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In this lecture, we will continue our derivation
of some model, which give up the possibility
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to better calculate packing density and diameter
of the yarn.
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In the last lecture, we finished this equation.
The pressure p which compressed fibrous material
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in the yarn is a function of packing… C
is some constant – we assume it; mu is packing
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density of the yarn; alpha is aerial type
of twist factor, twist coefficient; tau is
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relative finenesses of the yarn, so that the
ratio yarn finenesses by fiber finenesses.
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This equation was in the last lecture, derived
from geometrical relations inside of… By
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the pressure as a function of packing density,
we know from one of earlier lecture about
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the compression of fibrous material. We derived
the pressure – some k p times this ratio;
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where, mu is packing density; mu m is some
maximum value of packing density; not too
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far from 1; a is parameter usually equal to
1.
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And, we also mentioned that by solving more
difficult problem of two-dimensional homogenous
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stress, which can be assumed like this here
– comparative fiber bundle from all sides.
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We can obtain similar equation only some parameter
b more is here. So, we know the pressure as
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a function – some parameter times this ratio
based on the packing density mu. You have
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two equations of pressure: one equation is
going out from yarn geometry; second is going
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out from physical model, some generalization
of earlier one week model and so on. Evidently,
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these two right-hand sides must be equal.
Therefore, based on equivalency of right-hand
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sides, we can write this equation.
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This is the same as in the last slide – this
equation. Now, we make only on mechanically
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rearranging of such equation. For example,
on the place of alpha s, we give Z times square
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root of S; on the place of tau, we give capital
T by t, so that we obtain this equation; on
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the place of square root of S, we have square
root of T by rho; on the place of quantity
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t, fiber finenesses, we have S – fiber cross
section times rho. The other trivial equations,
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which we know usually from our lecture 1.
s – fiber cross section here is pi d square
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by 4; where, d is fiber diameter; also, trivial
n. So, we obtain right-hand side of this equation
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in such form. Therefore, we can write right-hand
side is same is equal to… right-hand side
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is repeated in this.
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Let us continue it. We obtain this here; we
can write it also in this formula; and, towards
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here, here is constant fiber diameter k p
– b, rho, pi – different parameters characterizing
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the material and technology, but no twist
and no fiber count, so that we can say that
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for given type of yarn, whole this expression
represents a common parameter, Q. Then, we
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can write our expression in the form mu power
to 2.5, because earlier was square root. Square
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root is also on the right hand side. Therefore,
from mu power to 3 is now mu power to 2.5
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is equal to Q, some characteristic parameter
of material times Z times T power to 1 by
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4 – a quarter square in Koechlin’s model.
On the left-hand side is packing density only
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as a variable; on the right-hand side is yarn
count power to quarter. In opposite to Koechlin’s
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expression, in which is square root 1 half.
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Using this equation in common physical dimensions,
we can iterate in such form. Now, here are
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the dimensions, which I can recommend for
practical application. What is the value Q?
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It is I said is a material parameter. Based
on our experiences, we can say that for different
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fibers and spinning technologies, the following
values we can recommend to you. In more details,
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it is based on special type of your fibrous
material, your situation in your spinning
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mill and so on. But, generally, you can use
this here. Often say the values for wool yarn
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are rough, because we had not too much experimental
material.
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You can see, for example, for cotton combed
yarn, 1.46 times 10 power to minus 7 for carded
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9.61 times 10 power to minus 8 and so on.
Combed and carded yarn have another, because
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another structure, so that they have another
values; for viscose yarn, for polyester yarns,
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I have only one. It is not produced as a combed;
it means it is here in the middle in this
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table, because also, the blends are used;
it can be blend if it is carded as far as
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combed yarn. Therefore, it is in the middle
here for open end, type BD; we have then this.
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This equation – because now, to derive this,
my speech, I want to comment this equation;
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I know my students comment this equation;
the meaning that application of this equation
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or this equation and its application is a
little difficult. Therefore, they started
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to call it a horrible Neckar’s equation.
