Journal of Food Engineering 124 (2014) 35–42

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Journal of Food Engineering

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Mathematical modeling of the viscosity of tomato, broccoli and carrotpurees under dynamic conditions

0260-8774/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jfoodeng.2013.09.031

⇑ Corresponding author at: SIK – The Swedish Institute of Food and Biotechnol-ogy, Box 5401, SE-402 29 Gothenburg, Sweden. Tel.: +46 10 516 6623; fax: +46 31833782 (L. Ahrné).

E-mail address: lilia.ahrne@sik.se (L. Ahrné).

Evelina Tibäck a, Maud Langton b, Jorge Oliveira d, Lilia Ahrné a,c,⇑a SIK – The Swedish Institute of Food and Biotechnology, Box 5401, SE-402 29 Gothenburg, Swedenb The Swedish University of Agricultural Sciences, Box 7051, SE-756 51 Uppsala, Swedenc Chalmers University of Technology, Department of Chemical and Biological Engineering, 41296 Gothenburg, Swedend Department of Process & Chemical Engineering, University College Cork, Cork, Ireland

a r t i c l e i n f o a b s t r a c t

Article history:Received 22 May 2013Received in revised form 22 September2013Accepted 28 September 2013Available online 5 October 2013

Keywords:RheopexyShearingThixotropyGellingIsothermalNon-isothermal

Different viscosity models were developed to describe the viscosity of unprocessed fruit and vegetablepurees under dynamic conditions. Temperature hysteresis cycles were carried out for three purees withdifferent structural characteristics (tomato, carrot, and broccoli), with heating and cooling phases from 10to 80 �C with isothermal (holding) phases at 10, 30, 60 or 80 �C. The apparent viscosity was measuredcontinuously with a rotational rheometer and the data was analyzed with time-independent and time-dependent models (quantifying rheopexy, thixotropy, or both). The results revealed clear thixotropicbehavior in tomato puree, attributed to shearing effects, and rheopectic in broccoli puree, attributed togel formation at the higher temperatures. Although carrot puree data from the isothermal periods couldbe quantified satisfactorily with no time dependency, analysis of the nonisothermal periods proved thatrheopectic effects also needed to be included.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction temperature and shear rate, the viscosity of a rheopectic fluid in-

A fruit or vegetable puree is a dispersed suspension of plant cellmaterials in a fluid phase containing soluble pectins. The pectins inthe fluid phase bind water and may form gels. Heat treatment in-duces pectin degradation as well as gel formation. Pectin degrada-tion can be caused either by enzymatic activity, b-elimination, oracid hydrolysis. Heating a fruit or vegetable puree also inducesleakage and solubilization of pectins from the cell walls into thefluid phase, which allows for gel formation. The pectin chains formgels through a three-dimensional network together with water,solutes and suspended particles. Consequently, thermal processingof fruit and vegetable purees can result in both pectin degradationand in gel formation (Duvetter et al., 2009).

A viscous material flows when a force is applied. For a Newtonianfluid, the viscosity remains constant at different shear rates. If the vis-cosity decreases with increasing shear rate it is said to be shear-thin-ning. A yield stress may be defined as the minimum stress required forinitiating flow, and reflects the deformation and breakage of networkstructures in the fluid (Rao, 1977). Time-dependent, non-Newtonianfluids can be categorized as rheopectic or thixotropic. At constant

creases with time while the viscosity of a thixotropic fluid decreaseswith time. Thixotropic (Armelin et al., 2006; Labanda et al., 2004)and rheopectic (Teyssandier et al., 2009) effects are widely studiedin polymer science, and food science has applied many of theconcepts developed there for various foodstuffs (Abu-Jdayil, 2003;Dolz et al., 2007; Hernandez et al., 2008). The rheological propertiesof a fruit or vegetable puree are influenced by the raw material com-position, particle concentration, hardness and morphology, particlesize distribution, particle–particle interactions, soluble pectins andpectins on the particle surfaces, temperature and shear conditions(Barrett et al., 1998; Hayes et al., 1998; Lopez-Sanchez et al.,2011). Fruit and vegetable purees generally behave as weak gelsand highly non-Newtonian liquids. Gelling of a fruit or vegetablepuree implies a rheopectic behavior, while degradation of thestructure in the puree causes a thixotropic behavior to take place.The most widely used method of analyzing and quantifying thesetime-dependent effects is to perform a so-called hysteresis cycle atconstant temperature, increasing the shear rate of the measurementto some limit and then decreasing it back to the original condition(Abu-Jdayil et al., 2004). The area defined by the two curves is ameasure of the hysteresis and hence of the extent of the rheopexyor thixotropy (Roussel, 2006). Time-dependent effects are modeledas a function of shear rate and time at constant temperature(Maingonnat et al., 2005). In this study, time-dependent effects in

36 E. Tibäck et al. / Journal of Food Engineering 124 (2014) 35–42

unprocessed fruit and vegetable purees were analyzed by perform-ing a temperature hysteresis cycle instead of a shear rate cycle. Tem-perature and time have a large impact on the rheological behavior offruit and vegetable purees, which in turn is important for pumping,stirring, and storage as well as for sensory properties.

