modelo matemático sobre la congelación de manzanas

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Postharvest Biology and Technology 25 (2002) 273–291

A mathematical model for the development of mealiness inapples

V. De Smedt a, P. Barreiro b , B.E. Verlinden a, E.A. Veraverbeke a ,J. De Baerdemaeker a , B.M. Nicolaı ¨ a, *

a Flanders Centre /Laboratory of Posthar est Technology , Catholic Uni ersity of Leu en, Willem de Croylaan 42 ,3001 Leu en, Belgium

b Laboratory De Propriedades Fı ´sicas de Productos Agrı ´colas , Escuela T .S . de Ingenieros Agrónomos , 28040 Madrid , Spain

Received 27 April 2001; accepted 16 October 2001

Abstract

Mealiness in apples ( Malus domestica Borkh.) is an internal quality defect which is characterised by a dry andcrumbly texture. It is related to the relative strength of the cell wall and the middle lamella. A mathematical modelhas been built to relate changes in the texture attributes juiciness, tensile strength and hardness, which are associatedwith mealiness, to the development of the turgor pressure of the tissue and the degree of hydrolysis of the middlelamella. The latter, in turn, are described in terms of properties which are meaningful from the physiological pointof view, such as starch content, soluble solids content, non-hydrolysed and hydrolysed middle lamella, water in thesymplast, and water in the apoplast. Biochemical reactions as well as water transfer processes are incorporated in themodel. The parameter values of the model are estimated using experimental data from a storage experiment. Themodel ts the three texture characteristics adequately. The correlation coefcients between the parameters were below0.96, which indicates that the model does not overt the data. © 2002 Elsevier Science B.V. All rights reserved.

Keywords : Apple; Mathematical model; Mealiness; Middle lamella

www.elsevier.com /locate /postharvbio

1. Introduction

Mealiness is an important internal quality

parameter of apple, which is characterised bytexture deterioration, resulting in soft, dry andmealy fruit. Mealiness reduces the quality of thefruit, and, hence, its commercial value, and in

apple develops during storage and depends on theair composition, temperature and relative humid-ity. It can be induced by applying room tempera-

ture (20 °C) and high relative humidity (95%;Barreiro et al., 1998b). Apples stored in refriger-ated but normal air conditions develop a mealytexture much faster than apples under ultra lowoxygen (ULO) conditions (De Smedt, 2000).Apart from the storage conditions, harvestingdate and fruit size also inuence the developmentof mealiness. Apples of larger size or of late

* Correponding author. Tel.: + 32-16-32-2375; fax: + 32-16-32-2955.

E -mail address : [email protected] (B.M.Nicolaı¨).

0925-5214 /02/$ - see front matter © 2002 Elsevier Science B.V. All rights reserved.PII: S 0925-52 14(01) 00185-5

mailto:[email protected]


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V . De Smedt et al . / Posthar est Biology and Technology 25 (2002) 273 – 291274

harvest are more susceptible to mealiness (Bar-reiro et al., 2000; De Smedt, 2000).

Hat eld and Knee (1988) and De Smedt et al.(1998) concluded that the cells in mealy applesbecome more rounded, and, as a consequence,the amount of apoplast is larger in mealy applesthan in fresh apples. Mealiness in apple is re-lated to the relative strength of the cell wallcompared with the strength of the middlelamella (Harker and Hallett, 1992; De Smedt etal., 1998).

Sensory analysis has shown that mealiness canbe characterised by a combination of a lack of crispiness, a lack of juiciness and a lack of hardness (Barreiro et al., 1998b). Crispiness ap-pears to be a characteristic which is lost soonafter harvest, whereas hardness and juiciness de-crease more gradually during storage. Barreiroet al. (1998a) were able to relate these threesensory characteristics to three instrumentallymeasured parameters. Hardness was measuredas the slope of the force deformation curve in acon ned compression test. Juiciness was deter-mined by placing a small lter paper under theapple sample during compression and measuringthe size of the wet spot immediately after com-pression. Crispiness was measured by perform-ing a shear-rupture or tensile test. Themaximum force reached on the force deforma-tion curve was used as a measure for crispiness.

Although qualitative information is availablewith respect to the development of mealiness inapples as a function of the storage conditions,to date no model is available for quantitativeprediction purposes. Such a model would beuseful for optimising storage conditions with re-spect to the development of mealiness. The ob- jective of this work was, therefore, to build amechanistic model which describes the changesof the middle lamella, the water transferthrough the tissue and their interaction at thecellular level as affected by the relative humidityfor both air and low oxygen storage. The modelexplains the development of the hardness, tensilestrength and juiciness of apple tissue. These me-chanical parameters have been shown to be di-rectly related to mealiness as perceived bysensory panels (Barreiro et al., 1998b).

2. Kinetic model development

2 .1. Model framework

Texture properties of apple such as mealinessare affected by the mechanical properties of thecell walls and middle lamellae, by the water statusand, in particular, turgor pressure of the cell (Pitt,1982). These properties change considerably dur-ing postharvest storage and affect each other. Forexample, a key transformation in apple is thehydrolysis of pectin which requires water as asubstrate. Water is available from inside the cellsand is also produced through respiration. It was,therefore, decided to include the following generalfeatures in the model,

respiration; changes of the middle lamella; transfer of water in the apple; relations between fruit texture attributes andthe middle lamella and cell turgor.For the purpose of this article it was assumed

that the apple can be considered as a homoge-neous object. The only independent variable left istime and, therefore, ordinary differential equa-tions are suf cient to de ne the model structure.It is noted that the model which is derived belowshould, therefore, be considered as a crude ap-proximation of the reality.

The model is based on a simpli cation of thehistological structure of the apple (Fig. 1). It isassumed that the apple consists of two compart-ments, the symplast, consisting of the entire net-work of cytoplasm interconnected byplasmodesmata, and the apoplast, consisting of the cell wall system and the intercellular space(Taiz and Zeiger, 1998). The symplast is separatedfrom the apoplast by a semi-permeable mem-

Fig. 1. Schematic representation of an apple.


