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LWT - Food Science and Technology 45 (2012) 246e252
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LWT - Food Science and Technology
journal homepage: www.elsevier .com/locate/ lwt
Mass transfer during the osmotic dehydration of West Indian cherry
Mirtes Aparecida da Conceição Silva a,*, Zaqueu Ernesto da Silva a, Viviana Cocco Mariani b,Sébastien Darche c
a Laboratório de Energia Solar, Universidade Federal da Paraíba, Cidade Universitária-Castelo Branco, 58051-970 João Pessoa, Paraíba, BrazilbDepartamento de Engenharia Mecânica, Pontifícia Universidade Católica do Paraná, 80215-901 Curitiba, Paraná, Brazilc Energetic and Environment Department, INSA Engineering School, 20 rue Albert Einstein, 69621 Villeurbanne Cedex, France
a r t i c l e i n f o
Article history:Received 13 October 2010Received in revised form25 July 2011Accepted 26 July 2011
Keywords:West Indian cherryDiffusion coefficientInverse problemOsmotic dehydration
* Corresponding author. Tel.: þ55 83 3216 7035.E-mail addresses: [email protected] (M.A.
(Z.E. Silva), [email protected] (V.C. Mariani),fr (S. Darche).
0023-6438 � 2011 Elsevier Ltd.doi:10.1016/j.lwt.2011.07.032
Open access under the El
a b s t r a c t
The main purpose of this work was to study water loss, solids gain, and weight and moisture reductionduring the osmotic dehydration process of the West Indian cherry (Malpighia punicifolia). The diffusioncoefficient of West Indian cherry was estimated by the inverse method using average moisture contents.Osmotic dehydration was examined for 12 h in a 65�Brix solution at temperature of 27 �C, withoutagitation, using a fruit:solution mass ratio of 1:4, 1:10, and 1:15. The kinetics and internal changesoccurring during the osmotic dehydration of West Indian cherry are reported. The product’s dryingkinetics was simulated using the diffusion model, and two optimization methods, LevenbergeMarquardtand Differential Evolution algorithm, were used to predict the diffusion coefficient. The results indicatedthat the two optimization methods performed similarly in estimating the diffusion coefficientadequately. The average calculated diffusion coefficient was 1.663 � 10�10 m2s�1, which is consistentwith values reported in the literature.
� 2011 Elsevier Ltd. Open access under the Elsevier OA license.
1. Introduction
In recent years, the West Indian cherry (Malpighia punicifolia)has been commercially exploited with good acceptance, particu-larly because of its high content of ascorbic acid (vitamin C) and itsnutritional characteristics associated with pleasant flavor andtexture. The ascorbic acid content in West Indian cherry is around8 mg g�1 in ripe fruits, 16 mg g�1 in half-ripe fruit and 27 mg g�1 inunripe fruit. In other words, its ascorbic acid content is approxi-mately 100-fold higher than that of oranges and 10-fold higherthan in guava, both of which are well-known for their high contentof vitamin C. The increasing production and consumption of WestIndian cherry, allied to the fact that it is a highly perishable fruit,make it urgent to develop alternative processing and conservationmethods. West Indian cherry offers several advantages over otherfruits, i.e., the high levels of ascorbic acid in its flesh enable WestIndian cherry to be industrialized and stored without causingsignificant nutritional changes. Several authors have studied thedrying of West Indian Cherry to achieve these objectives (Alves,
C. Silva), [email protected]@insa-lyon.
sevier OA license.
Antonio, Júnior, & Murr, 2004; Cerqueira, Lima, Teixeira, Moreira,& Vicente, 2009; Corrêa, Cacciatore, Silva, & Arakaki, 2008; Jesus,Scaranto, Jalali, & Silva, 2003; Marques, Ferreira, & Freire, 2007;Moreira et al., 2009).
