Tip:
Highlight text to annotate it
X
This lecture, we want continue about our earlier theme about the Yarn unevenness. In the end
of last lecture, we said something about the Huberty’s index of irregularity and we came
to the conclusion that, something is wrong in the Martindale’s concept of Sliver or
Yarn irregularity.
The question is why - the question is WHY it is because exists something. So, some or
institutions make some empirical corrections for example, so called Uster precisely company
in the Swiss land in a town name Uster but, it is known as Uster means and Uster company
they use for coefficient variation of local Sliver fineness. When I speak about the Sliver
is a little theoretical practically we use it at most for Yarn on. So, something like
some Sliver from fibers. They use such formula because logarithm of this is straight lines
and you know the graph from Uster which have the trend to be straight lines.
Interesting work is that the 1964 from Bornet which formulate the idea why the real Yarn
irregularity is higher than the calculated value based on Martindale’s concept, is
meaning that, as it is because fibers are not individual than in some bundles for similar
aggregates.
Nevertheless, it is verbal; nevertheless then he used quite empirical equation which is
this here. We accept generally the idea of Bornet that the influence the center of influence
is in some fiber aggregates and in opposite to him we derived the model exactly. So, let
us think at first step about the Martindale’s assumption. Martindale used the assumption
of fibers are straight and parallel. It is idealized we can say that yes is idealized
may be he assumed fibers have same length; you can say also idealized here – why? May
not, may be fibers are positioned randomly along the Sliver; it is evident but, he assumed
also and individually and this is not right. We have enough experimental results too in
general work that the fibers are not perfectly one beside the other, but they are moving
together in some fiber aggregates.
Therefore, we created the following model based on the idea which is shown in our picture.
The basic structural unit is a fiber this here from fibers is formed a Bundle a Bundle it
is an aggregation of several fibers which are together glued or knots. So, that the
separation of fibers in the individuals ones is not possible by technological process used.
So, they of they function as a for example, couple of such fibers function as 1 thick
fiber this is the bundle. From bundles are created clusters. Cluster is formed from aggregation
of several bundles. The clusters are only loosely bonded together and these could be
sometimes separated by applying suitable technologies. Bundle is too infinity for us made the fibers
together. Cluster is possible sometimes based on the
local relations in technological process divided to smaller clusters or not. Based on the situation
what we do in spinning mill for example, this is a Cluster and then the Sliver or Yarn.
Finally, it is formed from aggregation of several bundles. Thus, from this unit you
obtain the final structure Sliver the lower units create the higher unit randomly and
the creation are mutually independent. They are our assumptions difficult in short time
to have because it is not practically possible to each quantity as subscript right fiber
Bundle cross the Sliver then we use numbers which is related to fiber is 4 - number 4
which is related to hard bonded bundles is number 3 which related to clusters is have
number 2 and which is related to the Yarn or fiber have the number 1.
So, 1 is Yarn 2 is Cluster 3 is Bundle and 4 is fiber yes we imagine that we create from
fibers number 4 bundles number 3 then from bundles we create clusters number 2 and then
from Cluster we create Sliver or Yarn cross section number 1. In reality it is in opposite
we have some material and the this material we divide to clusters bundles individual fiber.
So, on , but let us imagine this structure this is our idea.
Now, the quantities number of the smaller parts in higher unit. So, that a number of
clusters in Sliver cross section equal q 2 1 2 is Cluster 1 is Yarn or sliver. How many
clusters in the moment is in Yarn or Sliver cross section, number of bundles in the Sliver
s q 3 1, number of bundles in yarn? So, and number of fibers in Yarn cross sections or
Sliver cross section it is q for 1 fiber Yarn. Our all the symbol was n number of fibers
in Yarn or Sliver cross section number of bundles what is the number of clusters in
Cluster evidently 1 how many Cluster in cross is given Cluster 1 number of Cluster number
of bundles in Cluster is q 3 2; bundles in Cluster Bundle is 3 Cluster 2 it is been necessary
to use this system of subscripts and so on. Where is the numbers of smaller parts? In
higher unit evidently, it is valid equations for mean numbers for example, here near mean
number of fibers in Cluster is mean number of fibers in Bundle times mean number of bundles
in Cluster based on such logic all this equation must be valid.