The objective of this work was to develop a mathematical mod-el to describe the viscosity of unprocessed fruit and vegetablepurees under dynamic conditions, in order to account for theshearing and/or gelling time and temperature dependent effects.The general model was applied to carrot, broccoli, and tomatopuree, and the results were interpreted on the basis of the struc-tural differences between the products.

2. Materials and methods

2.1. Preparation of purees

Carrots, broccoli, and tomatoes were purchased from a localsupermarket in Sweden.

Each vegetable was mixed with ionized water to facilitate theblending process (carrot/broccoli puree 40% vegetable and 60%water, tomato puree 70% fruit and 30% water). Four peeled carrotswere used for each carrot puree. In the top and bottom end of eachcarrot 2 cm was cut off and discarded and the carrots were cut in0.5 cm thick slices. Two broccoli heads were used for each broccolipuree. In the end of the stem 3 cm was cut off and discarded. Thebroccoli heads were cut lengthwise in half, one of the halves wascut into approximately 2 � 2 � 2 cm pieces, and 100 g of broccolipieces were used from each broccoli head. In the tomato purees4–5 tomatoes were used for each sample batch. The tomatoes werecut into 4 pieces each, and the seeds were removed. Vegetable/fruitpieces (carrot/broccoli: 200 g, tomato 120 g) were crushed togetherwith cold deionized water (carrot/broccoli: 300 g, tomato 280 g) ina kitchen blender (Moulinex, Ecully, Cedex, France) for a set time(carrot/broccoli 5 min, tomato 3 min). After crushing each samplewas immediately ultra-high-speed homogenized in an Ultra TurraxT25 Basic (IKA, Staufen, Germany, speed 24,000 min�1) for a settime (carrot/broccoli 3 min, tomato 2 min), while cooled on ice.After homogenization air was removed from the purees using a vac-uum pump under stirring while cooled on ice (carrot 10 min, broc-coli 30 + 30 min (samples were divided into two batches tofacilitate air removal, then remixed), tomato 60 min). The sampleswere kept chilled in a refrigerator until measurement. The pH ofeach puree was measured before viscosity measurement. For eachreplicate a new batch of puree was made from scratch.

Fig. 1. Examples of experimental temperature cycles at 10 �C, 30 �C, 60 �C, and80 �C and measured apparent viscosity for carrot, tomato and broccoli puree.

2.2. Rheological measurements

The rheological measurements were carried out in a stress-con-trolled rotational rheometer (Stresstech, Rheologica Instruments,Lund, Sweden) with a four-bladed vane geometry (diame-ter = 23 mm, height = 42 mm) and a serrated cup (diame-ter = 25 mm). The vane measurement system was calibratedagainst a concentric cylinder geometry (diameter = 25 mm) usinga Newtonian calibration oil (Viscosity standard, Brookfield Engi-neering Laboratories, Massachusetts, USA) and two shear-thinningpolymer solutions (Carbopol 981 NF; 0.25% in deionized water (pH7.14) and 0.5% in deionized water (pH7.00)). The cup was com-pletely filled (37 cm3) and a lid was used to minimize evaporativelosses. A constant shear rate of 50 s�1 was applied and viscosityvalues were recorded every 10 s. The heating rate was 2 �C/minand the cooling rate was 1 �C/min. In order to have a controlled his-tory of the samples prior to measurement, a pre-shear step at 10 �Cand 50 s�1 for 3 min was applied.

The purees were subjected to an isothermal period of 30 min at4 different temperatures, 10, 30, 60 and 80 �C. In addition, mea-surements were taken of the heating and cooling periods of the 3latter cases, following the product history from 10 �C up at a con-stant rate of 2 �C/min, and after the isothermal period, down to10 �C again, at a constant rate of 1 �C/min. This created a tempera-ture hysteresis cycle for the 3 different temperatures (illustrated inFig. 1). If there were no time-dependent effects, viscosity would re-main constant during the isothermal period and would be thesame when the sample goes back to 10 �C as it was in the begin-ning. As the heating and cooling rates were not the same, heatingand cooling times were different: 10, 25 or 35 min of heatingwhere the temperature was increased to 30, 60 and 80 �C, respec-tively, and 20, 50 or 70 min of cooling where the temperature wasdecreased back to 10 �C. These times are the programmed values,but for the data regression the actual times when the programmedtemperatures were reached were considered, in order to minimizethe error from sample to sample variability due to heating andcooling times. It is also noted that there was a small overshoot oftemperature experimentally when reaching the set-point, whichbecomes a source of experimental error, as the models assume thatonce the temperature reaches the isothermal set-point it immedi-ately remains constant. The temperature cycles are shown in Fig. 1,with experimental data from typical cycles, and showing the pro-gramme at 80 �C to help identify the overshoot.