Osmotic dehydration (OD) is a pre-treatment commonly appliedprior to air-drying. This technique consists in immersing the fruit ina hypertonic solution to remove part of thewater from the fruit. Thedriving force for water removal is the difference in osmotic pressurebetween the fruit and the hypertonic solution. The complex cellularstructure of the fruit acts as a semi-permeable membrane, creatingextra resistance to water diffusion within the fruit (Raoult-Wack,1994; Raoult-Wack, Lafont, Rios, & Guilbert, 1989; Simal, Bene-dito, Sánchez, & Rosello, 1998; Torreggiani, 1993). Osmotic dehy-dration changes the texture of fruit (Khin, Zhou, & Yeo, 2007;Mayor, Cunha, & Sereno, 2007; Prothon et al., 2001; Torreggiani &Bertolo, 2001), especially due to the dissolution of pectin and thebreakdown of cell tissue. The kinetics of OD processes is usuallyevaluated in terms of water loss, weight loss and solid gain (Fito &Chiralt, 1997) and depends mainly on the characteristics of the rawmaterial (Raoult-Wack, 1994) and on operational conditions, suchas the concentration, temperature (Barat, Chiralt, & Fito, 2001), andexposure time of the solution (Escriche, Garcia-Pinchi, Andrés, &Fito, 2000) and pressure (Barat et al., 2001; Fito & Pastor, 1994).
To improve the control of this unitary operation requires accu-rate models to simulate drying curves under different conditions.
M.A.C. Silva et al. / LWT - Food Science and Technology 45 (2012) 246e252 247
Drying involves four main transport phenomena: internal andexternal heat transfer, and internal and external mass transfer.However, the numerical solution of the corresponding four classicalpartial differential equations requires considerable computing time(Karathanos & Belessiotis, 1999). Researches frequently use simplemodels to simulate the food drying curves that can adequatelyrepresent experimental results (Akpinar, Midilli, & Bicer, 2003;Doymaz, 2004; Iguaz, San Martin, Maté, Fernandez, & Vírseda,2003; Senadeera, Bhandari, Yong, & Wijesinghe, 2003; Sogi,Shivhare, Garg, & Bawa, 2003). In this study, the mass transferprocess will be defined as a function of Fick’s law combined withthe microscopic mass transfer balance.
It should be noted that the production and consumption ofWestIndian cherry have increased in Brazil, and that there is a realpossibility for Brazil to export this fruit. Therefore, it is even moreimportant to carry out research on this fruit and to developedalternative processing technologies. The main objective of thiswork was to study water loss, solid gain, and weight and moisturereduction in West Indian cherry during the osmotic dehydrationprocess, using real average moisture contents to estimate thediffusion coefficient of West Indian Cherry based on the inversemethod. This paper describes the internal changes and the kineticsof moisture change and moisture transfer during the osmoticdehydration of West Indian cherry.
2. Materials and methods
Fresh West Indian cherry (M. punicifolia L.) and chemicalsproducts were purchased in a local market in João Pessoa (Paraíba,Brazil). The fruits were selected visually based on their similardegree of ripeness (same skin color), apparent fruit quality (flaw-lessness), firmness, and similar size. The fruit’s average radius wasapproximately 8.5 mm. The sample’s dimensions were measuredwith a Vernier caliper (SOMET) with 0.05 mm precision. Theaverage initial moisture content determined after blanching was91.7 kg kg�1 on a wet basis, determined by heating in a drying oven(LUFERCO, model 41181) at 65 �C for 24 h, following the 2002 AOACmethod. Other materials used in this work were obtained in thesame period in a local market too. The initial soluble solid contentdetermined by refractometry was 6.30�Brix. The water activity ofthe West Indian cherry (aw ¼ 0.989) was measured after blanchingat final dehydration time using a dew-point hygrometer (DecagonC-X2, Aqualab, USA, with 0.001 precision) at 27 �C.
Prior to their osmotic dehydration, the West Indian cherrieswereweighed and then blanched in boiling water for 1min in orderto increase the water permeability of the skin, followed by imme-diate cooling in a mixture of water and ice for 1 min to removeexcess heat. After blanching, the fruits were drained on absorbentpaper to remove excess water, weighed again, and immersed in anosmotic solution.
