short fibers accu:mulate is evident,
and for :m = 3. 0 a high concentration of fibers of length approxi:mately
40
1.0
m= 0.1
I
1.0
m= 0.5
I
I
I
-
I
I
I
I
I
.5
I
.5
I
I
I
I
I
I
I
Length
1.0
m= 1.0
t
I
Length
1.0
I
m= 3.0
I
I
I
I
I
I
.5
.5
Length
Figure 2.
Length
Output fiber weight distribution functions for various
degrees of breakage for input fibers of uniform
length (Eq. 18)
41
1
m= 0.1
1
m= 0.5
t
t
m= 3.0
1
m= 1.0
t
1
t
Figure 3.
Continuous distribution part of Eq. (18) for Figure 2
pres ented in probability density functions
42
a little less than
~
can be noted.
As n becomes larger the shift of
the peak to the left is expected with a higher ordinate, implying high
cumulation of short fibers.
In ordinary textile processes, the type of
short fiber cumulation for the case m = .5, more or less, may be
expected.
t
= 1.
For Figures 1, 2 and 3 above, the curves are actually for
However, the general shape and trend of the curves remain
alike for different values of initial fiber length t.
The fiber breakage model for an arbitrary initial fiber distribution Eq. (28) is also a mixed distribution.
It can be regarded as a
superimposition of the preceding model of uniform initial fiber length
for different fiber length groups, each being weighted by the initial
density of the particular length interval.
The graphical representation
can also be thought of as a weighted superimposition of fiber weight
distribution curves such as those of Figure 2 for various values of tis
and mls.
For this case, it must be noted that the various mls are
assumed to be functionally related to the various tIs such as shown in
Eq. (26), giving a set of new parameters a and
~
of the model.
It is interesting to note further the role of a and
practically suitable region for the parameter space.
concern associated with a and
~
~
and deduce a
The primary
is the role which these parameters
play in the expression for the probability of fiber breakage for a
th
particular length group, namely j
length group, i.
where a > 0, -co < 13 < + co by Eq. (27).
If 13 = 0,
IT.
J
e.,
TT.
J
=(l
- e
-at.13
is a constant
J ),
43
(1- e
-a
}forallt.'s.
J
For 13>0, TT.isamonotoneincreasingfunction
J
of t., and a monotone decreasing function of t. for 13 <
J
J
o.
Also, for
13 > 0, Q't~ is monotonically increasing with t., and monotonically
J
J
decreasing with t. for 13 < O.
The cas e of S < 0, for which both the
J
average number of breakage points occurring per fiber and the probability of fiber break is monotonically decreasing as fiber length
increas es, is clearly not in accord with the conventional concept of
fiber breakages.
13 < 0 implies that the longer the fiber the less the
fiber breakages, and vice versa.
look for the parameter space 13
breakages.
•
~
Thus, it is certainly reasonable to
0 and Q' > 0 for textile fiber
Furthermore, the cas e 13 > 0 is more probable .
44
ESTIMATION AND PRACTICAL USES OF PARAMETERS
The direct use of the fiber breakage model for uniform length
..
input fibers Eq. (18) may seldom be used in textile practice.
In
almost all cas es in practical applications the model for an arbitrary
input fiber weight distribution Eq. (28) is more pertinent, and is
applied through the matrix expression of Eq. (42).
model, in which two parameters a and
~
For the latter
are involved nonlinearly in
the model, a nonlinear iterative solution by the Gauss-Newton method
may be used for the estimation of the parameters.
Due to the com-
plexity of the model, there do not seem to be any straightforward
estimation procedures which do not require some form of iteration.
Since the model for uniform initial fibers can be regarded as a special
case of the model for an arbitrary input fiber weight distribution, as
discussed for Eq. (45), any estimation procedure for the latter may
apply to the former case.
However, a separate estimation of the
parameter m in the former model is also presented here, since a
simple and explicit expression for the estimate by the method of
moments is readily available and the estimate bears a descriptive
physical interpretation.
Method of Moments
This method consists in equating a convenient number of the
sample moments to the corresponding moments of the distribution,
45
which are functions of the unknown parameters.
For the model
Eq. (18) it is found that the first reciprocal moment of the weight
..
distribution function has a very simple linear expression in m.
If
w
.
1 momen t f or E q. (18) ,
E X (_1
X ) l'S the fl' r st reC1proca
t
EW(l)
X X
= ~
f
dt
w
x (x)
0
t
f
=
t
1
-m
d [ e -m 6t (x)] t (1 - e
)
x
0
0
m
t
= e -m ~x
J
1
dH (x/t)
x
c
1
Pr w (X
x
= x)
f
t
--x
;} e
t
[m (t - x) t 2t] dx
o
by use of Eqs. (17) and (20).
Hence, upon integration it is easily seen
that
EW(l)
X X
=1
tm
t
(46)
where t is the uniform initial fiber length and m is the parameter of
the model, the average number of break points occurring on a fiber of
the length t.
-w 1
Now, let EX (X) be the corresponding sample moment.
Since
the obs erved sample fiber weight distribution for the output fibers is
also available in the form of discrete densities as presented in Eq. (21),
EW (1) can be simply obtained as
X X
EW(l)
X X
=
r
~
i= 1
( :~)
(47)
46
where z.'s (i = 1, 2, 3, ... , r) are observed densities for the output
1
sample fiber weight distribution which correspond to the g. 's of
1
.
Therefore, equating Eqs. (46) and (47) and noting t is the
Eq. (42).
= t max = t
maximum fiber length, i. e., t
as before, the estimate
r
~ of m can be obtained as follows.
t
1
r
i
W
(.!..)
A
X X
m=--"";"'--1
=t r
(48)
EWXX
('!")
or explicitly
It is interesting to note that ~ becomes zero if no fiber
breakages occur, since if so z. = 1 at t. = t
1
at t
i
f
t
r
.
For any fiber breakages,
1
r
(or t) and z. = 0
l/E~ (~)
1
is always less than t
The estimate ~ is merely the ratio of the difference between the
initial fiber length and a diminishing length criteria
latter.
'l/E~ (~)
to the
It gives an even more descriptive interpretation when it is
recognized that
l/E~ (~)
is equivalent to EX (X), i. e., the mean for
•
the probability distribution function of fiber length, which is conventionally called the number mean length.
The equivalence of
l/E~ (~)
to EX (X) is shown in the proof used for Eq. (2) under the assumption
of the homogeneity of fibers.
In practice, it is simple to compute the
r
.
47
r
quantity !: (z./t.), once the output sample fiber weight distribution is
i=l 1 1
obtained in the conventional way.