2.3. Development of the viscosity models

Models were developed independently for 4 possible situations:time-independent, thixotropic, rheopetic and both thixotropic andrheopectic. However, for condensing the information, they can allbe described by a generic model, with some parameters switchingon or off the components associated to thixotropic and rheopeticbehavior.

The most usual structured kinetic models are n-order type ofmodels, with the rate of change proportional to a power n of thedeviation to equilibrium (Barnes, 1997). Assuming a first ordermodel for thixotropy, the rate is simply proportional to the devia-tion from equilibrium:

�dðl� l1Þdt

¼ ktðl� l1Þ ð1Þ

where kt is the thixotropy rate constant and l1 the value of viscos-ity for t ?1. This is also known as the stretched exponential model(Maingonnat et al., 2005).

To describe rheopectic behavior, where the apparent viscosity isevolving from a low to a high limit viscosity, a sigmoidal model

E. Tibäck et al. / Journal of Food Engineering 124 (2014) 35–42 37

would be required, such as Boltzmann’s model (Love, 2009). As thesigmoidal effect is expected to be relatively small in the tests per-formed here, it is unlikely that these parameters could be deter-mined with confidence. This was indeed verified with the actualdata regression. Pectin gels are weak and the rate of structureforming is low and can be related to the derivatives of apparentviscosity with time. Therefore, it can be assumed that only the on-set of gelling takes place, which is a stage where the viscosity in-creases exponentially with time. These conditions are far enoughfrom equilibrium to avoid the need for a sigmoidal model. Thus,a first order model for the rheopectic behavior of the viscositycould be proposed:

dldt¼ kgl ð2Þ

where kg is the rheopectic rate constant.Temperature affects the apparent viscosity and the two kinetic

rates kt and kg. For a fluid where viscosity does not change withtime, the effect of temperature on viscosity at a specific shear ratecan be expressed by an Arrhenius relationship (Dutta et al., 2006).However, the Arrhenius model does not allow solving some of thedifferential equations analytically under non-isothermal condi-tions. For this reason, an alternative model, known as Bigelow’smodel, was selected (Cunha et al., 1998). The temperature sensitiv-ity is quantified by a factor z which is equal to the number of de-grees of temperature required to change viscosity by one log cycle:

l ¼ l10e �T�10

zð Þ ð3Þ

Bigelow’s model is more commonly written with decimal logs,but natural logs were kept to simplify the expressions.

Similarly for the two rate constants:

kt ¼ kt;10 � eT�10

zt ð4Þ

kg ¼ kg;10 � eT�10

zg ð5Þ

where kt,10 and kg,10 are the thixotropic and rheopectic rates at10 �C, respectively, and zt and zg the respective temperature sensi-tivities of the rate constants, that is the number of degrees Celsiusfor one log cycle increase of the rate constant.

Considering that thixotropic and rheopectic effects occur simul-taneously but without interfering in each other gives the genericbasic equation:

dðl� l1Þdt

¼ ðkg � ktÞðl� l1Þ ð6Þ

Table 1Terms of the generic viscosity model Eq. (11).

Function Constant rateheating (h)

Isothermal (i) Constant rate cooling(c)

li l0,10 lh = last point ofheating curve

lc = last point ofholding phase

f1 e�bh t

z e�Th�10

z e�Thþbc ðt�tc Þ�10

z

f2 1 e�Th�10

z ð¼ f 1Þ e�Th�10

z

f3 e�kt;10 f5 f6 e�kt;10 f5 f6 e�kt;10 f5 f6

f4 ekg;10 f7 f8 ekg;10 f7 f8 ekg;10 f7 f8

f5ztbh e

Th�10zt

ztbc� e

Th�10zt

f6 ebh tzt � 1 t � th e

bc tzt � e

bc tczt

f7zg

bh eTh�10

zgzg

bc� e

Th�10zg

f8 ebh tzg � 1

t � th (=f6) ebc tzg � e

bc tczg

Note: in the experimental tests, for heating, bh = 2 �C/min, for cooling bc = �1 �C/min.

It is noted that in this case rheopexy would not predict an infi-nite viscosity like in Eq. (2), the limit viscosity for both joint effectsis the same, l1, which therefore may be lower or higher than l0,depending on the balance between the two effects.

Fig. 2. Experimental data obtained in the isothermal periods for (a) carrot, (b)tomato and (c) broccoli purees. s 10 �C, h 30 �C, 4 60 �C, } 80 �C. Different symbolshadings indicate the three different replicates.

