rolling of mozzarella cheese: experiments and simulations

Journal of Food Engineering 91 (2009) 269–279

Contents lists available at ScienceDirect

Journal of Food Engineering

journal homepage: www.elsevier .com/locate / j foodeng

Rolling of mozzarella cheese: Experiments and simulations

Evan Mitsoulis a,*, Savvas G. Hatzikiriakos b

a School of Mining Engineering and Metallurgy, National Technical University of Athens, 9 Heroon Polytechniou, Zografou, 157 80 Athens, Greeceb Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, BC, Canada V6T 1Z4

a r t i c l e i n f o

Article history:Received 12 April 2008Received in revised form 25 August 2008Accepted 3 September 2008Available online 12 September 2008

Keywords:RollingMozzarella cheeseViscoplasticityYield stressYielded/unyielded regionsHerschel–Bulkley fluidsSheet thicknessTorque measurements

0260-8774/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.jfoodeng.2008.09.003

* Corresponding author. Tel.: +30 210 772 2163; faE-mail address: [email protected] (E. Mitsou

a b s t r a c t

Mozzarella cheese is a visco-elastic-plastic shear-thinning material with a yield stress, which in a shearflow may be approximated by the Herschel–Bulkley model. The material was rolled in the SER exten-sional rheometer used as a rolling device with an aspect ratio of radius to minimum gap R/H0 = 4.3.For different feed thickness ratios and roll speeds, the exit thickness ratios and torques were measuredand found to increase substantially with roll speed and initial sample thickness. Two-dimensional finiteelement simulations based on the rolling geometry and the rheological data provide yielded/unyieldedzones, pressure, and stress distributions along the rolls. The results from the simulations are in goodagreement with the experimental torque values but underestimate (by an order of magnitude) the thick-ness of the exiting sheets. It is argued that the latter is due to strong viscoelastic effects, which manifestthemselves in free-surface flows, where (extrudate) swell becomes significant.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Rolling (or sheeting) between counter-rotating rolls is mainlyused by the food industry as a forming process for a wide rangeof products (Levine and Drew, 1990; Xiao et al., 2007). The rollingprocess is akin to calendering, which is used in many industries,such as the paper, plastics, rubber, and steel industries, for the pro-duction of rolled sheets of specific thickness and final appearance.The process is shown schematically in Fig. 1, where a material en-ters as a sheet of finite thickness 2Hf and exits as a sheet of reducedthickness 2H (Middleman, 1977). In contrast to calendering, wherethe aspect ratio of roll radius to minimum gap R/H0 > 100, rollinghas small aspect ratios, usually R/H0 < 10.

Interest in a better understanding of the process arises from thefact that rolling has been reported to have an impact on several as-pects of food’s behaviour in subsequent process steps and on theproperties of the final products (see, e.g., Engmann et al., 2005,and references therein for the special case of bread dough forwhich most rolling studies have been done). Although mozzarellacheese is not commercially processed by means of rolling, such apossibility is explored here. The motivation is that post-productionprocesses of cheese can be done to increase the shelf life of cheeseor to add value to the cheese by improving its texture, shape andoverall commercial attractiveness. Application of processing tech-

ll rights reserved.

x: +30 210 772 2251.lis).

niques that are widely used in the processing of commercial syn-thetic polymers might allow the development of novel foodproducts. For example: (a) co-extruding different types of cheesesusing an industrial extruder to manufacture a unique product; (b)shaping cheese using an extruder and dies with unique profiles tocreate cheese products with commercially attractive profiles; (c)rolling those cheese profiles to change their shape for commercialreasons. In addition, understanding the performance of cheese dur-ing these operations may be beneficial to the dairy industry as theymay identify alternative ways of more efficient and economicalways of continuous processing. In fact, a recent US patent illus-trates how extrusion process can be used to manufacture stringcheese (Cortes-Martines et al., 2005).

Experiments first are performed to examine the flowability andshapeability of mozzarella cheese in rolling. A two-dimensional(2D) flow analysis of rolling is then undertaken for viscoplasticmaterials using the continuous regularized Herschel–Bulkley–Papanastasiou model, which has shown good predictive capabili-ties of yielded/unyielded regions in other flows of viscoplasticmaterials (Mitsoulis et al., 1993). The parameters of the rheologicalconstitutive equations are obtained from Muliawan and Hatzikiria-kos (2007, 2008). The rheology of this material was found to becomplicated, exhibiting many phenomena, such as yield stressand time-dependent properties (Muliawan and Hatzikiriakos,2007). Obviously, the combination of process (rolling) and materialto be processed (mozzarella cheese) is a subject worthy of furtherinvestigation, since many unanswered questions still persist.

mailto:[email protected]

http://www.sciencedirect.com/science/journal/02608774

http://www.elsevier.com/locate/jfoodeng

Lo

Master Slave Drum Drum

2H

ω Rω

IntermeshingGears

Sample

Master Drum

Slave Drum

θ 2H0

θ

2Hf

+ +

Fig. 1. Schematic of the setup for the roll-forming experiments using theSentmanat extensional rheometer (SER); R is the radius of the SER drum/roller, xis the angular roll speed, Tq is the torque needed to roll one of the drums as theother is connected by means of intermeshing gears, 2H0 is the minimum gapbetween the rolls, 2Hf is the feed thickness of the sample, 2H is the exit thickness ofthe sample, L0 is the distance between the centres of the rolls, and h is the anglecorresponding to the arc between attachment and detachment points.