2.1. Osmotic dehydration
Osmotic solutions were prepared by dissolving a given amountof sucrose in distilled water to obtain a concentration of 65�Brix.Three different fruit-to-solution mass ratio were studied (1:4, 1:10and 1:15) to verify possible changes in sucrose concentrationduring the process. Each experiment was carried out in triplicate.The data presented in this paper correspond to the average of threedata sets obtained from different glass jars. The fruits wereimmersed whole into the osmotic solution in glass jars, which werethen covered with lids to reduce moisture loss of the syrup (27 �C),and left at room temperature during the experiment (for 12 h).
Fruits were removed from the jars at 1-h intervals, quicklyrinsed and gently blotted with tissue paper to remove excess
solution from the surface, then weighed and returned to theosmotic solution to continue the drying process. Each experimentwas carried out in triplicate. The water diffusivity of West Indiancherry during osmotic dehydration was calculated based on thefruit’s weights, according to Fick’s law of diffusion.
Water loss (WL), solid gain (SG) and Weight reduction (WR) ofthe sample was calculated based on its weight, moisture contentand sugar content, according to Eq. (1), (2) and (3), respectively:
WL ¼ wiXi �wf Xf
wi(1)
SG ¼ wf Xsf �wiXsi
wi(2)
WR ¼�wi �wf
wi
�� 100 (3)
where Xi is the fruit’s initial moisture content on kgmoisture/kg drymatter, Xf is its final moisture content on kgmoisture/kg drymatter,Xsi is the initial soluble solids content (�Brix), Xsf is its final solublesolid content (�Brix), wi is its initial mass (kg), and wf is its finalmass (kg).
2.2. Mass transfer model
The mechanisms of moisture transport during osmotic dehy-dration of fruit and vegetable tissues are complex and are notcompletely understood. It is usually assumed that water transfer,expressed by a diffusion coefficient Def, is controlled by differencesin moisture content. Based on experiments at a microscopic level,Ferrando and Spiess (2002) demonstrated the moisture diffusioncoefficient of several plant tissues is approximately of 10�12 m2s�1
whereas studies at a macroscopic level of carrot, coconut andpineapple in a sugar solution showed diffusion coefficients rangingfrom 10�10 m2s�1 to 10�9 m2s�1 (Rastogi & Raghavarao, 1995;1997). These differences in effective diffusion coefficients suggestthe existence of another mechanism.
Several empirical equations are used in the modeling of masstransfer kinetics during the osmotic dehydration process theseequations are useful for optimizing the process. Most of the modelsthat describe the process are based on the diffusion model of Fick’ssecond law for different geometries. To describe the transfer ofmass in food products subjected to osmotic dehydration, a diffusionmodel was developed based on the following assumptions:
� The product is represented by a sphere having a radius R;� Moisture transfer is predominantly one-dimensional;� The initial moisture content is distributed uniformly in theproduct;
� Shrinkage is negligible; and� The diffusion coefficient, Def, is considered constant andhomogeneous during drying, and it is the variable to be opti-mized in this work.
Thus, considering the West Indian cherry as a sphere withinitially uniform water and solid contents, the solution for Fick’sequation for a constant condition of the process is (Crank, 1975):
X � Xeq
Xi � Xeq¼ 6
p2
XNn¼1
exp�� n2,p2,Def ,
tr2
�(4)
where X is the average moisture content along the length of thediffusion pathway (kg moisture/kg dry matter), and the subscripts iand eq represent, respectively, the relevant initial and equilibrium
M.A.C. Silva et al. / LWT - Food Science and Technology 45 (2012) 246e252248
concentrations, r is the ratio (m), t is the time (s), and Def is thediffusion coefficient (m2 s�1), it is predicted by two optimizationmethods which are presented to follow.