In check language, it is a better; and therefore,
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if also there is some shortening for this
horrible Neckar’s equation. It is horrible
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Neckar’s equation number 1; later, it will
be horrible equation number 2 also. So is
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the life in students’ society.
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An example is here shown; when we know yarn
twist, yarn count and give a value of Q; for
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the yarn, we usually on the place of mu m,
this maximum value – its limit value of
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packing density – we use… It is our practical
experiences; its value 0.8; no 1; then, 0.8.
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Why? Because in each yarn also, very hard
twisted yarn; on the vicinity round surface
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of the yarn, the packing density is smaller.
Then, the mean value of packing density also
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in hard twisted yarn is a little smaller;
it is not too near to value one. Therefore,
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based on our experiences, we can recommend
to use on the place mu m – 0.1. And, for
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a, also based on our experiences, value – 1,
so that we obtain then this – this expression,
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which we use practically. When we know yarn
count, when we know yarn twist, no problem
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to calculate using such equation the packing
density; no problem; we can calculate right
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hand side of our equation.
And now, we need to solve the question – which
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mu on the left-hand side corresponds to our
right-hand side value. We have more possibilities
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– how to apply it in practice? You can prepare
tables of left hand side of our equation.
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So, mu and left-hand side value – like this
here; prepare such table. When we know value
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of right-hand side, it must be equal to left-hand
side; then, if we want this table, can say
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which of mu is corresponding to our equation.
Second version is use a numerical method.
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For example, interval splitting method, but
you need to know some basic tools from numerical
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mathematical to be able to and program it.
Maybe you are, but lots of people are not
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especially in textile industry. Therefore,
also, the first version for practical application
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is possible. When we know, the packing densities
of our yarns produced lie around a value – some
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value mu star; what I mean? When you produce
carded cotton yarns in your spinning mill,
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then you know that packing densities in all
of your yarn will be maybe from 0.4 to 0.5.
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So, you can say, it will be no too far from
value maybe 0.45 or 47.
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Your yarn will not have packing density 0.2,
for example. It is a question of… Maybe
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no far the products rowing, for example, so
that we can say my yarn are nearer packing
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density and need to some value mu star, which
I will choose based on my experiences. And
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then, it is possible to use an approximation.
The approximation function to our original
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function – this is our original function.
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The approximation, which is valid around our
packing density mu star, we obtain using following
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receipt. We calculate the value b; then, value
c; then, when we have this here, we calculate
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packing density mu using such expression.
It is very easy. Now, it is constant times
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twist times yarn count power to 1 by 4 whole
power to 2 by b minus 0.5. And, yarn diameter
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is square root of 4 times 2 by pi mu rho;
from D, we obtain this expression; practically,
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very easy. When we realize that it is constant
times T power to something times alpha power
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to something, that the type, which was derived
earlier as an empirical expression. Now, it
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is shown that it is an approximation – region
of approximation of theoretical equation.
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Here is an example, which for the approximation
for the carded yarns, is often possible to
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use this equation for calculation of packing
density. Then, diameter is evident. How is
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the relation between experimental result and
our model. The first graph show the relation
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on the is twist times yarn count power to
1 by 4 – quarter. And, the ordinate is packing
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density mu. Using Q of this value, we obtained
the original curve as this thick line; this
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is the thick line; varied our approximation
function is this thin line. And, we measured
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lot of carded cotton yarns; diameter of these
carded cotton yarns; and, we obtained from
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this measurement packing densities, which
characterize the points here on this graph.
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So, you can see that the thick line follows
the tendency of experimental values very well.
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But, because the yarns are only roughly for
this value – 1000 or something under 1000;
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from this value, no for very small packing
densities from… I do not know; this is 35
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for extremely small twist in the yarn, because
the yarns are not in whole region of whole
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area in this graph. The approximation curve,
which is precise only in our point mu star
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– mu equals mu star is enough good for whole
interval of yarns, for example.
