Oxygen Diffusivity in Avocado Fruit Tissue

ARTICLE IN PRESS Biosystems Engineering (2005) 92 (2), 197–206 doi:10.1016/j.biosystemseng.2005.06.001 PH—Postharvest Technology

Oxygen Diffusivity in Avocado Fruit Tissue Salvador Valle-Guadarrama1; Teodoro Espinosa-Solares1; Crescenciano Saucedo-Veloz2; Cecilia B. Pen˜a-Valdivia2 1

Departamento de Ingenierı´ a Agroindustrial, Universidad Auto´noma Chapingo (UACh), Chapingo, 56230, Texcoco, Mexico; e-mail of corresponding author: [email protected] 2 Programa de Fisiologı´ a Vegetal, Instituto de Recursos Gene´ticos y Productividad, Colegio de Postgraduados, Montecillo, 56230, Texcoco, Mexico (Received 14 July 2004; accepted in revised form 9 June 2005; Published online 15 August 2005)

The oxygen mass diffusion coefficient in pre-climacteric ‘Hass’ avocado fruits tissue at 20 1C was evaluated. A respiration–diffusion model was developed by using a non-steady–state mass balance routine, in combination with the second law of Fick and the Michaelis–Menten enzymatic kinetic theory. Diffusivity was determined by simulation trials where a set of quantities for this parameter was proposed during the model solution, until a predicted O2 partial pressure below the skin agreed with experimental information; the resulting value was 2
2  109 m2 s1. The effect of the diffusion on the respiration parameters is discussed. Oxygen internal concentration profiles were used to suggest a minimal permissible coating permeance for an adequate fruit modified atmosphere storage. r 2005 Silsoe Research Institute. All rights reserved Published by Elsevier Ltd

1. Introduction The avocado fruit has been studied from a modified atmosphere (MAP) storage-technology point of view. In several works, the presence of fermentative metabolism symptoms due to the reduction of O2 in the fruit surrounding medium has been reported (Yahia, 1993; Berrios, 2002; Hertog et al., 2003). Nowadays, it is well documented (Rajapakse et al., 1990; Perez & Beaudry, 1998; Zhang & Bunn, 2000; Lammertyn et al., 2001) that flesh and skin in fruits establish resistance to gas flow between external and internal layers and therefore concentration gradients between different points are induced. It has been determined (Valle-Guadarrama et al., 2002) that the O2 partial pressure underneath the skin in a ‘Hass’ avocado fruit can decrease to less than 5 kPa when the material is stored even in a natural condition at 20 1C, showing the presence of a notable concentration gradient in relation to the external medium. Since in a MAP system the O2 partial pressure in the fruit surrounding medium is reduced significantly in relation to normal conditions, there might be zones, inside the product, with aerobic conditions and others where the state is anaerobic. The study and application of MAP systems have been based, mainly, on the changes of gas concentrations in the surrounding medium, and the fruit has been 1537-5110/$30.00

considered frequently as a black-box model, where conditions inside the product remain unknown. With respect to the ‘Hass’ avocado fruit, it is known that the anaerobic compensation point (ACP), it means the O2 partial pressure at which the oxidative metabolism switches to a fermentative one has a value of 1
41 kPa at 20 1C (Valle-Guadarrama et al., 2004), but it is unknown if the O2 partial pressure at tissue level has decreased beyond that limit. A manner of examining these aspects consists of studying the factors that influence the O2 concentration profiles inside the tissue, specifically the fruit respiration rate and the O2 transport mechanism. In relation with the latter, two kinds of barriers to gas flow have to be considered: one of them is the skin, through which a gas is transported (from the external to the internal ambient) by a permeation mechanism that is affected, mainly, by the fruit cuticle structure and waxes deposited on it (Banks et al., 1995; Amarante & Banks, 2001; Lammertyn et al., 2001; Valle-Guadarrama et al., 2002); the other one is the pulp of the fruit, where the transport mechanism occurs by diffusion (Rajapakse et al., 1990; Abdul-Baki & Solomos, 1994; Zhang & Bunn, 2000; Lammertyn et al., 2001, 2003). Valle-Guadarrama et al. (2002) determined the O2 permeance for the skin of the avocado fruit. Although

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r 2005 Silsoe Research Institute. All rights reserved Published by Elsevier Ltd

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Notation Ask Aus cO2 DO2 GO2 gO2 gmax O2 KM K app M Kp K app p MO2 mfr mti P0ctmin P0shmin P0sk piO2 peO2 ptr

surface area of the skin, m2 surface area of a tissue element located underneath the skin, m2 molar concentration of O2 in tissue, kg[mol] m3 mass diffusivity coefficient for O2 in fruit tissue, m2 s1 consumption rate of O2 by volume unit, kg[mol] m3 s1 consumption rate of O2, kg[gas] kg1[fruit] s1 maximum O2 consumption rate, kg[gas] kg1[fruit] s1 Michaelis–Menten constant for O2, kg[mol] m3 apparent Michaelis–Menten constant for O2, kg[mol] m3 Michaelis–Menten constant for O2, Pa apparent Michaelis–Menten constant for O2, Pa molecular weight of O2, kg[gas] kg1[mol] fruit mass, kg tissue element mass, kg minimal permissible coating permeance, kg s1 m2 Pa1 minimal permissible shell permeance, kg s1 m2 Pa1 skin permeance, kg s1 m2 Pa1 internal partial pressure of O2, Pa partial pressure of O2 in the environment surrounding the fruit, Pa partial pressure of O2 in a node located at position r in a previous moment in time, Pa