The same is maximum numbers; similar equations maximum number of fibers in Cluster. It is
maximum number of fibers in Bundle times maximum number of bundles in Cluster, no it is logical;
on this logic all this equation are constructive. So, evident that it is valid. Now, to fineness
finenesses Sliver fineness we call in the moment t 1 our other our other symbol was
capital t. So, t 1 is same than earlier capital t co efficient variation of Sliver local fineness
is v 1 why 1 because structure number 1 Sliver early it was v capital t.
Cluster fineness of the Cluster is t 2 coefficient variation of this is v 2 coefficient variation
of fineness of clusters Bundle fineness moment the moment fineness of a Bundle some Bundle
is t 2 coefficient variation of finesses of from or bundles is v 3 and finally, fiber
fineness is t 4 fiber have subscript 4 it was identical as earlier small t and coefficient
variation of fiber fineness is v 4 identical of earlier symbol v t.
What we can write now? Fineness of the Sliver or Yarn is 0; if number of bundles sorry,
if number of clusters in the Yarn is equal to 0 no wonder and fineness of Sliver is some
of fineness of all bundles in clusters in all cross section from first to last. So,
that it is this way. But what is t 2 this fineness of Cluster is equal 0 if no fiber
is there the number of Bundle is 0 and the similarly sum of all fineness of bundles over
all bundles which are in the moment in cluster. The same what is the fineness of Bundle 1
one from fiber from fibers it is 0 if no fiber and it is sum of fineness of fibers or all
fibers in Bundle let us see this structures this and this all these 3 quantities t 1 t
2 t 3 are random quantities likewise we make in last lecture some small excursion to theory
of probability when you find some random quantity type y and we postulate how its coefficient
variation of such random quantity. All the 3 all the 3 are quantities type y.
Let us imagine now that each quantity t 1 t 2 t 3 have a it is random quantity and have
the binomial distribution now identical, but all 3 are type binomial distribution therefore,
in short here 3 times binomial distributions. We define such equation using this type of
expression what is this it is mean number bar by maximum number and this is divided
by maximum number now for v 1 square based on this formula.
For v 1 square we obtain this equation mean value maximum value for v 2 square we obtain
such equation similarly and for our v 3 square we obtain this equation the way how to obtain
it I think clear yes. So, how it is v 1 square v 1 square it is this here, but on the place
of v 2 square I can use this expression. So, it is what it is by q bar 2 1 and here
is plus this brackets you here plus and on the place of e 2 square I use this expressions.
So, it is this here it is bar q bar 3 2 and. So, seemly, but here is v 3 square and otherwise
of v 3 square I can use this expression it is here.
Coefficient variation of Sliver v 1 square of it is given by this expression after rearranging
th obtain is here after rearranging and using equation postulate earlier we obtain we go
for this q bar 4 3 times q bar 3 2 times q bar 2 1 is q bar 4 1 it is written earlier
where it is for example, here it is in some in some expression it is here. So, so that
we obtain this expression for Sliver local coefficient of variation and now let us accept
2 ideas the first this when you produce ring spun Yarn for example, then the drafting system
on the ring frame convert is a Sliver or finest Sliver based on count of final yarn.
Relatively higher number of fibers will be passing through the drafting system together
as a Cluster respectively when we produce a coarsen from a little coarser Sliver because
more fibers in the drafting part of ring spun ring frame when in another process rotor process
rotor spinning relatively higher numbers of fibers will be transported per unit tooth
of opening roller when we produce a coarsen or we use coarser sliver.
So, that when we produce Yarn having more fibers in the section may be the number of
clusters which are created earlier is higher and vice versa. So, therefore, we can postulate
or say some assumption of course, it is some simplification that number of fibers mean
number of fibers in 1 Cluster is proportional to the number of fibers in Yarn process section
and improve your coarsen the number of the number of fibers is cross section is high
and therefore, all of the clusters which are created are this Yarn will be bigger. So,
that you can write q 4 2 bar is p times q 4 1 bar in our relation where the parameter
p is a parameter of proportionality and it is if they have some sense it is a measure
of fiber individualization. It means if p is small then number of fibers in our Yarn
is created from clusters having small number of fibers if p is high then vice versa our
clusters we have lot of fibers think about this phenomenon we will use this equation
in our in our equation for v 1 square.