Nonlinear LeastSquares Estimation by Gauss-Newton Method
Consider a general nonlinear model in the form of
where Y is dependent variable,
XIS
are independent variables, and a's
1£ n is the number of
are unknown parameters to be estimated.
observations such that j = 1, 2, 3, ... , n, the above nonlinear model
can be put in a compact matrix form
Y =
.1 (X;
~)
(49)
where Y is (n XI), and also! is (n XI) column vector which is a
function of an (n X k) matrix X and (n X I) parameter vector!.
function! were linear in!, the least square estimation of
the linear estimation problem.
But, since the function
~
~
1£ the
is merely
is nonlinear
in!, an iterative solution of a system of nonlinear equations is
neces sary for the least squares estimation.
Let the observation of Y be
y. Since Eq. (49) is the expecta-
tion equation, the least squares equation can be written
where e is random error vector with E
for a known (n X n) nonsingular matrix V.
(~)
= .Q. and E
C~~')
= V
(J
2
For the general expression
48
of the variance-covariance matrix V .
2
(J
,
the least squares principle
is to estimate 8 to minimize
(51)
Hence, taking partial derivatives of 0 with respect to 8. (i = 1, 2, ... m)
1
and equating them to zero, we obtain
00
08
- D V-l [y-iJ
1
where
i =!
(Xi
= 0
(52)
8)
and
A
Eq. (52) is the normal equation to be solved for 8, the least
A
squares estimate of.!.
Since! appears in the normal equation non-
linearly, the ordinary linear estimation approach is not pos sible.
However, under the assumption that the first order Taylor series
A
approximation of
space 8
-0
~(Xi!)
around a starting point in the parameter
is adequate, then
(53)
or, more simply,
A
~
=
~
-0
into Eq. (52), we obtain
+ D0
"
(8 - 8
-
-0
).
By substitution of the above
i
49
- DI
o
v -l[ (1.
- ~
-0
) - D
0
"']
(8 - 8 )
-
-0
= -0
( 54)
The above system of m linear equations is the set of Gauss-Newton
equations and the linearized model Eq. (54) can now be solved for
(8'" - 8
-
-0
).
First compute
(55)
'" is the correction vector to be made on the starting vector
where .§..l
A
!
0
A
to get a new vector ! 1 =!o
A
+!
1 as the first iterated approxima-
A
tion to 8.
Then, letting ! l be another starting vector repeat the
iteration for a new correction vector '"
£.2 to obtain the second iterated
'" , !2'
'" and so on.
estimate of !
The iteration continues until the
successive corrections are effectively zeros; that is, successive
'" t (t = 1, 2, 3, ... ) stabilize, or the minimizing value Q of Eq. (51)
i
has been essentially realized.
The succes s in convergence by the above iterative method is
very much dependent on the surface of Q.
For nonlinear model the
contours of constant Q are usually distorted ellipsoids, the degree of
distortion depending on the severity of the nonlinearity of the model.
1£ the starting vector 8
-0
is too distant from the minimum or if the
contours of Q are very distorted, the convergence may sometimes fail.
In such a case, the method may be "modified (4, 6).
The correction
vector '"
£.t (t = 1, 2, 3, ... ) expressed by Eq. (55) replaced by a fraction
A
of it, k . ()
-t
with 0
~
k
~
1, to insure monotonic convergence in
50
successive iterations.
Such modified Gauss-Newton methods adopt
more analytical numerical iterations to choose a k (0 :s: k ,. l)
A
depending upon the correction direction 6
at each iteration stage.
-t
To apply the Gauss-Newton method to the fiber breakage model
for an arbitrary initial distribution, consider the matrix model of
Eq. (42).
The left hand column vector .& is the theoretical partition of
If the obs erved value
the densities of output fiber weight distribution.
of .&
is~,
then the least squares equation, corresponding to Eq. (50),
can be written in a simple vector notation as
~ =
!
(!..; (0',
S)' ) + !:
where t is the entire right hand side of Eq. (42) and is an (rXl) column
vector, and!.. is an (rX 1) vector containing the fixed initial fiber weight
distribution in discrete densities.
assumed to have E (!:) =
known.
.2. and
The random error vector
E (!:!:') = V (J
2
= I
2
(J
~
because V
(rxl) is
.
1S
not
However, it is preferable to use some approximation to V
through repeated experimental results for a given
i.
The initial
weight distribution!.. is normally determined by the experiment, but
here it is assumed to be measured without error.
The normal equation of Eq. (52) becomes
51
where
~
=[1]
A
Sf= 0'
=[1 (i.;
~)I)]
(0',
A
0'=0'
A
A
~=~
a=~
r
P
~
j=1
=
1j
f.
J
r
~ PZ· £.
j=Z
J J
P
rr
f
r
A
(rXl)
0'=0'
A
~=~
stage iteration (t
= 1,Z,
3, ... ) let the starting point be
(0'
t-
l' ~
t-
1) I .
Then for any t the Gauss-N:ewton equations, corresponding to Eq. (54)
can be written as
52
ex - ex
A
A
where ~t = ~
[
ex
= ex t _ l
t
_ ~
t-l]
t
t-l
and ~
= St-l'
o =
A
-t
[
,
D
t-l
and i
1 are D and i
-t-
respectively.
D'
D
t-l t-l
evaluated
Hence
]-1 [D't-l
(z - i
)
-t-l
]
Therefore, after the tth stage iteration, the iterates for ex and ~
become
A
+ £.t
(56)
The iterations are repeated until
Qt =
becomes a minimum.
[z- - -t
i ]' [z
i ]
- - -t
The choice of the first starting value (ex 0'
(57)
~ 0)
and the convergence criteria will be discussed in the chapter on
Application and Discussion.
Practical Uses of the Parameters
A
Once the estimates ex
and '13" are obtained, it is desirable to
deduce some criteria which give more direct physical meaning in
describing the breakage phenomena.
The following new measures are
readily available for some comparative studies.
Average Number of Breakage Points
The average number, m., of breakage points per fiber of length
J
t. is simply computed using Eq. (26),
J
i. e. ,
53
A
j = 1, 2, 3, ... , r
m.
J
As noted previously, m. represents the susceptibility to breakage of
J
the fibers of length L ..
J
The above m. 's may be inadequate to compare between
J
differ~nt
length groups, for m. 's for longer fiber groups may be expected to be
J
larger in magnitude.
For comparison between different length groups
the average number of breaks, per unit length; i. e.,
more informative.
'i. = :G1. It.,
J
J
are
It should be noted that >... is the average number of
J
breakage points associated with the initial fibers of the
group only.