270 E. Mitsoulis, S.G. Hatzikiriakos / Journal of Food Engineering 91 (2009) 269–279

2. Experimental

2.1. Materials

Best Buy mozzarella cheese (Lucerne Foods, Calgary, AB, Can-ada) was used as the material for this study. Table 1 summarizesits chemical composition. To assess the consistency of the cheese,

Table 1Compositional analysis of mozzarella cheese

Component Content Error Method

Carbohydrate 6.08 (wt%) – Calculateda

Fat 25.29 (wt%) ±0.1 (wt%) AOAC 991.36 (modifiedb)Moisture 42.44 (wt%) ±0.1 (wt%) AOAC 95.46Protein-total 22.6 (wt%) ±0.5 (wt%) AOAC 981.10 (modifiedc)Calories 3.42 (kcal g�1) – Atwated calculationCalcium 7.45 (mg g�1) – AOAC 985.35 (modifiedd)

a Carbohydrate content approximated as (100-wt% moisture-wt% protein-wt%fat-wt% ash).

b Extraction time at boiling was 3.5 h instead of 25 min, drying oven temperaturewas 100 �C instead of 125 �C.

c Weight of sample was 1.0 g instead of 2.0 g, digestion time was 2.5 h instead of45 min, acid used for titration was sulfuric acid instead of hydrochloric acid.

d Microwave digestion instead of ashing.

before and during the completion of the work, each batch of sam-ples procured were tested for their linear viscoelastic properties.Differences on these properties of the various batches (15 batches)were about +10%, which is acceptable in such a highly heteroge-neous material (Muliawan and Hatzikiriakos, 2007, 2008).

2.2. Equipment and methodology

Muliawan and Hatzikiriakos (2007, 2008) have studied in detailthe rheology of mozzarella cheese. Small-amplitude oscillatoryshear experiments, creep tests and capillary extrusion were per-formed to fully understand its rheology at room temperature. Itwas concluded that the material’s viscous behaviour can be de-scribed adequately by a Herschel–Bulkley viscoplastic model (seebelow). The material was found to flow under no-slip conditions.

The capillary extrusion experiments performed by Muliawanand Hatzikiriakos (2008) had as an objective to examine an alter-native way of processing mozzarella cheese. Rolling experimentsare performed in this work as a means of further assessing its pro-cessability. To our knowledge, there are no reports on the roll-forming properties of mozzarella cheese. In this work, some newroll-forming experiments were performed using the SentmanatExtensional Rheometer (SER, Xpansion Instruments, Akron, OH,USA). Descriptions of this rheometer can be found in Sentmanat(2003, 2004), and a simple schematic is shown in Fig. 1. The SERis designed to fit into the convective heating oven of a rotationalrheometer (Bohlin VOR, Malvern Instruments), so that measure-ments can be performed at elevated temperatures. However, sincethe rolling experiments were performed at ambient temperature(�25 �C), no temperature control was utilized. The gap betweenthe drums/rollers of the SER is fixed at a distance of2H0 = 0.241 cm, thus samples with different thickness were pre-pared to achieve different reduction ratios, Hf/H0. The rolling sam-ples had different widths (�5 to �13 mm), so that we could havecontrol on the contact area between the sample and the rollers.Thus, we could ensure that the rolling experiments were operatedwithin the acceptable range of operation of the torque transducerof the rheometer. The sample with the smallest thickness wouldbe the widest and vice versa. Six different linear roller speeds wereused. The steady thickness of each sample after it had passed therollers was also measured and recorded. Exit thickness value isthe average of measurements done at three different positionson the sample. At most a ±5% standard deviation was found fromthe average values. No spring-back was observed in any experi-ment. The recovery in thickness can be used as a measure of elas-ticity, i.e., the ability of the material to store energy when it passesthrough the rolls, which is being released after and manifests itselfas recovery in thickness.

Finally, microscopic images were produced to visually identifyany structural changes that occur within the cheese after roll form-ing. This was achieved by using a Cryo-Scanning Electron Micro-scope (SEM) (Hitachi S4700 SEM with Emitech K1250 CryoSystem, Pleasonton, CA, USA). The samples collected for imagingwere stored in an air-tight container before their microscopicimages were taken, and all these were taken within 24 h. Beforeimaging, the samples were flash-frozen using liquid nitrogen andthey were fractured to expose the internal surface. The macro-scopic images were taken at random locations within the extrudedand rolled samples and only representative images are shown here.

3. A rheological model for mozarella cheese

Fig. 2 summarizes the Bagley-corrected flow curves of mozza-rella cheese obtained by means of a capillary rheometer and dieshaving various diameters, D. All flow curves for three different dies

Apparent Shear Rate,

10-5 10-4 10-3 10-2 10-1 100 101 102 103 104

Wal

l She

ar S

tres

s,

w(k

Pa)

100

101

102

Fitting (H-B)Experiment

Mozzarella Cheese, T=25oC

y=1.958 kPa, n=0.28, K=3.229 kPa sn

y value from stress ramp test

1s

Fig. 2. The flow curve of mozzarella cheese at 25 �C and its representation by theHerschel–Bulkley viscoplastic model.

E. Mitsoulis, S.G. Hatzikiriakos / Journal of Food Engineering 91 (2009) 269–279 271

of various L/D ratios fall on a single line, defining uniquely theapparent flow curve at 25 �C and verifying the absence of slip.The experimental data have been fitted with the Herschel–Bulkleymodel of viscoplasticity, which has the form (Bird et al., 1982)

s ¼ Kj _cjn�1 _c� sy; for jsj > sy ð1aÞ_c ¼ 0; for jsj 6 sy ð1bÞ

where s is the shear stress, _c is the shear rate (=du/dy), sy is theyield stress, K is the consistency index, and n is the power-law in-dex. The values of the constants from the best fit are given in

Table 2Rheological parameters and physical constant for mozzarella cheese at 25 �C(Herschel–Bulkley model)

Yield stress, sy (kPa) 1.958Stress-growth exponent, m (s) 200Power-law index, n (�) 0.28Consistency index, K (kPa sn) 3.229Densitya, q (g cm�3) 1.062Heat capacityb, cp (J g�1 K�1) 2.8Thermal conductivitya, k (J cm�1 s�1 K�1) 0.00381Activation energy, Ea (J mol�1) 1.72 � 10+5

Temperature shift factor, bT (K�1) 0.208

a Tavman and Tavman (1999).b Heidenreich et al. (2007).

