Journal of Food Engineering 93 (2009) 52–58

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Journal of Food Engineering

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

Sorption isotherm and state diagram of grapefruit as a tool to improveproduct processing and stability

M.J. Fabra, P. Talens, G. Moraga, N. Martínez-Navarrete *

Food Technology Department, Institute of Food Engineering for Development, Universidad Politécnica de Valencia, P.O. Box 22012, 46071 Valencia, Spain

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

Article history:Received 16 September 2008Received in revised form 6 December 2008Accepted 28 December 2008Available online 1 January 2009

Keywords:Glass transition temperatureFreezing temperatureWater activityWater contentRelative humidityDifferential scanning calorimetryCryoscopy

0260-8774/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.jfoodeng.2008.12.029

* Corresponding author. Tel.: +34 96 3879362; FaxE-mail address: nmartin@tal.upv.es (N. Martínez-N

The sorption isotherm and state diagram of grapefruit have been constructed and modelled. Partially air-dehydrated and freeze-dried grapefruit was used to obtain samples in the range of freezable and non-freezable water content, respectively. The cooling curve and/or differential scanning calorimetry of thesamples allowed freezing and glass transition temperatures to be obtained. The sorption isotherm wasfitted to the BET (Brunauer, Emmett and Teller) and GAB (Guggenheim–Anderson–de Boer) modelsand the glass transition data to the Gordon and Taylor model. To predict the freezing curve, both general-ized Norrish and Robinson and Stokes’ equations and an empirical model have been used. At water con-tent greater than 0.19 g water/g sample, cooling the product to �31.2 �C and keeping it in frozen storageat below �50 �C will ensure its maximum stability. If only non-freezable water is present in the product,the more stable glassy state may be achieved at different temperature or water contents, which may bepredicted from the obtained state diagram.

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1. Introduction

Grapefruit is a citrus fruit which contributes to human healthwith especially high amounts of ascorbic acid and fibre (Peiróet al., 2006). The presence of flavonoids such as naringin, narirutinand hesperidin (Peiró-Mena, 2007) is also important, as their in-take seems to have a significant impact on the prevention of differ-ent kinds of cancer and cerebro and cardiovascular diseases,mainly due to their antioxidant activity and ability to inhibit cellproliferation (Kuo, 1996; Vanamala et al., 2006). Naringin isresponsible for the bitter taste of grapefruit, which limits its popu-larity among consumers. Nevertheless, fresh or processed grape-fruit may be conveniently mixed with some other foods toformulate attractive, healthy products. In this sense, grapefruitwith different water contents, from fresh to freeze-dried, can beconsidered and adequate storage conditions must be defined in or-der to ensure the best quality when finally used.

Water activity and water content, correlated through sorptionisotherms, have been considered relevant parameters to describefood stability. Sorption isotherms of foodstuffs are essential forthe design, modelling and optimisation of many processes suchas drying, mixing or storage. The typical shape of an isotherm re-flects the way in which the water binds the system (Wolf et al.,1985; Lahsasni et al., 2004). For a fixed water content, the weaker

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: +34 96 3877369.avarrete).

the water interactions, the greater the water activity and the prod-uct becomes more unstable. Water activity depends on the compo-sition, temperature and physical state of the compounds.

The physical state of food compounds is also related to the sta-bility. It is well known that the solid state is much more stable thanthe liquid one. Nevertheless, below the crystallization tempera-ture, glassy or rubbery non-equilibrium amorphous states maybe achieved, frequently in processes with high cooling or dehydra-tion rates and especially in highly concentrated systems. The glassystate resembles the characteristics of the solid state, and occurs be-low the glass transition temperature. Literature reports that foodscan be considered very stable in the glassy state, since the molec-ular mobility is greatly reduced. In this physical state, the com-pounds involved in deteriorative reactions take many months oreven years to diffuse over molecular distances and approach eachother to react (Slade and Levine, 1991). The rubbery state occursabove the glass transition temperature, where molecular mobilityincreases and viscosity decreases. Thus, the product becomes likea liquid, corresponding to a more unstable state. This is also anon-thermodynamic equilibrium state, which will evolve morequickly over time. In both cases, they will evolve up to the corre-sponding thermodynamic equilibrium state below the crystalliza-tion temperature, which is the solid one. The crystallizationtemperature decreases and Tg increases when the water contentdecreases. The state diagram of a food is the right tool to use to findout the relationships between these temperatures and the watercontent, allowing the control of phase transitions in order to design

Nomenclature

a parameter in Eqs. (9) and (10) shown in Table 1b parameter in Eqs. (9) and (10) shown in Table 1aw water activity�Brix soluble solid content in the liquid phase of the product

(g soluble solids/100 g liquid phase)C parameter in (1–4) shown in Table 1C0g solid content of the maximally cryoconcentrated matrixk parameter in Eqs. (5) and (6) shown in Table 1K parameter in (3) and (4) shown in Table 1Ki Norrish coefficient in (8) shown in Table 1.Tf initial freezing temperature (�C)T 0m initial melting temperature of the ice crystals surround-

ing the maximally cryoconcentrated matrix (�C)Tg glass transition temperature (�C)T 0g glass transition temperature of the maximally cryocon-

centrated matrix (�C)Tg(w) glass transition temperature of amorphous water (�C)

Tg(as) glass transition temperature of anhydrous solids (�C)we equilibrium water content (g water/g dry solids)wo monolayer water content (g water/g dry solids)W 0

g non-freezable water content (g water/g product)xI (T) mass fraction of ice (g ice/g product) formed at a deter-

mined temperature Txs mass fraction of soluble solids in the product (g solids/g

product)xw mass fraction of water (g water/g product)xccs

w mass fraction of water in the cryoconcentrated solution(g water/g cryoconcentrated solution) obtained whencooling the sample to temperature T

x0w initial mass fraction of water (g water/g product)

Xi solid molar fractionXw water molar fractionzs mass fraction of solids in the sample liquid phase (g sol-

uble solids/g liquid phase) = �Brix/100.