Table 2Model and goodness of fit parameters of the least squares regression of the apparentviscosity of purees with a time-independent model (Eq. (2)).

z (�C) Average R2 of the data set

10 �C 30 �C 60 �C 80 �C

Carrot 118.3 ± 1.2 0.38 0.49 0.94 0.79Tomato 21.4 ± 1.2 0.0 0.25 0.35 0.45Broccoli 50.4 ± 1.2 0.0 0.0 0.38 0.17

38 E. Tibäck et al. / Journal of Food Engineering 124 (2014) 35–42

For isothermal conditions when the puree is kept at a constanttemperature Th, replacing Eqs. (3)–(5) in Eq. (6) and integratinggives:

l ¼ l1;10e�Th�10

z þ li � l1;10e�Th�10

z

� �� ekg;10�e

Th�10zg �ðt�thÞ

� e�kt;10�eTh�10

zt �ðt�thÞ ð7Þ

where li is the viscosity at the beginning of the isothermal period.For a non-isothermal period, it must be noted that the viscosity

will change with time due to the joint effect of thixotropy itself andthe fact that temperature is changing. Thus, l1 is a variable. Eq. (3)becomes:

dldt� dl1

dt¼ kg;10e

T�10zg � kt;10e

T�10zt

h iðl� l1Þ ð8Þ

Replacing l1 by Eq. (3), which is also used to determine its deriv-ative, results in the following 1st order differential equation for anonisothermal period at a constant heating or cooling rate of b �C/s, with t being the time from the instant that the heating or coolingperiod began:

dldtþ kt;Toe

btzt l ¼ � b

zl1;Toe�

btz þ kt;Tol1;Toeð

1zt�1

zÞbt

� kg;Tol1;Toeð1

zg�1

zÞbt ð9Þ

where To is the temperature at the beginning of the non-isothermalheating or cooling period (thus, T = To + bt) and kt,To, kg,To and l1,To

are the thixotropic rate constant, rheopectic rate constant and thelimit apparent viscosity after infinite shearing and gelling at thattemperature, respectively. This equation does not have an analyticalsolution because the integrating factor is an exponential with an-other exponential in argument, and so the solution would then bea function that cannot be integrated analytically. However, thereis a simplifying assumption that would lead to an analytical solu-tion: considering that the effects of temperature and of thixotropyare independent of each other (which is the same as assuming apseudo-steady state resulting from the relaxation time of viscosityadjustments due to temperature being much smaller than thosedue to the thixotropic effect of shearing, that is, changes due to tem-perature are instantaneous compared to thixotropy). In that case,Eq. (6) can be written as:

�d ðl10 � l1;10Þe�

T�10z

h idt

¼ kte�T�10

zt � kge�T�10

zg

h iðl10 � l1;10Þ ð10Þ

where l10 is the apparent viscosity that the fluid would have if timedependent effects would stop and the sample would be broughtback to 10 �C (as temperature is being assumed to have no effecton thixotropy other than accelerating its rate). Developing thederivative with T for a constant heating or cooling rate, integratingand using Eq. (1) to obtain the viscosity at the temperature T, theapproximate solution with the pseudo-steady state assumption is:

l ¼ l1;10e�T0�10þbðt�tiÞ

z

þ li � l1;10e�T0�10

z

� �e�kt;10e

To�10zt

ztb e

btzt�e

btizt

� �e

kg;10eTo�10

zg zgb e

btzg�e

btizg

� �

ð11Þ

where To is the initial temperature of the linear nonisothermal per-iod and li the initial viscosity of that period.

A full cycle of heating, holding and cooling is connected by theend point of one period being the starting point of the next. Froman initial viscosity l0,10 for t = 0 there is a heating period defined byEq. (11) with b = 2 �C/min and ti = 0. When the T of the holdingphase of that particular experiment (Th) is reached, at an instant

of time denoted by th, the final point from Eq. (11) becomes anew initial viscosity for Eq. (7), which will then describe the evolu-tion of viscosity until the end of the holding phase, at the time in-stant tc when cooling begins. The last point of Eq. (7) becomes theinitial value for Eq. (11) again, now used with b = �1 �C/min andt = tc.

