Mathematical modelling of syneresis of cheese curd

Mathematics and Computers in Simulation 65 (2004) 165–175

Mathematical modelling of syneresis of cheese curd

E. Tijskens∗, J. De BaerdemaekerLaboratory for Agro-Machinery and Agro-Processes, Catholic University Leuven,

Kasteelpark Arenberg 30, B-3001 Leuven, Belgium

Abstract

A mathematical model for syneresis of cheese curd in one dimension is presented based on the theory of mechanicsof porous media. The simple mathematics leave room for surprising physical considerations and several interest-ing observations are made. Model parameters are estimated and validated using one-dimensional experiments.The agreement between experiment and simulation is reasonable given the speculative nature of the constitutiveequations. The model also reproduces the qualitative aspects as observed by MRI experiments.© 2003 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords:Syneresis; Deforming porous media; Mass transfer

1. Introduction

In order to predict and control the process of soft cheese production, cheese-making companies haveexpressed their interest in a general mathematical model for the drainage of beds of curd grains. From aphysical point of view the manufacturing of soft cheeses consists of: (1) coagulating milk to form a gel,thecurd, consisting of a protein (casein) skeleton, and an interstitial fluid, thewhey, (2) cutting the gelinto small pieces (typically 2 cm cubes), (3) pouring the curd grains in a perforated mould to let the wheyevacuate from the network. The entire process typically takes about 15 h. The evacuation of the wheyis in fact a two step process. The first step is the expulsion of whey by the shrinking curd grains. Thisprocess is calledsyneresis. It is a porous flow with the porous medium defined by the protein skeleton.We refer to this porous medium as themicro-scale porosity. Once the whey is outside the curd grain itenters a porous medium on a larger scale defined by the curd grains themselves. This is referred to as themacro-scale porosity. Here, the channels available for fluid flow are much larger than the channels in thecurd grains, but the volume fraction of these channels with respect to the total volume of the curd bedbecomes very small soon after the curd grains are poured into the mould. This is because the curd grainsare slightly more dense (about 4%) than the whey and they deform readily due to the low strength of

∗ Corresponding author. Tel.:+32-16-32-8595; fax:+32-16-32-8590.E-mail addresses:[email protected] (E. Tijskens), [email protected](J. De Baerdemaeker).

0378-4754/$30.00 © 2003 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2003.09.016

166 E. Tijskens, J. De Baerdemaeker / Mathematics and Computers in Simulation 65 (2004) 165–175

the protein skeleton, so they get packed. In addition, at contacts between curd grains there is curd grainfusion [1]. In this paper, we will further concentrate on syneresis, as it is a basic building block in theentire process.

The modelling of syneresis of cheese curd is a difficult problem for several reasons. From the physicalpoint of view, the problem is incompletely understood on nearly every scale of interest. On the atomisticscale most work is of physico-chemical nature, and has led to the hairy micelle concept[2,3]. Only roughmodels of the physics of gel-formation have been developed[4]. The microscopic geometry of the proteinskeleton has been revealed by confocal laser scanning images[5] and by transmission electron microscopy[6]. They show a sponge-like structure with channel diameters of about 5�m. A major difficulty of theproblem is that this skeleton is not in a stable state. The protein material is continuously rearranging itselftowards equilibrium. In doing so, it exerts a pressure on the contained fluid, which responds by trying toescape at the curd grain boundaries while the curd grains continue to shrink, and their porosity decreases.Thus, to some extent, the shrinking curd grain can be regarded as a “self-squeezing sponge”. Most workon syneresis of curd refers to the macro-scale and is experimental[7–11]. Appropriate continuum modelsfor syneresis of curd must treat the curd as a porous medium in which each of the components–that isthe protein network and the interstitial fluid—is represented by an overlapping continuum[12–14]. As aporous medium cheese curd is certainly a very unusual material, as it has a very high porosity—initiallyabout 90%—and is very deformable. For example, when put on a flat surface, a curd grain deforms almostinstant taneously under its own weight (probably by rupturing of the network). On the other hand, whenthe curd grain is surrounded by whey, the deformation is slow and almost completely driven by syneresis.However, on the time scale of the entire process, the deformation of curd grains certainly cannot beneglected, as the average porosity evolves from 90 to about 45% implying a volume change of about50%. Clearly, the problem is not only a problem of fluid flow, but also of deformation of the curd grains.This makes the syneresis problem also a difficult computational problem, as the governing equationswill constitute a time dependent nonlinear system of coupled PDEs with a free moving boundary andlarge deformations, and although the basic equations governing the system at the macro-scale are fairlyobvious—conservation of mass and momentum for the whey and the protein network—the selection ofconstitutive equations describing the material properties is subject to much uncertainty.

