Fabric Thickness Calculation

This solution calculates the approximate thickness of a fabric based on its yarn properties and construction details.

Yarn Diameter Calculation

First, we determine the effective diameter of the yarn using the given tex value, fibre density, and packing coefficient.

• Given:
  • Tex ($T_x$) = 38 g/km
  • Fibre Density ($\rho_f$) = 1.54 g/cm$^3$
  • Yarn Packing Coefficient ($k$) = 0.65
• Step 1: Calculate yarn volume per meter.
  • Mass per meter = $38 \text{ g/km} \times 1 \text{ km} / 1000 \text{ m} = 0.038$ g/m.
  • Volume per meter = Mass / Density = $\frac{0.038 \text{ g}}{1.54 \text{ g/cm}^3} \approx 0.024675 \text{ cm}^3$/m.
• Step 2: Calculate yarn cross-sectional area.
  • Area per meter = Volume per meter / Length = $\frac{0.024675 \text{ cm}^3}{100 \text{ cm}} = 0.00024675 \text{ cm}^2$.
• Step 3: Calculate yarn diameter ($d_{yarn}$).
  • Assume yarn cross-section area $A = \pi r^2$.
  • $r^2 = \frac{A}{\pi} = \frac{0.00024675 \text{ cm}^2}{\pi} \approx 0.00007854 \text{ cm}^2$.
  • Radius $r = \sqrt{0.00007854} \approx 0.00886 \text{ cm} = 0.0886 \text{ mm}$.
  • Yarn diameter $d_{yarn} = 2r \approx 2 \times 0.0886 \text{ mm} = 0.177 \text{ mm}$.

Fabric Thickness Estimation

The fabric thickness depends on the yarn diameter and how the yarns are packed. The condition "warp is jammed" suggests the warp yarns are tightly packed, significantly influencing the thickness.

• Given:
  • Ends per cm ($N_e$) = 30
  • Picks per cm ($N_p$) = 25
  • Calculated yarn diameter ($d_{yarn}$) $\approx 0.177$ mm
• Step 4: Estimate fabric thickness ($T$).
  • Fabric thickness is often related to yarn diameter. A common approximation, especially when yarns are tightly packed (like a jammed warp), suggests $T$ can be approximately $2.5$ to $3$ times the yarn diameter.
  • Using the approximation $T \approx 3 \times d_{yarn}$:
  $T \approx 3 \times 0.177 \text{ mm} \approx 0.531 \text{ mm}$.
• Step 5: Compare with options.
  • The calculated thickness $T \approx 0.531$ mm is approximately 0.5 mm.

Therefore, the approximate thickness of the fabric is 0.5 mm.