Table 3Geometric and experimental data in rolling of mozzarella cheese

Roll dimensionsRadius, R (cm) Minimum gap, 2H0 (cm)0.516 0.241Initial sample thicknesses# Experiment H1 H2Initial thickness, 2Hf (cm) 0.548 0.514Thickness ratio, Hf/H0 (�) 2.274 2.133

Roll speeds# Experiment U1 U2Speed, U (cm/s) 0.361 0.719

Characteristic variablesCharacteristic shear rate, �_c (s�1) 3 6Characteristic viscosity, �g (Pa s) 2119 1221

Dimensionless numbersBingham number, Bn (�) = syH0

n/(KUn) 0.446 0.368Reynolds number, Re (�) = qUH0/�g 2 � 10�6 8 � 10�6

Peclet number, Pe (�) = qcpUH0/kN 34 68Nahme number, Na (�) = bN�g U2/kN 0.015 0.035

Table 2. It is seen that the value of sy = 1958 Pa as determined byextrapolation, which compares favourably with the value ofsy = 1620 Pa found by a stress-ramp test, within its margins of error(Muliawan and Hatzikiriakos, 2007).

The above model gives rise to the dimensionless Bingham num-ber, Bn, defined as

Bn ¼ syHn0

KUn ð1cÞ

The Bn number is a measure of viscoplasticity. When sy = 0, Bn = 0,and a purely viscous fluid is recovered. At the other extreme, assy ?1, Bn ?1, and this corresponds to a purely elastic solid.

4. Rolling experiments

The rolling experiments were performed on the SER extensionalrheometer used as a rolling device. The dependent variables are thefinal sheet thickness and the torque, while setting the roll speed ofthe SER rheometer and the initial thickness of the entering sheet.The sheet width was in the range 0.5–1.27 cm and it was not con-trolled in any systematic way. Table 3 gives the roll dimensionsand the different test cases of roll speed (U1–U6) and initialthickness (H1–H6). Thus, each experiment can be defined as HiUj

(i = 1–6, j = 1–6). The experimental results for the average valueof the final thickness and the torque are shown in Figs. 3 and 4,respectively.

In Fig. 3 it is observed that the exit thickness is increased as theentry thickness increases and the Bingham number decreases. Thebiggest increase reaches more than 100% for Bn = 0.16 and Hf/H0 = 2.27. For a given roll speed, i.e., for a given Bn number, the exitthickness increases as the entry thickness increases. Also, for a gi-ven initial thickness, the final thickness increases as the roll speedincreases. If the roll speed increases 40 times, the relative finalthickness increases at most by 40%.

Fig. 4 plots the torque per unit width as a function of the rollspeed for several values of the dimensionless entry thickness ofsamples. The torque increases with increasing initial thicknessand roll speed (or with decreasing Bn number), a behaviour ex-pected for viscoplastic materials.

The operating window for the above results was determined asfollows. As shown in Table 3, measurements were made for initialthickness ratios 1.36 6 Hf/H0 6 2.27 and roll speeds 0.36 6 U (cm/s) 6 14.39. The upper and lower limits were determined mainlyby the capabilities of the device for measuring and registeringthe torque by its software. Going over these limits led to a veryweak signal, and the results were difficult to analyze. On the other

Roll length, L (cm) Sample width, W (cm)1.902 0.5–1.27

H3 H4 H5 H60.420 0.395 0.340 0.3281.743 1.639 1.411 1.361

U3 U4 U5 U61.445 2.864 7.192 14.385

12 24 60 120703 412 203 120

0.302 0.250 0.193 0.1593 � 10�5 9 � 10�5 5 � 10�4 2 � 10�3

136 269 677 13530.080 0.185 0.574 1.354

Entry Thickness, Hf / H0

1.0 1.5 2.0 2.5 3.0

Exi

t Thi

ckne

ss, H

/ H

0

1.00

1.20

1.40

1.60

1.80

2.00

2.20

2.40

2.60

Bn=0.16Bn=0.19Bn=0.25Bn=0.30Bn=0.37Bn=0.45

Mozzarella Cheese, 25oC

Fig. 3. Experimental results for the final exit thickness as a function of entrythickness for different roll speeds (hence Bingham numbers, Bn, see Table 3).

Roll Speed, U (cm / s)

0 2 4 6 8 10 12 14 16

Tor

que

per

unit

Wid

th,

/ W (

N-m

/ m)

0.00

0.50

1.00

1.50

2.00

2.50

Hf / H0=2.27

Hf / H0=2.13

Hf / H0=1.74

Hf / H0=1.64

Hf / H0=1.41

Hf / H0=1.36

Mozzarella Cheese, 25oC

Fig. 4. Experimental results for the torque as a function of roll speed for differententry thickness ratios Hf/H.

Fig. 6. Range of experiments in torque measurements during rolling of mozzarellacheese. The shaded area corresponds to stable operation, outlined by the limitationsof the measuring device and the observed surface defects.


hand, cutting samples with a thickness smaller than the lower lim-it tested resulted in difficulties and led to experimental errors.

For example, for Hf/H0 < 1.36 the material was too thin, and thecutting of samples proved especially difficult. Therefore, it was notpossible to assure sample homogeneity, which in turn would leadto experimental errors. Furthermore, in tests with Hf/H0 = 1.1, itwas observed that the torque measured was less than 10% of the

Fig. 5. Mozzarella cheese samples after rolling: (a) a combination of medium initial thicinitial thickness and high roll speed leads to the appearance of visible surface defects.

device capabilities, and therefore the measurements were limitedto Hf/H0 P 1.36.