3. Methods to estimate the diffusion coefficient
Many optimization methods have been proposed to solveinverse problems, based on the estimation of deterministic andstochastic parameters. Among the deterministic methods, theLevenbergeMarquardt method has been used successfully inseveral areas (Kanevce, Kanevce, & Dulikravich, 2005; Mejias,Orlande, & Özisik, 2003; Mendonça, Filho, & Silva, 2005). Amongthe stochastic methods, the Differential Evolution method has beenapplied less frequently to inverse problems, but has been used byKanevce et al. (2005); Mariani and Coelho (2009a); Mariani andCoelho (2009b); Mariani, Lima, and Coelho (2008); da Silva, daSilva, and Mariani (2009).
Optimization methods for estimating parameter are used toestimate the diffusion coefficient as a function of the minimizationof J by the sum of squared differences between the experimentalmoisture content and the moisture content computed witha diffusion model:
J ¼Xn1
�Xexp � Xcomp
�2 (5)
where Xexp is the experimental moisture content and Xcomp is thecomputed moisture content.
A set of moisture contents was obtained experimentally atdiscrete times during the unsteady drying process.
The literature uses different criteria to evaluate the quality of thefit obtained by the mathematical model and optimization methodsto simulate the experimental results. In this study, deviationsbetween measured and simulated moisture content were calcu-lated using the coefficient of determination in successive trials, asfollows:
R2 ¼ 1�P�
Xexp � Xcomp�2
P�Xexp � Xexp
�2 (6)
The Differential Evolution and LevenbergeMarquardt methodswere implemented.
3.1. Differential Evolution method
Differential Evolution (DE) is an evolutionary algorithmproposed by Storn and Price (1995). Although DE shares similaritieswith other evolutionary algorithms (EA), it differs significantly inthat the search process is guide based on information about thedistance and direction of the current population. DE uses thedifferences between randomly selected vectors (individuals) as thesource of random variations for a third vector (individual), referredto as the target vector. Trial solutions are generated by addingweighted difference vectors to the target vector. This process isreferred to as the mutation operator in which the target vector ismutated. A crossover step is then applied to produce an offspring,which is only accepted if it improves the fitness of the parentindividual. The basic DE algorithm is described in greater detailbelow with reference to the three evolution operators: mutation,crossover, and selection.
Mutation is an operation that adds a vector differential toa population vector of individuals, according to the followingequation:
ziðkþ 1Þ ¼ xi;r1ðkÞ þ fm,�xi;r2ðkÞ � xi;r3ðkÞ
�(7)
where i ¼ 1, 2, ., N is the individual’s population index, k is thegeneration, xiðkÞ ¼ ½xi1ðkÞ; xi2ðkÞ;.; xinðkÞ�T stands for the positionof the i-th individual of population of N real-valued n-dimensionalvectors, ziðkÞ ¼ ½zi1ðkÞ; zi2ðkÞ;.; zinðkÞ�T stands for the position ofthe i-th individual of a mutant vector; r1, r2 and r3 are mutuallydifferent integers and also different from the running index, i,selected randomly with uniform distribution from the setf1;2;.i� 1; iþ 1;.Ng, and fm > 0 controls the amplification ofthe difference between two individuals so as to avoid searchstagnation, and is usually taken from the range [0.1, 1].
Crossover applied to the population is employed to generatea trial vector by replacing certain parameters of the target vectorwith the corresponding parameters of a randomly generated donorvector. For each vector, zi(kþ1), an index rnbr() ˛ {1, 2, ., n} israndomly chosen using uniform distribution, and a trial vector,uiðkþ 1Þ ¼ ½ui1ðkþ 1Þ;ui2ðkþ 1Þ;.;uinðkþ 1Þ�T , is generated with
uijðkþ1Þ ¼�zijðkþ1Þ;se randbðjÞ�CR ou j ¼ rnbrðiÞ;xijðkÞ;se randbðjÞ>CR ou jsrnbrðiÞ; (8)
In the above equations, randb(j) is the j-th evaluation ofa uniform random number generation with [0, 1] and CR is in therange [0, 1].