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V . De Smedt et al . / Posthar est Biology and Technology 25 (2002) 273 – 291 275

brane, the plasmalemma. Passive (diffusive) trans-port of water between both compartments is pos-sible through the plasmalemma. The apoplast canexchange water with the environment via epider-mal transfer. The apple skin, with its protectivewax layer, is the major barrier for this transfer.The water loss of ‘Cox ’s Orange Pippin ’ applesduring a commercial storage period of 6 monthsat 3 °C and 90% RH is typically 5% or greater.

Relative humidity can be considered as a prop-erty of the environment, which affects the be-haviour of the apple. It is an input variable of themodel and is available to the postharvest technol-ogist to optimise the storage process. The effect of the ripeness stage of the apple on the relevantenzyme-mediated biochemical reactions was notconsidered in this paper, as only one batch of fruitwas used. Further, the temperature was assumedto be at its optimal value and its effect was notmodelled explicitly as no data were available fordifferent storage temperatures. Finally, experi-mental data were available for two storage condi-tions (controlled atmosphere (CA) and normalair), and the air composition was considered as acategorical variable. As a consequence, the oxy-gen concentration does not enter into the modelequations as such. The apple skin was chosen asthe system boundary.

2 .2 . Modelled processes

Respiration provides a source of symplastic wa-ter and, therefore, needs to be incorporated in themodel. In the symplast the respiration process isgoverned by two overall chemical reactions. Onedescribes the hydrolysis of starch to hexose sugarsand the other describes the oxidation of hexosesugars to provide energy. After harvest, applesstill contain a certain amount of starch dependingon the ripeness stage at harvest. During storagethe starch is hydrolysed resulting in an accumula-tion of sugar (Kays, 1991). To model this phe-nomenon, a hypothetical dissolution site isassumed for representing positions in the starchpolymers where hydrolysis occurs. The dissolutionsites (S) are dissolved in the symplastic water.Whenever, a dissolution site is hydrolysed, ahexose unit (H) is produced.

S + H 2Ok S

H (1)

During storage, hexose units are converted bymeans of molecular oxygen into carbon dioxideand water. As fermentation during commercialstorage is avoided at all costs, it is assumed thatthe conversion is governed by aerobic respiration

only.H + 6O 2

k H6CO 2 + 6H 2O (2)

Carbon dioxide diffuses through the apple tothe environment.

The hydrolysis of the middle lamella in theapoplast was considered as well, as this phe-nomenon affects the overall texture to a largeextent. During ripening and softening, the middlelamella, which consists mainly of pectins, dis-solves. The basic structure of pectic substancesconsists of chains of galacturonic acid residueslinked by (1 4) glycosidic bonds. A wide rangeof side chains can be found attached to the mainchain residues. Pectic substances can undergo avariety of chemical modi cations. These are reac-tions catalysed by different enzymes. Enzymescapable of catalysing the de-methoxylation andde-polymerisation of pectic materials are wide-spread in fruit and vegetables. The concentrationof many hydrolases, including polygalacturonases,increases with maturity (Van Buren, 1979; Ben-Arie and Kislev, 1979; Fisher and Bennet, 1991).

To simplify the model, a simple hydrolysis reac-tion is assumed in which galacturonic acid unitsare subsequently detached from the pectin chain.Similar to the conversion of starch, a hypotheticaldissolution site was assumed to represent (1 4)glycosidic bonds on the middle lamella were hy-drolysis occurs. The dissolution sites (L) are as-sumed to be dissolved in the intercellular waterbut are physically attached to the pectin matrixwhich is not in solution. Whenever a dissolutionsite is hydrolysed, a pectin monomer (P) isproduced.

L + H 2Ok L

P (3)

For this reaction, water is required. The rate isassumed to depend on the concentration of mid-dle lamella dissolution sites and the amount of water inside the apoplast.


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Fig. 2. Schematic representation of the model. Full arrows denote mass streams while dashed arrows denote in uences of conversionor ow rates denoted by two connected triangles.

Besides these chemical processes, the water ex-change processes between the compartments andthe environment are modelled. Gas exchange isnot incorporated and gas gradients between thecompartments and the environment are consid-ered to be zero. How the above processes relate toeach other is schematically presented in Fig. 2.

It is assumed that the symplast contains water(W s) and sugar substrate in the form of starchdissolution sites (S) and hexose units (H). Theapoplast contains middle lamella dissolution sites(L), solubilised middle lamella (P) and also somewater (W a ). The amount of molecules or ions insolution other than substrate and water (e.g. K + ,Cl − ) are not considered. Part of the water inside

the cell is transferred to the apoplast. This trans-fer is driven by a difference in water potentialbetween the two compartments. Finally, part of the water is lost to ambient. The transfer ratedepends on the relative humidity of the ambient.The concentration of these six chemical com-pounds, S, H, L, P, W s and W a are de ned as thestate variables of the model.

2 .3 . Model structure

The mathematical structure of the model willnow be developed in different steps based on thechemical kinetics of the selected processes andwater transfer phenomena by specifying mass bal-ances in terms of differential equations. This willresult in the core structure of the model whichdescribes the state of the system and how thesystem state changes depending on environmentalconditions. In a last step a relation between thestate of the apple (system) and measurable me-chanical properties related to mealiness will beestablished. Note that the rate of enzyme-medi-ated reactions is always assumed to be indepen-dent of the availability or activity of the involvedenzymes, as data of only one batch of fruit havebeen considered in this article. This assumptionmay not be valid when batches from differentharvest and maturity levels would be considered.

How the state of the apple changes over time ismodelled by setting up a mass balance for each of the six chemical compounds (S, H, L, P, W s, W a ).This results in a rst order differential equation


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for each state variable. The accumulation of acompound in the system is the result of fourterms: in ow and out ow through the systemboundaries and production and consumption in-side the system. First, the ow and conversionterms will be de ned and then they will be used todevelop the mass balances which will result in asystem of differential equations.

The rate of the hydrolysis of starch, S , isdened as the conversion of the amount of sub-strate (S) per mole reaction medium and persecond. Further, it is assumed that this rate isproportional to both the activity of dissolutionsites and water in the symplast. For simplicity it isassumed that the activity of a chemical compo-nent is approximately equal to its concentration.