This model has 7 parameters, with physical meanings that areclear and can be used to characterize the quality attributes of theproduct:

l0,10:

apparent viscosity of a puree just after preparation,before any thermal or shear effect modifies itsstructure, at 10 �C. Each product was tested with 12different samples and each would have its initialvalue. The initial experimental value was used in eachcase, this parameter was not allowed to vary in thedata regression.

l1,10:

apparent viscosity of a puree suffering all structuralchanges that can be caused by shearing and gellingeffects during testing, brought back to 10 �C

z:

the temperature sensitivity of viscosity unaffected byany kinetic effect (viscosity falls one log cycle for anincrease of z �C in temperature).

kt,10

the rate at which viscosity falls due to shearing at aconstant temperature of 10 �C (thixotropic rate).

zt

the temperature sensitivity of the thixotropic rate (therate increases by one log cycle for an increase of zt �Cin temperature).

kg,10

the rate at which viscosity increases due to gelling at aconstant temperature of 10 �C (rheopectic rate – notethat this is an extrapolation result to the referencetemperature, it does not imply that physically it ispossible to cause gelling at this temperature).

zg

the temperature sensitivity of the rheopectic rate (therate increases by one log cycle for an increase of zg �Cin temperature).

Developing any of the other models (thixotropic only, rheopec-tic only, or time independent) in a similar way would yield a sim-ilar generic expression:

l ¼ l1;10f1 þ ðli � l1;10f2Þf3f4 ð13Þ

where the functions denoted as f1 to f4 have the values defined inTable 1 for the models describing the 3 periods of an hysteresisexperimental cycle.

The model parameters were determined by least squaresregression, that is, for any set of data being fitted by a model, find-ing the minimum of the sum of squared residuals (SSR):

SSR ¼Xnp

j¼1

ðlj � lÞ2 ð14Þ

where j denotes a sampling time, np is the total number of points ofthe data being fitted lj the experimental value of viscositydetermined at that instant of time j, and l the corresponding model

Fig. 3. Experimental data and model predictions considering time-independentviscosity for (a) carrot, (b) tomato and (c) broccoli purees at s 10 �C, h 30 �C, 460 �C, } 80 �C. Grey data points are the averages of 3 experimental replicates andthe solid black line shows the average of the respective model predictions.

E. Tibäck et al. / Journal of Food Engineering 124 (2014) 35–42 39

prediction according to Eq. (13). It is possible to fit several sets ofdata jointly. The parameters that provide the best fit will be thosethat minimize SSR. The goodness of fit is quantified by the coeffi-cient of determination (R2), which is equal to the percentage ofthe variance of the data that is explained by the model. The 95%confidence intervals of each (cf0.95) are determined from:

cf0:95;k ¼ sek � t0:95;np�p ð15Þ

where k denotes a model parameter, p is the number of parametersof the model, t0.95,np-p is the one-sided t-Student distribution valuefor 95% confidence level and a number of degrees of freedom equalto the total number of data points minus the number of parametersof the model(s) being used and se is the standard error, which is gi-ven by:

sej ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiSSR� dk;k

np � p

sð16Þ

where dj,j is the element of the diagonal of the inverse of the Jaco-bian of the model in relation to its parameters, for the respectiveparameter j.

3. Results and discussion

3.1. Viscosity at isothermal conditions

The apparent viscosity as a function of time in the holding (iso-thermal) phase is presented in Fig. 2 for the 3 purees. It is easy toidentify the data from each temperature set due to the lag of thestarting time (which is the heating time, so the longer the lag ofthe initial point, the higher the temperature). There was a signifi-cant difference between replicates, although it can be seen that itis mostly due to the initial viscosity being different. Differencesof the initial viscosity of different samples are sometimes biggerthan the variations with time. This is likely due to variability inthe actual vegetable pieces used and natural variability in the pro-duction procedure over different days. For this reason, the initialviscosity was not considered as a model parameter; instead, the ac-tual initial viscosity of each sample determined experimentallywas taken as the initial point. The viscosity values were normalizedby dividing the viscosity value at each time/temperature with theinitial viscosity value which makes the replicates much more con-sistent (as will be shown later).

It is evident in Fig. 2 that the apparent viscosity of carrots didnot vary significantly with time during the holding periods at aconstant temperature, while that of tomato clearly falls with time,and the data for broccoli seem more confusing at first. Higher tem-peratures provide lower viscosities, as expected from Eq. (1), sothis would apparently suffice for carrots, while it is already evidentthat a time-independent model will be unsuitable for tomato. Theshearing effect observed in Fig. 2b was more significant the lowerthe temperature but it must be noted that when data records beginat the start of the isothermal period the sample has been shearedalready while heating, and for longer the temperature. Shearing(not temperature) is likely to be the reason why the lower the tem-perature of the data set, the greater the fall of viscosity of tomato inthe holding phase. A detailed analysis of the data for broccoli sug-gests a rheopectic effect: the samples are firming with time, morethe higher the temperature, and therefore the data at 80 �C show ahigher viscosity than at 60 �C, and the firming effect is still visibleduring the holding time at 80 �C. This suggests a significant gel for-mation caused by temperature. However, testing the samples im-plied shearing, which damages the structure formed in thepurees. Therefore, it is possible that gel formation occurred at tem-peratures below 80 �C as well, but not enough to overcome the ef-fects of the shearing. Lopez-Sanchez (2011) also remarked that atlow shear rates, there is a competition between building up struc-ture and flow, which makes it difficult to obtain a true flowmeasurement.