Now that the general system and its difficulties are sketched, the paper will proceed as follows. Inview of all the uncertainties on constitutive equations associated with the problem, it was decided thata model for syneresis should first consider a simplified situation before the problem should be attackedin three-dimensional. Thus, a one-dimensional syneresis problem is sketchedSection 2and a basicmathematical model is developed. The model is simple, yet it identifies accurately the physical propertiesthat come into play and that are needed in a three-dimensional model. Its development has increasedour insight in the syneresis problem significantly.Section 3describes an experimental realisation of thismodel. It is used to estimate the material properties, and to validate the model. The experiments showthat the basic model requires some adaptation and a physical explanation for this is provided. Finally,Section 4summarises the conclusions.

2. Syneresis modelling in one-dimensional

Consider the following problem in which thin cylindrical layers of curd are prepared in a large petridisk (Fig. 1). Because the curd is prepared in the petri disk it is attached to the walls, the whey can only

E. Tijskens, J. De Baerdemaeker / Mathematics and Computers in Simulation 65 (2004) 165–175 167

0

h (t>0)h (t=0)

1D behaviour

z

Fig. 1. A one-dimensional syneresis problem.

be expelled at the top surface. In the centre of the disk the fluid flow is directed along the vertical axisbecause of symmetry. Consequently, the syneresis is one-dimensional at the centre of the disk.

According to Coussy’s treatment of the mechanics of porous media[12], the curd grain is treatedas a system of two overlapping continua, theskeleton, representing the protein network, and thefluid,representing the whey. Here, both components are considered here as idealincompressiblefluids, i.e. theyareinviscid. Hence, the mass balance for the fluid, respectively the skeleton reads:

∂φ

∂t+ ∂φV

∂z+ ∂qfl

∂z= 0, (1)

−∂φ

∂t+ ∂ (1 − φ) V

∂z= 0. (2)

Here,φ is the local (micro-scale) porosity,V the velocity of the skeleton particle,qfl the volume flux ofthe fluid relative to the skeleton,t represents time andz is a spatial co-ordinate along the vertical axis,pointing upwards. InEqs. (1) and (2), the constant densities have been divided away. Summing bothexpressions yields the total mass balance:

∂V

∂z+ ∂qfl

∂z= 0. (3)

This equation expresses that the skeleton has to move at the same speed as the fluid, but in the oppositedirection. This is a direct consequence of the assumption that both components are incompressible. Theslow movement of fluid allows for modelling the fluid flow by Darcy’s law:

qfl = K

[−∂p

∂z− ρflg

]. (4)

In Eq. (4), the permeabilityK is modelled by the Carman–Kozeny equation:

K(φ) = κφ3

(1 − φ)2d2

m. (5)

Here,dm is the mean channel diameter, andκ depends on the microstructure. For constantκ anddm, thepermeabilityK(φ) is monotonously increasing with the porosityφ. As the curd grain shrinks, the porosityobviously decreases, and thinking of the curd grain as a “self-squeezing sponge”, one would also expectthe mean channel diameterdm to decrease, so that the permeability also decreases. This is not the case,however, as is shown in experiments with curd by Dejmek[15]. It was found that constrained curd—thatis curd which is practically not allowed to shrink because it is attached to the wall—shows an increasingpermeability with time. As the porosity of constrained curd is almost constant, this suggests thatdm must


be increasing with time. This is another consequence of the dynamical rearrangement of the skeletontowards an equilibrium state. Thus, we assume:

dm(t) = D(t)dm(0),

D(t) = 1 + D∞(1 − e−αt),(6)

whereD(t) satisfiesD(t = 0) = 1. If we absorb the constant factordm(0)2 in k we obtain