M.J. Fabra et al. / Journal of Food Engineering 93 (2009) 52–58 53

efficient processes to obtain high quality products and also to opti-mize storage conditions (Roos and Karel, 1991; Roos, 1995; Rah-man, 2006).

The freeze-drying of fruits is an industrial process that allowsdried products to be obtained by the sublimation of water at lowpressure. During this process, the formation of non-equilibriumamorphous states is usual (Roos, 1995). Several authors havediscussed the importance of the glassy-rubbery state in powderproducts as it relates to collapse, stickiness, caking and re-crystal-lization phenomena (White and Cakebread, 1966; Kasapis et al.,2004; Ohkuma et al., 2008; Wang et al., 2008). The formation ofsolute crystals will imply an increase in free water and thus inwater activity, with the consequent increase in the rate of deterio-rative reactions. In this sense, it is very important to know thewater content–water activity–glass transition temperature rela-tionships to determine storage temperature and relative humiditythat ensure the stability of freeze-dried products avoiding thechange from the stable glassy state to the rubbery one. In general,foods with a glass transition temperature higher than room tem-perature (25–35 �C) can be considered as stable (Aguilera et al.,1993).

For high water content products, such as fresh or concentratedfruit juices, frozen storage must be considered to ensure the max-imum stability, the amount of ice formed being dependent on stor-age temperature. As the glass transition temperature decreaseswhen the water content increases, in these kinds of products theglassy state is normally not achieved at the common processingand storage temperatures. Nevertheless, the cryoconcentration ofthe liquid phase of the product during ice formation implies the in-crease in the Tg, that will achieve a maximum value when, due tokinetic impediments, the maximum amount of ice crystals isformed in the system. The glass transition temperature of themaximally cryoconcentrated matrix is known as T 0g and the corre-sponding solid content is C0g. Its water content ð1� C0gÞ correspondsto the non-freezable water of the product, which is known as W 0

g.From this point of view, the maximum stability in frozen productswill be achieved when the freezable water is allowed to crystallizeand the temperature remains below T 0g. Cryostabilization and cryo-protection technologies are related to the possibility of increasingTg or W 0

g, respectively, by adding different solutes. For these rea-sons, the most important aspect in these kinds of foods is to knowthe temperature at which the ice begins to appear and the amount

of ice that will be formed at any temperature, as well as the T 0g andW 0

g.The state diagram of a product over the whole range of water

content gives full information about the temperature of the differ-ent phase transitions at any water content. Many authors havestudied state diagrams of the aqueous solutions of pure compo-nents and model systems such as sugars (Roos and Karel, 1991).Moreover, state diagrams have been reported for different fruitssuch as grape and strawberry (Sá and Sereno, 1994), apple (Baiet al., 2001; Sá et al., 1999), pineapple (Telis and Sobral, 2001) orpersimmon (Sobral et al., 2001) but no data have been found eitherfor citrus fruits in general or for grapefruit, in particular.

The aim of this work was to establish the state diagram and thewater sorption isotherm of grapefruit. This information may beuseful to optimize processing and storage conditions of differentgrapefruit products that may be used in the food industry, in therange of both freezable and non-freezable water content.

2. Materials and methods

Grapefruits (Citrus paradise var. Marsh) were purchased in a lo-cal market and selected on the basis of a similar soluble solid con-tent (9.6 ± 0.2 �Brix). They were peeled and homogenized with ahigh-shear probe mixer (Ultra-Turrax T25, IKA). A part of thehomogenizate was partially dehydrated at three different watercontent levels using a modified household oven at 3.9 m/s and50 �C (Contreras et al., 2008), in order to obtain different grapefruitsamples in the range of freezable water content. On the other hand,the rest of the homogenizate was frozen at�40 �C in an ultrafreezer(ARC -45/87, Dycometal) at a potency of 3.2 kW. After that, it wasfreeze-dried at �50 �C and 10�2 Pa (Lioalfa 6–50, Telstar Industrial)and conditioned at different water contents, as described below,presumably in the range of non-freezable water content.

Fresh and partially dehydrated samples were analyzed as towater and soluble solid content, water activity and initial freezingpoint. Three replicates of each analysis were carried out. Mass frac-tion of water was obtained by vacuum drying the sample at 60 �Cto constant weight (Vacioterm, Selecta). �Brix was measured in anAbbé Atago 3-T refractometer. To obtain aw a dew point hygrome-ter (Aqualab CX-2, Decagon Devices) was used. The initial freezingtemperature was obtained from the change in the slope of the cool-ing curve obtained using a cryostatic bath (Refrigerated Circulator

54 M.J. Fabra et al. / Journal of Food Engineering 93 (2009) 52–58

9101, PolyScience) at �18 �C. Samples were placed into a glass testtube, immersed in the cold refrigerated bath with a thermocoupleand subjected to permanent stirring. The temperature of each sam-ple was recorded every 5 s.