It is obvious that a time-independent model would fit the datavery poorly for broccoli at 80 �C and tomato at 10 and 30 �C. How-ever, the fits at other temperatures and of carrots may actually be

Table 3Model and goodness of fit parameters of the least squares regression of the apparent viscosity of purees with time-dependent models. Parameters with statistical significance arehighlighted in bold, and without statistical significance in italic.

Parameter Carrot Tomato Broccoli

Rheopectic Both Thixotropic Both Rheopectic Both

l0,10

(Pa s)a0.0961 0.1140 0.2331

l1,10

(Pa s)=l0,10 0.0963 ± 0.0002 0.05233 ± 0.0006 0.03918 ± 0.00004 =l0,10 0.2339 ± 0.0003

z (�C) 100.3 ± 0.1 101.7 ± 0.4 21.17 ± 0.44 84.69 ± 0.10 86.84 ± 1.5 91.33 ± 0.32kt (s�1) �0 6.26 � 10�8 ± 1 � 10�3 5.121 � 10�4 ± 1.3 � 10�6 0.0328 ± 0.0039 �0 2.53 � 10�14 ± 7 � 10�4

zt (�C) n.a. 84.4 ± 3.7 � 107 3.00 � 105 ± 9 � 107 193 ± 108 n.a. 84.4 ± 5 � 1011

kg (s�1) 2.227 � 10�4 ± 6 � 10�6 2.367 � 10�4 ± 1 � 10�3 �0 0.0324 ± 0.0039 4.025 � 10�4 ± 1.9 � 10�5 4.27 � 10�4 ± 7.6 � 10�4

zg (�C) 112.678 ± 0.001 116.402 ± 0.002 n.a. 373.9 ± 6 � 10�5 109.09 ± 0.01 110.8 ± 2 � 10�3

Total SSR 0.00857 0.00847 0.0819 0.0326 0.1397 0.1194

Average R2 of data sets10 �C 0 0 0.82 0.97 0 030 �C 0.93 0.90 0.92 0.89 0.97 0.9560 �C 0.96 0.97 0.90 0.97 0.96 0.9880 �C 0.98 0.98 0.82 0.96 0.99 0.99

n.a. Not applicable to the model.a This is the average of the experimental values of all data for each puree, it is not a model fit parameter. It is shown to analyze the difference to l1,10.

40 E. Tibäck et al. / Journal of Food Engineering 124 (2014) 35–42

hiding rheopetic and/or thixotropic effects, with the apparent vis-cosity being an integrated result of temperature, shearing and gel-ling effects.

3.2. Viscosity at non-isothermal conditions

3.2.1. Time-independent modelIt is already known from the isothermal data that a time-inde-

pendent model is unsuitable for tomato and broccoli puree, butconstant viscosities (influenced only by temperature, as per Eq.(1)) could be used for carrot puree. However, if the parametersare apparent, they will not explain well the viscosity changes un-der non-isothermal conditions. Table 2 shows the data fittingparameters with the time-independent model (Eq. (1)). This corre-sponds also to the generic model (Eq. (13)) with kt,10 and kg,10 equalto 0 and l1,10 = l0,10. Each replicate was fitted separately, but forthe sake of clarity only the averages of the experimental dataand of the model predictions are presented. It is quite clear thata time-independent model is unsuitable even for carrot puree(Fig. 3). This is an interesting conclusion, because if carrot pureehad been evaluated on its own and with only isothermal data, agood fit would have been obtained: challenging the system withnon-isothermal experiments however reveals that the isother-mally-obtained parameters obtained are apparent.

3.2.2. Time-dependent viscosity models, considering either thixotropyor rheopexy

Fig. 2 actually makes it obvious that significant gelling occurredin both carrot and broccoli purees because the viscosity after heat-ing at 80 �C was above the initial point. Therefore, it is not surpris-ing that even for carrots Eq. (1) is not able to fully describe theviscosity behavior. The opposite occurs with tomato. Therefore aviscosity model taking in account the rheopectic effects was testedfor carrot and broccoli puree, and with thixotropic effects for toma-to puree. Table 3 shows the model and goodness of fit parametersof the models. The fit for carrot and broccoli puree is quite accept-able, with errors between model and experimental data of the or-der of magnitude of differences between replicates, but for tomatothe model is not acceptable. The model fits are clearly biased at thehigher temperatures and the coefficients of determination are alltoo low. The coefficients of determination of the data sets in Table 3are not very good for the data at 10 �C for the other 2 purees also,

but those data generally vary very little throughout the entire30 min, and therefore the spread is relatively very large whichmakes the R2 unable to quantify a goodness of fit meaningfully.Differences between model and data are much smaller than be-tween replicates at that temperature. Fig. 4 shows the model andthe normalized viscosity (value measured at time t divided bythe initial value) for carrot, tomato and broccoli at 80 �C.