S = k S[S][Ws] (4)

The rate of oxidation of sugars, H , is dened asthe conversion of the amount of hexose substrate,H, per mole reaction medium and per second.This rate depends on the physiological state andtemperature. As the oxygen concentration is be-lieved to be constant in controlled air conditionsand hexose is in practice not a limiting substrateduring a normal storage period, zero-order kinet-ics is assumed:

H = k H (5)

Note that the oxygen dependency is absorbedin the rate constant k H and is different for CAand normal air conditions.

The rate of the hydrolysis of the middle lamella,L , is dened as the conversion of the amount of

substrate, L, per unit amount of reaction mediumand per second. The rate is assumed to be propor-tional to both the concentration of dissolutionsites and water in the apoplast.

L = k L [L][W a ] (6)

During commercial storage, apples lose a cer-tain amount of water depending on several factorssuch as variety, ripeness stage, fruit size, storagetemperature and relative humidity in the coolchambers. Water is lost to the apoplast when thewater potential inside the symplast is higher thanin the apoplast. Water is consumed in the hydrol-ysis of starch and produced in the respiration

reaction. According to Nobel (1991), the waterpotential ( ) is dened as:

= p + (7)

with p (Pa) the hydrostatic potential or turgorpressure potential, and (Pa) the osmotic poten-tial. It is assumed that the matric potential is zero.

If the water potential ( 1) of compartment 1 isdifferent from the water potential 2 of compart-ment 2, and if both are separated by means of asemi-permeable membrane, then water will betransferred from the compartment of the highestpotential to that with the lowest potential. Ac-cording to Fick ’s rst law, the ux of water fromcompartment 1 to compartment 2 is proportionalto the difference in water potential. The transferof water from the symplast to the apoplast isdriven by the water potential difference according

to:rW s = hsA s* ( s − a ) (8)

with rW s the transfer rate of water from the sym-plast to the apoplast, hs a mass transfer coef cientfor water from the symplast to the apoplast, A s*,the total cell membrane area in the symplast and

s and a , respectively, the total water potentialof the water in the symplast and apoplast.

The symplastic osmotic potential , s dependson the water activity according the following

equation (Nobel, 1991).

,s =RT

V Wln aW s, (9)

with aW sthe water activity in the cell, R the gasconstant (8.314 J /mol K), T the absolute tempera-ture (K) and V W the partial molar volume of water (18 × 10 − 6 m 3/mol). For sugar solutionsand high moisture food products ( aw 0.75), awcan be related to the concentration of the solventexpressed as a mole fraction (Nobel, 1991):

aW s = [Ws] (10)with the activity coef cient. For dilute solutions(less then 0.1 mole /l of solvent, Taiz and Zeiger,1998) the activity coef cient is close to 1. Al-though in the case of apples there is more than 0.1mole of sugar per litre of solvent, the activitycoef cient is assumed to be 1. Hence, the osmoticpotential in the symplast is given by:


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, s =RT

V Wln[W s] (11)

From Eq. (7), it is clear that the turgor poten-tial is given by:

p, s = s − , s (12)

The osmotic potential in the symplast,

, s , isgiven by Eq. (11) and the total water potential,s, is assumed to be linearly dependent on the

water content of the tissue:

s = a (W s + W a )M W − b (13)

where, ( W s + W a )M W is the water content of thetissue expressed as a proportion of the freshweight. As the turgor pressure cannot be negative,the following relations can be established:

p, s = 0, , s s

p, s = s − , s , otherwise (14)

The osmotic active species in the apoplast aregalacturonic acid monomers. The osmotic poten-tial in the apoplast is given by:

, a =RT

V Wln aW a =

RT

V Wln[W a ] (15)

The turgor potential of the apoplast is, bydenition, equal to zero.

Water is also transferred from the apoplast toambient with a rate depending on the relativehumidity of the environment. The lower the rela-tive humidity, the more water is lost to ambient.There is also in ow of water to the apoplast fromthe symplast. The transfer rate of water from theapoplast to ambient is, in analogy to, rW s propor-tional to the difference in water potential:

rW a = ha A a* ( a − ) (16)

with the water potential of ambient surround-ing the apple, ha a mass transfer coef cient fromthe apoplast to ambient, and A a* the total applesurface area. can be rewritten:

=RT

V Wln( rel ) (17)

with rel the relative humidity of the environmentsurrounding the apple.

All conversion and ow rates are de ned andwill now be used to develop the mass balances of the six chemical compounds resulting in a systemof differential equations. The change of starchdissolution sites, S, in time is proportional to thehydrolysis rate of starch, S , and the amount of molecules, ( nS + nH + nW S), in the symplast.

dnSdt

= − S(nS + nH + nW s) with nS(t = 0) = nS 0(18)

The amount of hexose in the symplast compart-ment increases because it is produced from starchand it decreases because of respiration:

dnHdt

= ( S − H )(nS + nH + nW s) with

nH (t = 0) = nH0

(19)

An analogous derivation for the change of pectin substrate and pectin monomers can bemade which results in:

dnLdt

= − L (nL + nP + nW a ), with

nL (t = 0) = nL 0 (20)

dnPdt

= P (nL + nP + nW a ), with nP (t = 0) = nP 0(21)

The water balance in the symplast has threeterms, water produced during respiration of sug-ars; water consumption in the hydrolysis reactionof starch; and water loss through theplasmalemma.

dnW sdt

= (6 H − S)(nS + nH + nW s) − rW s, with

nW s(t = 0) = nW s0 (22)

The water balance in the apoplast has threeterms as well, water consumption in the hydroly-sis reaction of pectin in the middle lamellae, waterinow from the symplast and water loss toambient.

dnW adt

= − L (nL + nP + nW a ) + rW s − rW a , with

nW a (t = 0) = nW a0 (23)


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These six differential equations can be rewrittenas follows.

