Fabric Engineering

Step 1: Understanding Peirce’s geometry.
Peirce’s geometry is a mathematical model used to describe the geometry of woven fabric structures, particularly the path and interlacement of yarns.
Step 2: Role of equations and variables.
In Peirce’s geometric model, the fabric structure is defined using a set of simultaneous equations that relate yarn diameter, spacing, crimp, and fabric thickness.
Step 3: Standard formulation.
Peirce’s geometry consists of 12 governing equations involving 8 independent variables. This makes the system over-determined and requires simplifying assumptions.
Step 4: Conclusion.
Hence, the correct number of equations and variables in Peirce’s geometry is 12 and 8 respectively.

Step 1: Understanding terry towel construction.
Terry towels are designed to have high absorbency, which is achieved by creating looped pile structures on the fabric surface.
Step 2: Type of pile used.
In terry towels, the pile is formed by warp yarns, not weft yarns. These warp pile yarns form loops on the fabric surface.
Step 3: Yarn arrangement.
A terry fabric typically consists of one series of warp yarns (ground and pile warp together) and two series of weft yarns (ground wefts) to bind the pile securely.
Step 4: Conclusion.
Hence, a terry towel is a warp pile structure with one series of warp and two series of weft yarn.

Step 1: Definition of strength utilization.
Strength utilization is defined as the ratio of fabric strength to the sum of the strengths of individual yarns used in the fabric.
Step 2: Effect of fabric structure.
In woven fabrics, yarns are crimped and interlaced, which prevents all yarns from bearing load simultaneously and uniformly.
Step 3: Practical implication.
Due to yarn crimp, friction, and uneven load sharing, the effective strength of yarns in fabric is less than their individual strength.
Step 4: Conclusion.
Therefore, the strength utilization of yarn in a woven fabric is always less than 1.

Step 1: Understanding the relationship between picks and shaft rotation.
In a shuttle loom, one pick is inserted for every one complete revolution of the bottom shaft. Therefore, the rotational speed of the bottom shaft is equal to the number of picks per minute.
Step 2: Converting picks per minute to revolutions per second.
Given picks per minute = 240
Step 3: Converting revolutions per second to angular velocity.
Angular velocity is given by:
Step 4: Identifying the value of .
Comparing with , we get:
Step 5: Conclusion.
Hence, the value of is 8.

Step 1: Understanding shuttleless weaving systems.
Shuttleless weaving systems insert weft yarn without using a conventional shuttle. Different systems have different limitations in terms of fabric width due to the method of weft insertion.
Step 2: Analyzing the options.
(A) Air jet: Uses compressed air for weft insertion and is limited in width due to air dispersion over long distances.
(B) Water jet: Uses water to carry the weft yarn and is also restricted in width and fabric type.
(C) Projectile: Uses a small metal projectile to carry the weft across the shed. It allows very large weaving widths, even exceeding 500 cm.
(D) Rapier: Uses flexible or rigid rapiers and has moderate width limitations compared to projectile systems.
Step 3: Conclusion.
Projectile weaving systems can operate at the greatest fabric widths among shuttleless looms.

Step 1: Understanding the terms.
Wales per inch ( ) represents the number of wales measured in the finished fabric after shrinkage. Fractional shrinkage ( ) indicates the proportion by which the fabric dimensions reduce during finishing. Machine gauge ( ) refers to the number of needles per inch on the knitting machine before shrinkage occurs.
Step 2: Relating fabric shrinkage to machine gauge.
Due to shrinkage, the wale density in the finished fabric increases. Therefore, the wale density on the machine must be lower than the final measured value. The relationship between machine gauge and finished wale density is:
Step 3: Rearranging the formula.
To find the machine gauge, rearrange the equation:
Step 4: Conclusion.
Hence, the correct expression for machine gauge in terms of wales per inch and shrinkage is .