Fig. 5 shows samples of the material after rolling. In Fig. 5a acombination of medium initial thickness and roll speed leads to asmoothly rolled sample. Regarding the upper limit of stable oper-ation, it was observed that for values of initial thickness Hf/H0 > 2.27, the thickness was such that the material would not passthrough the gap, resulting in alteration of the sample appearance,as shown in Fig. 5b. Regarding the roll speeds, it was observed thatfor speeds lower than 0.36 cm/s, the material would not passthrough the gap, while for speeds higher than 14.39 cm/s, meltfracture was observed. On the other hand, SEM analysis has shownthat there are no noticeable structural changes in the mozzarellacheese due to rolling (Muliawan and Hatzikiriakos, 2007).

Fig. 6 shows the range of stable operation (shaded area) forrolling this particular mozzarella cheese, as obtained from theabove-mentioned observations. It is noted that the region of stableoperation is specific to the aspect ratio R/H0 = 4.28 used in thiswork.

5. Mathematical modelling

5.1. Governing equations

The flow is governed by the conservation equations of mass,momentum, and energy under creeping flow conditions for anincompressible viscous fluid in a two-dimensional cartesian coor-dinate system. These are

kness and roll speed leads to a smoothly rolled sample; (b) a combination of small


r � �u ¼ 0 ð2Þ

0 ¼ �rP þr � s¼ ð3Þ

qcp�u � rT ¼ kr2T þ s¼

: r�u ð4Þ

where �u is the velocity vector, P is the pressure, s¼

is the extra stresstensor, q is the density, cp is the heat capacity, k is the thermal con-ductivity, and T is the temperature.

The constitutive equation that relates the stresses to the veloc-ity gradients is the generalized Newtonian fluid and is written as

s¼¼ g _c

¼¼ Kj _cjn�1 þ sy

j _cj ½1� expð�mj _cjÞ��

_c¼

ð5Þ

where g is the apparent viscosity given by the Herschel–Bulkley–Papanastasiou model (Mitsoulis et al., 1993). In the above, m isthe regularization parameter (with units of time), which controlsthe exponential growth of stress, and j _cj is the magnitude of therate-of-strain tensor, _c

¼¼ r�vþr�vT , which is given by

j _cj ¼ffiffiffiffiffiffiffiffiffi12

II _c

r¼ 1

2_c¼

: _c¼� �� 1=2

ð6Þ

where II _c is the second invariant of _c¼

. Similarly we obtain the sec-ond invariant of the stress tensor IIs.

The criterion to track down yielded/unyielded regions is for thematerial to flow (yield) when the magnitude of the extra stresstensor |s| exceeds the yield stress sy, i.e.

yielded : jsj ¼ffiffiffiffiffiffiffiffiffi12

IIs

r¼ 1

2s¼

: s¼n o� �1=2

> sy ð7aÞ

unyielded : jsj ¼ffiffiffiffiffiffiffiffiffi12

IIs

r¼ 1

2s¼

: s¼n o� �1=2

6 sy ð7bÞ

Contours of |s| = sy separate the yielded from the unyielded re-gions and are drawn a posteriori from the numerical solution.

Table 2 shows the rheological parameters sy, K and n, and phys-ical properties for mozzarella cheese at 25 �C. It is noted that sinceno measurements were made for the thermal properties of thematerial, the values for the physical properties were taken fromthe literature.

5.2. Dimensionless numbers

In calendering (and hence in rolling), the following dimension-less parameters are introduced (Middleman, 1977)

x0 ¼ xffiffiffiffiffiffiffiffiffiffiffiffi2RH0p ; ðaÞ y0 ¼ y

H0; ðbÞ h0 ¼ h

H0¼ 1þ x2

2RH0; ðcÞ

P0 ¼ PK

H0

U

� �n

; ðdÞ k2 ¼ Q2UH0

� 1 ðeÞ ð8Þ

where k is a dimensionless flow rate (or leave-off distance), Q is thevolumetric flow rate per unit sheet width, and the rest of the sym-bols are defined in Fig. 1.

Dimensionless numbers are also introduced to assess the rela-tive importance of the various terms in the conservation equations.We choose as a reference temperature, T0, the ambient tempera-ture, a characteristic length equal to half the minimum gap be-tween the rolls, Y0, and a characteristic speed equal to the rollspeed, U. A characteristic shear rate is then

�_c ¼ UH0

ð9Þ

and a characteristic viscosity for a material that obeys the Herschel–Bulkley model is

�g ¼ gð�_c; T0Þ ¼ K�_cn�1 þ sy

�_c¼ K

UH0

� �n�1

þ syH0

Uð10Þ

The temperature-dependence of the viscosity is given by

gT ¼ g0 exp½�bTðT � T0Þ� ¼ g0 expEa

Rg

1T� 1

T0

� �� ð11Þ

where bT is the shift factor (1/K), Rg is the ideal-gas constant(=8.13 J/K mol), Ta is the activation energy (J/mol), T is the absolutetemperature (K), and T0 is the absolute reference temperature (K).Combining the above (11) yields

bT ¼Ea

RgTT0ð12Þ

The value given by Muliawan and Hatzikiriakos (2007) for Ea atT0 = 60 �C is 1.72 � 105 J/mol (4.11 � 104 cal/mol). Therefore forT = 25 �C we get bT = 0.208 K�1, a constant, which was used in thecalculations.

With these characteristic variables it is possible to define thefollowing dimensionless numbers: Reynolds, Re, Bingham, Bn, Pec-let, Pe, Nahme, Na. Their definitions and range of values for the roll-ing experiments are given in Table 3. We observe negligible Renumbers (inertialess, creeping flow), low Bn numbers (low visco-plasticity), high Pe numbers (dominant convective heat transfer),and low Na numbers (little viscous heating, almost isothermalflow). More information about the meaning of these numbers inmaterials processing is given by Mitsoulis et al. (1993).

