Calculating Random Fibre Contacts

Expected number of fibres in cube of side $\lambda$

$$n_{crowd} \ \frac{6 \lambda^3}{\pi \lambda^3} = \frac{6}{\pi}\ n_{crowd}$$ (2)

Number of intersections with one fibre

$$\frac{[\lambda\overline{\sin\theta}]\times 2 [width]} {[Area \ of \ base \ of \ cube]}\times \frac{6 n_{crowd}}{\pi}$$ (3)

Here, $\lambda\overline{\sin\theta}$ is the mean projected length of one fibre onto cube base:

$$\lambda\overline{\sin\theta} = \frac{\lambda}{4\pi}\int_{\p... ...{2\pi} \sin^2\theta \ d\theta \ d\phi = \frac{\pi \lambda}{4}$$ (4)

Hence

$$n_{contacts}=\frac{6}{\pi} \ n_{crowd}\ \frac{\pi \omega}{2... ...\frac{3\omega }{\lambda} \ n_{crowd} = \frac{3}{A} \ n_{crowd}$$ (5)

Variance of fibre contacts per fibre
We expect the number of contacts per fibre to vary from fibre to fibre as a Poisson random variable, so has variance equal to the mean in the random case; variance exceeding the mean for flocculated.

For more information see Dodson [5].