For adsorption experiments, the freeze-dried grapefruit wasplaced, at 25 �C, in hermetic chambers at different relative humid-ities (RH) which were achieved by placing different saturated saltsolutions (LiCl, KCH2CH3, MgCl2, K2CO3, Mg(NO3)2, CuCl2, NaCl) in-side them. Relative humidities ranged between 11 and 75% (Green-span, 1977). Three replicates of about 1.5 g were placed in eachchamber. The samples were weighed periodically till a constant va-lue was reached (assumed when the difference between two con-secutive weights was lower than 0.001 g), thus ensuring aw of eachsample to be equal to the corresponding RH/100. In each equili-brated sample, xw and Tg were analyzed. To this end, equilibratedsamples (�10 mg) were placed into DSC pans (Seiko Instruments,P/N SSC000C008), sealed and analyzed using a DSC 220 CU-SSC5200 (Seiko Instruments). The heating rate was 5 �C/min andthe temperature range varied between �100 and 100 �C, depend-ing on sample water content. The water content after the thermalanalysis was determined by drying the sample in the pans (previ-ously holed) in a vacuum oven (<100 mm Hg) at 60 ± 1 �C untilconstant weight. After drying, the pans were again thermally stud-ied to determine the Tg of the anhydrous sample.

T 0g and T 0m were also analyzed in triplicate by DSC. To this end, asample of fresh grapefruit was placed into a sealed DSC pan andcooled from room temperature to �35 �C (at 2 �C/min), held at thistemperature for 30 min and then cooled to �100 �C (at 10 �C/min),thus ensuring the maximum ice crystallization (Moraga et al.,2004). Afterwards, the heating curve was registered till 40 �C at aheating rate of 5 �C/min.

3. Results and discussion

3.1. Sorption isotherm

Many food products exhibit a different behaviour for adsorptionand desorption processes. Nevertheless, when a food product issubjected to water content changes in a range that implies waterloss or gain, the use of a ‘‘working isotherm”, as suggested by Labu-za (1984), may be useful to predict any drying or humidificationcycle undergone by the food. In this work, it was of interest to ob-tain useful data over the whole water content range of grapefruitand so, all the experimental we–aw values were considered to con-struct the ‘‘working sorption isotherm” of grapefruit (Fig. 1). The

6

7

8

0

1

2

3

5

0 0.25 0.5 0.75 1aw

we

(g w

ater

/g d

ry s

olid

s)

Fig. 1. Water sorption isotherm of grapefruit at 25 �C. Experimental points (�) andfitted BET (x) and GAB (—) models.

observed behaviour was typical of products with high-sugar con-tents which adsorb relatively small amounts of water at low-wateractivities but present a sharp increase in the amount of sorbedwater at higher water activities, due to the prevailing effect of sol-ute-solvent interactions associated to sugar dissolution (Hubingeret al., 1992; Saravacos et al., 1986; Tsami et al., 1990). Similar re-sults have been obtained with other fruits such as pineapple (Telisand Sobral, 2001), persimmon (Sobral et al., 2001), strawberry(Moraga et al., 2004), guava, mango and pineapple (Hubingeret al., 1992) or grape (Gabas et al., 1999).

Experimental sorption data were fitted to GAB (Guggenheim–Anderson–de Boer) (Van den Berg and Bruin, 1981) and BET (Bru-nauer, Emmett and Teller) (Brunauer et al., 1938) models. The BETequation (Table 1, Eq. (1)) is the only one that stems from a ther-modynamic approach to the sorption process (Iglesias and Chirife,1976) and the obtained parameters have a physical meaning, wo

corresponding to the amount of water needed to surround the foodsurface with just one layer of water molecules and C relating to theheat liberated in the monolayer sorption process. The concept ofmonolayer water content was found to be a reasonable guide tovarious aspects of interest in dried foods, related to the limit fromwhich the rate of deteriorative reactions began to increase signifi-cantly. Nevertheless, this equation is not adequate at high wateractivity levels, when dissolution phenomena become more impor-tant than sorption. The GAB model (Table 1, Eq. (3)) has been suc-cessfully used by many investigators to fit sorption data over awide range of aw.

Fig. 1 shows the predicted sorption isotherm obtained fromboth fitted models (Table 1, Eqs. (2) and (4)). The limit for BETmodel application in grapefruit was 0.68 water activity, thus indi-cating the limit from which solute–solvent interactions becomemore important in this fruit. Determination coefficients (r2) ofBET and GAB models were 0.913 and 0.996, respectively. The samemonolayer water content was obtained in both cases, this being0.10 g water/g dry solids. Regarding the C parameter, it differs from1.48 to 1.52 depending on the use of the BET or GAB model. The Kparameter of the GAB model, related to the enthalpy of water mul-tilayer sorption, was 0.98. As deduced from the C values, the ob-tained sorption isotherms may be classified as type III, accordingto Brunauer’s classification (Brunauer et al., 1940) in agreementwith that found for other fruits (Lim et al., 1995; Maskan andGögüs, 1998; Roos, 1993; Vázquez et al., 1999).

3.2. State diagram

Fig. 2 shows some experimental DSC curves obtained in sam-ples presumably without freezable water content. Fig. 3 showsthe experimental DSC curve of the maximally cryoconcentratedfresh grapefruit sample, allowing the evaluation of T 0g and T 0m. Ina DSC curve, a step change in the heat flow implies a change inthe heat capacity of the sample, which occurs during a glass tran-sition. This is clearly observed in Fig. 2 for the different samples.Fig. 3 shows a typical thermogram of a sample containing freezablewater content where, despite the glass transition, there appears anendotherm associated to the ice melting. In this figure, two glasstransitions, one condensed to the ice melting endotherm, can beobserved. As described by Rahman (2004), this may be the resultof the formation of a solute-unfrozen water unequilibrated phasetrapped around or within the ice crystals. This phase relaxed at ahigher temperature than the bulk unfrozen phase, resulting inthe two-step glass transition. From this point of view, the Tg atthe lowest temperature was assumed as the T 0g of the sample. Itwas characterized at the mid point of the transition, resulting ina value of �50.0 ± 0.3 �C. Related to the endotherm, the beginningof the ice melting, evaluated as indicated in Fig. 3, corresponds toT 0m, this being �31.2 ± 0.9 �C for grapefruit.

