Question

A square jammed plain cotton woven fabric is produced from 10 Ne yarn of circular cross-section. Assuming yarn density \(=0.91~\text{g/cm}^3\), find the number of threads per inch (rounded to the nearest integer).

Hint:

To estimate jammed square plain weave: 1) find \(d\) from linear density and fibre density; 2) use \(\text{EPI}=\text{PPI}=1/(\sqrt{3}\,d)\). This quickly yields a value near 50 for 10 Ne cotton.

Answer 1

Correct Answer: 49

Step 1: Convert count and find yarn diameter.
10 Ne \(\Rightarrow\) tex \(= \dfrac{590.5}{\text{Ne}} = \dfrac{590.5}{10} = 59.05~\text{tex}\).
Mass per cm: \(m = \dfrac{59.05}{1000}\,\text{g/m} = 0.05905\,\text{g/m} = 0.0005905\,\text{g/cm}\).
Cross-sectional area: \(A = \dfrac{m}{\rho} = \dfrac{0.0005905}{0.91} = 6.4945 \times 10^{-4}~\text{cm}^2\).
Assuming circular section, diameter \(d = \sqrt{\dfrac{4A}{\pi}} = 2.874 \times 10^{-2}~\text{cm} = 0.011316~\text{in}\).
Step 2: Use square jammed limit for plain weave.
For a square, jammed plain weave with equal warp and weft diameters, Peirce geometry gives the limiting setting:
\[ \text{EPI} = \text{PPI} = \frac{1}{\sqrt{3}\,d}. \] Hence, \(\text{threads/inch} = \dfrac{1}{\sqrt{3} \times 0.011316} \approx 51.0\).
Final Answer: 51 threads per inch