GATE (TF) Textile 2016 Question Paper Solution | GATE/2016/TF/46

Question 46 (Textile Engineering & Fibre Science)

The surface area per unit volume (mm^-1) of a circular polyester fibre of 1.5 denier fineness and 1.38 g/cm³ density, ignoring the fibre ends, is __322 : 323__.

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Given in the question:
Fineness of polyester fibre=1.5 denier

Density of fibre=1.38 g/cm³

Surface area per unit volume=?

Formula:

$Volume of cylinder(fibre,V)=\pi \times r^2 \times h$

Where, r is the radius of the cylinder and h is the height of the cylinder

$Surface area of fibre(S)= 2\pi\times r \times h$

$Surface area per unit volume=\frac {Surface area of fibre}{Volume of fibre}$

$Surface area per unit volume=\frac {2\pi\times r \times h}{\pi \times r^2 \times h}$

$Surface area per unit volume=\frac {2}{r}$

and

By formula:

$Denier=\pi r^2\times 9000\times100\times density of fibre$

$Denier=3.14 \times r^2 \times 9\times10^5 \times 1.38$

$1.5=3.14 \times r^2 \times 9\times10^5 \times 1.38$

$1.5=38.99\times10^5 \times r^2$

$\frac{1.5\times 10^-5}{38.99}= r^2$

$0.0385\times 10^-5= r^2$

$0.00385\times 10^-4= r^2$

$\sqrt 0.00385\times 10^-2= r$

$0.062\times10^-2=r$

$r=0.062\times10^-2 cm$

$r=0.062\times10^-1 mm$

$r=0.0062 mm$

Now,
$Surface area per unit volume=\frac {2}{r}$

= $\frac{2}{0.0062}$
=322.58 mm

Surface area per unit volume=322.58 mm^-1 (Answer)