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_eis 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}