J
It is apparent that if ~
= 1,
~.
J
=~
t. length
J
for all j.
Even if the comparison of different lots or types of fibers can be
achieved by comparing respective m. or >... within a given length
J
J
group, it is also desirable to have a similar criteria for quick comparison in an over-all sense.
Consider a new criteria
>.. = (over-all number of breakage points occurring on all fibers)
(over-all length of all fibers)
r
1£ n. is the initial number of fibers in the t. length group and ~ n = N
J
J
j=l j
is the total initial number of fibers, then clearly
N (f.lt.)
J _J~_
n. = .....,.,.._i:lJ
r
~ (f.l t.)
j= 1 J
J
for f. is initial fiber weight density for the fiber length
J
(j
= 1,
t.
J
2, 3, ... , r) and by Eq. (2) (f.lt.)!';' (f.lt.) is the probability
J J
j= 1 J J
54
t..
density for the fiber length
Hence X. defined previously can be
J
written
~m.n.
J J
x. =
~t.n.
J J
and it is simple to see by use of the expression above for n. that
J
A
X.
=
~~. (f.
J
~
J
f.
J
It.)
J
A
m.
r
A
= ~--l f = ~ f. X..
t. j
j= 1 J J
J
The physical meaning of X. is immediate.
It is simply a weighted
average of X..' s, weighted by the discrete densities of the initial fiber
J
weight distribution f. 's.
J
However, it must be noted that two different
fiber lots may exhibit quite different responses to a breakage system
.
In
terms
0 fA
m.
J
an d'"
x.. within each length group even if both types show
J
identical values for
'"x..
Probability of Fiber Breakage
Having estimated the m.ls, it is now possible to answer the two
J
fundamental questions rais ed at the beginning of this study.
The
probabilities of n-point breakage (or breakage into (n+l) segments) of
a fiber of the
t. length group can be found through the Poisson law of
J
Eq. (4) for n = 1, 2, 3, 4, . . .
the
The probability of no break of a fiber of
t. length group u . is simply e -m j and the probability of at least
J
one break is (1 - e
oJ
-m'
n=o
~
n=l
co
~
co
J) which is equal to
u . ::;: 1 fo r all j ::;: 1, 2, ... r.
nJ
u
..
Clearly,
nJ
Direct comparisons of u .'s between
nJ
55
different length groups or within a saIne length group for two different
lots of fibers are Ineaningful.
All thes e probabilities indicate a
particular Inode of breakage behavior of the fibers, the cause of which
Inay be due to inherent properties of the fibers or to external effects
of the breakage systeIn.
Though the above u . 's indicate the probabilities of n-point
nJ
breakage in each length group, it is reasonable to have such probabilities which take over-all fibers into account.
r
By weighing u . by
nJ
f. (j = 1, 2, . .. r), we cOInpute U = ~ f. u . for n = 0, 1, 2, . ..
J
n
j= 1 J nJ
quantity U
n
The
can be physically interpreted as the relative weight pro-
portion of initial fibers which undergo n-point breakage irrespective
co
of length differences.
Obviously,
~
n=1
U
n
= 1.
All the discussed criteria above, In. , >..., >.., u ., and U can
J
nJ
n
J
be Ineaningfully used in practice with good descriptive physical interpretations, especially for cOInparative studies of different fibers or
different breakage systeIns with respect to randoIn fiber breakage.
FurtherInore, it is the particular interest of this study to deduce u .' s
nJ
froIn experiInental data.
56
APPUCATION AND DISCUSSION
Though the applications of the derived fiber breakage models are
..
extensive in the field of textile operations, the us e of the model will be
exhibited only with two different lots of cotton fibers for demonstration
.
purposes.
The comparison of the two lots of cotton will be made with
respect to the modes of fiber breakage which represent inherent
properties of the fibers.
Experimental Data
The two different lots of cotton are labeled "Cotton H" and
"Cotton L.
11
Both lots are the same variety of cotton (Robinsonville,
Mississippi cotton), and are from the same locality.
The two differ
only in the level of moisture control in the ginning operation.
Cotton H
received high moisture control (low temperature) and cotton L low
moisture (high temperature) in the drying tower.
Both cottons
received the same sequence of controls thereafter in the remaining
ginning sequence (elaborate overhead cleaner and two lint cleaners).
Then, both lots of fibers were proces sed through opening, picking and
carding under the same fiber processing conditions.
The sample fiber weight distributions before and after fiber
processing are experimentally determined by the Suter- Webb Array
method (1) which gives relative proportions of fiber weight in 1/8"
length invervals.
All fibers in a length interval are regarded as having
57
a length of the midpoint of the interval, i. e.,
for j = 1, 2, ... , rand
group.
t
r
t.
J
= 1/16 + (j
- 1)/8
is the midpoint of maximum fiber length
The observations for a bale sample (before the process) and a
card sliver sample (after the process) are shown in Table 1.
To give
a better picture for the changes in fiber weight distributions due to
fiber breakage during the process, the data is presented in diagrams
as shown in Figures 4 and 5.
The data should be presented in the form
of discrete densities erected at each
t.,
J
but in order to get clear
visual contrast between input and output data they are shown in histograms as used in practice.
The diagrams clearly show a reduction in
long fiber groups and an increas e in short fiber groups due to fiber
breakage.
Such phenomena seem to be more pronounced for cotton L
with implication of more fiber damage.
Iteration of Parameters
Since the data represents the case for an arbitrary initial distribution, the application of Eq. (28) through the matrix form Eq. (42) is
appropriate and it requires the nonlinear iterative solution in estimating the parameters
Q(
and f:3.
The least squares iterative procedure by
the unmodified Gauss-Newton method was performed by use of IBM
System 360/Mode1 30.
Fortran IV.
The computer program was written in
e
e
Table 1.
1716
Cotton
H
Before .0122
(Bale)
After
(Card)
L
..
.
e
Observed data for the changes in fiber weight distributions
due to the fiber breakage
(Cotton H and Cotton L)
3/16--5/16
.0187
,
.0308
-7/l6-9/16
.0342
.0366
Length Group (Inches)
11716 13/16 15716 177(6
.0456
.0685
.1011
.1669
.0125.0205.0418.0467.0654.0592.0801.1121.1539
19716
21~237i625716
.2517
.1683
.0555
.0099
.2074.1487.0494.0022
Before .0148
(Bale)
.0266
.0465
.0397
.0467
.0564
.0761
.1173
.1599
.2347
.1357
.0437
.0019
After
(Card)
.0278
.0636
.0712
.0582
.0882
.0990
.1272
.1306
.1755
.1178
.0251
.0013
.0148
\.J1
00
59
..