In the same time now we come back to our traditional symbols as possible. So, that on the place
v 1 v is back v t on the place v 4 v small t on the place q bar 4 1 number of fibers
in mean number of fibers in Yarn of fibers cross section small n bar it is t by t and
so on. Using this substitution plus this assumption
we obtain such equation and symbolically we can write that this part have the name a capital
and sorry, this our part here rename a capital a and this part p times the p times this here.
It is it is quote and the symbol B when a is this here and B is this here we obtain
coefficient variation and this from as and if n bar is T by T T bar by t bar worked in
the final equation in such form. It is a difference between, in relation to
Martindale’s equations because we are here the part B which we write it is which is multiplicative
when number of fibers generally say we it is also theoretically possible in this moment
that 1 plus v v square t i can be smaller than it can be higher than our part a theoretically
can be. So, that theoretically we can also to obtain the unevenness or you know irregularity
similar than limit value based on binomial sliver. It is only theoretical example in
the practice nobody watch it, but theoretically it is possible more easier with this when
we use for us easier distribution when on the place of binomial we give the Poisson
distribution we derived it by Martindale’s concept.
What is in the Poisson distributions also 3 times of coarse distribution of fineness
of bundles clusters and yarn, but e type for each this Poisson distribution.
Then in Poisson distribution the maximum number of possible cases is limited to infinity isn’t
it. So, that by Poisson in Poisson version q 4 3 max as well as q 3 2 max as well as
q 2 1 max is limited to infinity then our part a using this limit have this form you
know have this form and our part B after this limiting is equal to p a parameter which say
us something about the I mentioned it where measure of fiber individualization is not
it. So, on the formally the final equation stay
be same have the same shape by the by what is more what is more in relation to Martindale
have here 1 plus v square t now we have here plus q bar 4 3 number of fibers in Bundle
it means in a in a structure in which the fiber stay fixed together glued knotted. So,
that is it. So, that it is not possible to our technologies to process it divide.
Yes how is the practical result it is another experiment because earlier it is better for
us than today’s the last variation from Uster statistics why in the modern spinning
process the draw frames are used this some mechanism of regulations they can very intensively
regulate the irregularity or the unevenness of Sliver therefore, also the Yarn have smaller
values We studied this Yarn unevenness as a pure
random process with some standard regulation of reason that for I use the result as an
example result from statistics 1907 1997. The basis may be the graphical explanation
on the on the abscissa here is mean Yarn count it is for Yarn carded cotton Yarn count in
tex is here on the ordinate it is coefficient of variation according coefficient of variation
of local Yarn fineness Yarn count. The Uster reserves are the thin lines here
you know this Uster graphs, but now it is not linear because I have linear the abscissa
scale this scale can be abscissa here is linear. In its standard Uster graph this abscissa
this graphic symbol logarithmic therefore, in some logarithmic graph or logarithmic graph
if you obtain linear relation graphically this result logarithm therefore, the Uster
result are the thin lines here and because you know Uster graphical interpretation, they
use little thick red lines I showed also dotted lines how is the thickness of this red line
in the pictures from yes. Using our equations and having two 2 parameters
a and B suitable parameters we obtain the thick curves here set of thick curves. Here
you can see that in practical used region our curves are linked practically inside of
thickness of Uster red line. So, our model can very good to approximate
or no approximate to interpret better the experimental results published by Uster yes
therefore, because this influence because this bundles and so on. The real unevenness
is higher than the ideal unevenness according Martindale.
Here some for 50-50 percentage Uster curve 50 percentage some graph which show how it
is in different types of yarns here is q 4 3 as a function of as a which is a minus 1
minus v square t and q bar 4 3 it is a how is the number of fibers mean value of numbers
fibers, in the fiber bundle do not worry, but it is starting from 0.
We speak about the Poisson distribution. So, Poisson is, in Poisson distribution also, number equals
0 is possible as far as the binomial distribution only in re Yarn when we have some part of
fiber in which number is 0 then it is break there is some knot here.