5.3. Lubrication approximation theory (LAT)

In calendering, where the aspect ratio R/H0 > 100, it is assumedthat the process is dominated by shear flow between the rolls andaround the nip, so that only the axial velocity u and its derivativedu/dy are considered (Middleman, 1977). Furthermore, the rollcurvature is approximated by a parabola (see Eq. (8c), above),which is a very good approximation for large aspect ratios. Theboundary conditions are

P0 ¼ dP0=dx0 ¼ 0; at x0 ¼ k ð13aÞP0 ¼ 0; at x0 ¼ �x0f ð13bÞ

Note that Eq. (13a) is referred to as the Swift condition and isinstrumental in the solution of the problem. The above are thekey assumptions for the solution of the problem according toLAT. The problem can then be solved either for a given k (thedetachment point) and finding �x0f (the attachment point), or thereverse.

Once k or�x0f are known, the sheet thicknesses are readily avail-able according to

HH0¼ 1þ k2 ð14aÞ

Hf

H0¼ 1þ x02f ð14bÞ

For Herschel–Bulkley fluids, Sofou and Mitsoulis (2004a,b) havegiven an analytical expression and a numerical solution for fullparametric studies of the power-law index, n, and the Binghamnumber, Bn.

The solution process includes important operating variables,such as the maximum pressure, the roll-separating force, and thetorque exerted by the rolls. The roll-separating force per unitwidth, F/W(n, Bn), is defined by

FWðn;BnÞ ¼

Z k

�x0f

PðxÞdx ð15Þ

while the torque for both rolls, Tq(n, Bn), is defined by:

Tqðn;BnÞ ¼ 2WRZ k

�x0f

sxy

y¼hðxÞdx ð16Þ


where W is the sheet width. Eqs. (15) and (16) are based on LAT andimply that extra normal stresses are negligible and that roll curva-ture can be neglected in the integration, a severe approximation inrolling.

Fig. 7. Flow domain and boundary conditions for the two-dimensional FEM analysis.respectively, while qn = �kdT/dn denotes the heat flux.

Fig. 8. Finite elements meshes used in the computations. Upper half shows a

Fig. 9. Flow variables (H1U6 case). Kinematic variables: (a) velocity vectors

5.4. Method of solution

The constitutive equation for the viscoplastic fluids must besolved together with the conservation equations and appropriate

The subscripts n and t denote components normal and tangential to the surface,

mesh with 2000 elements (M2), lower half mesh has 500 elements (M1).

with magnitude j�Uj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ v2p

(UBAR) (cm/s), (b) streamlines, w (STR).


boundary conditions. Fig. 7 shows the solution domain and bound-ary conditions for the symmetric problem for flow from left toright. Because of symmetry, only one half of the flow domain isconsidered.

All lengths are scaled with the minimum gap H0, all velocitieswith the roll speed U, and all pressures and stresses with K(U/H0)n. Also the stress-growth exponent m gives rise to the dimen-sionless stress-growth exponent M = mU/H0. Based on previousexperience (Mitsoulis, 2008), the value of m = 200 s is set as amaterial property and is kept constant in all runs.

The numerical solution is obtained with the Finite ElementMethod (FEM), using the program UVPTHSLIP, originally developedfor multilayer flows (Hannachi and Mitsoulis, 1993; Mitsoulis,2005), which employs as primary variables the two velocities,pressure, temperature, and free-surface location (u-v-p-T-h formu-lation). It uses 9-node Lagrangian quadrilateral elements withbiquadratic interpolation for the velocities, temperatures andfree-surface location, and bilinear interpolation for the pressures.The free surface is found in a coupled way as part of the solutionfor the primary variables.

Fig. 10. Flow variables (H1U6 case). Dynamic variables (in MPa) and isother

Data for the rolling geometry is given in Table 3. We have usedtwo meshes, which are shown (put together for brevity) in Fig. 8.Mesh M1 (lower half) has 500 elements, while mesh M2 (upperhalf) has 2000 elements, and is produced by subdividing each ele-ment of M1 into 4 sub-elements. Mesh M2 gives 8241 nodes and26643 unknown degrees of freedom (DOF) with 41 points in thetransverse direction. The less dense mesh M1 with 500 elementswas used primarily for preliminary runs to gain experience withtwo-dimensional rolling flows. The final runs for this work wereperformed with M2 and the results were found to be within 2% be-tween the two meshes. Details about mesh convergence studiesand mesh independence of the results can be found in a thesis(Sofou, 2008).

6. Results and discussion

We present results from the two-dimensional, creeping, non-isothermal analysis of rolling of mozzarella cheese. Typical resultsare given for the field variables for the H1U6 case, which can be

ms (in �C): (a) pressure, H (H), (b) sxx (N{{), (c) sxy (N{Y), and (d) T (T).


described as the most ‘‘difficult”, having the highest roll speed U6and the biggest entry thickness H1.

6.1. Contours

We begin the presentation of the flow field through contours ofthe kinematic variables in Fig. 9 and of the dynamic and thermalvariables in Fig. 10. There are 11 contours drawn with the highestlegend value being the maximum and the lowest legend valuebeing the minimum of each variable.

Fig. 9a shows the velocity vectors based on the magnitude of thevelocity. Not all vectors are shown due to their multitude.The velocity vectors at entry and exit are constant to reflect theplug velocity profiles there due to the existence of the free surfaces.The velocity vectors take their maximum values after the nip re-gion. Fig. 9b shows the streamlines (contours of the stream func-tion, w, which has been made dimensionless and normalized bythe exit value wC = UCHC). The streamlines are all open and equallyspaced away from the rolls (two-dimensional planar flow), there isno recirculation, but the squeezing of the material to pass throughthe minimum gap becomes obvious, and the streamlines bend to-wards the rolls to reflect the change from free-surface flow to dragflow by the rolls.

