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 d_p is independent of fiber diameter and dependent on fiber length l.

Fig: randomly oriented fibres

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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 A_c occupied by all fibres croosing per unit area of the plane is:

A_c = ∑_n=2^n→∝(n-1)P(n)

= ∑_n=2^n→∝(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 + ce^c - e^c]

= e^-c + c - 1

= c - (1 - ξ)    (using equation i)

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

Fig: area of one fibre

The area occupied by one fibre crossing is given by: 

a_c = (^π⁄₂ d) * d = ^π⁄₂ d² = ^π⁄₂ d²        -----------(i)

then number of fibre crossing per unit area of the plane is:

n_c = ^A_c/_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:

n_p = (n_c - n)e^-c

mean cross-sectional area of pore is:

a_p = ξ/(n_c-n)^-e-c

= 1/n_c-n

= 1/n_c(1-n/n_c)

≅ 1/n_c

= 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 d_p ,then

a_p = Πd_p²⁄4 = ^π⁄_(nl)² 

Fig: pore diameter

=> d_p² = ⁴⁄_(nl)² 

=> d_p = ²⁄_nl

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