Therefore, it is going from 0 in a special in a cited where it is in a cited publication
in here it is the publication in which it is written in more details. It is shown how
it is non break Yarn, but what is interesting from this theoretically it is possible to
apply Poisson model; Poisson’s distribution on these are on the left is not possible.
It must use binomial distribution. So, you can see that this side the Poisson distribution
can be used for each Yarn is model we can formally use this, but we do not obtain relation
to the model right results you can also see that the parameter B and a are different type
of yarns and we will on the times see watch this diagram you can see that it is for example,
interesting differences between ring Yarn and rotor yarn.
In rotor Yarn the Cluster the number of fiber in Cluster are relatively small as shown this
experimental experiences the values here are relatively small smaller than in ring Yarn
another thing is the bundles this smaller number of fibers in the Bundle is higher.
Please do not imagine number of fibers in bundles is not too high value it can be 1.7
for Rotarian example or 1.3 1.2 number of fibers it means sometimes 1 fiber sometimes
2 fibers together sometimes 3 fibers together and 4 fibers together is a chance as in lottery.
Imagine so, but it can play very significant role for unevenness here in Rotarian the Cluster
the Cluster is from is from small quantity of bundles, but the bundles are for example,1.8
fibers mean value of fibers in a position to ring frame on which we obtain 1.1 or 2
something. It is in a rink Yarn; it is vice versa in rink Yarn because drafting process
make the fine Sliver which is twisted and because drafting process the Cluster have
the tendency be have more bundles bind in Bundle is only a little more than 1 fiber
for its 1.1 1.2based on our experimental. So, that you can see that we can also divide
we can also divide the 2 influences 1 influence is the fibers are based on another technological
principle permanently together glued knotted than this and the second the no clusters which
can be when our process is good can be divide and it bring us when we analyze our data from
industry based on this it is theoretically it is very easy because these equations are
the these equations is a what is it is a linear equation or you can use the linearised statistical
regression and from set of data, I do not know from your spinning mill to obtain the
parameters a and B for your Yarn it is evident.
Now, when a is too you now you have some Yarn which is not good the irregularity is too
high based on this analyses or you can compare you can compare your data from where’s I
do not know that the data which are here for example, for from Uster and when a is too
high when a the quantity a is in your yarns too high what it means a is in the case of
Poisson distribution 1 plus v square t this is a Martindale, but plus q bar 4 3 number
of fibers in our bundles when you have too much fibers in bundles knot, do not be distracted
by bundles. What you can do? You can try only. You can try and you can find on the market
another type of material which have knot so, intensive tendency to create such bundles
because, this is the influence of the material. When you will leave, but value of your Yarn
unevenness and based on this analysis you obtain to them that B is false. B is by Poisson’s
model equivalent to p. Then, no forces sorry, that p is too high; it means you have too
big you know, these bundles than the clusters our clusters you have 2 bigger clusters. Then,
what can you do? Because, to obtain better unevenness, big fight in your spinning mill
all people must watch why we must rearrange machines - the carding machines; the surface
of the of the all cylinders must be sharp the all this rearranging of machine machines
must be optimum and so on. You can start to make some big fight in your
spinning mill; in the first case not in the first case, you can only try and/or find on
the market another type of your I do not know cotton fibers which have knot; so, high tendency
have 2 fibers together. In the second case, you must organize some
symbolic big fight in your spinning mill and there is a practical end for our theoretical
model. Principally, we use the same idea - the Martindale nevertheless.
We apply it to more general idea; you know from individual fibers immediately yarn. From
individual fibers we create bundles; from bundles clusters and from them, clusters the
Yarn; these are the difference - logical difference to Martindale. You can say how we obtain this
system? It was worked and my first version of model was only 3 steps from fibers to clusters
and from clusters to yarn. But, when I compare it with our own experimental
research as well the Uster diagrams and specially for this fine part of the Yarn, it was not,
the curve was not enough precise; was not enough. Well, I think what is the logical
sense of this? No good results in one part of the case and we came to them that logically
it must exist something like this bundles, some fibers, some fiber bundles which are
based on our principle use technological principle used permanently together; not possible to
divide and then finish our lecture.