.....----
.25
Before (Bale)
-----
After (Card)
~---
.20
----
----
--f-
---~---
1----
.05
r----
-
>----
..... --I----
----
1
3
16
16
5
16
7
16
I
9
16
11
16
13
16
15
16
17
16
. . -.- J
19
16
21
16
23
16
25
16
Length Group
Figure 4.
Observed data for the changes in fiber weight distributions
due to fiber breakage in the process (Cotton H)
60
..
.25
-
Before (Bale)
After (Card)
-----
.20
~---
,
s::
....0
~
I-t
.15
0
p..
---- ----
0
I-t
111
----
j:!
....OIl
Q)
~
,...---
. 10
------
,---~---
.05
1
16
Figure 5.
.-
~---
3
5
7
16
16
16
9
16
11
13 15
17
16
16 16
16
Length Group
19
21
23
==
25
16
16
16
16
Observed data for the changes in fiber weight distributions
due to fiber breakage in the process (Cotton L)
61
Initial Values for a and
~
Where there is no analytical way to determine a starting point in
..
the parameter space, an intelligent guess by the experimenter is
normally used.
However, for this model a half and half approach is
available; that is, it is pos sible to find a starting value of a by the
method of moments if a good guess of
~
is available.
It was discussed previously for the derivation of Eq. (46) that
only the first reciprocal moment of Eq. (l8) is readily available in a
simple expres sion involving the parameter m.
holds for the model Eq. (28).
E
A similar situation
1£ we let the first reciprocal moment be
w 1
G
(X), the subscript G denoting the output fiber weight distribution
function G
w
' then using Eq. (28)
X
t
=
r xl
J
dG
w
(x)
X
o
~
r
=
where t
r
r
+ ~
:z
f. e
j=l J
j=l
is maximum fiber group length.
f.
J
(
1 - e
-at.) Jt
J
o
r
.!. d H
x
c
(x/t.)
J
Upon similar manipulation
used for Eq. (46) it can be found that the above expression yields
f.
r
r
1
EW(_)=~~
+a ~
G
X
j=l t j
~
f.t.
j= 1 J
-1
J
Since f.'s are discrete densities for the initial fiber weight distribuJ
tion as expres s ed in Eq. (21),
r
~ (f.l t.) is simply the first reciprocal
j=l J J
62
m.om.ent of the initial fiber weight distribution, E;
(~).
Sim.ilarly,
r
~ f. t~ -1 =
j=l J
E
Hence,
W
(1.)
G X
or
QI
J
W
= E F (~)
=
I
+ QI E; (X 13 - )
E W (1.) _ E W (1.)
G X
F X
(58)
E w (X13 -1)
F
It is interesting to note that Eq. (58) reduces equivalent to Eq. (48) if
initial fiber weight distribution is of uniform. length.
For a degenerate
distribution for F~ (t) such that dF~ (t) = 1 for t = t
r
dfwL(t)
=0
for t
-I
t
r
, Eq. (58) becom.es Qlt 13 - 1
r
= E GW
and
(1.) __1_
X
t
and
r
Eq. (48) follows by taking Qlt 13 = m. .
r
r
Eq. (58) can be used to estim.ate
QI
o
for a given 13
noted previously, 13 can be safely restricted to 13 > O.
13
o
=1
was chosen and
data as
QI
o
= 0.1015
QI
0
= 13 0
.
As
For sim.plicity
's were com.puted by Eq. (58) from. sam.ple
and 0.1342 for the cottons Hand L, respectively.
Convergence in Iterations
The results of the nonlinear iterations are presented in Tables 2
and 3 for cottons Hand L, respectively.
six iteration trials are shown.
For each case the results of
Each trial represents different choice
for the starting point in the param.eter space.
All trials exhibit a nice
convergence to the sam.e final estim.ates in the respective cases.
T rial I is with the starting point obtained in the pr evious section,
QI
63
Table 2.
Convergence in the nonlinear iteration of the parameters
(Cotton H)
Trial 4
Tria13
3.0
.75
.7993
.8191
.8299
.8311
.01
.1970
.1861
.1854
.1850
.1850
0
1
3
5
10
11
.00357143
.00050303
.00049483
.00049475
.00049474
.00049474
0
1
3
5
10
15
.001
.0043
.2034
.1882
.1847
.1850
15.0
8.317
.5198
.7297
.8412
.8309
Trial 6 b
Trial 5
.3
.03
.2026
.1887
.1856
.1850
0
1
3
5
10
15
9.0
7.485
.4678
.7151
.8146
.8308
.00363466
.00351342
.00050925
.00049564
.00049475
.00049474
0
1
3
5
10
16
.05636740
.00189699
.00050986
.00049592
.00049477
.00049474
alnitia1 value determined by Eq. ( 58),
bTrial without restriction of
~>
3.0
.8266
.2354
.1789
.1854
.1850
~
1.0
-5.55
1.924
.9623
.8186
.8310
.15614880
.02945238
.00150681
.00049650
.00049476
.00049474
= 1.
O.
t = Stage of iteration.
Note:
O!
t
,~
t
= Estimates after the tth iteration, Eq. (56).
Q = Sum of squares of deviations with
t
O!
t
and 13 ' Eq. (57).
t
64
Table 3.
Convergence in the nonlinear iteration of the parameters
(Cotton L)
Trial 4
Trial 3
0
1
3
5
10
16
3.0
.75
1.147
1.268
1.368
1.356
.01
.2866
.2727
.2680
.2635
.2642
.00734721
.00125856
.00119926
.00119340
.00119219
.00119217
0
1
3
5
10
16
.001
.0198
.2527
.2592
.2632
.2642
Trial :5
0
1
3
5
10
14
.3
.2519
.2513
.2615
.2644
.2642
9.0
4.383
1.461
1.412
1.350
1.356
.00737937
.00651065
.00120009
.00119376
.00119224
.00119217
Trial 6 b
.03304649
..00416914
.00119953
.00119264
.00119218
.00119217
0
1
3
5
10
15
3.0
.3
.2498
.2599
.2646
.2640
alnitia1 value determined by Eq. (58),
bTrial without restriction of
Note:
15.0
3.75
1.587
1.459
1.377
1.356
1.0
-3.181
1.587
1.445
1.345
1.358
.12655686
.01660386
.00120069
..00119335
.00119219
.00119217
S = 1.
S > o.
= Stage of iteration.
S E'
f
h
th.
.
at' t = shmates a ter t e t lterahon, Eq. (56 ).
