# Formulating equations for calculating & controlling width, WPI, CPI & GSM of circular weft knitted fabric Introduction:

Fabrics are produced mainly by knitting technique and weaving technique. Knit fabric production is easier and quicker than weaving technique. Though knitting production is simple, its technical parameters calculation is complicated and difficult than weaving.  Technical parameters involved in the knit fabrics are: loop structure, loop/stitch length, CPI ,WPI, GSM, shrinkage, spirality, fabric width etc. In reality knitted loops in the fabric is curvilinear in shape but this work has done necessary analysis to consider it like a circle, following thata circle geometry and corresponding formulae has been used in formulating some parameters of circular knitted single jersey fabric. It is very difficult to maintain& control circular knitted fabric parameters due to severalvariables (like yarn type, yarn quality, yarn twist, fabric structure, machine speed, machine rotation, yarn tension, take up tension & many more) related with circular knitting production and influences of them are unavoidable, so the parameters can be varying more than acceptable limit. But still the derived formulae focus practicalenvironmentand help to calculate parameters directly and easily.

Methodology:

The first attempt was to get a technical equation to calculate fabric diameter or width for factory use and for that several ways was followed, both searching books and trying to formulate new one. To have new equation, the proposed ideawas using circle for loops shape, then the same idea was used to calculate others parameters. All the equations were then checked with large number of factory data.

Related Terminologies & Definitions:

Knitting

Knitted structures are progressively built-up from row of intermeshed loops. The newly fed yarn is converted into a new loop in each needle hook. The needle then draws the new loop head first through the old (fabric) loop, which has retained from the previous knitting cycle. The needles, at the same time, release, (cast off or knock over) the old loops so that they hang suspended by their heads from the feet of the new loops whose heads are still held in the hooks of the needles. A cohesive knitted loop structure is thus produced by a combination of the intermeshed needle loops and yarn that passes from needle loop to needle loop.There are so many variables in knitting production like, yarn related (i.e. yarn composition, count, twist direction, twist per inch (TPI) etc.); machine related (i.e. machine diameter, machine RPM, rotation direction, no of feeder, VDQ pulley diameter etc.) & fabric related (i.e. fabric type, construction, stitch length, stitch density etc.) and this can vary fabric parameters at large.

Wales Per Inch

Wales per inch (WPI)indicate the total numbers of wales in one inch of linear fabric. WPI is one of the important parameters to determine GSM of fabric and other fabric properties.

Courses Per Inch

Coursesper inch (CPI)indicate thenumbers of coursesin one inch of linear fabric.  In the same manner, CPI is one of the important parameters to determine GSM of knit fabric and other fabric properties.

Stitch length

Stitch length is theoretically the single length of yarn which include one needle loop and half of a sinker loopbetween that needle loop and the adjacent needle loop.The length is measured in millimeter (mm). Itinfluencesfabric dimension  and other properties including weight.

Fabric Width

Where,  d = Cylinder diameter, G = Gauge of machine GSM

GSM means the weight in gram per square meter of fabric.
Fabric area density can be calculated by the following formula, Here, T = Tex, S = Stitch density,
l= Stitchlength
GSM is one of the primary parameters for determining fabric quality and GSM depends on WPI, CPI, yarn count (Ne) and stitch length.The GSM cutter isvery popular and easy usable GSM testing instrument used in most of the knittingfactories.The construction of this cutter is very simple. It is circular disk of 100 square cm area with sharp blade attached to its edge. So 100 square cm of fabric can easily cut by it and weighted at the electric balance to get GSM Reading.
Technical formula used for GSM calculation, Yarn Diameter Calculation:

In indirect system of yarn count, higher the yarn count finer the yarn Analysis:

The shape of a loop is not circular, but the yarn for that loop or stich length can be taken as equal to the circumference of a circle (Figure 1). The point is that, loop shape is not circle but can be assumed as a circle and that will help to formulate some equations. So, it has been assumed that every loop is equivalent to a circle and then circumference of that loop will be equal to Stitch Length. Width of Fabric:

Fabric width is formed by courses that are laying crosswise (loops in horizontal direction).
So, Width=loop Dia* no of loops
Width=loop Dia* no of course
Width=loop Dia* no of needles According to this equation the width calculation should be easy and loop should stay like the image cited below:
But every circle share part of circle of both sides. Fabric width from this equation will be less than actual width because the loop circles placed consecutively shares both sides with preceding and successive loops (Figure 3). Now let’s make relation between values from this equation & actual values and see how much closer they stay.