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But, it is not good for following experiment.
Here the difference is very high my colleague
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earlier colleague in research institute mister
Zalaba measured also the diameters of followings.
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His empirical equations of rowing diameter;
the result of it are shown in these two short
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curves here. You can see that it is very far
from our approximation, but very near to our
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theoretical curve. So, our theoretically derived
curve is valid in acceptable comparison – this
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experiment in whole region from rowing to
twisted yarns. The approximation equation
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– when you use the characteristical value
mu star this. For yarn, is not possible to
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use for rowing and opposite. Write this relation
of packing density; here is approximation
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by original – this ratio. You can see that
in this region, it is very small difference.
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We derived an equation for calculation for
the packing density inside of the yarn. The
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equation in which the influence of compression
or generalized to one week’s; one week’s
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equation was used. The same equation is possible
to rearrange and to obtain the second equation,
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which can help us to find the best; not the
best, but, the good value of twist of the
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yarn twist – suitable yarn twist.
Let us start now with the rearranging of our
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equation to the second form, which is good
for such work. Schematically, this is a yarn
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cross section; diameter of our idealized yarn
is D. But, the axis of peripheral fibers are
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lying on the little smaller diameter. In the
moment I call it D dash, we will find the
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ratio D dash by D. This ratio found mister
Schwarz at first. Therefore, it is known in
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later age as the Schwarz’s constant. Let
us imagine first step of our ideal; let us
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imagine – the yarn is limit packing density;
then, the fibers are mutually in contact.
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The part of this structure seems like this
here – D minus D dash. The diameter D is
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this; diameter D dash is this here; D minus
D dash in this case is equal to 1 half of
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fiber diameter, so that this is 1 half on
the other side too.
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So, both together, D minus D dash is D. C
D is D dash by D. So, it is this here; then,
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it is this here; using on the place of D,
4 times S by pi mu; mu limit with Now, we
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speak about the hypothetical yarn having limit
value of packing density. So, then, after
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rearranging, on the place of S, pi D S square
by 4 is rearranged. So, it is this here – d
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by D S – you can see from lecture 1; it
is 1 by square root of tau. So, we obtain
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this here. And finally, because mu lima, we
know it is something over 0.9; we obtained
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1 minus 0.952 by square root of tau. And,
because it is a little rough theory, we can
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too high. I do not say approximately that
C D is 1 minus 1 by square root of tau.
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When we use tau as T by small t, C D is 1
minus square root of small t by capital T.
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It was derived for yarn having limit packing
density. And, when we go back to our real
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yarn, packing density is more. We can say
that all in our dimensions can be elongated
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in the same percentage, so that also, in our
real yarn roughly, this ratio can be same,
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so that this quantity C D – we will use
as an expression for our Schwarz’s constant.
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On the diameter D dash, where a lying axis
of our peripheral fibers – the axis of peripheral
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fibers have a little other angle than beta
D – our earlier beta D. This angle is beta
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dash; and what its value tangents beta dash,
is 2 pi D dash by 2 is radius; and, Z D dash
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after rearranging this, so that it is kappa
times Schwarz’s constant. Now, let us rearrange
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our earlier equation 1 – this equation.
Let us rearrange. This is our earlier equation
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– Z times T power to a quarter; T power
to a quarter is here; Z – it is pi D Z by
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pi by D; it is multiplied and divided by pi
D; D – we know is square root of 4 T by
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pi mu rho. So, we obtained this expression.
After graphically arranging this expression
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is black symbols – kappa from this expression
is tangents beta dash by C D. So, after I
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obtained this here on the right hand side,
left-hand side is this here. Using C D based
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on our earlier derivation, we obtained this
expression. And, Q times tangent square beta
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dash times rho may be S by 4 pi – means
this quantity.
