Random Fibre Suspensions: Crowding and Contact Numbers

Fibre mass density c_mass
Fibre volume density c_vol
Fibre length $\lambda,$ diameter $\omega,$ mass/length $\delta.$

  

Figure 2: Representation of the crowding number.

Kerekes and Schell [21] developed further the concepts of Mason [26,24,25,27] and introduced a number (they called it a factor) to represent the average state of fibre crowding in a suspension. Crowding number:
Expected number of fibres
in sphere of diameter $\lambda$

$$n_{crowd}=\frac{\pi}{6} \frac{c_{mass} \lambda^2} {\delta} = \frac{2}{3}A^2 c_{vol}$$

Contact number:
Expected number of fibre contacts per fibre

$$n_{contacts}= \frac{3\omega n_{crowd}}{\lambda}= \frac{3 n_{crowd}}{A} =2A c_{vol}$$

  

Figure 3: Crowding number $n_{crowd}=\frac {\pi \bar{\lambda}^{2} c}{6 \delta}$ plotted against mean fibre length $\bar{\lambda}$ and mean coarseness $\delta,$ for $c=0.5\%.$

Contacts per unit mass of fibre:

$$\frac{n_{contacts}}{\lambda\delta} = \frac{\pi\omega}{2\delta^2} c_{mass}$$

Which emphasises the importance of $\delta.$

For more information see Dodson [5].