The primary variable of the pressure (Fig. 10a) takes its maxi-mum value at the attachment point (singular point) and then de-creases towards the symmetry line and the exit. Similar

Fig. 11. Progressive reduction of the unyielded zones (shaded) predicted by the Herschel(see Table 3).

behaviour is observed for the secondary variable of elongationalstress sxx (=�syy due to a two-dimensional planar incompressibleflow) (Fig. 10b), where there are regions in the nip with high val-ues. Similar patterns are also observed for the shear stress sxy

(Fig. 10c), where at the attachment and detachment points (singu-larities) there are maxima according to FEM, but also high shearstress values are obtained in the nip region.

Finally, Fig. 10d shows the isotherms. It is noted that the max-imum values are obtained near the isothermal walls (thermalboundary layers due to shearing at the rolls). However, the maxi-mum T-value is just 0.02 �C higher than the roll and entry sheettemperature of 25 �C, which corroborates the isothermal characterof the flow process.

In all the above figures of the flow field, the flow domain is seendeformed to accommodate the upstream and downstream freesurfaces. It is seen that upstream there is more deformation, asthe material ‘‘swells” to satisfy the stress-free boundary conditionsimposed at the free surface. Thus, starting from the attachmentpoint as an ‘‘anchor” with Hf/H0 = 2.274 and moving to the left to-wards the inlet of the incoming sheet, the sheet swells to 2.501 atentry, giving a relative change of 10% for this particular geometryand set of conditions. This is reminiscent of the Newtonian planarextrudate (or die) swell values of 18%, which viscoplasticity bringsdown as the Bn number increases (Mitsoulis, 2007). On the otherhand, the swelling of the exiting sheet, defined as the thickness ra-tio between points C and D (see Fig. 7), is negligible, being 1.7%.

–Bulkley–Papanastasiou model. Different roll speeds and a given entry thickness Y1


Therefore, viscoplasticity serves to reduce the admittedly smallswelling in rolling, which is also the case for the benchmark extru-date swell problem from long extrusion dies (Mitsoulis, 2007).

6.2. Yielded/unyielded regions

Because of the yield stress, viscoplastic fluids have the charac-teristics of both viscous fluids and plastic solids. The yield line,y0, separates the two regions, called yielded and unyielded, respec-tively. Several cases of viscoplastic flows have been analysed in theliterature and have shown explicitly these regions (Abdali et al.,1992). In calendering, such regions have been shown only sche-matically by Gaskell (1950) and more recently numerically bySofou and Mitsoulis (2004a,b). As mentioned in the latter works,the interesting yielded/unyielded regions found by LAT are errone-ous, so the two-dimensional analysis here is instrumental in find-ing out where these regions lie.

With regard to Fig. 9, we observe that the entering and exitingsheets, some distance from the attachment and detachment points,have a constant thickness and move with a plug velocity as a solidplastic sheet. Therefore, they constitute unyielded regions (shaded)before coming into contact with the rolls and after leaving them.But apart from these regions, it is not a priori clear where the uny-ielded regions between the rolls might be.

Fig. 12. Progressive increase of the unyielded zones (shaded) predicted by the Herschel–(see Table 3).

We present here results for these regions for the Y1U cases, i.e.the thickest sheet for all roll speeds. It was found that due to thesmall range of Bn (see Table 3), the detachment points, k, for whichthe exit speed was equal to the roll speed U, did not change appre-ciably, ranging between 0.26 for H1U1 to 0.25 for H1U6.

The results are shown in Fig. 11, where the extent and shape ofyielded/unyielded regions become evident as the roll speed U in-creases (or as Bn decreases). The regions are distinguished intoshaded (unyielded) and clear (yielded), making use of the criterionthat the separating line is the contour with a value|s| = sy = 1958 Pa. The unyielded regions decrease with increasingspeed. Mainly there are unyielded regions in the entry and exitareas, without their appearance between the rotating rolls. Theunyielded regions at entry and exit are due to the plug velocityprofile there. The unyielded regions are truly unyielded regions(TUR), since the velocity is nonzero, but its derivatives are zero(no deformation). The unyielded regions are continuous and moreextended for small roll speeds, but they are reduced and becomediscontinuous, giving rise to diminishing ‘‘islands” in the entryregion as the speed increases (cf. H1U5 and H1U6). However, atexit these unyielded regions remain solid. These islands are appar-ently unyielded regions (AUR), and in all probability they are aconsequence of using the regularization by Papanastasiou in theHerschel–Bulkley model. The present results, and especially the

Bulkley–Papanastasiou model. Different entry thicknesses and a given roll speed U1


shapes of unyielded regions, are similar to the ones obtained fromthe non-isothermal extrusion of pastes using the Herschel–Bulk-ley–Papanastasiou model (Mitsoulis et al., 1993).

The results for different entry thicknesses and a given roll speed(U1) are presented in Fig. 12. They show the extent and shape ofyielded/unyielded regions as the entry thickness Y decreases.Since now the roll speed is constant (hence the Bn number), thereare no changes in the unyielded regions at the exit, as expected.However, at entry there is an increase of the unyielded regionsas the thickness of the entering sheet decreases, since the free sur-face has negligible curvature away from the rolls, which results in aplug velocity profile there, hence in unyielded regions.

It is interesting to note that plots H1U1 and H1U2 in Fig. 11shows dark patches on the roll downstream of the nip, at x/H0

�0.5. Similar zones appear in all the plots on Fig. 12. It is not clearwhether these spots correspond to the ‘‘sticking point” in rolling.More detailed work will be needed to clarify this point.

