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].