K(φ) = kφ3

(1 − φ)2D(t)2. (7)

We proceed by decomposing the fluid pressure inside the curd grain in a hydrostatic componentph anda syneresis component or overpressureps:

p = ps + ph,

ph = p(z1) −∫ z

z1

gρ(z) dz′,(8)

where the reference levelz1 is taken to be the curd–whey interface. Hence, we obtain

qfl = K

[−∂ps

∂z+ (ρ − ρfl)g

]. (9)

It is unclear which density one has to take forρ. Two interpretations are possible. Either the fluid doesnot feel the weight of the skeleton and its hydrostatic pressure is caused by its own weight only, in whichcaseρ = ρfl and the second term inEq. (9)vanishes:

qfl = −K∂ps

∂z. (10)

At the other extreme, the fluid does feel the weight of the skeleton, i.e.ρ = ρcurd, and

qfl = K

[−∂ps

∂z+ (1 − φ)�ρg

]. (11)

Here, we defined

�ρ ≡ ρsk − ρfl ≈ 0.1 kg/m3. (12)

The extra driving force

f skw = (1 − φ)�ρg (13)

is identical to the driving force determining the settling of a system of fluid particles in suspension. Weshall therefor refer to it as the “settling” force. The obvious time evolution of a settling system is that theheavy particles accumulate at the bottom while the lighter interstitial fluid is pushed upwards. The firstcase, described byEq. (10), can be recovered fromEq. (11)by setting∆ρ = 0.

An argument in favour of the caseρ = ρfl, is that a curd grain submerged in whey can be observed not tocollapse under its own weight, as it does in free air. The observed deformation of a curd grain submergedin whey is due to syneresis (shrinking). The computations (see below) show that the magnitude of thisdriving force generally exceeds the syneresis component by two orders of magnitude, except where


the syneresis pressure gradient,∂ps/∂z, is very steep. The inclusion of the gravity term in simulationsaccordingly leads to unrealistically high predicted syneresis rates. It may, however, be argued that thisis due to the skeleton being treated as an inviscid fluid, while in reality its viscosity exceeds that of thefluid by several orders of magnitude, causing a finite resistance to deformation and balancing the gravityforces. That is, the settling force is compensated by the viscous friction of the skeleton and either both ornone have to be included.

For the syneresis pressureps, the following assumption seems acceptable:

ps = βδv − δv∞

δv(14)

for an infinitesimally small volumeδv. Here,δv∞ is the equilibrium value ofδv. By taking account ofthe definition of porosity and the incompressibility assumptionEq. (14)can be expressed in terms ofporosity:

ps = βφ − φ∞

1 − φ(15)

Finally, integratingEq. (3)to obtainV(z), and substituting the result intoEq. (1)we obtain a scalar partialdifferential equation in three-dimensional:

∂φ

∂t+ qfl(z0)

∂φ

∂z+ ∂(1 − φ)qfl

∂z= 0 (16)

wherez0 is an arbitrary reference level. It should be noted thatEq. (3) can only be integrated in theone-dimensional case. In two or more dimensions,Eq. (3)can no longer be integrated to obtainV(z), anda momentum balance becomes necessary to close the system of equations.Eq. (16)is subject to boundaryconditions

φ(z1, t > 0) = φ∞ (17)

qfl(z = 0, t > 0) = 0 (18)

and initial condition

φ(0 ≤ z ≤ z1, t = 0) = φ0 (19)

By letting the reference levelz0 coincide withz = 0, which is the impermeable boundary (qfl = 0) at thebottom of the petri disk, the middle term inEq. (16)vanishes. The partial differentialEqs. (16)–(19)isdiscretised using a standard Galerkin finite element method (see, e.g.[16]). As the problem is nonlinear itis transformed into a series of linear PDEs by means of a Newton–Raphson scheme[17]. The Jacobiansneeded for the Newton–Raphson scheme are computed exactly and efficiently by automatic differentiationusing Fastder++ [18,19].

