Estimating Fibre Properties

From transmission images of thin paper, or of shallow fibre suspensions of concentration c_mass, image analytic software counts fibres and projected contacts, measures variances of density [5].

Number of projected contacts per fibre:

$$n_{contacts}^{projected}= \frac{12 \omega n_{crowd}}{\pi \lambda}= \frac{2\omega\lambda}{\delta} c_{mass}$$ (29)

Plot n_contacts^projected against c_mass; slope $2\omega\lambda/\delta$

Decay of variance of areal density with size of inspection zones in random networks can yield averages of fibre properties.
Small-scale measurements:

$$Var(x) \approx \bar{\beta} \frac{\delta}{\omega}(1-\frac{x} {\pi \omega}(1+\frac{1}{A})) \ \ \ \ {\rm for} \ x\leq \omega$$ (30)

Large-scale measurements:

$$x^2 Var(x) \rightarrow \lambda\delta \bar{\beta} \ \ \ \ {\rm as} \ x\rightarrow\infty$$ (31)