Table 1 shows that the actual width and the width from the equation always maintain approximately same ratio among themselves.If this ratio can be considered as a constant actual width of the fabric can easily be derived. So the equations can be written as considering R as a constant as it shows in the Table 1. The value of R for Single jersey is 1.25-1.30 and for 1*1Rib is about 1.00 and for others fabric type’s value of R can be calculated.  Variation of R is related withtype of fabric construction, needles number, yarn types and composition, machine tension, speed, and tightness factor.

WPI:
WPI can be calculated from this equation of width.
No of needles=no of wales, and width is expressed in inch.
So, No of wales per inch=Total no of wales or no of needles/Width of the fabric in inches.
t is to be mentioned that actual WPI can be less than the WPI found from this equation if any needles are inactive. Course Per Inch (CPI):

A knitted fabric is a combination of courses in horizontal direction with continuous yarn. A wale is formed having courses as shown in figure 5. A loop contains a tail yarn of previous loop and a head yarn of next loop [From figure 5], so effective height of every course is (d1-2dy) mm. So, Course height is (d1 – 2dy) mm or { (d1 – 2dy)*39.36/1000 } inch
Let, n=no of courses required for 1 inch of fabric or CPI This equation can be used to have idea of CPI or to determine required no of feeder/course. GSM:

The equation for GSM can be formulated by using equation of WPI & CPI formulated here.
The general formula for calculating GSM is  Figure 6 shows the variation of Actual GSM and GSM from derived equation for  single jersey grey fabrics made of 100% cotton at standard condition. Figure 7 shows a relative presentation of actual and calculated GSM when required GSM is 130.The analyzed data shows that the error percentage for calculated GSM is below 5% in most of the time. And this variation of GSM is related with so many factors like: color, yarn count, tightness factor, relaxation state, andprocessesoccurred onthe fabric.

The equationof GSM contains the constant ‘R’ &statistical analysis for the values of R for 100% cotton single jersey fabric foundfrom analyzed data shows 1.46% standard deviation when mean value is 1.273 at table 4. On the other hand factory data for 120 GSM single jersey fabric productions from 100% cotton yarn shows 5.46% standard deviation, which shows that equation derived here will give less error then the existing ways of controlling GSM. Results:

The equations for some knitting parameters of circular knitted fabric are Where,  R=constant that depends on fabric types.
D=Machine Diameter and G= Machine gauge
Sl=Stitch Length (mm)& Ne=Yarn count (English count)

Knitted fabric is dimensionally unstable fabric. Here value of ‘R’ can be different for different types of
variables. So, a constant ‘R’ value (for S/J R=1.25-1.3) must be measured by maintaining a standard procedure & standard parameters of all variables. A further study should be done to identify the constant value of ‘R’ for all basic types of knitted fabric.

Conclusion & Remarks:

All the equation derived here got two specific variables, Stitch length and Yarn Count. The two which are mainly responsible for determining parameters like width, WPI, CPI & GSM. The existing equations for calculating parameters may need more variables or need to be calculated after production. All the equation here is to have a prediction about WPI, CPI, Width and GSM, and process of formulation is so easy to understand that nontechnical & technical persons both will find it easy to absorb. The values from these equations show less standard deviation than the present situation of production. The error can be less if ‘R’ value is more precise & standard procedure can be followed.

References:

1. Knitting technology By Devid J. Spencer.
2. Handbook of Technical Textileby Horrocks&Anand.
3. Analysis of Specification of basic single jersey Fabric,Supervised by Shah Alimuzzaman.
4. Understanding textiles for a Merchandiser by Shah Alimuzzaman
5. Data courtesy: EchoTex Ltd. Bangladesh.
6. An Introduction to Weft Knitting, J. A. Smirfitt, Merrow Publishing Co. Ltd., England
7. Circular Knitting, C. Iyer, B. Mammel and W. Sehach, Meisenbach Bamberg
8. Knitting – Reference Books of Textile Technologies (e-book), Carmine M. &Paola Z., Fondazione, ACIMIT, Milano, Italy
9. Knitting : Fundamentals, Machines, Structures and Developments, N.Anbumani, New Age International Publisher, New Delhi
10. Knitting Technology, Prof. D. B. Ajgaonkar, Universal Publishing Corporation,Mumbai

Acknowledgements:
1. Prof. Masud Ahmed, Dean, Faculty of Textile Manufacturing Engineering, Bangladesh University of Textiles.
2. Dr. MominulAlam, Head, Fabric Manufacturing Engineering Department, Bangladesh University of Textiles.
3. KaziSourov, Lecturer,Fabric Manufacturing Engineering Department, Bangladesh University of Textiles.
4. SaidurRahman, Bangladesh University of Textiles.

This project is one of the Textile Talent Hunt, (TTH 12012) research projects. to know more about TTH 