In Eq. (4), the concentration of the dissolutionsites S is de ned as the molar fraction:

[S]= nS

nS + nH + nW s(24)

The concentrations of intracellular water andhexose units are de ned in a similar way. Eq. (18)can then be written as:

dnSdt

= − k SnSnW s

nS + nH + nW s(25)

This equation holds for a certain apple with acertain fresh weight. The fresh weight, mapple, 0 , isdened as the weight immediately after harvest. Itis advantageous to introduce the following concen-

trations expressed in mols per kg fresh weight:

S = nS

mapple,0, H =

nHmapple,0

, W s = nW s

mapple,0(26)

Substitution into Eq. (25) yields:

dS dt

= − k SS W s

S + H + W s, with S (t = 0) = S 0

(27)

In a similar manner these substitutions werecarried out for the other ve differential equationsresulting in:

dH dt

= − k H (S + H + W s) + k SS W s

S + H + W s,

with H (t = 0) = H 0 (28)

dLdt

= − k L L W a

L 0 + P 0 + W a, with L (t = 0) = L 0

(29)

dP

dt

= k L L W a

L 0 + P 0 + W a, with P (t = 0) = P 0

(30)

dW sdt

= 6k H (S + H + W s) − k SS W s

S + H + W s

− hsAs( s − a), W s(t = 0) = W s, 0 (31)

dW adt

= hsAs( s − a ) − ha Aa ( a − )

− k L L W a

L 0 + P 0 + W awith

W a (t = 0) = W a , 0 (32)

Besides storage conditions, mealiness is alsoaffected by other parameters such as harvest timeand fruit size. Larger apples and later picked applesare more susceptible to the development of a mealytexture. Since those parameters in uence the systembut can not be in uenced by the system themselves,they are considered to be inputs of the system. Theinitial values of the symplast properties are afunction of harvest time and fruit size. For example,late harvested apples have a higher amount of soluble solids and a lower amount of starch thanearly harvested apples.

2 .4 . Output relations

In this third step, the relations between the stateof the apple and measurable mechanical propertiesare proposed. The desired outputs of the model aretensile strength [ N ], compressive hardness (N /mm)andjuiciness(mm 2).Apartfromthose,solublesolidscontentandweightlossarealsoconsideredasoutputvariables.

Whena load isapplied toapple tissuewitha strongmiddle lamella, the cell walls break preferentially,liberating juice and thereby giving a sensation of

crispinessand juiciness.In contrast,when themiddlelamella is dissolved and weak, the tissue will yieldalong the middle lamella. The proportion of brokencells (broken cell index, B ) can, therefore, beconsidered as the amount of undissolved middlelamellarelativetothetotalamountofmiddlelamella(dissolved + undissolved).

B = L

L 0 + P 0(33)

Juicinessis modelled starting from this brokencell

index. The juice released during the con ned com-pressiontestispartlyoriginatingfrominsidethecellswhen they break and partly from the water in theapoplast.

J = c (W a + BW s) (34)

with J juiciness measuredas the spot on a lterpaper.When the middle lamellae of the apples are still

strong, the cell walls will break when a tensile force


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is applied to the apple tissue. If the middle lamellaeis weak, then the cells will tear apart along theirmiddle lamellae. The tensile strength can, there-fore, be described by the following relationship:

F maxt = BF max, wallt + (1 − B )F max, Lt (35)

with F maxt the tensile strength of the tissue mea-sured as the maximum force in a tensile test, andF max, wallt and F max, Lt the tensile strength of the cellwalls and middle lamellae, respectively. Unlikecompressive strength, the tensile strength of the cellwall increases with increasing potential, and hencewith increasing turgor potential (De Baerdemaekeret al., 1978). A linear relationship between cell wallstrength and turgor has been assumed here:

F max, wallt = d + e p, s (36)

Although in mealy tissue the tensile strength of the middle lamella may somewhat increase withincreasing turgor potential because the contactsurface area between neighbouring cells may in-crease slightly, it is assumed here that this effect isof minor importance and F max, Lt is considered tobe a constant parameter.

Compressive hardness (N /mm) is related to theelasticity modulus ( E , (N /m 2)). A generalisation of the cell model of Nilsson et al. (1958) is proposedfor relating the cell E -modulus to the turgorpressure and the amount of middle lamella dissolu-tion sites.

E = E 0 + f p, s + gL (37)

The relation between the E -modulus (Pa) of asimple uniaxial compression and the compressivehardness (N /mm) is given below (De Baerde-maeker and Segerlind, 1976):

S cc =Apl p

1 −(1 + )(1 − 2 )

E (38)

where S cc is the hardness measured as the slope of the force – deformation curve in a con ned com-pression test, Ap (2.27 × 10

− 4 m 2) is the crosssection of the probe used for the test, l p is thelength of the probe (17 mm) and the Poisson ’sratio of the material. This can be written as:

S cc =Apl p

E (39)

with = 1 − /(1 + )(1 − 2 ). The constant canbe incorporated in the parameters E 0 , f and g Eq.(37) and will be omitted from now on.

The percentage of soluble solids (SSC, ( °Brix)) inthe juice is de ned as the overall concentration of hexose units and pectin monomers:

SSC =HM

H+ PM

P(W s + W a )M W 100 (40)

with M H , M P and M W the molar mass of hexose,pectin monomer and water, respectively.

The weight loss (%) was calculated as follows:

m =mapple, 0 − mapple

mapple, 0100 = 1 −

mapplemapple, 0

100

(41)

with mapple, 0 the mass immediately after harvestand mapple the mass after a certain period of

storage. The proportion of the fresh weight at acertain time was estimated by adding the differentcompounds of the apple:

mapplemapple, 0

= SM S + HM H + LM L + PM P + W sM W

+ W a M W + Ash (42)

with M S , M H , M L , M P and M W the molar mass of a dissolution site of starch, the molar mass of ahexose molecule, the molar mass of a dissolutionsite of the middle lamella, the molar mass of a

pectin monomer and the molar mass of water,respectively.The amount of ash (Ash) is considered not to

change during storage and calculated from theconstraint:

1 =mapple, 0mapple, 0

= S 0M S + H 0M H + L 0M L + P 0M P

+ W s, 0 M W + W a,0 M W + Ash (43)

3. Materials and methods

3 .1. Storage experiment

The parameter estimation was based on the datafrom a storage experiment carried out on ‘Cox ’sOrange Pippin ’ apples. Apples were picked on thecommercial harvest date and stored for 6 monthsunder the following conditions,


