# Rieter

### Derivation of the twist equation

#### Index

Fig. 63 – Number of turns of twist in yarns of different thicknesses

If the two yarns are illustrated on a somewhat larger scale, the situation of Fig. 63 is obtained  [20]. The following relationships can be derived:

$\frac {h}{H} = \frac {d_I}{d_{II}}$   and   $\frac {d_I}{d_{II}} = \frac {T_2}{T_1}$
T = Twist in the yarn.

The mass of a yarn is given by

$m=V \times \sigma$
V = volume
σ = specific mass

Since the volume is given by

$V = A \times\ L$,
A = surface area in cross section
L = length

and the area

$A = {d^2} \times \frac { \pi}{4}$

the mass of the yarn is

$m = {d^2} \times \frac { \pi}{4} \times L \times \sigma$

The masses of the yarns I and II are:

$m_1 = \frac {{d^2}_I \times \pi}{4} \times L \times \sigma$
$m_2 = \frac {{d^2}_{II} \times \pi}{4} \times L \times \sigma$

If these masses are inserted in the count formulas of the English system, the following results are obtained:

$Ne_I = \frac {L}{m} = \frac {L}{ \frac {{d^2}_I \times \pi}{4} \times L \times \sigma} = \frac {4}{{d^2}_I \times \pi \times \sigma}$

$Ne_{II} = \frac {L}{m} = \frac {L}{ \frac {{d^2}_{II} \times \pi}{4} \times L \times \sigma}$

$= \frac {4}{{d^2}_{II} \times \pi \times \sigma}$

Here the yarn counts are related by the formula:

$\frac {Ne_I}{Ne_{II}} = \frac { \frac {4}{{d^2}_I \times \pi \times \sigma}}{ \frac {4}{{d^2}_{II} \times \pi \times \sigma}} = \frac {{d^2}_{II} \times \pi \times \sigma}{{d^2}_I \times \pi \times \sigma}$

which reduces to

$\frac {Ne_I}{Ne_{II}} = \frac {{d^2}_{II}}{{d^2}_I}$

The diameters are related by the formula:

$\frac {{d^2}_{II}}{{d^2}_I} = \frac {Ne_I}{Ne_{II}}$     i.e.    $\frac {d_{II}}{d_I} = \frac { \sqrt{Ne_I}}{ \sqrt{Ne_{II}}}$

but, since also

$\frac {d_{II}}{d_I} = \frac {T_1}{T_2}$     we therefore have    $\frac {T_1}{T_2} = \frac { \sqrt{Ne_I}}{ \sqrt{Ne_{II}}}$

Expressed in an alternative form:

$\frac {T_1}{ \sqrt{Ne_I}} = \frac {T_2}{ \sqrt{Ne_{II}}} = \frac {T_3}{ \sqrt{Ne_{III}}} = \frac {T_n}{ \sqrt{Ne_n}}$
$= Constant = \alpha$

This constant can be arbitrarily designated, for example, as  $\alpha$, and the following generally valid formula can then be derived:

$\frac {T}{ \sqrt{Ne}} = \alpha_e.....T = \alpha_e \sqrt{Ne} = turns/inch$

The twist coefficient $\alpha_e$is derived in accordance with the English count system, and for cotton yarns it takes the following values:

Yarn type
Short staple
Medium staple
Long staple
Knitting
-
2.5-3.0
2.1-2.6
Weft
3.3-3.8
3.0-3.5
2.5-3.0
Semi-warp
3.7-4.0
3.5-3.8
3.0-3.4
Warp
4.0-5.0
3.8-4.5
3.4-3.9

For the other count systems, the following formulas apply:

Turns per meter:

$T/m = \alpha_m \times \sqrt{ \frac {100}{tex}}$
$= \frac { \alpha_{tex}}{ \sqrt{tex}}$

Conversion factors are:

$T/inch = T/m \times 0,0254$

$\alpha_e = \alpha_m \times 0,033$

$\alpha_e = \frac { \alpha_{tex}}{958}$