Table 1Models used for fitting the experimental data.

Model Expression Eq.

BET (Brunauer et al., 1938) we ¼ wo �C�awð1�awÞ�ð1þðC�1Þ�awÞ (1)

Linearized awð1�awÞwe

¼ 1wo �C þ

C�1wo �C aw (2)

GAB (Van den Berg and Bruin, 1981) we ¼ wo �C�K�awð1�K�awÞ�ð1þðC�1Þ�K�awÞ (3)

Linearized awwe¼ 1

wo �C�K þC�2wo �C aw þ Kð1�CÞ

wo �C a2w (4)

Gordon and Taylor (1952) Tg ¼ð1�xwÞ�TgðasÞþk�xw �TgðwÞ

ð1�xw Þþk�xw(5)

Linearized Tg ¼ TgðasÞ þ k xwðTgðwÞ�TgÞð1�xwÞ (6)

Robinson and Stokes (1965) � log awaw ¼ 0:004207DTf þ 2:1 � 10�6DT2f where DTf = 0�Tf (7)

Generalized Norrish (Norrish, 1966) ln aw ¼ ln Xw � ðP

iK1=2i XiÞ2 (8)

Empirical Eq. based on Tchigeov (Fikiin, 1998) Tf ¼ a1þ b

lnð2�xw Þ(9)

Linearized 1T ¼ 1

a þ ba � 1

lnð2�xwÞ (10)

Soluble solids fraction in the product xs ¼ ð zs1�zsÞxw (11)

Mass balance to obtain mass fraction if ice xIðTÞ ¼ x0w�xCCS

w ðTÞ1�xCCS

w ðTÞ(12)

-90 -70 -50 -30 -10 10 30 50 70

0.1 mW

End

o H

eat f

lux

aw=0.112

aw=0.225

aw=0.320

aw=0.432

aw=0.500

aw=0.675

Tg

T (ºC)

Fig. 2. DSC thermograms showing glass transition of grapefruit samples equili-brated at different water activities.

-6

-5

-4

-3

-2

-1

-70 -50 -30 -10 10 30

T (ºC)

ΔH

T ’g Tm’

-45

-35

-25

-15

-5

5

-70 -50 -30 -10 10 30

T (ºC)

Hea

t flu

x (m

W)

Hea

t flu

x (m

W)

Fig. 3. DSC thermogram of fresh grapefruit sample. The inner graph shows thecomplete information and the outer one an amplified scale to analyze thecharacteristic T 0g and T 0m of the sample.

Table 2Values of the glass transition temperature (onset and mid point Tg) for freeze-driedgrapefruit samples equilibrated at different water activity (aw) and water content(xw).

aw xw (g water/g sample) Onset Tg (�C) Mid-point Tg (�C)

0 38.0 ± 1.3 42.0 ± 0.30.113 0.018 ± 0.004 12.0 ± 0.8 23.3 ± 0.20.230 0.039 ± 0.011 8.1 ± 1.4 13 ± 0.20.330 0.058 ± 0.001 �8.3 ± 0.6 �3.1 ± 0.20.430 0.083 ± 0.002 �17.0 ± 1.2 �11.0 ± 0.20.520 0.118 ± 0.001 �33.0 ± 0.6 �27.2 ± 0.20.680 0.151 ± 0.055 �54.1 ± 0.5 �47.5 ± 0.30.755 0.232 ± 0.002 �60.0 ± 1.2 �53.1 ± 0.4

-140

-120

-100

-80

-60

-40

-20

0

20

40

60

00.250.50.751

xw (g water/g product)

T (

ºC)

Wg’

Tm’

Tg’

Eq. (7)+(8) Tf curve Eq. (9) x Experimental

Eq. (5) 2p Eq. (5) 1p Tg curve Experimental

Fig. 4. State diagram of grapefruit. Experimental and predicted values using theequations described in Table 1 (freezing temperature (Tf) curve predicted using Eqs.(7) and (8) considering all the solutes of grapefruit; glass transition temperature(Tg) predicted using Eq. (5) fitted with one (1p) or two (2p) parameters; T 0g: glasstransition temperature of maximally cryoconcentrated fruit liquid phase; T 0m:melting temperature of ice crystals surrounding the maximally cryoconcentratedfruit liquid phase; W 0

g: non-freezable water content).

M.J. Fabra et al. / Journal of Food Engineering 93 (2009) 52–58 55

Table 2 summarizes the onset and mid-point temperatures ofthe glass transition for the rest of the analyzed samples, which oc-curred over a temperature range of 8–22.6 �C. Midpoint Tg wasconsidered as the characteristic of the glass transition (Roos,1995). In Table 2 it stands out the Tg of the sample withaw = 0.755, which is lower than T 0g. This indicates the presence offreezable water content in this sample. If a partial ice formation oc-curred during the cooling process, the evaluated Tg will correspondto a sample with an unknown amount of water present in theamorphous matrix. This fact could not be checked in this case, asthe sample was only heated up to �20 �C during DSC analysis.

For this reason, the experimental Tg value analyzed for this samplewas not considered in the study.