3.2.3. Time-dependent viscosity model considering both thixotropyand rheopexy

As tomato puree is not described satisfactorily by a model withthixotropic effects only, a model considering both thixotropy andrheopexy was tested. Table 3 also shows the parameters obtainedwith this model. It can be seen that the data regression of the car-rot and broccoli purees provides the same results, minus differ-ences that can be simply due to natural error and variability. Thetomato puree is now much better explained with this model.Fig. 5 shows the fits at 80 �C.

From the parameters kt, kg and their z-values, what can be sta-ted with confidence is that the difference between the two ratesvaries from 0.0034 s�1 at 10 �C to 0.0102 s�1 at 80 �C, as shownin Fig. 6. This difference varies with temperature much less thana Bigelow (or Arrhenius) model would suggest, being almost linear.It is noted that at 10 �C this should effectively be the thixotropicrate only, as there is no gelling at such a low temperature (gellingrates at 10 �C are extrapolations to the reference temperature).Thus, the thixotropic rate may well be increasing exponentially,but with gelling beginning to set in, it subtracts and the net effectis a lower impact of temperature in the apparent viscosity of thetomato puree.

3.3. Interpretation of the model results from the puree properties

Carrot and tomato purees have been reported to be week gelswith a yield stress and shear-thinning behavior (Rao and Cooley,1983; Yoo and Rao, 1995; Lopez-Sanchez et al., 2011). It can be as-sumed that broccoli puree would behave in a similar way.

Heating of fruit and vegetable purees involves both pectin deg-radation and gel formation. Pectin degradation can be causedeither by enzymatic activity, b-elimination, or acid hydrolysis. Thispectin degradation would imply a thixotropic behavior, which was

Fig. 4. Normalised experimental viscosity of carrot (a), tomato (b) and broccoli (c)puree at 80 �C, and average model predictions at considering rheopectic effects.Open data points show the experimental and the full line marks the average modelcurve.

Fig. 5. Normalised experimental data of tomato puree and average model predic-tion considering both thixotropic and rheopectic effects at 80 �C. Open data pointsshow the experimental data with a different symbol denoting different replicatesand the full line marks the average model curve.

Fig. 6. Difference between the thixotropic (shearing) and rheopectic (gelling) ratespredicted by the general model for tomato puree.

E. Tibäck et al. / Journal of Food Engineering 124 (2014) 35–42 41

very clear in the case of tomato, but not statistically significant incarrot and broccoli.

Heating fruit and vegetable purees promotes gel formation, andtherefore, rheopectic behavior. The pectin chains form gels througha three-dimensional network together with water, solutes and sus-pended particles. Two mechanisms are involved in pectin gel for-mation. Pectins with high degree of methoxylation form gels inthe presence of high amounts of soluble solids and under acidicconditions. Pectins with low degree of methoxylation form gelsin the presence of calcium ions at a wide range of pH values. Inthe purees, pectinmethylesterase activity at the elevated tempera-

tures lowers the degree of methoxylation and promotes gelformation.

The time-dependent viscosity model indicated both thixotropicand rheopectic behavior for tomato puree. Pectin degradation ex-plains the thixotropic behavior. The pH of tomato puree was low(4.3) allowing for acid hydrolysis of pectins to take place, whichmay explain the higher rate of thixotropy in tomato puree. The vis-cosity model also predicted a rate of rheopexy due to gelling ofpectins, which was much higher for tomato. In carrot and broccolipuree, the viscosity model predicted no thixotropic behavior, asthe limit viscosity was about the same as the initial viscosity. Thiswas not surprising, since polygalacturonase occurs only in smallamounts in carrots, limiting enzymatic pectin degradation (Fachinet al., 2004).

4. Conclusion

Different viscosity models were evaluated to describe therheological properties of fruit and vegetable purees. A time-independent model (Arrhenius plot) could be used describe theviscosity under isothermal conditions easily for carrot puree,however, under non-isothermal conditions a time-dependent

42 E. Tibäck et al. / Journal of Food Engineering 124 (2014) 35–42

model is needed for all products tested. Under isothermal condi-tions viscosity of tomato showed clearly thinning effects and broc-coli firming effects at higher temperatures. Therefore, time-dependent models considering thixotropic and rheopectic behav-ior, either separately or combined, were evaluated. Rheopecticbehavior can be explained by gel formation and was sufficient todescribe satisfactorily the apparent viscosity behavior of carrotand broccoli puree, while thixotropic behavior, considered to bedue to pectin degradation and shearing of the purees, was not ob-served with statistical confidence in these two purees and wasextensive in tomato puree, but not sufficient to describe thechanges in viscosity measured. Evidence of both shearing and gel-ling in tomato puree was therefore clear. It is however noted thatunder such conditions, differentiating the gelling and shearing con-stants with a single shear-rate test is intricate.

