Calculating Plain Weave Fabric Weight with Crimp

This solution explains how to calculate the weight of a plain weave fabric per square meter (g/m²), considering the yarn counts (in tex), the density of warp (ends per cm) and weft (picks per cm), and the percentage of crimp for both warp and weft yarns.

Understanding the Variables

• Warp Count: 30 tex (grams per kilometer)
• Weft Count: 20 tex (grams per kilometer)
• Ends per cm (EPI): 40 (Number of warp yarns across the fabric width per centimeter)
• Picks per cm (PPI): 30 (Number of weft yarns along the fabric length per centimeter)
• Warp Crimp: 10% (The waviness of warp yarns reduces their effective length in the fabric)
• Weft Crimp: 10% (The waviness of weft yarns reduces their effective length in the fabric)
• Target Unit: Grams per square meter (g/m²)

Step-by-Step Calculation

Step 1: Calculate Warp Weight per Square Meter

First, we determine the total length of warp yarn present in one square meter of fabric. The crimp needs to be accounted for, as it means the actual yarn length is longer than the fabric dimension it spans.

Calculation:

• Number of warp ends in 1 meter (100 cm) width: $ \text{Ends per meter} = \text{Ends per cm} \times 100 \text{ cm/m} = 40 \times 100 = 4000 \text{ ends/m} $
• Actual length of warp yarn needed to span 1 meter (100 cm) of fabric width, considering 10% crimp: $ \text{Effective Length per end} = \frac{\text{Fabric Width}}{\text{1 - Warp Crimp \%}} = \frac{100 \text{ cm}}{(1 - 0.10)} = \frac{100}{0.90} \approx 111.11 \text{ cm} $
• Total actual length of warp yarn in 1 square meter: $ L_{warp} = (\text{Ends per meter}) \times (\text{Effective Length per end in meters}) $ $ L_{warp} = (40 \text{ ends/cm}) \times \frac{100 \text{ cm}}{(\text{1 - 0.10})} \times (1 \text{ m fabric width}) = 40 \times \frac{100}{0.90} \text{ cm/m} \times 1 \text{ m} $ $ L_{warp} \approx 4444.44 \text{ cm/m} = 44.444 \text{ meters} $
• Convert warp yarn count from tex (g/km) to g/m: $ \text{Warp Count (g/m)} = \frac{30 \text{ tex}}{1000 \text{ m/km}} = 0.030 \text{ g/m} $
• Calculate the weight of warp yarn per square meter: $ W_{warp} = L_{warp} \times \text{Warp Count (g/m)} $ $ W_{warp} = 4444.44 \text{ m} \times 0.030 \text{ g/m} \approx 133.33 \text{ g/m}^2 $

Step 2: Calculate Weft Weight per Square Meter

Similarly, we calculate the total length and weight of the weft yarn in one square meter of fabric.

Calculation:

• Number of weft picks in 1 meter (100 cm) length: $ \text{Picks per meter} = \text{Picks per cm} \times 100 \text{ cm/m} = 30 \times 100 = 3000 \text{ picks/m} $
• Actual length of weft yarn needed to span 1 meter (100 cm) of fabric length, considering 10% crimp: $ \text{Effective Length per pick} = \frac{\text{Fabric Length}}{\text{1 - Weft Crimp \%}} = \frac{100 \text{ cm}}{(1 - 0.10)} = \frac{100}{0.90} \approx 111.11 \text{ cm} $
• Total actual length of weft yarn in 1 square meter: $ L_{weft} = (\text{Picks per meter}) \times (\text{Effective Length per pick in meters}) $ $ L_{weft} = (30 \text{ picks/cm}) \times \frac{100 \text{ cm}}{(1 - 0.10)} \times (1 \text{ m fabric length}) = 30 \times \frac{100}{0.90} \text{ cm/m} \times 1 \text{ m} $ $ L_{weft} \approx 3333.33 \text{ cm/m} = 3333.33 \text{ meters} $
• Convert weft yarn count from tex (g/km) to g/m: $ \text{Weft Count (g/m)} = \frac{20 \text{ tex}}{1000 \text{ m/km}} = 0.020 \text{ g/m} $
• Calculate the weight of weft yarn per square meter: $ W_{weft} = L_{weft} \times \text{Weft Count (g/m)} $ $ W_{weft} = 3333.33 \text{ m} \times 0.020 \text{ g/m} \approx 66.67 \text{ g/m}^2 $

Step 3: Calculate Total Fabric Weight

The total weight of the fabric per square meter is the sum of the warp weight and the weft weight.

Calculation:

• Total Fabric Weight per square meter: $ W_{total} = W_{warp} + W_{weft} $ $ W_{total} = 133.33 \text{ g/m}^2 + 66.67 \text{ g/m}^2 = 200.00 \text{ g/m}^2 $

Final Result

Based on the standard calculation methods for textile fabrics, the weight of the plain weave fabric is approximately 200.00 grams per square meter. (Note: This calculated result differs significantly from the provided options.)