GATE (TF) Textile 2010 Question Paper Solution | GATE/2010/TF/51

A shirting cotton cloth is made from the following particulars :
Warp crimp, C₁ = 10%
Warp yarn dia, d₁ = 0.01627 cm
(T₂/T₁)^1/2 = 0.8
K₁ = 11.3
K₂ = 9.1 |
Fabric area density = 91.15 g/m²
where T is yarn tex and K is the cover factor is SI unit denoted by (threads per cm)(tex)^1/2 X 10^-1 .

Question 51 (Textile Engineering & Fibre Science)

The estimated specific volume of the fabric in cm³/g will be

(A)	2.62
(B)	0.62
(C)	3.62
(D)	1.62

[Show Answer]

Given in the question-

Warp crimp, C1 = 10%

Warp yarn dia, d1 = 0.01627 cm

(T2/T1)1/2 = 0.8

K1 = 11.3

K2 = 9.1
Fabric area density = 91.15 g/m2

Fabric areal density=91.15/10000 gm/cm2

Fabric areal density=0.009115 gm/cm2

Firstly we will calculate tex of warp yarn-

By formula-

$Dia of warp yarn (d_1)=4.44 \times 10^-3 \times \sqrt \frac{tex of warp yarn(T_1)}{Density of cotton fibre}$

$0.01627=4.44 \times 10^-3 \times \sqrt \frac{T_1}{1.54}$

$16.27 \times 10^-3=4.44 \times 10^-3 \times \sqrt \frac{T_1}{1.54}$

$16.27 =4.44 \times \sqrt \frac{T_1}{1.54}$

$\frac{16.27}{4.44}|=\sqrt \frac{T_1}{1.54}$

$3.66=\sqrt \frac{T_1}{1.54}$

By doing square both side-

$3.66^2=(\sqrt \frac{T_1}{1.54})^2$

$13.29= \frac{T_1}{1.54}$

$13.29 \times 1.54=T_1$

$T_1=20.47$ Tex

Now, given-

$(\frac{T_2}{T_1})^1/2 = 0.8$

By putting value of T1

$(\frac{T_2}{20.50})^1/2 = 0.8$

Square both side-

$\frac{T_2}{20.5} = 0.8 \times 0.8$

$T_2 = 0.64 \times 20.5$

$T_2=13.12$ tex

cover factor(K)=(threads per cm)(tex)1/2 X 10-1 .

So that,

$Cover factor(K_2)=(threads per cm_2)\times(T_2)1/2 \times 10^-1$

$9.1=(threads per cm_2) \times (13.12)^1/2 \times 10^-1$

$9.1=(threads per cm_2)\times 3.62 \times 10^-1$

$9.1=(threads per cm_2)\times 0.362$

$\frac{9.1}{0.362}=(threads per cm_2)$

$25.14=threads per cm_2(PPCM_2)$

$Threads per cm_2(PPCM_2)=25.14$

$Threads per cm_2(PPCM_2)=25.14 \times 2.54$

$Threads per Inch_2(PPI_2)=63.86$

$Formula of fabric thickness(t)=h_1+d_1=h_2+d_2$

Where h is the crimp height and d is the diameter of the yarn.

$Formula of fabric thickness(t)=h_1+d_1$

$Formula of fabric thickness(t)=\frac{136\times \sqrt C_1 percentage}{1000 \times PPI_2} \times 2.54 +d_1$

$Formula of fabric thickness(t)=(\frac{136\times \sqrt 10 }{1000 \times 63.86})\times 2.54 +0.01627$

$Formula of fabric thickness(t)=(\frac{136\times 3.16 }{1000 \times 63.86}) \times 2.54 +0.01627$

$Formula of fabric thickness(t)=(\frac{429.76}{63860}) \times 2.54 +0.01627$

$Formula of fabric thickness(t)=0.006729 \times 2.54 +0.01627$

$Formula of fabric thickness(t)=0.01709 +0.01627$

$Fabric thickness(t)=0.033$ cm

Specific volume of the fabric(cm3/gm)=fabric thickness/fabric area density

Specific volume of the fabric(cm3/gm)=0.033/0.009115

Specific volume of the fabric(cm3/gm)=3.62 (Ans)

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