References

Abu-Jdayil, B., 2003. Modelling the time dependent rheological behavior of semi-solid foodstuffs. Journal of Food Engineering 57, 97–102.

Abu-Jdayil, B., Banat, F., Jumah, R., Al-Asheh, S., Hammad, S., 2004. A comparativestudy of rheological characteristics of tomato paste and tomato powdersolutions. International Journal of Food Properties 7 (3), 483–497.

Armelin, E., Marti, M., Rude, E., Labanda, J., Llorens, J., Aleman, C., 2006. A simplemodel to describe the thixotropic behavior of paints. Progress in OrganicCoatings 53 (3), 229–235.

Barnes, H.A., 1997. Thixotropy – a review. Journal Non-Newton Fluid 70, 1–33.Barrett, D.M., Garcia, E., Wayne, J.E., 1998. Textural modification of processing

tomatoes. CRC Critical Reviews in Food Science and Nutrition 38 (3), 173–258.Cunha, L.M., Oliveira, F.A.R., Brandão, T.S., Oliveira, J.C., 1998. Optimal experimental

design for estimating the kinetic parameters of the Bigelow model. Journal ofFood Engineering 33, 111–128.

Dolz, M., Hernandez, M.J., Delegido, J., Alfaro, M.C., Muñoz, J., 2007. Influence ofxantham gum and locust bean gum upon flow and thixotropic behaviour of foodemulsions containing modified starch. Journal Food Engineering 81 (1), 179–186.

Dutta, D., Dutta, A., Raychaudhuri, U., Chakraborty, R., 2006. Rheologicalcharacteristics and thermal degradation kinetics of beta-carotene in pumpkinpuree. Journal of Food Engineering 76 (4), 538–546.

Duvetter, T., Sila, D.N., Van Buggenhout, S., Jolie, R., Van Loey, A., Hendrickx, M.,2009. Pectins in processed fruit and vegetables: Part I – stability and catalyticactivity of pectinases. Comprehensive Reviews in Food Science and Food Safety8 (2), 75–85.

Fachin, D., Smout, C., Verlent, i., Nguyen, B.L., Van Loey, A.M., Hendrickx, M.E., 2004.Inactivation kinetics of purified tomato polygalacturonase by thermal and high-pressure processing. Journal of Agricultural and Food Chemistry 52 (9), 2697–2703.

Hayes, W.A., Smith, P.G., Morris, A.E.J., 1998. The production and quality of tomatoconcentrates. Critical Reviews in Food Science and Nutrition 38 (7), 537–564.

Hernandez, M.J., Dolz, J., Delegido, J., Cabeza, C., Dolz, M., 2008. Thixotropic behaviorof salad dressings stabilized with modified starch, pectin and gellan gum.Influence of temperature. Journal of Dispersion Science and Technology 29,213–219.

Labanda, J., Marco, P., Llorens, J., 2004. Rheological model to predict the thixotropicbehaviour of colloidal dispersions. Colloids Surfaces A 249 (1–3), 123–126.

Lopez-Sanchez, P., Nijsse, J., Blonk, H.C.G., Bialek, L., Schumm, S., Langton, M., 2011.Effect of mechanical and thermal treatments on the microstructure and therheological properties of carrot, broccoli and tomato dispersions. Journal of theScience of Food and Agriculture 91 (2), 207–217.

Love, B.J., 2009. Revisiting Boltzmann kinetics in applied rheology. Plastics ResOnline [serial online]. 10.2417/spepro.001588. Available from SPEPRO(4spepro.org). Posted Nov 30, 2009.

Maingonnat, J.F., Muller, L., Leuliet, J.C., 2005. Modelling the build-up of athixotropic fluid under viscometric and mixing conditions. Journal of FoodEngineering 71 (3), 265–272.

Rao, M.A., 1977. Rheology of liquid foods – a review. Journal of Texture Studies 8,135–168.

Rao, M.A., Cooley, H.J., 1983. Applicability of flow models with yield for tomatoconcentrates. Journal of Food Processing Engineering 6, 159–173.

Roussel, N., 2006. A thixotropy model for fresh fluid concretes: theory, validationand applications. Cement Concrete Research 36, 1797–1806.

Teyssandier, F., Ivankovic, M., Love, B.J., 2009. Modelling the effect of the curingconversion on the dynamic viscosity of epoxy resins cured by an anhydridecuring agent. Applied Polymer Science 115 (3), 1671–1674.

Yoo, B., Rao, M.A., 1995. Yield stress and relative viscosity of tomato concentrates:effect of total solids and finisher screen size. Journal of Food Science 60 (4), pp.777–779,785.