The typical time evolution of the porosity profile along the central axis of the experimental set-up(seeFig. 1) obtained with this model is shown inFig. 2, without inclusion of the settling force, and inFig. 3, with inclusion of the settling force. The difference between both is obvious. As the settling forcedominates the syneresis pressure by approximately two orders of magnitude, settling is the dominantprocess, yielding a positive slope throughout most of the domain, and an almost vertical slope at thecurd–whey interface (Fig. 3). Thus, the porosity decreases towards the bottom. Without settling force, the


Fig. 2. Typical time evolution of the porosity profile when the settling force is neglected (problem parameters:h(t = 0) = 0.02,kβ = 10−10, φ0 = 0.9, φ∞ = 0.5, H(z) = D(t) = 1, all quantities in SI units).

Fig. 3. Typical time evolution of porosity profile with the settling force included (same problem parameters as inFig. 2).


Fig. 4. MRI image of a system of syneresing curd grains after 2 hours and 37 minutes. The bottom of the mould is perforated. Arecipient is positioned below the mould. Curd grain contours are visible as low humidity areas. The arrows show the directionof fluid transport at the boundary.

profile shows a decrease towards the top of the curd–whey interface and a much slower time evolution.Only the latter is in agreement with MRI observations[20]. Fig. 4 clearly shows that syneresing curdgrains develop a decreasing porosity towards the curd–whey interface. Curd grain contours are visibleas low humidity areas. These low humidity areas seem to develop rather uniformly all around the grainsshowing that the porosity profile across a curd grain is quite independent of the orientation of the interfacewith respect to the vertical axis.

3. Parameter estimation and validation

Experiments of a one-dimensional model system for syneresis of curd have been carried out by Lodaiteet al. [21]. Thin layers of curd (initial height,h0 = 3, 6.5, 10.5, 20 and 35 mm) were prepared in alarge petri disk. After preparation the curd surface is dry, and surface tension counteracts the syneresispressure. As soon as the surface is wetted, syneresis proceeds. When the surface is not wetted, theskeleton will still rearrange itself, but at a constant porosity. This process is referred to asageing. Theheight of the curd layer was measured by means of a laser displacement sensor (seeFig. 1). The shrinkage,defined as

h(t) − h(t = 0) (20)

is plotted in Fig. 5. In one dimension, the shrinkage is related to the porosityφ by the volumebalance:

h(t) − h(0) =∫ h(t)

0φ(t) dh′ −

∫ h(0)

0φ(0) dh′. (21)

Eq. (16)is entirely local in nature and therefore predicts the initial behaviour to be independent of thethickness of the curd layer. This is demonstrated inFig. 6, where the slope of all curves at timet = 0


0 1000 2000 3000 4000 5000 6000 7000 8000-6

-5

-4

-3

-2

-1

0x 10

-3

time [s]

shrin

kage

[m]

0.0030

0.0065

0.0105

0.0200

0.0350

Fig. 5. Experimental shrinkage as defined byEq. (20)of the curd layer as a function of time for different initial heights (m),h(t = 0) = {0.003, 0.0065, 0.0105, 0.020 and 0.035} at pH= 6.0.

is identical. The experiments, however, shown inFig. 5, show a quite different picture. There, the slopeof the curve at timet = 0 clearly varies systematically with the initial height of the curd bedh(t = 0).This thickness dependence is thought to originate from the fact that the skeleton is attached to the bottomof the petri disk and therefore cannot pursue its normal time evolution. This is called thewall effect. To

0 1000 2000 3000 4000 5000 6000 7000 8000-8

-7

-6

-5

-4

-3

-2

-1

0x 10

-3

time [s]

shrin

kage

h-h

0 []

h040

Fig. 6. Effect of the initial height on the shrinkageh(t) − h0 in the absence of a wall effect (H(z) = 1). From top to bottom:h(t = 0) = {0.01, 0.02, 0.04 and 0.08} (kβ = 5 × 10−11, other model parameters as inFig. 2).


