Understanding Knitted Fabric Properties: Stitch Length and Linear Density

This question asks us to determine how the ratio between the tightness factor and the areal density of plain knitted fabrics changes when the stitch length ($\ell$) varies, while the packing density and yarn material (cotton) remain constant, and the linear density ($T$) of the yarn differs.

Key Textile Terms Explained

• Stitch Length ($\ell$): This is the length of yarn used to form a single loop or stitch in the knitted fabric.
• Linear Density ($T$): This measures the mass per unit length of the yarn (e.g., in Tex or denier). A higher $T$ means a thicker or heavier yarn.
• Packing Density ($\delta$): This represents the ratio of the volume occupied by the yarn fibers to the total volume of the fabric. It indicates how compactly the yarn is arranged in the fabric structure. The question states this is constant.
• Tightness Factor (TF): This is a measure of how tightly the yarn is knitted into the fabric. A common definition relates it to the stitch length ($\ell$) and the yarn diameter ($d$), often expressed as $TF \propto \ell/d$.
• Areal Density ($\rho_A$): This is the mass of the fabric per unit area (e.g., in g/m²). It's often referred to as fabric weight.

Deriving Proportional Relationships

To find the relationship, we need to establish how the Tightness Factor (TF) and Areal Density ($\rho_A$) depend on the given variables, particularly stitch length ($\ell$) and linear density ($T$).

• Yarn Diameter and Linear Density: The diameter ($d$) of a yarn is related to its linear density ($T$). Since the mass of a yarn segment is proportional to its volume (Area $\times$ Length), and the cross-sectional Area ($A$) is proportional to $d^2$, we have $T \propto A \propto d^2$. Therefore, the yarn diameter is proportional to the square root of the linear density: $d \propto \sqrt{T}$.
• Tightness Factor (TF): Using the common definition $TF \propto \ell/d$, and substituting the relationship $d \propto \sqrt{T}$, we get the proportionality for the Tightness Factor as:

  $TF \propto \frac{\ell}{\sqrt{T}}$

• Areal Density ($\rho_A$): Areal density relates to the mass of yarn within a unit area of fabric. It can be expressed as $\rho_A = \delta \times t$, where $t$ is the fabric thickness. Since the packing density ($\delta$) is constant, the areal density is directly proportional to the fabric thickness:

  $\rho_A \propto t$

• Fabric Thickness and Stitch Length: For a given yarn and constant packing density, the thickness ($t$) of a knitted fabric is generally influenced by the loop structure. An increase in stitch length ($\ell$) typically leads to a larger loop structure, which often results in a thicker fabric. Thus, we can assume the fabric thickness is proportional to the stitch length:

  $t \propto \ell$

• Combining for Areal Density: Substituting the proportionality $t \propto \ell$ into $\rho_A \propto t$, we find that the areal density is proportional to the stitch length:

  $\rho_A \propto \ell$

Calculating the Ratio of Tightness Factor to Areal Density

The question asks for the proportionality of the ratio $\frac{TF}{\rho_A}$. Using the derived relationships:

• $TF \propto \frac{\ell}{\sqrt{T}}$
• $\rho_A \propto \ell$

Now, we calculate the ratio:

$ \frac{TF}{\rho_A} \propto \frac{(\frac{\ell}{\sqrt{T}})}{(\ell)} $

Simplifying this expression:

$ \frac{TF}{\rho_A} \propto \frac{\ell}{\sqrt{T}} \times \frac{1}{\ell} $

The $\ell$ terms cancel out:

$ \frac{TF}{\rho_A} \propto \frac{1}{\sqrt{T}} $

Conclusion

The ratio between the tightness factor and the areal density of the fabrics is proportional to $\frac{1}{\sqrt{T}}$.