The Tg–xw data of samples without freezable water were fittedto the linearized Gordon and Taylor model (Table 1, Eq. (5)), con-sidering Tg(w) as �135 �C (Roos, 1995). The linearized equation (Ta-ble 1, Eq. (6)) was fitted with no intercept, assuming theexperimental 42 �C value as Tg(as), with a very close agreement be-tween experimental and predicted values (Fig. 4). In this case, theobtained empirical constant k value was 4.66 (R2 = 0.980, standard

56 M.J. Fabra et al. / Journal of Food Engineering 93 (2009) 52–58

error of the estimate (SE) = 4.3). When the model was fitted byestimating both Tg(as) and k parameters these being 34.53 �C and4.11, respectively (R2 = 0.989, SE = 3.7), a close fit was also ob-tained, except in the case of the anhydrous sample (Fig. 4). In thiscase, the predicted Tg(as) was lower than the experimental one.From this point of view, if the value of the anhydrous sample isneeded, it is advisable to determine it experimentally. The kparameter obtained for grapefruit was similar to that obtainedwhen working with other fruits such as strawberries (k = 4.14,Moraga et al., 2004), apple (k = 3.59, Bai et al., 2001), blueberryand blackberry (k = 4.02 and 4.12, respectively, Khalloufi et al.,2000).

The experimental Tg(as) and the k parameter obtained from theGordon and Taylor fitted model were used to predict W 0

g of thisproduct, taking into account the experimental T 0g value. The ob-tained value was 0.19 g water/g maximally cryoconcentrated ma-trix. Similar values have been reported for strawberry (Roos,1987), apple (Bai et al., 2001) and pineapple (Telis and Sobral,2001), these being 0.214, 0.264 and 0.220, respectively.

Fig. 5 shows, as an example, the cooling curve obtained for freshgrapefruit and the procedure used to determine Tf in samples withfreezable water content. In a similar way as that proposed by Rah-man et al. (2005), Tf was considered as the equilibrium tempera-ture observed in the cooling curve. Table 3 shows experimentalvalues of water content, water activity, �Brix and initial ice forma-tion temperature of fresh and partially dehydrated samples. In thecase of Tf–xw relationships, no common models have been de-scribed to fit experimental data in foods. Bai et al. (2001) suggestthe use of the theoretical Clausius–Clapeyron equation but it islimited to ideal solutions. A modification of this equation was pro-posed by Sablani et al. (2004) by introducing parameters for non-ideal behaviour. Two different approaches have been followed inthis work to predict the freezing curve. On the one hand, the liquid

-5

0

5

10

15

20

25

0 100 200 300 400Time (s)

T (

ºC)

Tf

Fig. 5. Cooling curve of fresh grapefruit. The arrow indicates the initial freezingtemperature.

Table 3Experimental (mean ± standard deviation) and predicted values of water content (xw, g wa(aw) and initial freezing temperature (Tf, �C) of samples considered in the study in the ran

Experimental values

xw �Brix aw Tf (�C)

0.876 ± 0.003 9.6 ± 0.2 0.993 ± 0.003 �0.9 ± 0.10.839 ± 0.002 13.2 ± 0.2 0.987 ± 0.002 �1.2 ± 0.20.712 ± 0.004 18.0 ± 0.2 0.975 ± 0.004 �2.5 ± 0.10.647 ± 0.002 25.0 ± 0.3 0.964 ± 0.003 �3.3 ± 0.10.190d �31.2 ± 0.9e

a Following the indicated equations (see Table 1).b Considering sucrose, glucose, fructose and citric acid as soluble solids of grapefruit.c Considering sucrose as the only soluble solid of grapefruit.d Water content of the maximally cryoconcentrated matrix (predicted value from Gore Experimental T 0m value.

phase of grapefruit samples was considered as a solution of water,sugars and citric acid and the classical equations used to predictthe behaviour of solutions were applied. On the other hand, anempirical equation based on the Tchigeov model (Fikiin, 1998)was also used.

When the liquid phase of the grapefruit is considered as a solu-tion, the Robinson and Stokes equation (Table 1, Eq. (7)), obtainedfrom a thermodynamic approach to solid-liquid equilibrium, maybe useful to predict Tf of samples with different aw. On the otherhand, if the predominant solutes in the liquid phase of the productare sugars and/or citric acid, as in fruits, the Generalized Norrishequation (Table 1, Eq. (8)) may be used to predict aw of sampleswith known water and soluble solid content. The combined useof both equations allows us to obtain a predicted Tf–xw relationshipin a product of known composition; the more that is known aboutthe present compounds and their ratio, the greater the exactitudein the prediction.

In order to estimate the soluble solid composition of the liquidphase of the product with different water contents, the followingapproaches were considered. For a determined xw, the solid massfraction will be (1�xw). The ratio of soluble to insoluble solids forgrapefruit was obtained from the experimental xw and �Brix ofthe fresh sample (Table 3). The mass fraction of soluble solids inthe product was calculated from Eq. (11) (Table 1), this being0.093. In this way, the ratio of soluble to insoluble solids was0.75:0.25.

As described by Peiró-Mena (2007), who worked with 10 �BrixStar Ruby grapefruit samples, the main soluble solids in this fruitare sucrose, fructose, glucose and citric acid, in mass ratios of45.5, 21.2, 18.0 and 15.3, respectively. If the composition of the dif-ferent soluble solids is not known, the simplest approach will be toconsider the major one present in the fruit (in this case sucrose) asthe only one. Table 3 shows predicted Tf values for the experimen-tally considered samples. When only the major solute (sucrose)was considered, important differences were obtained betweenexperimental and predicted values. When the different soluble sol-utes and their ratios in grapefruit were taken into account, a muchcloser fit between experimental and predicted Tf was observed.

