###### Question 46 (Textile Engineering & Fibre Science)

Two types of polyester staple fibres of fineness 3 and 6 denier and having the same length are mixed in a ratio of 2:3 by weight. The mean fibre fineness (denier) of the mix (rounded off to 2 decimal places) is __**4.25 to 4.35**__.

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**Given in the question**:

Fineness of first polyester fibre=**3 denier**

Fineness of second polyester fibre=**6 denier**

Both have same length in the ratio **2:3** by weight

The mean fibre fineness of the mix=**?****Calculation:**

Let us assume the total weight of the mix(**w**) =**100 gm**

Then,

Weight of the first polyester fibre in the mix (**w _{1}**)=100 x (2/5)

**w**=

_{1}**40**

**gm**

Weight of the second polyester fibre in the mix (

**w**)=100 x (3/5)

_{2}**w**=

_{2}**60 gm**

Here,

**L**

_{1}=L_{2}Where ,

**L**Length of first polyester fibre

_{1}=and

**L**Length of second polyester fibre

_{2}=**By definition of denier**– Denier=weight in gm/9000 meter length

**From first polyester fibre**:

Weight of 9000 meter Length fibre=3 gm

1 gm weight have=9000/3

Then,

40 gm weight have=9000 x 40/3

**L**

_{1}=120000 mtr**From second polyester fibre:**

Weight of 9000 meter Length fibre=6 gm

1 gm weight have=9000/6

Then,

60 gm weight have=9000 x 60/6

**L**

_{2}=90000 mtr**Total length of mix (L)=L**

_{1}+L_{2}L=120000+90000

**L=210000 mtr**

**By definition of denier**–

We have,

Weight 210000 mtr length of polyester =100 gm

Then ,

Weight of 1 mtr length of polyester=100/210000

Weight of 9000 mtr length of polyester=100 x 9000/210000

**Fineness of the mix=4.28 denier**(Answer)

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