Q = Sum of squares of deviations with a and S ,
t
t
t
t
Eq. (57).
65
being obtained by the method of moments through Eq. (58) with
~
=1 .
Other trials are executed merely to confirm trial 1 and further to see
the behavior of convergence from other arbitrary starting points.
The probability of fiber breakage plays the key role in the model
and also we are interested in discriminating any two different sets of
breakage probabilities which, within the realm of reasonable practicality, differ only slightly.
Thus, the convergence criteria for
iteration in this study might best be stability of successive estimates
of the breakage probabilities in each fiber length group.
It was found
that generally the iteration stage where those probabilities remain
stable up to 10- 3 in two consecutive stages correspond to the second
or third consecutive stabilization of Q up to 10-
8
.
Therefore, the
stage of the third cons ecutive stability of Q up to 10- 8 was taken as the
point of the convergence.
In Tables 2 and 3 the last stage underlined
is the above defined stage number of convergence.
For trial 1, 15
and 14 were respective convergence stages for the cottons Hand L.
However, even from other arbitrarily chosen parameter points convergence was achieved with almost the same number of iterative
stages.
It is interesting to note that some trials are even better in
convergence of Q than trial 1 for cotton H.
the 16 th stage iteration.
All trials converge within
However, the method of determining the
initial values by Eq. (58) is certain to insure starting the iteration
near the minimum.
Only a suitable choice of
~
is necessary, but
~
66
between. 5 and 4 would be appropriate for similar types of fiber
breakage, judging from the behavior of the probability of fiber
breakage in this region.
For simplicity
~
=1
is preferred.
For
more severe breakage in which very accelerated fiber breakages for
longer fiber groups are suspected, a larger 13 is the logical choice.
Depending on the initial parameter point, the correction vector
sometimes leads the iterates into a < 0 and
~
< O.
Since a > 0 by
Eq. (27), the program was set to reduce the starting a to 0'/10 for the
relocation of the starting a whenever a correction value for a leads
the point into negative a region.
This effect can be seen for (0 -1)
stage of trial 5 in Table 2 and of trial
indicates
-<Xl
phenomena.
<
~
<
<Xl,
but
~
6 in Table 3.
Eq. (27) also
> 0 is more logical for the fiber breakage
Hence, the similar command for the relocation of
taking ~I 4, whenever ~ became negative, was also used.
such a negative
both cases.
~
~,
The effect for the
~
However,
command can be seen for (0-1)
For both a
such relocations occur when the starting point is too far away
from the region of the convergence point.
The relocation is merely to
move the starting point in the general direction of the correction
vector.
by
would not affect the iteration as shown for trial 6 in
stage of trial 3 in Table 2 and trials 3 and 4 in Table 3.
and
~
67
Discussion of Results
Estimated Output Distributions
A
A
For the estimated parameters ex and 1:1, the estimated output
fiber weight distributions in discrete densities are shown together with
the obs erved ones in Table 4.
Also a direct visual comparison of the
observed and the theoretical values for the two cottons are made in
Figures 6 and 7.
The fit of the theoretical curve to the observed is
seen to be good from a practical standpoint in both cases.
In the
actual fiber process, there is some loss of fibers during the process,
especially short fibers.
However, unintended loss of fibers from
over-all fiber groups is also present and the losses may be balanced
out in terms of the weight proportion.
The over estimation for very
short fiber groups is surely on the safe side and more acceptable.
Interpretation on Probabilities of Fiber Breakage and Multi- segment
Breakage
A
A
Using the estimated parameters ex and 1:1, the estimates for m.,
J
x.., x.,
J
u ., and U are computed, all of which were defined and
nJ
n
discussed previously in the section of the Practical Uses of the
Parameters.
The entire deduced information is compiled in Tables 5
and 6, respectively, for the two cottons.
These results constitute the
unique part of this study which gives a realistic interpretation of
interesting fiber breakage phenomena.
..
e
e
Table 4.
1/16
3/16
..
e
Obs erved and estim.ated output fiber weight distributions
(Cotton H and Cotton L)
5/16
7/16
9/16
Length Group, Inches
11/16
13/16
15/16 17/16
19/16
21/16
23/16
25/16
-Cotton H
Observed
.0125 .0205 .0418
.0467
.0654
.0592
.0801
.1121
.1539
.2074
.1487
.0494
.0022
Estim.ated
.0159
.0432
.0489
.0529
.0618
.0812
.1065
.1558
.2153
.1384
.0445
.0077
a
Difference .0034 .0074 .0014
.0022
-.0125
.0026
.0011
-.0056
.0019
-.0079
-.0103
-.0049
.0055
.0279
Cotton L
Observed
.0148
.0278 .0636
.0712
.0582
.0882
.0990
.1272
.1306
.1755
.1178
.0251
.0013
Estim.ated
.0197
.0396 .0642
.0612
.0696
.0782
.0927
.1203
.1432
.1825
.0981
.0295
.0012
a
Difference .0049
.0118 .0006
-.0100
.0114 -.0100
-.0063
-.0069
.0126
.0070
-.0194
.0044 -.0001
aDifference = (Estim.ated - Observed).
0'
00
69
.
•
.25
•
-----
Observed
Estimated
i----
.20 -
" 0.1850
a=
"
13
>-
= 0.8312
""- -
.-::: .15
----
...-l
• .-4
..c
rd
..c
o
I-t
r----
~
.10 f-
....
---
---
1----
- - ----
.05
--1
16
i----
J
3
5
7
16
16
16
9
16
11
16
13
16
15
16
17
16
19
16
21
16
23
16
25
16
Length Group
Figure 6.
Observed and esti:mated output fiber weight distributions
(Cotton H)
70
••
.25
-----
Observed
Estimated
.20 ~
....--~---
'" 0.2640
0'=
'"S = 1.358
---- ---.10 -
----
----
----
r---
~---
.05
... ---
f----
~---
1
16
Figure 7.
3
5
7
16
16
16
9
16
11
13
15 17
16
16
16 16
Length Group
19
16
21
16
23
16
25
16
Obs erved and estimated output fiber weight distributions
(Cotton L)
..
e
e
Table 5.
Lengt~
Group (Inches)
13716 15/16 17716
57167716
9716
11716
m·
J
.018
.046
.070
.093
.115
.135
.156
.175
A'J
.295
.245
.225
.213
.204
.197
.192
(1 - u .)
oJ
.018
.045
.068
.089
.108
.127
U o = .831
Uoj
.982
.955
.932
.911
.892
U 1 = .153
u 1j
.018
.044
.066
.085
U 2 =·015
U2j
.001
.002
U 3 =·001
U3j
U4 =
U4j
-
-
U5 =
U5j
A = .188
(1 - U o )
= .169
Code:
mj
Aj
A
(l - Uoj)
Uoj
Unj
(1 - U o '
U0
Un
=
=
=
=
=
=
=
=
=
=
--
Probabilities of fiber breakage and average number of break points
in each length group and over-all (Cotton H)
1/16-3716
Q'= .1850 13= .8312
-.