To apply the empirical procedure based on the Tchigeov model(Table 1, Eq. (9)), experimental Tf–xw and T 0m �W 0

g data were con-sidered and fitted to the corresponding linearized equation (Table1, Eq. (10)). The two obtained empirical parameters, a and b, were4.427 and �0.696 (R2 = 0.9996), respectively. The predicted valuesin this case (Table 3) were also very similar to the experimentalones, with only slightly more marked differences at the water con-tent corresponding to W 0

g.Fig. 4 shows the state diagram of grapefruit, which shows

experimental and predicted Tf and T 0m values over all the freezablewater content range, together with the experimental and predicted

ter/g sample), �Brix (g soluble solids/100 g liquid phase in the sample), water activityge of freezable water content.

Predicted valuesa

awb Eq. (8) Tf

b Eq. (7) Tfc Eq. (7) Tf Eq. (9)

0.991 �0.91 �0.60 �0.890.988 �1.23 �0.82 �1.210.975 �2.61 �1.80 �2.520.966 �3.54 �2.50 �3.380.754 �28.78 �29.50 �24.05

don and Taylor model).

M.J. Fabra et al. / Journal of Food Engineering 93 (2009) 52–58 57

Tg–xw relationships. If the water content of the product recom-mends its frozen storage, temperatures must be maintained at be-low Tg to avoid ice recrystallization and improve grapefruit qualityafter thawing. In this sense, and from an economic point of view,the most desirable process will be that which allows the maximumice formation during cooling and the storage at temperatures be-low T 0g. Nevertheless, as T 0g ¼ �50 �C, the use of cryostabilizers witha high molecular weight, such as maltodextrines (Roos, 1995), maybe recommendable to increase the Tg of the product and approxi-mate it to that of commercial freezers.

The freezing curve of the state diagram can also be applied toevaluate the amount of ice formed in foods according to their ini-tial water content and the product’s freezing point, which is extre-mely important in the quality preservation of frozen fruits. Fig. 6shows the mass fraction of ice in frozen grapefruit as a functionof temperature for, both, the fresh sample (87.6% water content)and the partially dehydrated sample (70% water content). Eq.(12) (Table 1), deduced from the mass balances at each tempera-ture in the curve, was used to estimate these values. To this end,the freezing curve predicted when using Eqs. (7) and (8) was used.

In Fig. 6, the different initial ice formation temperature depend-ing on the sample’s initial water content can be observed. It is alsopossible to observe that the greatest amount of ice is formed at thebeginning of the freezing process and that not all the water of thesample will be converted into ice. In fact, at �28.75 �C, 84.6 and63 g ice/100 g sample are formed in fresh and partially dehydratedsamples, respectively, which imply that 97% and 90% of the waterpresent in the product will be frozen. On the other hand, it can beobserved that if the fresh fruit is stored at�18 �C, the amount of iceformed at equilibrium is 82.5 g ice/100 g product (94 g ice/100 gwater) whereas only 58.5 g ice/100 g product (84 g ice/100 gwater) is formed if the sample is previously dehydrated till 70%water content. The smaller amount of ice formed per gram of waterpresent in the dehydrated sample will presumably be related witha better product quality after thawing.

When no freezable water exists in the product, it becomes morepractical to know about the Tg–xw relationships. Collapse and thedevelopment of stickiness during the drying of fruit juices is prob-ably the main technical obstacle to obtain free-flowing powder andto maintain it when handling, in order to improve its rehydration(Karathanos et al., 1993). As described by Roos (1995), the powderswill be able to support their own weight when thermal and waterplasticizations are minimized by keeping the product temperaturebelow the Tg. As can be observed in the grapefruit state diagram(Fig. 4), the critical water content (CWC) that takes the productfrom the glassy to rubbery state at room temperature (25 �C) is2.23 g water/100 g product. As the usual water content of freeze-

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-30 -25 -20 -15 -10 -5 0

T (ºC)

x I (

g ic

e/g

sam

ple)

Fig. 6. Mass fraction of ice (xI) in frozen grapefruit as a function of temperature.Fresh, 87.6% water content (––), and partially dehydrated, 70% water content (- - -),fruit. Arrows indicate the amount of ice formed in each case at �18 �C.

dried or spray-dried products ranges between 3–5%, there aretwo possibilities to ensure powder quality: increasing the Tg ordecreasing the storage temperature. As commented on above, add-ing high molecular weight solutes (i.e. maltodextrins) will increasethe Tg of the product. On the other hand, the chilling storage offreeze-dried grapefruit at 5 �C increases CWC to 5.37 g water/100 g product (Fig. 4).

3.3. Glass transition temperature- water activity-water contentrelationship

Freeze-dried products may suffer a glass transition with smallchanges in temperature or water content. An increase in the watercontent will occur if the powder is exposed to an atmosphere witha RH/100 higher than its aw. Modelling water plasticization andwater adsorption phenomena together will allow food stability tobe predicted in various processing and storage conditions. The Gor-don and Taylor and GAB models may be used to obtain a modifiedstate diagram showing depression of Tg with increasing aw. Fromthis diagram, the critical water activity (CWA) that depresses theTg below the storage temperature may be obtained. If the sorptionisotherm is also plotted in the diagram, the corresponding CWC ofthe product will be obtained.

Fig. 7 shows the Tg–aw–xw relationships of freeze-dried grape-fruit in the range of non-freezable water content. In this case, ifthe product is commercialized at 25 �C, the CWA that takes theproduct from the glassy to rubbery state is 0.140 (CWC 2.23 gwater/100 g sample). This means that storing the powder at a RHof over 14% will lead to the rubbery state. Above these values, col-lapse and stickiness of grapefruit powder, as well as the crystalliza-tion of amorphous compounds, could take place, the higher thestorage temperature, the more likely it is that this will occur. Fromthis result, it can be deduced that the product must be adequatelypackaged in a low water vapour permeability material. Knowinghow water sorption behaves at different temperatures will allowus to analyze the convenience of, for example, the chilled storageof dried powder.