Table 1Estimated model parameters for the model described byEqs. (16)–(19) and (22), accounting for the wall effect (all quantities inSI units)

kβ = 2.20× 10−11 φ∞ = −0.00699D∞ = 0.304 α = 4.38× 10−4

H0 = 0.0989 λ = 88.6

account for such a wall effect, the constitutive equation for the permeability was adapted as

K′ = H(z)κφ3

(1 − φ)2D(t)2

H(z) = H0 + (1 − H0)(1 − e−λz)

(22)

The effect of the functionH(z) is to diminish the increase of the mean channel diameter towards thebottom of the petri disk, where the skeleton is attached to the glass wall.

The model parameters (seeTable 1) have been estimated indirectly for two of the five experiments(h(t = 0) = 0.00105 and 0.0020) by minimising the error function√∑

i

(hsim,h0=0.0105(ti) − hexp,h0=0.0105(ti))2 +

√∑i

(hsim,h0=0.020(ti) − hexp,h0=0.020(ti))2 (23)

The remaining three experiments (h(t = 0) = 0.0030, 0.0065 and 0.035) are used for validation. Theresults are shown inFig. 7. They are remarkably good considering the speculative nature of the constitutive

0 1000 2000 3000 4000 5000 6000 7000 8000-6

-5

-4

-3

-2

-1

0x 10

-3

time [s]

sh

rin

ka

ge

h-h

0[m

]

0.0030

0.0065

0.0105

0.0200

0.0350

Fig. 7. Comparison of experimental (seeFig. 5) and simulated shrinkage curves. The simulated curves (dashed lines) have beenobtained using the parameters ofTable 1. The curves with initial heightsh(t = 0) = {0.00105 and 0.0020} have been used forparameter estimation. The remaining curves for validation.


equations and the simplicity of the model. It also shows that the difference in syneresis rate for differentinitial thicknesses is probably due to the external influence of the bottom wall rather than due to a nonlocaldriving force. The slightly negative value forφ∞ is considered an indication thatφ∞ should in fact equalzero, although in practice the porosity is never observed to approach zero. The reason for this is that, astime goes by, the skeleton becomes firmer due to ageing, and is no longer able to rearrange. The functionH(z) approaches unity aszgoes to infinity. In practice, using the estimated parameters fromTable 1, H(z)∼ 0.95 forz = 0.035 m. Thus, the wall effect is estimated to penetrate 3.5–4.5 cm into the curd.

4. Conclusions

The one-dimensional syneresis of curd can be modelled by considering the skeleton and the fluid as anideal fluid (incompressible and inviscid). The fact that the skeleton actually has a very high viscosity, iscompensated by neglecting the settling forcef sk

w (13). The governingEqs. (16)–(19)permit a reasonablygood prediction of the experiments, given the speculative nature of the constitutive equations for thepermeability (7) and (22) and the syneresis pressure (15). If the curd is attached to the wall, this has asignificant effect on the ability of the skeleton to dynamically rearrange and on the syneresis process. Thepenetration depth of this wall effect is estimated to be in the range 0.035–0.045 m.

Despite the simplicity of the model, it clearly identifies all the important properties that will be neededto develop a full three-dimensional model.

The model is in fact overly simple, yet it accurately identifies the physical properties that come into playand that will be needed in a full three-dimensional model of syneresing curd grains. A first improvementwould be to treat the skeleton as a viscous fluid, and incorporate the settling force.

Acknowledgements

This research was sponsored by the European Community under contract Fair-CT-1096. The authorsacknowledge Professor Peter Dejmek and Dr. Karin Östergren of Lund University (Sweden) and Dr.François Mariette of Cemagref Rennes (France) for numerous interesting discussions and for providingthe experimental data.

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Mathematical modelling of syneresis of cheese curd

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