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Proved that mean pore diameter of nonwoven fabric dp is independent of fibre diameter d and dependent on fibre length l.

Assume all fibers were randomly deposited in an elementary plane then prove that the mean pore diameter of the nonwoven fabric dp is independent of fiber diameter and dependent on fiber length l.

Know textile
Fig: randomly oriented fibres
questionsoftextileblog
Know textile
Let the fibers be distributed randomly in an elementary plan of unit area and the probability of n fibers that present per unit area is given by Poisson distribution

P(n) = e-c * cn/n!    , where c is total projected area fibres per unit area of the plane (total coverage).

Evendently, c=nld, 

where  l= fibre lingth

and d=fibre width (diameter)

The fraction of area covered by one fibre is:

P(0) = e-c = ξ        -----------(i)

Clearly, (1-ξ ) is the fraction of the unit area covered by fibres.

The total area Ac occupied by all fibres croosing per unit area of the plane is:

Ac = ∑n=2n→∝(n-1)P(n)

= ∑n=2n→∝(n-1)e-c * cn/n!

= e-c[c2/2! + 2c3/3! + ... + (n-1)*cn/n! + ...]

= e-c[c0/0! + c1/1! + 2c2/2! + ... + ncn/n! + ... - (c0/0! + c1/1! + ... + cn/n! + ...)]

= e-c[c0/0! + (c1/1! + c2/1! + ... + cn/(n-1)! + ...) - (c0/0! + c1/1! + ... + cn/n! + ...)]

= e-c[c0/0! + c(c0/1! + c1/1! + ... + cn-1/(n-1)! + ...) - (c0/0! + c1/1! + ... + cn/n! + ...)]

= e-c[1 + cec - ec]

= e-c + c - 1

= c - (1 - ξ)    (using equation i)

thus, the total area of fibre crossing Ac is difference between the total coverage c and fraction of the unit area covered by fibres (1 - ξ).

Know textile
Fig: area of one fibre
The area occupied by one fibre crossing is given by: 

ac = (π⁄2 d) * d = π⁄2 d2 = π⁄2 d2        -----------(i)

then number of fibre crossing per unit area of the plane is:
nc = Ac/ac
= (e-c + c - 1)/Πd2/2
= 2(e-c + c - 1)/Πd2
= 2/Πd2[(c0/0! - c1/1! + c2/2! - ...) + c1/1! - c0/0!]
= 2/Πd2[c2/2! + c3/3! + ...]
≅ 2/Πd2 * c2/2!
= c2/Πd2
= (nld)2/Πd2
= (nl)2/Π                      -----------(ii)
Since fibre crossing form the vertices of polygons in the plane. Each polygon simulates a pore.
So according to Kallmes and Corte,
the expression for pores is:
np = (nc - n)e-c

mean cross-sectional area of pore is:

ap = ξ/(nc-n)-e-c

= 1/nc-n

= 1/nc(1-n/nc)

≅ 1/nc

= π /(nl)2 -----------(iii)          (using equation ii)

we consider the mean cross-sectional area of pore is equal to area of a circle of diameter dp ,then

ap = Πdp2⁄4 = π⁄(nl)2 

Know textile
Fig: pore diameter

=> dp2 = 4⁄(nl)2 

=> dp = 2⁄nl


Hence, it is proved that mean pore diameter of nonwoven fabric dp is independent of fibre diameter d and dependent on fibre length l.

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unknown said…
helpful👏🏻
10 May
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