Analysing Flocculated Paper

Analysis of radiographic images of papers yields estimates of floc grammage and size distribution, from variance decay with scale [5].

$$Var(x) \approx \bar{\beta} \bar{G} \int^{\sqrt{2}x}_0(1-\fr... ...}{\pi \bar{D}}) \ b(r,x) \ dr, \ \ \ \ {\rm for} \ x\ll \bar{D}$$ (42)

The pdf b(r,x) for pairs of points in a square of side x, is known analytically [4] and we obtain:

$$Var(x) \approx \bar{\beta} \bar{G} (1-\frac{2x}{\pi \bar{D}}), \ \ \ \ {\rm for} \ x\ll \bar{D}$$ (43)

For more details see Dodson [5] and Farnood et al. [16].

Kerekes and Schell [22] observed:
Long fibre fraction $\bar{D}\approx 11$mm;
Short fibre fraction $\bar{D}\approx 2.3$mm
Expect from above:

$$\frac{Var(x)^{Long}}{Var(x)^{Short}}\approx 1+\frac{x}{5}, \ \ \ \ {\rm for} \ x\ll\bar{D}{\rm mm}$$ (44)

Figure 10 shows how the two parameters $\bar{D}$ and $\bar{G}$ were used to discriminate among the effectiveness of different refiner geometries from the point of view of formation potential and tolerance of the fibres for adverse forming conditions.

  

Figure 10: Formation Diagram for potential of a single pulp refined in three ways, from radiography of sheets made under four forming conditions. Same image analysis procedure for optical transmission bitmaps of shallow fibre suspensions. Spherical flocs project grammage G proportional to diameter D. Experimentally: $\sigma_D~\approx~1.2~\bar{D}^{2/3};$ analogous to pore sizes.

Farnood et al. [17] provide graphical representations of a wide range of simulated papers. They used lognormal disk diameter distributions parametrized by the mean $\bar{D}$, which represents the scale of the variability in the structure, and the mean disk areal density, $\bar{G},$ which represents the intensity or contrast in the areal density distribution. A tiling of such structures for typical ranges of parameters found for commercial papers is shown in Figure 11.

  

Figure 11: Simulated structures from lognormal distributions of disks with $\bar{G}$ representing intensity and $\bar{D}$ representing scale.