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3 °C, 95%RH, normal air composition; 3 °C, 95%RH, CA (2% O 2, 0.7% CO 2).A total of 100 apples was stored at each condi-

tion. Tensile strength was measured as the maxi-mum force in a ring tensile test ( F maxt ) performedon a universal testing machine (UTS TestsystemeGmbH, Ulm, Germany; Verlinden and DeBaerdemaeker, 1997). Ring-shaped samples withan outer diameter of 17 mm, an inner diameter of 9 mm and a thickness of 11 mm were pulled apartby means of two half ring-shaped cylinders overwhich the ring-shaped sample could slide. Thesample was precharged with a force of 1 N andthen stretched with a constant deformation speedof 20 mm /min until rupture. Hardness was mea-sured as described by Barreiro et al. (1998a). Itwas de ned as the slope of the force – deformationcurve from a con ned compression test ( S cc ) car-ried out on a universal testing machine (UTSTestsysteme GmbH) on cylindrical probes of 17mm height and diameter. A maximum deforma-tion of 2.5 mm was applied at a compressionvelocity of 20 mm /min. Deformation was immedi-ately removed at the same velocity. The probeswere con ned in a disk, which had a hole with thesame size as the probes. A lter paper (Albet no.1305, 77.84 g /m 2) about the size of the disk wasplaced beneath it in order to recover the juiceextracted during the compression test. The juicecontent was determined by measuring the area of the spot accumulated in the lter paper after asteady spot size was obtained.

The soluble solids content was measured usinga digital refractometer (PR-101, Palette Series,ATAGO CO, Ltd., Japan). Apples were weighedimmediately after harvest and after storage. Allmeasurements were done right after harvest andevery subsequent 6 weeks.

3 .2 . Determination of the EMC cur e of appletissue

Measurements of the apple tissue water poten-tial were carried out on three ‘Cox ’s Orange Pip-pin ’ fruit obtained from a local supermarket.From each fruit, 20 cylinders of 2 cm height and1 cm diameter of apple tissue were recovered andweighed on an analytical balance (Sartorius,

BP160P, Germany) after removing excess mois-ture with paper. To determine the water potential,the apple tissue cylinders were equally divided andimmersed in four solutions with different aW ,established by four different mannitol concentra-tions (0.2, 0.4, 0.6 and 0.8 M). After overnightequilibration at 1 °C, the weight of the cylinders,dipped dry on paper, was measured again atregular time intervals, until a stable weight wasobtained. The intercept with the X -axis of the plotrepresenting the weight difference at equilibriumagainst the mannitol concentration gives the man-nitol concentration for which the apple tissueneither gains or looses weight. The water potentialof the apple tissue can then be calculated from thefollowing equation (Nobel, 1991).

=

RT

V W ln aW − RT [M] (44)

where is the water potential of the apple tissue,aW is the water activity of the mannitol solutionand [M] the mannitol concentration (mol /m 3) cor-responding with this aW . The EMC curve wasobtained by plotting the water content against thewater potential of the apple tissue. The watercontent was de ned as the relative amount of water after drying to constant weight at 105 °C.

3 .3 . Parameter estimation

The model has six state variables, S , H , L, P ,W s and W a with six initial conditions, S 0 , H 0 , L0 ,P 0, W s, 0 and W a, 0 and ve output variables, F maxt ,S cc , J , SSC and m.

As a rst step, literature data have been used toestablish approximate initial values for the statevariables. Table 1 indicates the approximate com-position of apples as percentage of fresh weight(Belitz and Grosch, 1997). No indication of theapproximate amount of starch was given by theseauthors, while Herrero and Guardia (1992) men-tioned 2 – 3% of fresh weight as the maximumstarch content. As a consequence, the state of themodel represents around 97% of the fresh weightof the apple. Based on this information the initialvalues for the state variables shown in Table 2were used in the model calculations.


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Table 1Approximate compositions of apples (Belitz and Grosch, 1997)

Fresh weightFresh weight(%) (moles /kg)

85 47.2Total water(W s, 0 + W a, 0 )

Total sugars 11.1 0.6(S 0+ H 0)0.6Titratable acid2.1Insoluble matter

0.03Pectin 0.60.3Ash

Others 0.3 Fig. 3. Water content of apple tissue samples in equilibriumwith four mannitol solutions expressed as water potential.Symbols denote mean values of 15 samples. Error bars denote95% con dence interval of the means.Additional information from the literature was

used to calculate values for some of the modelconstants (listed in Table 2) or to assign realistic

starting values to the parameters to be estimatedthrough tting.Aa was estimated assuming an apple to bespherical with a radius of 37.5 mm, and anaverage weight of 170 g, corresponding to avalue of 0.10395 m 2/kg fresh weight.As was estimated based on Reeve (1953) whomeasured the total surface are of cell wall areaper unit volume for different apple varieties.

The calculated number was 6.45 m 2/kg fresh

weight.Hertog et al. (1997) modelled the degradationof starch and oxidation of sugars in potatoes.Rate constants for starch and sugar degrada-tion differed from season to season and werebetween 0.0161 and 0.0033 (1 per day) and8.39 × 10 − 5 and 1.53 × 10 − 2 (1 per day) for,respectively, starch and sugar.The EMC curve of ‘Cox ’s Orange Pippin ’ appletissue samples in the range of 0 to − 2 MPawas experimentally determined and shown in

Fig. 3; a and b were calculated by tting Eq.(13) to these data; a had a value of 45.25 MPaand b a value of 40.72 MPa.The relative humidity in the storage rooms was

95% RH, the temperature 3 °C.A total of 12 parameters ( k S , k H , k L , hs, ha , c, d ,

e, F tmax, L , E 0, f and g ) remain to be estimated bytting the model solution to the experimentaldata. The model was tted by means of a softwarepackage developed by Verlinden et al. (1996).Only ve parameters were allowed to change forthe two storage conditions, namely k S , k H , k L , hsand ha . The reason behind this was that the threerate constants may be affected by the air composi-tion in the cool rooms; the two transfer coef -cients might also be different for the two storageconditions because they were stored in cool roomsof different design, and hence air ow pattern andvelocity.