Selection is the procedure involved in producing betteroffspring. To decide whether or not the vector ui(kþ1) should bea member of the population comprising the next generation, it iscompared with the corresponding vector xi(k). Thus, if f denotes theobjective function under minimization, then
xiðkþ 1Þ ¼�uiðkþ 1Þ; if f ðuðkþ 1ÞÞ � f ðxiðkÞÞ;xiðkÞ; otherwise
(9)
In this case, the cost of each trial vector ui(kþ1) is comparedwith that of its parent target vector xi(k). If the cost, f, of the targetvector xi(k) is lower than that of the trial vector, the target isallowed to advance to the next generation. Otherwise, the targetvector is replaced by a trial vector in the next generation.
Storn and Price (1997) proposed ten different strategies for DEbased on the individual being disturbed, the number of individualsused in the mutation process and the type of crossover used. Thestrategy described above is known as DE/rand/1, meaning that thetarget vector is randomly selected, and only one difference vector isused. This strategy is considered here. This study used fm ¼ 0.2,CR ¼ 0.8, population size 30, and 30 generations.
3.2. LevenbergeMarquardt method
The LevenbergeMarquardt (LM) Method was derived bymodifying the ordinary least squares norm and is a combination ofthe Gauss and Steepest Descent methods. Based on the criterion ofordinary least squares, the iterative formula has the followingexpression (Press, Teikolsky, Vetterling, & Flannery, 1992, p.963),
bðkþ1Þ ¼ bðkÞ þ
Jk�T
Jk þ lkUkm
��1Jk�T
Y � Xbk��
(10)
where k is the number of iterations, l is a positive scalar calleddamping parameter, Um is a diagonal matrix, and J is the sensitivitycoefficient matrix defined as JðbÞ ¼ vXT ðbÞ=vb.
The purpose of the matrix term lkUkm in Eq. (10) is to damp
oscillations and instabilities caused by the ill-conditioned nature ofthe problem by making its components larger than those of JTJ, ifnecessary. The damping parameter is set large in the beginning ofthe region around the initial guess used for the exact parameters.With this approach, the matrix JTJ does not have to be non-singularat the beginning of iterations and the LevenbergeMarquardt
30
35
40
M.A.C. Silva et al. / LWT - Food Science and Technology 45 (2012) 246e252 249
method tends toward the steepest descent method, i.e., a fairlysmall step is taken in the direction of the negative gradient. Theparameter lk is then gradually reduced as the iteration procedureadvances to the solution of the parameter estimation problem, atwhich point the LevenbergeMarquardt method tends toward theGauss method. The iterative procedure begins with an initial guess,b0, and at each step the vector b is modified until:
bðkþ1Þi � bðkÞi
bðkÞi
þ x
< d; for i ¼ 1;2;3. (11)
where d is a small number (typically 10�3) and x (<10�10). The LMmethod is quite a robust and stable estimation procedure whosemain advantage is a good rate of convergence (Fguiri, Daouas,Borjini, Radhouani, & Aïssia, 2007).
Both optimization methods, LM and DE, are applied to minimizethe Eq. (5), denominated objective function. Such equation dependsof the moisture content, X, calculated from Eq. (4). Note that in Eq.(4) the diffusion coefficient is considered constant but it is known.To obtain such coefficient using for example DE method, firstdifferent values for diffusion coefficient are generated randomlybetween at fixed interval then in these coefficients are appliedmutation and crossover operations as explained in Eqs. (7) and (8)generating new solutions (new coefficients). The previous and newdiffusion coefficients are evaluated through of Eq. (4) providinga set of moisture content which will have its objective functionevaluated by Eq. (5), and so the optimization process continuesuntil the objective function to be minimized.