6.3. Detachment point

As mentioned above, the detachment point, k, was found bymaking use of the roll speed U at exit. The values of k were foundto range between 0.23 and 0.26, which correspond to Y/Y0 = 1.053–1.068, in the limited range of Bn numbers. As shown

Roll Speed, U (cm / s)

0 2 4 6 8 10 12 14 16

Torq

ue p

er u

nit W

idth

, /

W (N

-m/ m

)

0.00

0.50

1.00

1.50

2.00

2.50

Hf / H0=2.27Hf / H0=2.13Hf / H0=1.74Hf / H0=1.64Hf / H0=1.41Hf / H0=1.36

sim. Hf / H0=2.27sim. Hf / H0=2.13sim. Hf / H0=1.74sim. Hf / H0=1.64sim. Hf / H0=1.41sim. Hf / H0=1.36

Fig. 13. Torque per unit width as a function of roll speed. Comparison betweenexperiments and simulations based on the two-dimensional FEM and integration ofstresses according to Eq. (18).

Fig. 14. Schematic diagram of the sheet thickness at the detachment point k (as pexperimentally). Swelling is observed, which begins right after detachment. The flow up tdetachment is a free-surface flow without shear or normal stresses.

in Fig. 3, the experimental Y/Y0 values range between 1.3 and2.1 (an order of magnitude higher), and they depend both on theroll speed and the thickness of the entering sheet. Therefore, otherphenomena are responsible for the experimental swell, with mostimportant those of viscoelasticity, as shown to exist in the rheolog-ical characterization of mozzarella cheese (Muliawan and Hatziki-riakos, 2007). However, viscoelastic effects are beyond the scope ofthe present work.

6.4. Torque

As a first attempt at calculating the torque, Eq. (16) was usedaccording to LAT, which integrates simply the shear stresses, sxy,along the roll surface. The LAT results when compared with theexperimental ones of Fig. 4 shows that qualitatively were similarboth in shape and ordering, so that the torque increases withincreasing roll speed and entering sheet thickness. However, theLAT results underestimated the experimental ones by an order ofmagnitude. Therefore, it became obvious that integrating simplythe shear stresses produced unrealistic torque values.

The two-dimensional FEM results have shown that in rollingwith small aspect ratio R/H0 all stresses have values of the same or-der of magnitude, and therefore they cannot be neglected in theforce and torque calculations. In contrast to LAT and its Eqs. (15)and (16), not only the pressure and the shear stresses have to benumerically integrated along the roll, but also the normal stresses.Therefore the equation for the torque substitutes the shear stresssxy with srh, in a cylindrical coordinate system, with r being the ra-dial coordinate and h the azimuthal coordinate:

Tq ¼WFdR ¼ 2WRZ h2

�h1

srhRdh ð17Þ

where Fd is the drag force, srh is the tangential stress on the roll, h isthe local angle at each point on the roll, and the integration takesplace from the attachment point with an angle �h1 to the detach-ment point with angle h2. Substituting into Eq. (17) and taking intoaccount the usual trigonometric equalities, we obtain for the torque:

Tq ¼ 2WRZ xD

�xE

ðsyx cos hþ ryy sin hÞdxþ 2WRZ yD

yE

ðrxx cos h

þ sxy sin hÞdy ð18Þ

where (�xE, �yE) are the coordinates of the attachment point E and(xD, yD) are the coordinates of the detachment point D. Note that thetotal stresses are: rxx = �P + sxx, ryy = �P + syy, and because of sym-metry of the stress tensor, syx = sxy. Details about the derivation andintegration can be found in a thesis (Sofou, 2008).

The results from the above integrations are given together withthe experimental data in Fig. 13. In contrast to LAT, there is nowgood agreement between simulations and experiments, which isdue to integrating all stresses and pressure, since these have thesame order of magnitude, confirming the two-dimensional charac-ter of the rolling process.

redicted by the simulations) and the final sheet thickness Y/Y0 (as measuredo detachment is confined, with wall boundary conditions (‘‘shear flow”), while after


The contradictory findings of getting the torque correct but notthe final thickness can be explained by the nature of the two typesof flow in the rolling process and referring to Fig. 14. Namely, theflow with wall boundary conditions (between attachment anddetachment points), which may be called (perhaps improperly)‘‘shear or shear-dominated flow”, and the flow with free-surfaceboundary conditions, for the entering and exiting sheet. The latteris known to produce large swelling in a viscoelastic material, dueto the relaxation of all stresses, especially the normal stresses,upon exiting from a shear or shear-dominated flow (Tanner,2000). Apparently, this is the case here, so that viscoelasticity man-ifests itself as a sudden recovery of the stressed state of the mate-rial, as it passes between the rotating rolls. On the other hand, thestresses in the ‘‘shear” flow are well captured by the model, whichis better suited for such types of flow. Similar results were found inrolling of bread dough (Mitsoulis and Hatzikiriakos, 2008). Thework of Kempf et al. (2005) for elastic recovery during bread doughsheeting may be instrumental in capturing this enhanced swelling.Finally, the works by Goh et al. (2003, 2004, 2005) have clearlyshown the importance of viscoelastic properties of cheese, whichare paramount in the determination of its behaviour in extensionalflows.

7. Conclusions

Rheological measurements of mozzarella cheese showed thatit possesses yield stress and shear-thinning behaviour, which ina shear flow is well described by the Herschel–Bulkley model ofviscoplasticity. Rolling experiments in a device with R/H0 = 4.3showed that the exiting sheet thickness increases appreciablywith increasing roll speed and entry thickness, as does thetorque.

Modelling based on the lubrication approximation theory(LAT) showed that the latter is not valid for rolling, but onlyfor calendering (R/H0 > 100). Two-dimensional FEM simulationsshowed that the process is essentially isothermal and that uny-ielded regions are limited to the incoming and outgoing sheetsleaving basically all material yielded between the rolls. This isdesirable, so rolling is a good process that eliminates unyieldedregions in viscoplastic fluids, unlike extrusion (Mitsoulis et al.,1993). The results for the torque are in good agreement withthe experiments due to the proper integration of all stressesand pressures on the roll surface, coming from the two-dimen-sional simulations. However, the exit thicknesses areseverely underestimated by the viscoplastic model used. Itwas argued that strong viscoelastic effects are responsible forthe latter.

