The expression model
Due to the distinction between inter-aggregate and intra-aggregate liquid, we need two continuity equations with a source and sink term, respectively, at the RHS. Thanks to the use of the material coordinateω , we arrive at simple continuity equations for the inter-aggregate oil
and for the intra-aggregate oil
respectively, in which q in the source/sink terms at the RHSs denotes the release, per aggregate volume (in s-1), of inter-aggregate oil from the aggregates. We have dropped the convective term in Eq. (1.4), since as long as the liquid stays within the aggregate pores, its velocity (relative to the solids) is zero. In Eq. (1.4), the total solidosity occurs within the time derivative. Our biporous model essentially differs from the simple single continuity equation ∂e/∂t = ∂u/∂ω used by Sørensen et al. [24] and Kamst et al. [21].
The (local) flux u depends on the (local) pressure gradient in the liquid phase and is assumed to obey Darcy’s law with permeabilityk . The convective term of Eq. (1.4) is then rewritten:
The liquid pressure balances the stress in the deforming filter cake (see e.g. , Olivier et al. [9]) :
while an elastic modulus E connects the solids pressureps with the logarithmic strain:
with δ standing for the thickness of the filter cake and the subscript 0 denoting initial values, before cake deformation sets in. Applying the chain rule twice, using Eqs. (1.7) and (1.8), and eliminating the total solidosity s results in
We should realize that in a non-linearly elastic medium the elastic modulus depends on the filter cake strain itself, i.e. E =E (e 1, e 2). These manipulations turn the (seemingly) convective term of Eq. (1.4) into a diffusive term. Such a diffusive term is not uncommon: see e.g. , Tosun [26], Sørensen et al. [24], Kamst et al. [21], and Olivier et al. [9]. As a matter of fact, the basic idea can already be found in the classical Terzaghi paper dated as early as 1923 [6].
Substituting Eq. (1.9) into Eq. (1.4) and re-writing the solidositiess and s 2 in terms ofe 1 and e 2 results in
while Eq. (1.5) can be rewritten as
The next step is to find an expression for the release rate q . Different from Mrema and McNulty [19], we assume the flux out of the aggregates is Darcian, with a permeability k 2 =k 2(e 2) associated with the aggregates, through the specific area a = 6/da for the spherulitic aggregates of constant average size da . The pressure gradient can be transformed as above, resulting in
The above Eqs. (1.10) and (1.12) contain the cake propertiesk 1, k 2 and E which all are dependent on the pertinent the pertinent void ratios. We need empirical correlations for these parameters. As, according to Tien and Ramarao [23], the Kozeny-Carman relation is not valid under consolidating conditions, we use the Meyer and Smith [27] correlation
For k 1, we use void ratioe 1 and aggregate size da , while k 2 needs e 2 and the typical diameter dc of the individual crystals that build the agglomerate. Fitting an exponential function through data for strainmeasured at varying constant load ppresults in an expression of the type
Using Eq. (1.8) then results in the expression
The eventual set of the two partial differential equations fore 1 and e 2 then is
in which
Ce is a type of diffusion coefficient, in the consolidation literature denoted as a modified consolidation coefficient [9, 24]. While this coefficient in a real-life expression process is varying with position and in time, in many papers (e.g. , [14], [28]) it is treated as a constant: this simplifies solving the consolidation equation which is a second-order partial differential equation. Kamst et al. [21], however, appreciate the consolidation coefficient (also) depends on local cake porosity and compressibility. The review paper by Olivier et al. [9] cites a number of authors (among which [5]) who all use similar relationships for diffusivity or consolidation coefficient. Our expression forCe in Eq. (1.18) is essentially different from earlier proposals due to the biporous character of our fat crystal slurry as a result of which it includes both the intra-aggregate and the inter-aggregate solidosities. In addition, our consolidation equation, Eq. (1.16), contains a source term which to the best of our knowledge is a novelty. Finally, our model looks much simpler than Lanoisellé’s.
In more general terms, our expression model is a rheological model composed of two dashpots in series parallel to a spring. The double porous nature of the fat crystal aggregate filter cake is represented as a series of two dashpots described with the Meyer & Smith correlation for the permeability (rather than the Kozeny-Carman relation). The spring is due to the elastic modulus that can be determined experimentally with a constant load test.