4. Results and discussion
The effects of osmotic dehydration on physical and chemicalproperties of West Indian cherry are presented in Table 1. Theexperimental results described in Table 1 showed that, during theprocess, the fruit’s moisture content decreased approximately 16 kgmoisture/kg dry matter, its soluble solid content increased almost20�Brix, and water activity decreased next to 0.015, these valueswere calculated by the difference between initial and final values ofmoisture content, soluble solid content and water activity,respectively, according to the values shown in Table 1. Note that theaim of osmotic dehydration is to reduce water activity, increasesugar gain, and remove water from fruits (or vegetables) toincreasing their storage stability.
The effects of osmotic dehydration treatment over time wereevaluated in terms of the evolution of moisture content, water loss(WL), solid gain (SG), and also weight reduction (WR) (Derossi, de
Table 1Physical-chemistry characterization of West Indian cherry after osmoticdehydration.
Samples:solution
Average values
Moisture content kg moisture/kg wetsample
Initial 91.70 � 0.011:4 76.36 � 0.111:10 75.63 � 0.031:15 75.77 � 0.02
Soluble solids content (�Brix) Initial 6.30 � 0.101:4 20.60 � 0.201:10 24.30 � 0.101:15 26.20 � 0.2
Water activity Initial 0.989 � 0.0011:4 0.980 � 0.0011:10 0.979 � 0.0011:15 0.974 � 0.001
Pilli, Severini, & McCarthy, 2008; Sacchetti, Gianotti, & Dalla Rosa,2001; Spiazzi & Mascheroni, 1997). Figs. 1e3 present, respectively,the water loss, solid gain and weight loss over time during osmoticdehydration.
The results in Fig. 1 indicate that water loss increased withprocessing time and was almost equal at the ratios of 1:10 and 1:15during the first 2 h. Increasing water loss in other fruits was alsoobserved by El-Aouar and Murr (2003) - papaya, and Corzo andGomez (2004) - melon. Water loss between 2 and 9 h from thebeginning of the osmotic process was higher at the 1:10 ratio thanat the 1:15 ratio, probably due to the concentration of sucrose in thefruit’s outer layer, which acts as an additional resistance to watertransfer between fruits and solution. This finding is in agreementwith the observations of Teles et al. (2006). On the other hand, atthe ratio 1:4, water loss occurred slowly due to the dilution of theosmotic solution. Fig. 4 illustrates the variation in the concentrationof the osmotic solution (SS) for the three ratios studied here. ItFig. 2, note that the use of a higher fruit:solution ratio increased thesolids incorporation rate, which is consistent with the findings ofLima et al. (2004). Note, also, that the solid content increased overprocessing time.
The fruit’s average solids gain at the end of the osmotic processfor all ratios investigated was 10e12�Brix, while the water loss wasapproximately 20 kg kg�1 for 1:4 ratio and 35 kg kg�1 for otherratios. A comparison of the data in Figs. 1 and 2 indicates that thevalues of solid gain were much lower than those of water loss. Thisfinding is significant since the main objective of osmotic dehydra-tion is to achieve maximal water loss with a minimal solid gain.
Fig. 3 shows the evolution of weight loss over time. The weightandwater loss curves showed the same behavior (Fig.1), i.e., weightand water losses were proportional. Weight loss appeared toincrease with osmotic dehydration processing time, but showeda tendency to stabilize over time as the system approached equi-librium. This behavior has been studied by several researchers(Córdova, 2006; Lenart, 1996; Moura, Masson, & Yamamoto, 2005;Raoult-Wack, 1994; Santos, 2003).
An analysis of Figs. 1e3 clearly indicates that the curves of the1:10 fruit:solution ratio showed the most uniform behavior withevery parameter studied here. Thus, it can be stated that the use ofthis ratio ensures a constant concentration of the solution duringthe entire osmotic process, which is consistent with the work ofFerrari, Rodrigues, Tonon, and Hubinger (2005).
0 2 4 6 8 10 120
5
10
15
20
25
Time (h)
Wat
er L
oss
(kg.
kg- 1)
Fig. 1. Water loss as function of time for osmotic dehydration of West Indian cherrysamples for 3 replications in different concentrations of solutions: solid, ratio of fruit tosolution 1:4, dotted, ratio of fruit to solution 1:10, dashed, ratio of fruit to solution 1:15.