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I will call under the symbol R, because here
was on right-hand side mu and this mu power
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to 2.5 for some left-hand side. Therefore,
it is another exponent; it is 1.5 only. So,
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we obtained our equation in this form, where
R is here. In this moment, it is no more than
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only a rearrange form of our earlier equation;
nothing new; only mathematically rearranged;
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the new will come. We can study how is the
quantity R. Let us think, the yarns from the
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same material and simultaneously for same
or analogical end-use. We will study two special
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cases. Case 1 – the same technology and
different yarn counts; example – carded
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cotton technology, but one time 20 tex; one
time 40 tex; one time 60.5 tex and so on.
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The second case, which we will study, is different
technologies and the same yarn count; I said
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same material and similar use. For example,
cotton material, cotton yarn – 20 tex; carded
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version, combed version, open end yarn version
– different technologies.
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We said this is our equation – often equation.
Case 1 – if the case, same technology and
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different yarn counts; Koechlin said that
for different yarn counts is good when the
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yarn have a geometrical proportions, when
we accept the geometrical similarity. Therefore,
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corresponding angles shall be same. This Koechlin’s
idea – 200 years old is very good also in
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these days; it is very good idea, but we must
think now about the beta D angle on the surface
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of the yarn. Nevertheless, on the angles,
which have the axis of peripheral fibers have
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angle beta dash, so that this assumption which
may be very good have… We must interpret
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angle beta dash for the whole of this Yarn
must be constant, because beta dash is constant
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rho; say material for pi; total constant,
Q for given material is constant. So, R is
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constant. Resulting recommendation for such
yarn from same technology and different yarn
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counts – R shall be constant; for suitable
twisting, R must be constant.
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Let us have the second case – different
technologies and same yarn count. The contact
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density – density of contacts – number
of contact per volume unit should be constant,
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because of the mutual influence fiber to fiber
and so on. We said we use this yarn for similar
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analogical use and so on, because number of
contacts shall be same, because number of
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contacts is parameter times mu square; we
know it from earlier lectures. Then also,
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packing density shall be same. So, let us
twist carded, combed as well as open-end yarn;
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same yarn count, so that it will have same
packing density. Then, left-hand side of our
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equation is constant – same packing density
mu. Denominator of right-hand side is we compare
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yarns having same yarn count; it is from same
material. So, denominator is constant too.
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And, because this equation is valid, also
R must be constant evidently. Understandable?
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Left-hand side – we say is good for suitable
twist to have packing density constant; how
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will have left-hand side of our equation.
Therefore, all this left-hand side is constant.
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Denominator, because all yarn counts are same,
is constant too and because this is equivalency.
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Therefore, R must be constant also. So, R
is constant. In both cases, which are quite
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different we obtained the same result; R should
be constant.
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So, let us generalize this knowledge and say
that for given material independently to yarn
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count and independently of type of technology,
I speak about the technologies using twist
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for another group – fiber to fiber together
and obtained some linear product; spinning
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technology used twist; no bonding silver or
something. For each material
and use of our yarn, the quantity R must be
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constant. What is good value? You can see
in this table; for example, for cotton fiber
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- long staple – 2.1 tex power to 1 half;
medium staple – 2.7 and so on.
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So, we have couple of equations. This is the
first and this is the second. This couple
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called my students horrible Neckar’s equations.
In Czech language, [FT] In a shortening, it
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is [FT] S n R and to one time, one, one student
will come to my assistant and ask she, please
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she for some consultation. And, my assistant
say yes and what the theme of which was…
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where is your problem. And, she said I want
to have the consultation about [FT] Horrible
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So, it is very known in my university;is very
known. Professor Ishteyak can comment it more.
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He knows it very well; yes. How is possibility
how to apply these couple of equations? Usually,
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when Q and R… Maybe when you do not have
more precise values, you can use some values
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from my tables. So, you know Q and R. Then,
we know t and rho – fiber fineness and specific
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mass; that the mass density of all these materials.