Table 2Constants and estimated initial values

Molecular massesInitial values of state variables

S 0 = 0.10985 mol /kg M S = 162× 10− 3

kg /molH 0 = 0.58 mol /kg M H = 180× 10

− 3

kg /molL 0 = 0.0257 mol /kg M L = 176× 10

− 3

kg /molP 0 = 0.0040457 mol /kg M P = 194× 10

− 3

kg /molM W = 18× 10

− 3W s, 0 = 45 mol /kgkg /mol

W a, 0 = 3.22 mol /kg

Other constants used in the model calculationsAp = 0.227 × 10

− 3 m 2 l p = 0.017 mR = 8.314 J /mol KAa = 0.10395 m 2 /kgT = 276 KAs = 6.45 m 2/kg

a = 45.25 MPa V W = 18× 10− 6

m 3 /molb = 40.724 MPa rel = 0.95


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The integration of the differential equations wasdone with a variable order, variable step Gear-method (NAG integration routine D02EBF).Parameters were tted with a least squares proce-dure (NAG minimisation routine E04FDF). TheNAG routine E04YCF was used to calculate thevariance – covariance matrix from which the corre-lation matrix of the estimated model parametersand their standard deviation was calculated.

4. Results and discussion

Table 3 lists the estimated values of the ttedmodel parameters. The correlation matrix of thesemodel parameters was calculated and is listed inTable 4. It was clear that the correlation betweenall model parameters was relatively low whichindicated that the model was not over-parameter-ised. The only exceptions were g and E 0 which hada correlation coef cient of − 0.96. It is not clearfrom the model structure why this was the case.

The rate constants for the degradation reactions of starch, sugar and middle lamella were higher forapples stored in normal air composition comparedwith apples stored in CA conditions. This seemsreasonable as the latter are known to have aretarding effect on respiratory activity. Also themass transfer coef cient from the symplast to theapoplast was higher for the apples stored in normalair conditions.

The difference in mass transfer coef cients fromthe apoplast to the environment between the twostorage conditions may be caused by the fact thatbecause of experimental restrictions two coolrooms of different design were used. This impliesthat the air ow pattern and velocity eld inside therooms might be different as well which in turnmight affect the mass transfer coef cients consider-ably. Also, the accuracy of the hygrometers may bedifferent in the two cool rooms.

The con dence interval of the estimate of k H wasfor both storage conditions extremely large andincluded zero. This means that there was notenough information in the experimental data toestimate this parameter. This was also the case forhs for the CA-stored apples. Likewise, F tmax, L wasnot signi cantly different from zero. This indicatedthat the tissue tensile strength does seem to dependon the cell wall strength only. In other words, thecell wall always seems to be weaker than the middlelamella, if present.

In Fig. 4, the experimental data of the veoutput variables were plotted against storage timetogether with the simulated model values. Thesymbols were the averages of 20 measurements.The 95% con dence interval of the mean wasgiven by vertical bars. By examining the gure itcan be seen that the model tted the data verywell, although the model slightly underestimatedthe tensile strength (crispiness) in the case of apples stored in air (Fig. 4e). Juiciness and hard-ness were estimated more adequately (Fig. 4c anda). According to the model, the soluble solids forthe apples stored in normal air composition kepton increasing after 100 days while apples stored inCA conditions reached a more or less constantvalue (Fig. 4a). This could not be veri ed byexperimental measurements because the measur-ing technique did not allow the juice to be sam-

Table 3Model parameter values with 95% con dence interval esti-mated by tting

Model Estimate 95% con dence intervalparameter

Air storage CA storageconditions conditions

(0.757 0.153)k S (1/s) (0.543 0.542)× 10− 6 × 10− 6

k H (1/s) (0.0265 131) (4.20 30400000)× 10− 12 × 10− 18

(0.0702 0.0134)k L (1/s) (0.0260 0.00682)× 10− 6 × 10− 6

ha (mol /m 2 .Pa.s) (1.08 0.157) (0.628 0.108)× 10− 12 × 10− 12

hs (mol /m2.Pa.s) (34.5 21.0) (0.231 2.51)

× 10− 15× 10− 15

c (mm 2 kg /mol) 20.6 0.946d (N) 13.2 2.19e (m2) (18.7 4.98) × 10− 6

0.885 3.10F tmax, L (N)(1.84 0.566) × 106E 0 (Pa)0.895 0.758 f (− )

g (Pa /mol) (97.9 30.3) × 106


12/19


T a b

l e 4

C o r r e l a t

i o n m a t r i x o

f t h e

t t e d m o

d e l p a r a m e t e r s

k H ( C

A )

k L ( A

i r )

k L ( C

A )

h s ( A

i r )

h s ( C

A )

h a ( A

i r )

h a ( C

A )

F t m

a x ,

L

E 0

f

g

d

c

e

k S ( C

A )

k S ( A

i r )

k H ( A

i r )

− 0 . 0 3

− 0 . 7 0

− 0 . 2 9

0 . 1 3

0 . 0 1

− 0 . 5 3

− 0 . 0 0

− 0 . 1 9

− 0 . 2 5

0 . 0 4

0 . 1 6

0 . 0

1

− 0 . 3 7

0 . 0 3

1

k S ( A

i r )

0 . 0 0

− 0 . 0 2

0 . 0 0

− 0 . 0 0

− 0 . 0 3

0 . 0 0

− 0 . 1 6

− 0 . 0 0

− 0 . 0 0

− 0 . 0 0

− 0 . 0 0

0 . 0 0

k S ( C

A )

0 . 0

0

− 0 . 0 1

− 0 . 0 0

0 . 3 1

0 . 0 0

1

0 . 0 6

− 0 . 2 6

− 0 . 0 0

− 0 . 2 6

k H ( A

i r )

− 0 . 0 0

0 . 0 3

0 . 0 4

− 0 . 0 3

− 0 . 0 1

0 . 0

2

0 . 0 5

− 0 . 0 2

1

0 . 0 1

0 . 2 4

0 . 1 1

0 . 0 2

− 0 . 1 2

− 0 . 0 1

− 0 . 1 3

0 . 0 0

0 . 0 1

− 0 . 0 3

0 . 0 4

0 . 0 1

0 . 0

3

0 . 0 5

− 0 . 0 2

1

k H ( C

A )