0 2 4 6 8 10 1258
59
60
61
62
63
64
65
Time (h)
Sol
uble
Sol
id (
Brix
)
Fig. 4. Soluble solid concentration as function of time for osmotic dehydration of WestIndian cherry samples for 3 replications in different concentrations of solutions: solid,ratio of fruit to solution 1:4, dotted, ratio of fruit to solution 1:10, dashed, ratio of fruitto solution 1:15.
0 2 4 6 8 10 120
2
4
6
8
10
12
14
Time (h)
Sol
id G
ain
( B
rix)
Fig. 2. Solid gain as function of time for osmotic dehydration of West Indian cherrysamples for 3 replications in different concentrations of solutions: solid, ratio of fruit tosolution 1:4, dotted, ratio of fruit to solution 1:10, dashed, ratio of fruit to solution 1:15.
M.A.C. Silva et al. / LWT - Food Science and Technology 45 (2012) 246e252250
The initial moisture content of West Indian cherry was11.05 � 0.01 kg moisture/kg dry matter. Fig. 5 presents some of thedrying curves for different fruit:solution rations. The equilibriummoisture content of the dried cherries was calculated based on thechanges in their weight. The calculated equilibrium moisturecontent was 1.089 � 0.150 kg moisture/kg dry matter. The equi-librium moisture was determined from three samples for eachration studied. These samples were dehydrated for 12 h, then oven-dried until they reached a constant weight, following the 2002AOAC method. In all the conditions, there was a period ofdeclining moisture content characterized by a rapid drop in thedrying rate. This indicates that the main mechanism of watertransport was diffusion and that the diffusion equation can beemployed to analyze drying data.
The moisture content of West Indian cherry decreased expo-nentially over time, from 11.05 to 3.10 kg moisture/kg dry matterafter 12 h, which is in agreement with previous research (Derossi
0 2 4 6 8 10 120
5
10
15
20
25
30
Time (h)
Wei
ght R
educ
tion
(kg.
kg1)
Fig. 3. Weight reduction as function of time for osmotic dehydration of West Indiancherry samples for 3 replications in different concentrations of solutions: solid, ratio offruit to solution 1:4, dotted, ratio of fruit to solution 1:10, dashed, ratio of fruit tosolution 1:15.
et al., 2008; Spiazzi & Mascheroni, 1997) on other fruits. Expo-nential changes were also observed in weight reduction, solid gainand water loss of West Indian cherry, as shown in Figs. 1e3.
Table 2 shows a statistical analysis of water loss, solid gain andweight loss at the fruit:solution rations under study. As can be seenin this table, the fruit:solution ratio of 1:10 showed the loweststandard deviation (SD) values, except for the solid gain, whoselowest SD occurred with the 1:4 ratio. This can be explained by theeffect of solution dilution at the 1:4 ratio. The profiles presented inFigs. 1e4 reflect the above described patterns.
The high coefficients of determination obtained by the Lev-enbergeMarquardt method and Differential Evolution method(R2 > 0.958) indicated the goodness of fit of experimental data toEq. (4), see Table 3. The Def values varied from approximately1.558 � 10�10e1.771 � 10�10 m2 s�1 for West Indian cherry. Thesevalues are within the range of Def (10�12e10�8 m2 s�1) normally
0 2 4 6 8 10 123
4
5
6
7
8
9
10
11
12
Time (h)
Moi
stur
e co
nten
t (kg
moi
stur
e/kg
dry
mat
ter)
Fig. 5. Moisture content (kg moisture/kg dry matter) as function of time for osmoticdehydration of West Indian cherry samples in different concentrations of solutions:solid, ratio of fruit to solution 1:4, dotted, ratio of fruit to solution 1:10, dashed, ratio offruit to solution 1:15.
Table 2Statistical analysis of osmotic dehydration process.