On the other hand, the CWC at 25 �C is much lower than themonolayer moisture content obtained from the BET and GAB mod-els, which indicates that wo is not a water content that assuresquality preservation during the storage of dried grapefruit, as hasbeen reported by Moraga et al. (2004) and Roos (1993) forstrawberry.

-100

-80

-60

-40

-20

0

20

40

60

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

aw

Tg

(ºC

)

0

0.05

0.1

0.15

0.2

0.25

0.3

xw

CWA 25

CWC25

(g

wat

er/g

pro

duct

)

Fig. 7. Temperature - water activity (—) and water activity - water content (——)relationship of grapefruit in the range of non-freezable water content. CWA: criticalwater activity; CWC: critical water content.

58 M.J. Fabra et al. / Journal of Food Engineering 93 (2009) 52–58

4. Conclusions

In products such as grapefruit, the freezing curve at differentwater contents may be predicted by using Generalized Norrishand Robinson & Stokes equations, although the solute compositionof the sample is needed to obtain accurate results. When this is un-known, the proposed empirical equation based on the Tchigeovmodel may also be applied and in this case some experimentalTf–xw data are needed. This curve, together with the Tg–xw relation-ship predicted by the Gordon and Taylor model, allows the statediagram to be constructed, which is useful to establish the process-ing and storage conditions that will ensure the maximum stabilityof different grapefruit-products. If the water content of the productis greater than 19.0 g water/100 g sample (aw = 0.687), frozen stor-age will be recommended. In this case, cooling the sample to –31.2 �C will allow the maximum ice formation and the frozen stor-age at temperature lower than –50 ± 0.3 �C will be necessary to en-sure the glassy state of the remaining amorphous matrix and so itsmaximum stability. The partial dehydration of the sample plays acryoprotective role. If non-freezable water is present in the sample,the more stable glassy state may be achieved at different temper-ature-water content relationships. At room temperature (25 �C),the low CWA and CWC values of the product indicate how packag-ing is needed to ensure stability. Both for freezing or freeze-drying,strategies to increase the Tg will improve the product’s stabilityduring storage.

Acknowledgements

The authors thank the Ministerio de Educación y Ciencia andthe Fondo Europeo de Desarrollo Regional (FEDER) for the financialsupport throughout the project AGL2005-05994. Author M.J. Fabrathanks Spanish Ministry of Culture and Education for a FPU grant(AP2005-3562).

References

Aguilera, J.M., Levi, G., Karel, M., 1993. Effect of water content on the glasstransition and caking of fish protein hydrolyzates. Biotechnology Progress 9,651–654.

Bai, Y., Rahman, M.R., Perera, C.O., Smith, B., Melton, L.D., 2001. State diagram ofapple slices: glass transition and freezing curves. Food Research International34, 89–95.

Brunauer, S., Emmett, P.H., Teller, E., 1938. Adsorption of gases in multimolecularlayers. Journal of American Chemistry Society 60, 309–320.

Brunauer, S., Deming, L.S., Deming, W.E., Teller, E., 1940. On a theory of the van deWaals adsorption of gases. Journal of American Chemistry Society 62, 1723–1732.

Contreras, C., Martín-Esparza, M.E., Chiralt, A., Martínez-Navarrete, N., 2008.Influence of microwave application on convective drying: effects on dryingkinetics, and optical and mechanical properties of apple and strawberry. Journalof Food Engineering 88, 55–64.

Fikiin, K.A., 1998. Ice content prediction methods during food freezing. A surveyof the eastern European literature. Journal of Food Engineering 38, 331–339.

Gabas, A.L., Telis-Romero, J., Menegalli, F.C., 1999. Thermodynamic models forwater sorption by grape skin and pulp. Drying Technology 17 (4&5), 961–974.

Gordon, M., Taylor, J.S., 1952. Ideal copolymers and second-order transitions ofsynthetic rubbers. I. Non-crystalline copolymers. Journal of Applied Chemistry2, 493–500.

Greenspan, L., 1977. Humidity fixed points of binary saturated aqueous solutions.Journal of Research of the National Bureau of Standards, 81–89.

Hubinger, M., Menegalli, F.C., Aguerre, R.J., Suarez, C., 1992. Water adsorptionisotherms of guava, mango and pineapple. Journal of Food Science 57 (6), 1405–1407.

Iglesias, H., Chirife, J., 1976. BET monolayer values in dehydrated foods. Comparisonwith BET theory. Lebensmittel Wissenschaft and Technologie 9, 123–126.

Karathanos, V., Anglea, S., Karel, M., 1993. Collapse of structure during drying ofcelery. Drying Technology 11 (5), 1005–1023.

Kasapis, S., Mitchell, J., Abeysekera, R., MacNaughtan, W., 2004. Rubber-to-glasstransitions in high sugar/biopolymer mixtures. Trends in Food Science andTechnology 15 (6), 298–304.

Khalloufi, S., El-Maslouhi, Y., Ratti, C., 2000. Mathematical model for prediction ofglass transition temperature of fruit powders. Journal of Food Science 65, 842–845.

Kuo, S.M., 1996. Antiproliferative potency of structurally distinct dietary flavonoidson human colon cancer cells. Cancer Letters 110 (1/2), 41–48.

Labuza, T.P., 1984. Moisture Sorption: Practical Aspects of Isotherm Measurementand Use. American Association of Cereal Chemists, St. Paul, MN.

Lahsasni, S., Kouhila, M., Mahrouz, M., 2004. Adsorption-desorption isotherms andheat of sorption of prickly pear fruit (Opuntia ficus indica). Energy Conversionand Management 45, 249–261.