1

0 . 4 1

0 . 4 2

− 0 . 0 2

− 0 . 2 4

0 . 0 1

0 . 2 8

0 . 3 6

− 0 . 0 6

− 0 . 2 3

− 0

. 0 1

0 . 5 0

− 0 . 0 5

k L ( A

i r )

1

0 . 1 7

− 0 . 0 5

− 0 . 0 9

− 0 . 0 2

0 . 0 7

0 . 0 9

− 0 . 2 7

0 . 0 7

0 . 2

1

0 . 6 2

− 0 . 2 0

k L ( C

A )

1

− 0 . 0 1

− 0 . 6 8

0 . 0 0

0 . 1 0

0 . 1 3

− 0 . 0 4

− 0 . 0 8

h s ( A

i r )

0 . 0

1

0 . 2 0

− 0 . 0 3

1

0 . 0 0

− 0 . 1 1

− 0 . 0 0

h s ( C

A )

− 0 . 0 0

0 . 0 1

− 0 . 0 1

− 0

. 0 1

− 0 . 0 2

0 . 0 1

1

0 . 0 0

− 0 . 0 7

− 0 . 1 0

0 . 0 2

0 . 0 6

0 . 0

0

− 0 . 1 0

0 . 0 1

h a ( A

i r )

1

0 . 0 1

0 . 0 1

0 . 0 1

− 0 . 0 1

− 0

. 0 1

0 . 0 3

0 . 0 1

h a ( C

A )

1

0 . 1 8

0 . 0 3

− 0 . 1 5

− 0

. 7 9

0 . 1 5

0 . 5 1

F t m

a x ,

L

1

0 . 4 9

− 0 . 9 6

E 0

− 0

. 0 7

0 . 1 9

0 . 0 3

1

− 0 . 6 3

− 0

. 0 9

f

− 0 . 1 4

0 . 0 7

1

0 . 0

9

− 0 . 0 7

− 0 . 0 5

g

1

0 . 0 8

− 0 . 7 4

d c

1

− 0 . 1 0

e

1

( C A ) , C A s t o r a g e c o n

d i t i o n s ;

( A i r ) ,

a i r s t o r a g e c o n

d i t i o n s .


13/19


Fig. 4. Change of the measured output variables during storage. Error bars denote 95% con dence intervals of the mean of 20measurements (note, the value of soluble solids after 84 days of air storage is the mean of only three measurements).

pled any more once the apples became rathermealy, for the normal air storage condition. How-ever, this prediction was plausible because of theconcentration effect one can expect from of theconsiderable weight loss (Fig. 4b). The modeltted the weight loss well.

Fig. 5 shows the apple behaviour during stor-

age. During the rst 50 days of storage, starchwas degraded (Fig. 5a) in favour of hexose andthe concentration of the latter increased corre-spondingly in this period (Fig. 5b). Degradationand production were faster in normal air than inCA storage conditions. After about 100 days allstarch was gone and the concentration of hexosedecreased. However, this consumption of hexose

was very limited and almost not noticeable be-cause respiration rates of fruit stored in theseconditions was very low. The pectin in the middlelamellae was degraded gradually during the wholestorage period (Fig. 5c) but with a far higher ratein normal air conditions, resulting in less middlelamellae and more solubilised pectin at the end of

the storage period compared with CA storage(Fig. 5d). Fig. 5e and f show the water state in theapple. In the case of CA-stored apples a consider-able decrease in apoplastic water content wasobserved, while the symplastic water content re-mained almost constant throughout the wholestorage period. In the normal air-stored apples, aconsiderable amount of symplastic water was lost


14/19


through the apoplast to ambient and this explainedthe large weight loss of the apples. Note that theapoplastic water content represented a dynamicequilibrium between water in ux from the sym-plast and water loss to ambient. The mass transfercoef cients from the apoplast to ambient was forboth CA and air conditions, much smaller thanthat from the symplast to the apoplast, whichcon rmed that the skin was the major barrier withrespect to moisture transport. The large differencein symplastic to apoplastic mass transfer coef cientbetween CA and air storage might be due to thefact that in the latter case the tissue was moremealy, and hence the cells were more rounded anddetached from each other (De Smedt et al., 1998).The surface area available for water exchange tothe apoplast was in this case considerably larger.

Fig. 6 shows four intermediate variables thatare functions of the state and which determine theoutput variables of the model. Fig. 6a shows theosmotic potential in the symplast which wasclosely related to the water concentration (see Fig.5e). Fig. 6b, shows the water potential in theapoplast. The water potential decreased corre-sponding with a reduction in water concentration,and levelled off after about 150 days of storage.This level was lower in CA conditions despite alower mass transfer rate from apoplast to ambi-ent. The change in turgor pressure is shown inFig. 6c. Because of the high water loss, accountingfor most of the weight loss, turgor pressure wasalready lost after 12 and 20 days in normal airand CA storage, respectively. The turgor pressurewas responsible for the sudden drop in tensile

Fig. 5. Change of the state variables during storage.


15/19


Fig. 6. Change of some important intermediate variables during storage.

strength and to a lesser extent in hardness duringnormal air storage. Whereas, in CA storage thesymplastic water concentration seemed to be con-stant, there was a clear loss of symplastic waterunder air storage (Fig. 5e). Although this seemedto indicate complete plasmolysis, this was notnecessarily true as the apple has been considered asa bulked object in this article. Most probably theloss of symplastic water occurred just beneath theskin where complete turgor loss and even plasmol-ysis may be expected, whereas the rest of the applewas still turgid. A re nement of this model whichwould include spatial effects such as water diffu-sion in the tissue might elucidate this further. Fig.6d shows the change of the broken cell index whichwas mainly responsible for the change in juiciness.

Sensory experiments which were carried outsimultaneously with the instrumental measure-ments described in this article (De Smedt, 2000),showed that the apples stored in normal air com-position were more mealy than those stored inCA. According to the model this could be ex-plained through an accelerated degradation of

starch and a more pronounced dissolution of themiddle lamella.

