Figure 2 - The two stages of a filtration and expression process
separated by the random close packing (see the middle figure)
representing the transition from filtration to expression.
The topic of cake filtration goes back to the paper by D’Arcy, as early
as 1856, on the flow of water through sands and sand stones, and was
then further investigated in the 1920s and 1930s in papers by Terzaghi
[6], Ruth [7] and Carman [8]. Ruth [7] referred to the
“widespread idea that the mechanism of filtration is one of such
extreme variability that the engineer may perhaps never hope to find law
and order in its operation”. Since then, the topic has challenged many
experimentalists and modelling researchers: the review by Olivier et al.
[9] in 2007 already cites 159 papers. An updated review of the topic
is beyond the scope of the current paper. We will just focus on our
novel expression model and discuss where it differs from earlier models.
The basic filtration equations due to Ruth [7], still in use today,
relate filtrate volume as a function of time to pressure drop (over
filter cake and filter medium) in terms of specific resistance and
volume of the filter cake. Terzaghi [6], interested in consolidation
of clay due to a load on top, assumed that layer (or cake) thickness,
compressibility and permeability remain constant. Tiller et al. [10]
combined Darcy’s law for the flow through a porous medium with the
notions of solids pressure and consolidation, which not only are
relevant to soil mechanics but also to the filter cake of current
interest. The common models for constant pressure filtration lead to a
quadratic relationship between filtration time and filtrate volume
[11, 12]. Stickland et al. [5] reviewed deviations from such a
quadratic behaviour. Owolarafe et al. [13] reported about a model
for expressing oil from oil palm fruit on the basis of Darcy’s law for a
cylindrical geometry and supplemented with several empirical relations.
Shirato et al. [14] distinguished between primary consolidation and
secondary consolidation (due to creep), releasing the assumption of
instantaneous mechanical equilibrium made in the Terzaghi model. Venter
et al. [15] successfully applied the Shirato model to the expression
behavior of cocoa liquor from finely grinded cocoa nibs. The Shirato
model was also used by Abduh et al. [16] for studying the expression
of rubber seed oil from dehulled rubber seeds in a hydraulic press.
Buttersack [17] developed a two-zone model. In the first zone, with
a void fraction between the initial value and a threshold value, the
solids-solids interaction is ignored. When and where the water content
falls short of the threshold value, a second zone consisting of an
solids network with increasing elasticity modulus is formed.
Filtration and consolidation are not regarded as subsequent stages, but
are assumed to occur alongside each other to a extent varying in
time. This elastic network may be associated with the dense sphere
packing for a filter cake composed of spherical particles. His model
gave satisfactory results for press-dewatering of materials such as
protein, sawdust, semi-solid clay and sugar-beet tissue.
In an increasingly sophisticated approach, Lanoisellé et al. [18]
studied pressure filtration of cellular material (as applied in various
agro-food processes) and pointed out that for cellular filter cakes the
expression step is much more complex than for mineral cakes. This was
already appreciated by Mrema and McNulty [19] who built their model
of oil expression from oil seeds upon three elements: (1) the oil flow
through the cell wall pores; (2) the oil flow in the inter-kernel voids;
and (3) consolidation of the oil seed cake. More or less similarly,
Lanoisellé’s “Liquid-Containing Biporous Particles Expression Model”
describes liquid transport within a network of three different volume
fractions of a cake: extra-particle, extracellular and intracellular
with different behavior. The resulting system of three complex partial
differential equations is solved for a constant imposed pressure and
allows for the calculation of the total layer settlement as well as the
deformation of the separate extra-particle, extracellular and
intracellular volumes. The more recent paper by Petryk and Vorobiev
[20] uses a similar model to describe the expression of soft plant
materials. However, in both papers, the cellular material properties are
very different from those of the fat crystal aggregates of current
interest while the pressures applied are much higher than in a
filtration process of edible fat crystals.
Kamst et al. [21, 22] modified the old empirical non-linear
viscoelastic model due to Nutting (1921) to describe the compressibility
of palm oil filter cakes which are highly compressible and viscoelastic.
In addition, these authors used a strain hardening model to accommodate
the effect of the pressure history of the filter cake. These models,
combined with an empirical relation for the permeability, made up a
novel expression model. The numerical implementation was done with a
finite difference scheme exploiting an exponential grid and a variable
time step. This model ignores the Kozeny-Carman equation, just like Tien
and Ramarao [23] question the applicability of the Kozeny-Carman
equation to consolidating cakes, after Grace already did the same in
1953.
Kamst’s expression model predicts a pressure of 4.7 bar above which the
solid fat content (SFC) does not increase anymore. Another finding of
the Kamst model – relevant for us – was that applying a constant
pressure, compared to a time-dependent pressure profile with the same
end pressure, does not lead to a higher eventual SFC, although the
option of applying different pressure-time profiles was not studied.
Further, some of Kamst’s experiments and simulations exceed the time
scales of our process by an order of magnitude. Most importantly,
however, their model ignores the biporous nature of the filter cake (in
their case, palm oil), while the double porosity is a very attractive
element of Lanoisellé’s model, given the fat crystal slurries of current
interest.
After filling of a filter chamber (during which some liquid already may
leave the chamber), the first step is filtration (see figure 2, top): as
long as the inter-aggregate porosity is smaller than the random close
packing ε rcp (=0.64). When pressurization
continues, the stage of expression or consolidation is entered in which
the aggregates get compressed and squeezed (see figure 2, bottom). The
expression model we developed and describe in this paper builds on the
above three elements described by Mrema and McNulty [19] and on
Lanoisellé’s biporous model [18] while considering the typical
behaviour and physical properties of the edible fat crystal aggregates
of current interest and the pressure levels of the pertinent expression
process. The crystal aggregates will therefore be considered as
additional sources of oil when squeezed in the expression stage. For the
sake of simplicity, we will consider a flat cake with (essentially) 1-D
transport of liquid, as a result of a unidirectional pressure applied at
the right-hand side of the cake, towards a filter cloth at the left-hand
side through which the liquid leaves the cake. We do assume that the
agglomerates stay intact, i.e. do not break up when squeezed.