Lim, L.T., Tang, J., He, J., 1995. Moisture sorption characteristics of freeze driedblueberries. Journal of Food Science 60 (4), 810–814.

Maskan, M., Gögüs, F., 1998. Sorption isotherms and drying characteristics ofmulberry (Morus alba). Journal of Food Engineering 37, 437–449.

Moraga, G., Martínez-Navarrete, N., Chiralt, A., 2004. Water sorption isotherms andglass transition in strawberries: influence of pretreatment. Journal of FoodEngineering 62, 315–321.

Norrish, R.S., 1966. An equation for the activity coefficients and equilibrium relativehumidities of water in confectionery sugars. Journal of Food Technology 1, 25–39.

Ohkuma, C., Kawai, K., Viriyanattanasak, C., Mahawanich, T., Tantratian, S., Takai, R.,Suzuki, T., 2008. Glass transition properties of frozen and freeze-dried surimiproducts: effects of sugar and moisture on the glass transition temperature.Food Hydrocolloids 22 (2), 255–262.

Peiró, R., Dias, V.M.C., Camacho, M.M., Martínez-Navarrete, N., 2006. Micronutrientflow to the osmotic solution during grapefruit osmotic dehydration. Journal ofFood Engineering 74, 299–307.

Peiró-Mena, R., 2007. Cambios en los micronutrientes y compuestos fitoquímicosasociados al procesado osmótico de frutas y su estabilidad en un productogelificado. Ph.D. Polythecnic University, Valencia (Spain).

Rahman, M.S., 2004. State diagram of date flesh using differential scanningcalorimetry (DSC). International Journal of Food Properties 7 (3), 407–428.

Rahman, M.S., 2006. State diagram of foods: its potential use in food processing andproduct stability. Trends in Food Science and Technology 17, 129–141.

Rahman, M.S., Sablani, S.S., Al-Habsi, N., Al-Maskri, S., Al-Belushi, R., 2005. Statediagram of freeze-dried garlic powder by differential scanning calorimetry andcooling curve methods. Journal of Food Science 70 (2), E135–E141.

Robinson, R.A., Stokes, R.H., 1965. Electrolyte Solutions. Butterworths Publication,Ltd., London.

Roos, Y.H., 1987. Effect of moisture on the thermal behaviour of strawberriesstudies using differential scanning calorimetry. Journal of Food Science 52 (1),146–149.

Roos, Y.H., 1993. Water activity and physical states effects on amorphous foodstability. Journal of Food Engineering 24 (3), 339–360.

Roos, Y.H., 1995. Phase Transitions in Food. Academic Pres, San Diego, CA.Roos, Y., Karel, M., 1991. Plasticizing effect of water of thermal the behaviour and

crystallization of amorphous food models. Journal of Food Science 56 (1), 38–43.

Sá, M.M., Sereno, A.M., 1994. Glass transitions and state diagrams for typical naturalfruits and vegetables. Thermochimica Acta 246, 285–297.

Sá, M.M., Figueiredo, A.M., Sereno, A.M., 1999. Glass transitions and state diagramsfor fresh and processed apple. Thermochimica Acta 329, 31–38.

Sablani, S.S., Kasapis, S., Rahman, M.S., Al-Jabri, A., Al-Habsi, N., 2004. Sorptionisotherms and state diagram for evaluating stability criteria of abalone. FoodResearch International 37, 915–924.

Saravacos, G.D., Tsiourvas, D.A., Tsami, E., 1986. Effect of temperature on the wateradsorption isotherms of sultana raisins. Journal of Food Science 51 (2), 381–383,387.

Slade, L., Levine, H., 1991. Beyond water activity: recent advances based on analternative approach to the assessment of food quality and safety. CriticalReviews in Food Science and Nutrition 30 (2–3), 115–360.

Sobral, P.J.A., Telis, V.R.N., Habitante, A.M.Q.B., Sereno, A., 2001. Phase diagram forfreeze-dried persimmon. Thermochimica Acta 376, 83–89.

Telis, V.R.N., Sobral, P.J.A., 2001. Glass transitions and state diagram for freeze-driedpineapple. Lebensmittel Wissenschaft and Technologie 34, 199–205.

Tsami, E., Marinos-Kouris, D., Maroulis, Z.B., 1990. Water sorption isotherms ofraisins, currants, figs, prunes and apricots. Journal of Food Science 55, 1594–1597, 1625.

Van den Berg, C., Bruin, S., 1981. Water activity and its estimation in food systems:theoretical aspects. In: Rockland, L.B., Stewart, G.F. (Eds.), Water Activity:Influences on Food Quality. Academic Press, New York, pp. 1–43.

Vanamala, J., Reddivari, L., Kil-Sun-Yoo, Pike, L.M., Patil, B.S., 2006. Variation in thecontent of bioactive flavonoids in different brands of orange and grapefruitjuices. Journal of Food Composition and Analysis 19 (2–3), 157–161.

Vázquez, G., Chenlo, F., Moreira, L., Carballo, L., 1999. Desorption isotherms ofmuscatel and aledo grapes, and the influence of pretreatments on muscatelisotherms. Journal of Food Engineering 39, 409–414.

Wang, H., Zhang, S., Chen, G., 2008. Glass transition and state diagram for fresh andfreeze-dried Chinese gooseberry. Journal of Food Engineering 84 (2), 307–312.

White, G.W., Cakebread, S.H., 1966. The glassy state in certain sugar containing foodproducts. Journal of Food Technology 1 (6), 73–82.

Wolf, W., Spiess, W.E.L., Jung, G., 1985. Sorption Isotherms and Water Activity ofFood Materials. Elsevier Sciences Publishing Co., New York.