Fig. 7 shows the effect of relative humidity onthe output variables of the model by three modelsimulations in which the relative humidity was setto 90, 95 and 98%. It is clearly seen from Fig. 7b,that juiciness was not sensitive to changes inrelative humidity. This was in accordance with thecommon knowledge that a wrinkled apple thathas experienced considerable water loss, can stillbe juicy. The largest effect was observed withrespect to tensile strength (Fig. 7c) and hardness(Fig. 7a) which increased with increasing relativehumidity, mainly due to the fact that turgor pres-sure was maintained at non-zero for a longerperiod. Obviously weight loss was sensitive torelative humidity as well (Fig. 7d). A lower rela-tive humidity resulted in a higher weight loss. Thiswas most pronounced in normal air storage.

To illustrate that the high weight loss observedin the experiment with normal air conditionsmight be due, at least partly, to high air velocitiesand so to a high mass transfer coef cient because


16/19


of the different cool rooms which were used be-cause of practical constraints, a simulation wascarried out in which ha for normal air storageconditions was set to the same value as the onefor CA storage. The results are shown in Fig. 8.Clearly the effects on hardness, juiciness andcrispiness were marginal (Fig. 8a, c and e). How-ever, the weight loss (Fig. 8b) was much lowerthan the one which was actually observed, andcoincides during the rst 100 days with the weight

loss observed in CA conditions. This resulted in amuch more reasonable weight loss at the end of the storage period. Since less water was lost, thesoluble solids concentration was smaller whichresulted in lower SSC values (Fig. 8d). The weightloss under CA at 90% RH was about 6% andsmaller than expected from practical experience.This may be due to extrapolation of the modelbeyond its validity range or may indicate somedef ciences of the model.

Fig. 7. Model simulation results to illustrate the effect of different relative humidity conditions on the change of four outputvariables during air and CA storage.


17/19


Fig. 8. Model simulation results to illustrate the effect of mass transfer coef cient of water transfer through the apple skin to theambient.

5. Conclusions

The mathematical model which has been devel-oped does describe the mealiness-related texturechanges of apple during CA and air storage rea-sonably well. However, only a limited amount of experimental data were available to check thefeasibility of the model. More experimental datawould be needed to check all hypotheses madeduring the construction of the model. Further-more, it would also be useful to include detailedmeasurements of the state variables of the model,such as the apoplastic and symplastic water, the

turgor pressure and the pectin content. Also, avalidation of the model using an independent dataset is mandatory.

The model is built by assuming that the temper-ature is constant and optimal for the storage of apples. When apples are stored under temperatureconditions which are higher than optimal, thedevelopment of mealiness will be accelerated. Thiswould in uence the parameters of the system,notably the rate constants and the water poten-tials. As the oxygen concentration is constantduring the storage period except at the start of thestorage, it can readily be assumed that different


18/19


oxygen concentrations only in uence the respira-tion rate constant. A more complete model inwhich these dependencies are explicitly modelledwould allow us to predict and, hence, optimise thedevelopment of mealiness under arbitrary storageconditions.

6. Notation

a coef cient Eq. (13) (Pa)aw water activity ( − )As speci c cell membrane area (m 2/kg)

total cell membrane area (m 2)A s*speci c apple surface area (m 2/kg)Aaapple surface area (m 2)A a*cross section of the probe used forApthe con ned compression test (m 2)proportion of ash of the appleAshfresh weight ( − )

b coef cient (Eq. (13)) (Pa)B broken cell index ( − )

coef cient Eq. (34) (mm 2 kg /mol)cC m mannitol concentration (mol /m 3)

coef cient Eq. (36) (N)d e coef cient Eq. (36) (m 2)E elasticity modulus of the apple

esh (Pa)E 0 coef cient Eq. (37) (Pa) f coef cient Eq. (37) ( − )

tensile strength (N)F maxt

F max, wallt tensile strength of the cell wall (N)F max, Lt tensile strength of the middle

lamella (N) g coef cient Eq. (37) (Pa /mol)hs mass transfer coef cient of water

transfer from the symplast to theapoplast (mol /m 2 Pa s)mass transfer coef cient of waterhatransfer from the apoplast to theambient (mol /m 2 Pa s)hexose unitHmoles of hexose per kg freshH weight of apple tissue (mol /kg) juiciness (mm 2)J

k rate constant (1 /s)l p length of the probe used for the

con ned compression test (mm)

L dissolution site on the pectinchains of the middle lamellaemoles of dissolution sites of theLmiddle lamella per kg fresh weightapple tissue (mol /kg)

mapple mass of the apple (kg)m weight loss of the apple (%)

M Mannitolmolecular mass (g /mol)M number of moles (mol)npectin monomerPmoles of pectin monomers per kgPfresh weight of apple tissue (mol /kg)ow rate (mol /s)runiversal gas constant, 8.3143 (J /Rmol K)

S dissolution site on starch chainsmoles of dissolution sites on starchS per kg fresh weight of apple tissue(mol /kg)hardness (N /mm)S ccPercentage of soluble solids in theSSC juice [°Brix]

t Time (s)T Temperature (K)

Reaction rate (1 /s)V Molar volume (m 3/mol)

Water moleculeWMoles of water per kg fresh weightW of apple tissue (mol /kg)

Greek symbolsactivity coef cient ( − )

parameter ( − )relative humidity ( − )rel

Poisson ’s ratiototal water potential (Pa)

p turgor potential (Pa)

osmotic potential (Pa)mole fraction[ ]

Subscriptsinitial condition0Hexose unitH

L middle lamellaP pectin monomer


19/19


symplastSS starch

waterWapoplastA

ambient

Acknowledgements

The nancial support from the E.U. (FAIRproject CT95-0302) and the Flemish Ministry forScience and Technology is gratefully acknowl-edged. Author Els Veraverbeke is a doctoral fel-low of the IWT. We would like to thank thereviewers for their exceptionally lucid contribu-tion which led to some substantial improvementof the model.

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modelo matemático sobre la congelación de manzanas

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