-
19716
21716
23716
25716
.195
.213
.232
.250
.268
.187
.183
.180
.177
.174
.172
.144
.161
.177
.192
.207
.221
.235
.. 873
.856
.839
.823
.808
.793
.779
.765
.102
..l18
.133
.147
.160
.172
.184
.195
.205
.004
.006
.008
.010
.013
.016
.018
.021
.024
.027
-
-
-
.001
.001
.001
.001
.002
.002
.003
-
-
-
.
Average number of break points per fiber of tj length group.
Equivalent average number of break points per unit length of a fiber of -tj length group.
Equivalent average number of break points per unit length over-all.
Probability of fiber breakage given tj length group.
Probability of no breakage given tj length group.
Probability of n-point fiber breakage (n = 1, 2, 3, ... ) given t j length group.
Probability of fiber breakage over-all.
Probability of no breakage over-all.
Probability of n-point fiber breakage over-all (n = 1,2, 3, ... ).
Negligibly small.
-.J
I-'
e
..
e
Table 6.
ex = .2640 l3 = 1.358
Probabilities of fiber breakage and average number of
in each length group and over-all (Cotton L)
Length Group (Inches)
11/1613/1615/1617/16
1/16
3/16
5/16
7/16
9/16
m.
J
.006
.027
.054
.086
.121
.159
.199
.242
~.
.098
.145
.174
.197
.215
.231
.245
(1 - u .)
oJ
.006
.027
.053
.082
.114
.147
U o =·775
Uoj
.994
.973
.947
.918
.886
U 1 = .192
u
.006
.027
.052
.079
U 2 = .029
U2j
U3j
-
.001
U 3 = .003
=
U4j
-
U5 =
U5j
~
= .256
J
(1 - U o )
= .225
U4
Code:
1j
mj
-
bre~k
e
'.
points
19/16
21/16
23/16
25/16
.287
.33~
.382
.432
.484
.258
.270
.281
.291
.301
.310
.181
.215
.249
.284
.317
.351
.384
.853
.819
.785
.751
.716
.683
.649
.616
.107
.136
.163
.190
.215
.239
.261
.281
.298
.003
.006
.011
.016
.023
.031
.040
.050
.061
.072
-
-
-
.001
.001
.004
.003
.004
.006
.009
.012
-
-
-
-
-
-
-
-
.001
.001
.001
= Average number of
= Equivalent average
break points per fiber of t j length group.
~.
number of break points per unit length of a fiber of t j length group.
~J = Equivalent average number of break points per unit length over-all.
(l - u .) = Probability of fiber breakage given t· length group.
u~~ = Probability of no breakage given tj l~ngth group.
u ~ = Probability of n-point fiber breakage (n = 1, 2, 3, ... ) given tj length group.
(1 - U~) = Probability of fiber breakage over-all.
U o = Probability of no breakage over-all.
Un = Probability of n-point fiber breakage over-all (n = 1,2,3, ... ).
= N egligib1y small.
-.J
N
73
A comparative diagram for the relation between fiber length and
probabilities of fiber breakage is pres ented in Figure 8.
.
It can be
seen that the probability of fiber break is nonlinear with respect to
fiber length.
Though the demonstration examples used here did not
exhibit severe nonlinearity, it is most likely that some fibers or
studies for other types of breakage operations may present severe
nonlinearity in the relation of fiber breakage probability and length.
The assumption of linear breakage probability with respect to the fiber
length that all other workers have so far used seems too restrictive.
The weakness of cotton L is now quantitatively assessed by the
higher probability of breakage and m. 's (or
J
group.
~
.' s) for each length
J
The equivalent average number of break points per unit length
base within each length group
two cottons.
~.
shows reversal trends between the
J
It is theoretically due to the fact that 13 > 1 for cotton L
and 13 < 1 for cotton H.
For a quick general comparison, we note that
the equivalent average number of break points per unit length of fiber,
~,
is . 188 and. 256 for cottons Hand L, respectively, indicating
more severe breakage for cotton L. 1£ 13
= 1,
clearly
~.
J
=~
for all j.
The probabilities of n-point breakage (n = 1, 2, 3, ... ) or (n
+ 1)
segment breakage of fiber for each length group t. are shown in
J
Figures 9 and 10 for both cottons.
For both cases, one-point break
(two segments) is predominant (u
's) but sizeable two-point break
(three segments) still present (u
1j
2j
's), especially for cotton L.
74
..
.5
.4
...,>.....
~
,Ll.
3
ro
,Ll
o
d:
.2
.1
1
16
Figure 8.
357
16 16 16
9
16
11
13 15
17
16
16 16 16
Length Group
19
16
21
23
25
16
16
16
Comparison of probabilities of fiber breakage
with respect to fiber length between cottons Hand L
75
·
~
1.0
·
•
.9
.8
.7
u.
oJ
.6
u
...,>-
~
.,-4
,.Q
1j
U2j
.,-4
.5
u
rd
,.Q
3j
0
I-t
0..
.3
--------------
.2
.1
--
135
16
16
16
---- - - - - - -
7
9
11
13
15
17
19
21
23
25
16
16
16
16
16
16
16
16
16
16
Length Group
Figure 9.
Probabilities of no fiber breakage, u oj_' and multi-point
breakages, u .: n = 1, 2, 3 (Cotton H)
nJ
76
1.0
.9
.8
.7
.6
Uoj
>-
+>
.~
...-t
.~
..0
u
.5
1j
U2j
(Ij
..0
0
u
11l'"' .4
3j
.3
---- ----
.2
.1
1
3
5
16
16
16
7
16
9
16
---- -- --. --- --------- -11
16
13
16
15
16
17
16
19
21
16
16
23
16
25
16
Length Group
Figure 10.
Probabilities of no fiber breakage, Uoj' and multi-point
breakages, u nj : n = 1, 2, 3 (Cotton L)
77
Three-point break (four segments) is almost nil for cotton H and about
0.01 for longer fiber groups of cotton L.
It can be fairly stated that
the breakages producing more than four segments were almost
negligible.
..