Samples:solution
Averagevalues
Standarddeviation
Variance
Water Loss (kg kg�1) 1:4 25.868 1.054 1.1101:10 34.924 0.331 0.1101:15 38.140 1.518 2.304
Solid gain (�Brix) 1:4 11.359 0.121 0.0151:10 11.942 0.171 0.0291:15 12.215 0.256 0.066
Weight reduction (kg kg�1) 1:4 13.796 0.723 0.5231:10 24.927 0.600 0.3601:15 29.317 1.900 3.609
Table 3Values of diffusion coefficient of West Indian cherry samples during osmoticdehydration.
LevenbergeMarquardt Differential evolution
Samples: Solution Def (m2/s) J R2 Def (m2/s) J R2
1:4 1.558 � 10�10 1.426 0.979 1.5604 � 10�10 1.411 0.9791:10 1.760 � 10�10 2.725 0.958 1.7711 � 10�10 2.569 0.9601:15 1.657 � 10�10 2.490 0.960 1.6691 � 10�10 2.326 0.963
M.A.C. Silva et al. / LWT - Food Science and Technology 45 (2012) 246e252 251
expected for dehydrated foods (Azoubel & Murr, 2004; Corrêa,Cacciatore, Silva, & Arakaki, 2006; Gely & Santalla, 2007).
This variability in diffusion coefficient depends on the experi-mental conditions and procedures used for the determination ofthe moisture diffusivity, as well as on the data treatment methods,the product’s properties, composition, physiological state, andheterogeneity of its structure. For instance, Corrêa et al. (2006)obtained Def values between 2.78 � 10�10 and 8.42 � 10�10 m2s�1
for West Indian cherry samples osmotically dehydrated at a sol-ution:sample ratio of 3:1 for 60�Brix of sucrose solution, during24 h of osmotic dehydration.
Fig. 6 show experimental moisture distribution during theosmotic treatment for the fruit:solution ratio 1:15 studied here,similar results for the other cases were obtained. The distributionbehavior corresponds to the model calculated with diffusion valuesestimated by two methods: LevenbergeMarquardt and Differential
0 2 4 6 8 10 123
4
5
6
7
8
9
10
11
12
Time (h)
Moi
stur
e co
nten
t (kg
moi
stur
e/kg
dry
mat
ter)
Fig. 6. Experimental and predicted values of moisture content (kg moisture/kg drymatter) of West Indian cherry samples during osmotic dehydration for ratio of fruit tosolution 1:15: solid, LevenbergeMarquardt, circle, Differential Evolution, dashed,Experimental.
evolution. Thus, this figure demonstrates that the experimentaldata fit the mathematical model and that the estimation methodsused here are efficient.
5. Conclusions
An analytical solution of Fick’s second law for diffusion in spheregeometry successfully determined the effective water diffusioncoefficient of West Indian cherry during osmotic dehydration fordifferent fruit-to-solution mass ratios. The values found here aresimilar to values reported in the literature, obtained with othertechniques and for other dehydrated foods. Based on these results,it can be concluded that water loss, solid gain, and weight lossincreased during dehydration and were higher at increasing ratios.Thus, an osmotic solution at a fruit:solution ratio of 1:10 is the bestconfiguration studied, enabling us to affirm that this proportionensures the constancy of the solution’s concentration throughoutthe osmotic process. Effective diffusivity values estimated by Lev-enbergeMarquardt and Differential Evolution algorithms were ofsame order of magnitude as those reported in the literature forother fruits under similar conditions. The R2 values (Table 3)demonstrate that two optimization methods performed similarlyunder the various experimental conditions applied to this study.The inverse method applied to the estimation of thermophysicalproperties is a very attractive technique because of its accuracy andrapid estimation of parameters.
Acknowledgments
The authors would like to thank CNPq (The National Council forScientific and Technological Development of Brazil) for theirsupport through the process (141522/2007-0, 568221/2008-7,475689/2010-0, and 302786/2008-2/PQ).
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