And, we also know the required yarn count
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T. How to evaluate, how to calculate the future
quantity of our yarn? That from point 1 to
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calculate right-hand side of equation 2; is
it possible? Yes, is out problems. Point second
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– find the packing density mu as a root
of equation 2 are worth find a mu, because
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the left-hand side was given the same number
as our earlier calculated right-hand side
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using numerical method tables and so on. Then,
we know mu. Point 3 – calculate the left-hand
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side on the equation 1. We know mu; we can
calculate left-hand side value. And, point
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4 – find the suitable yarn twist Z as a
root of equation 1. We know all right-hand
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side, Q yarn fineness. So, Z is trivial to
explicitly evaluate it; is not it? Now, Z
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square is what? This by this and by Q and
T power to 1 quarter times; the square is
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1 one half; is not it? So, it is trivial to
obtain Z from this equation. So, we obtained
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Z as a suitable twist of the yarn. And, calculate
yarn diameter using packing density, which
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we derived from the equation from lesson 1.
So, it is the way how to practically use this
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couple of expressions.
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An example is here. Numerical example you
can home study and verify that; I calculate
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it. The problem, which can you have, is the
problem in practical application in evaluation.
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In point 2, find the packing density mu as
a root of equation 2; find the mu on left-hand
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side, because we valid an equivalency, is
known right-hand side. You can use tables;
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you can use numerical method.
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And, you can use also approximation. Similarly,
we spoke about the approximation of our first
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equation, but it is possible to approximate
also our second equation. The result is here.
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It is presented as a result. Both approximations
are derived using following way. We said in
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some point, the first section – it was mu
star; in some point, mu is the up most approximation.
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The approximation equation must give the same
value than the original. And, the second assumption
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for approximation is or the second necessity
is the first derivative of approximation function.
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And, the first derivative of original function
in our point must be same. Based on these
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two equations: equivalency of value and equivalency
of derivatives, we obtained the final equations,
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which are present; I am presenting now here.
So, we can calculate the following.
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We must evaluate the quantities of Z star,
T star and mu star of middle yarn from equation
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1 and 2. One yarn – let us say one Yarn
I will calculate based on original equations
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include… I do not know tables or numerical
method and so on; only one yarn. One typical
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yarn, which is in the middle of area of my
activity in my spinning mill; this yarn is
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average or middle yarn; values have stars
here. This is
the result, which I presented. Now, we evaluate
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the value b, because we know mu star is possible;
then, capital b also possible; Z x I also
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possible; all these three are helping quantities
in our way. Then, q – it is very important;
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and, alpha q according to this equation. Having
these quantities, we can formulate the approximation
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equation that the yarn twist is alpha q times
T power to q.
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An example – numerical example shows that
the correspondence between… this is numerical
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example and this is the graphical interpretation
of two curves. One is alpha – recommended
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alpha as a function of yarn count; two lines
– one is original theoretical equation;
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the second is approximation. Using that in
large interval, both versions are no identical,
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but practically same; very good for application.
That is all for today. I presented you some
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model of packing density; I presented you
a couple of horrible Neckar’s equation.
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And, in short, also the way – how to apply
it in practice in practical calculation? I
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did not explain one moment; I spoke about
yarn diameter, but the question is what is
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it yarn diameter, because yarn diameter – where
is the end radius for yarn diameter? Where
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is the maximum radius in the yarn? And, where
started arial – this sphere of hairiness
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sphere? In reality, it has no strong borders;
diameter is every time a little… the question
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of our convention. And, we need to solve it
together; the modeling of external part of
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the yarn body, which is sphere of hairiness.
Our next lecture – we will study the models
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of hairiness. And, in connection with this
model – we will find; we will explain in
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more detail the question about the yarn diameter
too, but it will be in my next lecture.
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Thank you very much for your attention.