Also, 'a quick assessment can be readily made by U
the probabilities for n-point fiber breakage over-all.
n
IS,
For both cases,
the probabilities of three or more break points can be said to be
negligible in the over-all sense irrespective of fiber length groups, as
indicated by U 'so
n
However, in general, this will depend on the
severity of fiber breakage in any system, and truncation of any higher
order-point breakage (n > 1) is not desirable unless the order is very
high.
The over-all probability of fiber breakage (l - U ) for cotton H
o
is found to be O. 17 while the same for cotton L is O. 23, indicating an
over-all higher probability of fiber breakage for cotton L.
Remarks on the Model
When et is estimated by method of moments for given
S=
1, the
resulting et shows a very interesting feature in connection with the
work done by Tallant et al. (9).
w
since EG(l)=l.
B ut b Y us e
0
For
S = 1,
Eq. (58) simply becomes
EW ( .!.)
,
f t h e equa I lty
F X
__
E
F
1
(X)
used in
the proof of Eq. (2) the above becomes
r-
et =
1
1
(59)
78
where E
(X) and E
G
F
(X) are the first moments of the probability
distributions of output fibers and input fibers, respectively.
course, E
G
(X)
>
0 and E
F
(X)
>
Of
0 since the random variable "fiber
length " was defined as positive random variable in the definition for
It is interesting to note that Eq. (59) is the same expression
Eq. (2).
as Tallant's parameter ST (S in the article) which is the constant of
proportionality in the breakage function 'IT (x) when it is as sumed that
the relation between probability of fiber breakage and length is linear,
i. e.,
'IT
(x) = ST x.
Since E
G
(X) :s;; E
F
(X),
in Eq. (59) is clearly
Ct
larger than or equal to zero, which is consistent with the theoretical
domain for
Ct
(Eq. 27) except the trivial case of
Ct
= 0 of no fiber
breakages.
Tallant et al. define
that 0 :s;; 'IT (x) :s;; 1.
o :s;; ST
1
:s;; ;
'IT
(x) as probability of fiber breakage such
This implies, when 'IT (x) = ST x is assumed,
In turn, it means if maximum fiber length is x
ST must be under restriction of 0 :s;; ST :s;;
1
x max
max
> 0
However, if ST
is
"
d as E 1 (X) - E F(X)
1
"a1
to b e eshmate
, 13
may attam
va ue greater
T
G
1
1,"
than
w h enever E 1(X) ~ E 1(X) +
SInce t h
e"
rIg h t SI"d e
G
F
x max
x max
of the inequality is a fixed number for a given fiber population and
E
G
(X) may theoretically become as small as possible.
It seems that
the domain for the estimate is inconsistent with the domain of the
definition of 13 T in the model.
79
The probability of fiber breakage for a given fiber length t. ,
~
J
-at·
-at·
(1 - u .) or (1 - e
J), becomes (1 - e
J) for ~ = 1. It is then
oJ
-at·
simple to see that (1 - e
J) can be approximated by at for a very
j
t
<Xl
n
small value of a since (1 - e -a j) = at. _ ~ (-atj)
It indicates
J n=2
n!
that if a is very small the probability of fiber break is approximately
linear with respect to fiber length for
model.
~
~
= 1 as a special phas e of the
It should be recalled that the estimate of a by Eq. (59) and
= 1 was used in this model as the starting point of the least squares
nonlinear iterative solution of the respective least squares estimates.
Any point of convergence from the iterations whose ~
J1
implies a
better fit of the estimated curve to the observed at the estimated point
than the case of
~
= 1 in the sense of least squares.
The derived models Eqs. (18) or (28) are applicable to any
system in which a random size reduction of an essentially one dimensional substance is taking place.
For the use of Eq. (18) the input
distribution must possess a degenerated distribution or the form in
Eq. (3).
The use of Eg. (28) requires presentation of the arbitrary
initial weight distribution in the form of Eq. (21) which may be a more
practically feasible way of expressing the initial distribution by
experiment when its exact theoretical continuous distribution is not
available.
For such a cas e, the experimental determination of output
weight distribution may also take the form of discrete densities.
Then,
the estimation can be achieved through the more realistic model in the
80
matric form of Eq. (42) and various quantitative characterizations
pertaining to the length reduction phenomena can be deduced as
demonstrated in this chapter for textile fiber breakage.
Unlike the
determination of the textile fiber length distribution in terms of the
fiber weight distribution, if the determination in practice is available
in terms of probability distributions it may be simply converted to a
weight distribution by Eq. (2) under the assumption of homogeneity of
the one dimensional substances, and the model developed here can be
readily applied.
81
SUMMAR Y AND CONCLUSIONS
Fiber processing through a mechanical process causes inevitable fiber breakage in the process.
The mode of fiber breakage in
the breakage system represents the inherent fiber property with
respect to fiber weakness, or the harshness of the mechanical process
with respect to fiber damaging characteristics.
An effective com-
parative study can be made in terms of various modes of fiber
breakage between different types of fibers in a given fiber process or
between different breakage systems for a given type of fiber.
Models
for analytical characterization of any fiber breakage process are
essential for such a study.
For a realistic interpretation of fiber breakage phenomena, two
fundamental questions were raised: Given a fiber group of any particular length, (l) What is the probability that a fiber of the group breaks
at all?
(2) What is the probability that a fiber breaks into s segments
~
All the existing models by the previous workers (2, 7, 8, 9)
(s
2)?
are not appropriate to clarify such basic questions due to the different
approaches to the problem under the as sumptions of linear probability
of fiber breakage with respect to fiber length and two segment breakage
per fiber if it breaks.
To elucidate such questions, a new fiber
breakage model is desirable in which multi-segment fiber breakage is
taken into account.
82
The fiber weight distribution is defined (Eg. 1) to ascertain the
legitimacy of handling fiber weight distribution as a conventional
probability distribution function.
The fiber weight distributions are
us ed throughout this study to accommodate the expe rim ental data which
are obtainable in weight proportions by the existing array method (1)
in practice.
A duality of the weight distribution function and the
probability distribution function of fiber length is shown under the
assumption of homogeneity of fibers.
A fiber breakage model for uniformly random fiber breakage
when the initial fibers are uniform in length is derived first (Eg. 18)
under the assumptions:
(l) each fiber behaves independently in the
breakage system, (2) the number of breakage points N occurring on an
initial fiber due to the breakage process follows a Poisson law with
parameter m
(Eg. 4), (3) when many break points occur on an initial
fiber a prior probability is attached to each breaking point on the initial
fiber of length
t
such that each break point chooses a position along the
fiber with uniform probability 1/t, irrespective of the breakage
s eguence (Egs. 6 and 7), and (4) homogeneity of fiber diameter and
density, irrespective to length.
