Some basics
The volume reduction of a fat crystal aggregate upon compression, or
squeezing, implies the aggregate must release oil, as the intrinsic
densities of the oil and fat may be taken constant. On the analogy of
Lanoisellé’s biporous model [18], we therefore distinguish between
the inter-aggregate liquid (surrounding the crystal aggregates) with
volume fraction ε 1 and the liquid contained
inside the aggregates with porosity ε 2. We
presume that the pores inside the aggregates are smaller than the
inter-aggregate pores by at least an order of magnitude. We define the
complements of the above liquid volume fractions: the inter-aggregate
solidosity (or packing fraction) s 1 = 1-ε 1 , the aggregate solidositys 2= 1- ε 2 and the total
solidosity s =s 1s 2. As very common in
filtration and expression paper, we work in terms of void ratios
denoting the volume of liquid per volume of solids; in our case, the
inter-aggregate void ratio e 1 =ε 1/s 1 and the
(intra-)aggregate void ratio e 2 =ε 2/s 2. The latter void
ratio denotes the aggregate pore volume per solid fat volume. The total
void ratio e denotes the combined volume of intra-aggregate and
inter-aggregate liquid per volume of solids. Owing to the above s= s 1s 2 , we can write
All volume fractions, solidosities and void ratios vary spatially and in
time. Our model aims at resolving them.
Following e.g., Terzaghi [6], Sørensen et al. [24], Kamst et al.
[21] and Landman and White [12], we apply the Lagrangian or
material coordinate ω , defined by
where x denotes the spatial coordinate in the direction of the
flow towards the cloth filter, with x =0 at the high-pressure end.
This transformation simplifies the equation(s) – see e.g. ,
Smiles [25] – and facilitates the numerical solution. In the
material coordinate system, the only flow is that of the liquid relative
to the solids. The liquid flux passing the solids is denoted by uand is related to the linear liquid and solids velocities by
In exploiting this material coordinate system we deviate from the
analyses presented by Lanoisellé et al. [18] and Petryk and Vorobiev
[20].