The present results are a first step towards a better under-standing of rolling and calendering of such complicated materialsas foodstuff, including cheese, bread dough, etc. The analysismust definitely move towards inclusion of viscoelasticity, on topof viscoplasticity, by using the recently proposed visco-elastico-plastic model (Sofou et al., 2008). Such work in currently underway.

Acknowledgements

Financial support from NSERC of Canada and NTUA in the formof a grant for basic research, under the code name ‘‘KARATHEODO-RI”, is gratefully acknowledged.

References

Abdali, S.S., Mitsoulis, E., Markatos, N.C., 1992. Entry and exit flows of Binghamfluids. J. Rheol. 36, 389–407.

Bird, R.B., Dai, G.C., Yarusso, B.J., 1982. The rheology and flow of viscoplasticmaterials. Rev. Chem. Eng. 1, 1–70.

Cortes-Martines, E., Schroeder, A., Wolfschoon, A., Schmid, T., Mehnert, D., 2005.Mozzarella-type string cheese product. US Patent Application No.20050208193.

Engmann, J., Peck, M.C., Wilson, D.I., 2005. An experimental and theoreticalinvestigation of bread dough sheeting. Food Bioprod. Proc. 83, 175–184.

Gaskell, R.E., 1950. The calendering of plastic materials. J. Appl. Mech. 17, 334–337.Goh, S.M., Charalambides, M.N., Williams, J.G., 2003. Large strain time dependent

behaviour of cheese. J. Rheol. 47, 701–716.Goh, S.M., Charalambides, M.N., Williams, J.G., 2004. Determination of the

constitutive constants of non-linear viscoelastic materials. Mech. Time-Depend Mater. 8, 255–268.

Goh, S.M., Charalambides, M.N., Williams, J.G., 2005. Characterization of thenonlinear viscoelastic constitutive properties of mild cheddar cheese fromindentation tests. J. Texture Stud. 36, 459–477.

Hannachi, A., Mitsoulis, E., 1993. Sheet coextrusion of polymer solutions and melts:comparison between simulation and experiments. Adv. Polym. Technol. 12,217–231.

Heidenreich, S., Langer, T., Rohm, H., 2007. Heat capacity of cheese: determinationor calculation? J. Therm. Anal. Calorim. 89, 815–819.

Kempf, T.P., Labuza, T.P., Levine, L., 2005. Elastic recovery of a wheat flour doughduring sheeting. Cereal Foods World 50, 316–320.

Levine, L., Drew, B.A., 1990. Rheological and engineering aspects of the sheeting andlaminating of doughs. In: Faridi, H.M., Faubion, J.M. (Eds.), Dough Rheology andBaked Product Texture. Van Nostrand Reinhold, NY, USA.

Middleman, S., 1977. Fundamentals of Polymer Processing. McGraw-Hill, New York.Mitsoulis, E., Abdali, S.S., Markatos, N.C., 1993. Flow simulation of Herschel–Bulkley

fluids through extrusion dies. Can. J. Chem. Eng. 71, 147–160.Mitsoulis, E., 2005. Multilayer film coextrusion of polymer melts: analysis of

industrial lines with the finite element method. J. Polym. Eng. 25, 393–410.Mitsoulis, E., 2007. Annular extrudate swell of pseudoplastic and viscoplastic fluids.

J. Non-Newtonian Fluid Mech. 141, 138–147.Mitsoulis, E., 2008. Numerical simulation of calendering viscoplastic fluids. J. Non-

Newtonian Fluid Mech. 154, 177–188.Mitsoulis, E., Hatzikiriakos, S.G., in press. Rolling of bread dough: experiments and

simulations. Food Bioprod. Proc. doi:10.1016/j.fbp.2008.07.001.Muliawan, E.B., Hatzikiriakos, S.G., 2007. Rheology of mozzarella cheese. Int. Dairy

J. 17, 1063–1072.Muliawan, E.B., Hatzikiriakos, S.G., 2008. Rheology of mozzarella cheese: extrusion

and rolling. Int. Dairy J. 18, 615–623.Sentmanat, M.L., 2003. Dual windup extensional rheometer. US Patent No.

6,578,413.Sentmanat, M.L., 2004. Miniature universal testing platform: from extensional melt

rheology to solid state deformation behaviour. Rheol. Acta 43, 657–699.Sofou, S., 2008. Calendering of viscoplastic materials. Ph.D. Thesis, School of Mining

and Metallurgical Engineering, NTUA, Athens, Greece.Sofou, S., Mitsoulis, E., 2004a. Calendering of pseudoplastic and viscoplastic sheets

of finite thickness. J. Plast. Film Sheeting 20, 185–222.Sofou, S., Mitsoulis, E., 2004b. Calendering of pseudoplastic and viscoplastic sheets

using the lubrication approximation. J. Polym. Eng. 24, 505–522.Sofou, S., Muliawan, E.B., Hatzikiriakos, S.G., Mitsoulis, E., 2008. Rheological

characterization and constitutive modeling of bread dough. Rheol. Acta 47,369–381.

Tanner, R.I., 2000. Engineering Rheology, second ed. Clarendon Press, Oxford.Tavman, I.H., Tavman, S., 1999. Measurement of thermal conductivity of dairy

products. J. Food Eng. 41, 109–114.Xiao, W.F., Charalambides, M.N., Williams, J.G., 2007. Sheeting of wheat flour dough.

Int. J. Food Sci. Technol. 42, 699–707.

rolling of mozzarella cheese: experiments and simulations

Documents