The model Eg. (18) is a mixed weight distribution function of
fiber length for the output fiber aggregates.
The newly generated
fiber weight distribution function is expressed in terms of the input
fiber weight distribution and the parameter of the model m, the
83
average number of break points occurring on an initial fiber, which
represents the severity of fiber breakage.
The parameter m is esti-
mated by method of moments (Eq. 48).
Most textile fibers, however, exhibit a nondegenerate distribution.
The fiber weight distribution is determined by the array method
and expressed in a discrete distribution due to the nature of the testing
technique involved.
Therefore, the first model (Eq. 18) for the
uniform initial fibers is extended to the cas e of an arbitrary initial
distribution of the form of Eq. (21). Then, the second model Eq. (28)
is derived under the assumption that the average number m. of break
J
points occurring on an initial fiber of length .t. is a function of the
J
fiber length .t. such that m.
J
J
= Q'.t~
J
for j
= I,
2, 3, ... , r, allowing
flexibility of the probability of fiber breakage with respect to length.
The model Eq. (28) is also a mixed distribution function in terms of
the discrete initial fiber weight distribution and the parameters
~.
Q'
and
Then, the model is converted into a compact matrix expression
(Eq. 42) which is a more realistic model to accommodate the experimental data for both output and input fiber weight distributions available in practice.
An axiomatic interpretation of fiber breakage
process naturally results from the matrix model.
The first model
(Eq. 18) is essentially a special case of Eq. (42) when the output fiber
weight distribution is partitioned and the input fiber weight distribution
is regarded a degenerate one (Eg. 45).
84
Since the model is nonlinear in the parameters ot and
~,
a non-
linear least squares iterative solution by the Gaus s -N ewton method is
proposed for the estimation of the parameters.
Upon the estimation of
ot and ~, various important criteria associated with average number of
break points and probability of fiber break can be deduced, which lend
realistic interpretations of fiber breakage phenomena.
Application of the model Eq. (42) is demonstrated on two different lots of cotton (H and L).
Cotton H received a high moisture
control and cotton L a low moisture in ginning, then both were processed through a textile processing system.
are estimated (Tables 2 and 3).
The parameters ot and ~
Various related criteria defined in
the section on "Practical Uses of the Parameters' l are found, and the
two fundamental questions raised earlier are completely elucidated as
compiled in Tables 5 and 6 for the two different modes of fiber
breakage.
Applicability of the model is well exhibited through good fit
of the estimated output distributions to the observed cases (Figures 6
and 7).
An analytical characterization and a realistic interpretation
of fiber breakage phenomena through the use of the model is thus
demonst rat ed.
Based on the applied examples, it is found that the probability of
fiber breakage is nonlinear with respect to fiber length (Figure 8).
The nonlinearity may be enhanced with more severe fiber breakage or
some different modes of breakages.
As to the probability of
85
multi-segment breakage, the probabilities of obtaining four or more
segments are found to be negligibly small in the over-all sense (U 's
n
in Tables 5 and 6).
Two segment breakage is, of course, predominant,
but sizable three segment fiber breakage is found to be still present,
especially for longer fiber groups.
Needless to say, these will be
dependent on different breakage systems or different fibers.
What-
ever the mode of fiber breakage for any particular system is expected,
it will be fully interpreted through the us e of the model regarding the
relation of the probability of fiber breakage and length and the
probability of multi-point breakage.
Therefore, it can be concluded
that the assumptions of linearity for the relation of fiber breakage
probability and length and of two segment breakage per breaking fiber,
mostly used by the other workers, are too restrictive in light of the
flexibilities of the established model.
Because of such shortcomings of existing models in lending
more realistic interpretations to fiber breakage phenomena, this
dissertation was particularly aimed at deriving a probability model
which would elucidate the subject, allowing a more flexible relation
between the probability of fiber breakage and length and multi-segment
fiber breakage.
Such a model has been derived and useful applica-
bility has been shown, giving clarification to the set-out fundamental
questions relating to fiber breakage phenomena.
86
In the nonlinear least squares estimation of the parameters for
the model Eq. (42), the observed densities
~
for the output fiber
weight distribution function were assumed to have variance matrix 10
2
This implies equal weighting of the z. in the least squares estimation.
1
.
However, it may be preferable to use a weighting matrix other than
the identity, 1.
The form of the proper weighting matrix was not
found theoretically.
An alternative is to us e a sample variance-
covariance matrix obtained by repeated runs of the fiber breakage
process for a given initial input fiber distribution.
Although the standard Gauss -N ewton method was us ed for the
examples in this dissertation, it may be desirable to program the
iterative procedure based on the modified Gauss-Newton approach (4,0).
Such a program would be able to take care of more peculiarities in the
surface of sum of squares of deviations and would insure monotonic
convergence of the iterative process.
However, with the data used
here, the ordinary Gaus s.N ewton method was adequate.
,.
87
LIST OF REFERENCES
1.
A. S. T. M. Committee D-13. 1966. A. S. T. M. Standards on
textile materials. American Society for Testing Materials.
Philadelphia, Pa.
2.
Byatt, W. J., and Elting, J. P. 1958. Changes in the weight
distribution of fiber lengths of cotton as a result of random fiber
breakage. Tex. Res. J. 28:417-421.
3.
Feller, W. 1961. An Introduction to Probability Theory and Its
Applications. Vol. I, 2nd Ed. John Wiley & Sons, Inc., New
York.
4.
Hartley, H. O. 1961. The modified Gauss-Newton method for the
fitting of nonlinear regression functions by least squares.
Technometrics 3(2):269-280.
5.
Kendall, M. G. and Stuart, A. S. 1952. The Advanced Theory of
Statistics. Vol. 1. Distribution Theory. Charles Griffin & Co. ,
Ltd., London.
6.
Marquardt, D. W. 1963. An algorithm for least squares estimation of nonlinear parameters. J. Soc. Indust. Appl. Math.
2(2):431-441.
7.
Meyer, R., Almin, K. E., and Steenberg, B. 1966. Length
reduction of fibers subject to breakage. Brit. J. Appl. Phys.
17:409-416.
8.
Shapiro, H. N., Sparer, G., Gaffney, H. E., Armitage, R. H.
and Tallant, J. D. 1964. Mathematical aspects of cotton fiber
length distribution under various breakage models. Tex. Res. J.
34:303-307.
9.
Tallant, J. D., Pittman, R. A., and Schultz, E. F., Jr. 1966.
The changes in fiber-number length distribution under various
breakage models. Tex. Res. J. 34:729-737.