prtþ1

partial pressure of O2 in a node located at position r in a later moment in time, Pa ptrþ1 partial pressure of O2 in a node located at position (r+1) in a previous moment in time, Pa ptr1 partial pressure of O2 in a node located at position (r1) in a previous moment in time, Pa ptot total pressure; atmospheric pressure, 77 993 Pa R ideal gas constant, 8314 Pa m3 kg1[mol] K1 r sphere radius, m rO2 exchange rate of O2 through the fruit skin, kg[gas] kg1[fruit] s1 T temperature, 1C t time, s tc time interval during O2 consumption rate evaluation, s V volume of a fruit tissue element, m3 Vc free volume in a container during O2 consumption rate evaluation, m3 vr, vf, vy linear velocities of O2 associated with convective movements in the radius, azimuth and latitude directions, m s1 fin ini yO2 ; yO2 initial and final concentrations, respectively, in a container during O2 consumption rate evaluation, volume fraction Z geometric parameter, s m3 Dr thickness of a spherical shape tissue element, m Dt simulation time interval, s e effectiveness factor, dimensionless f, y azimuth and latitude, respectively, of a point on the surface of a sphere, radians rti fruit tissue density, kg m3

2.1. Mathematical model development

particular spheres produced a tissue spherical band of thickness dr in m which was considered an analysis element. For a particular element the O2 entrance rate was considered equal to the sum of the rates at which the gas exits, accumulates and is consumed. This O2 mass balance was applied on two kinds of analysis items: an internal tissue element where the O2 mechanism transport occurs only by diffusion and the element located in the region immediate underneath the skin, which is affected by both, a permeation process imposed by the skin at its external surface and a diffusional transport that occurs throughout the element itself and in the transport to the adjacent one.

The fruit was described as a unit of concentric spheres, each one located at a given radius r in m from the centre. The difference in volume between two

2.1.1. Oxygen transport mechanism in the fruit tissue According to Bird et al. (1960) the representation, in polar spherical coordinates, of an O2 mass balance

this parameter is useful to predict the gas concentration underneath that layer, it cannot be used to characterise deeper regions and because of this the knowledge of the O2 mass diffusivity coefficient (O2-MDC) is necessary. In this context, the objective of this work was to estimate the O2-MDC in ‘Hass’ avocado fruits tissue to be able to predict the internal O2 concentration profiles and allow the improvement of storage systems of this material for longer periods.

2. Mathematical model

ARTICLE IN PRESS OXYGEN DIFFUSIVITY IN AVOCADO FRUIT TISSUE

where the transport mechanism occurs by diffusion is expressed by Fick’s second law in the form described by
 qcO2 qcO2 1 qcO2 1 qcO2 þ vr þ vy þ vf r qy r sin f qf qt qr 

 1 q 2 qcO2 1 q qcO2 r sin y ¼ DO2 2 þ 2 r qr r sin y qy qr qy  2 1 q cO2 ð1Þ  G O2 þ 2 r sin2 y qf2 where: subscript O2 represents oxygen which diffuses in the fruit tissue; r, y and f are the distance from the centre (radius in m), latitude and azimuth (both in radians), respectively, of a point located on the surface of the sphere; v represents the linear velocity of the gas in m s1, associated with a convective movement in the direction pointed out by the corresponding term subscript; cO2 is the oxygen molar concentration in kg[mol] m3, in the diffusional environment; DO2 is a mass diffusivity coefficient for the gas (O2-MDC) in m2 s1, GO2 represents the O2 consumption rate for each volume unit in kg[mol] m3 s1. By analogy with the heat equation (Carslaw & Jaeger, 1959), when there are no convective movements (it means when vr, vy and vf are zero), initial and boundary conditions are uniform and consecutive analysis elements correspond to concentric spheres (as it has been supposed in this case), the concentration gradients through the latitude and the azimuth directions are negligible or zero, and the oxygen concentration (cO2) depends only on the radial coordinate r and time t; this way, Eqn (1) is transformed into Eqn (2):  2
 qcO2 q cO2 2 qcO2 ¼ DO2 þ (2)  G O2 r qr qt qr2 The term for consumption rate of oxygen for each volume unit (GO2 in kg[mol] m3 s1) was expressed by means of the enzymatic kinetic theory of Michaelis– Menten (Peppelenbos & van’t Leven, 1996; Hertog et al., 1998) where a process without inhibition and ideal gas behaviour were considered (Eqn 3): 


max  gO2 cO2 mti mti G O2 ¼ gO2 ¼ (3) M O2 V K M þ cO2 M O2 V The involved variables are the O2 consumption rate gO2 in kg[gas] kg1[fruit] s1; the Michaelis–Menten constant K M in kg[mol] m3 for O2, the maximum rate 1 1 of consumption gmax O2 in kg[gas] kg [fruit] s ; the O2 3 concentration cO2 in kg[mol] m and the O2 molecular weight MO2 in kg[gas] kg1[mol]. Moreover, if cO2 and K M are expressed in terms of O2 partial pressures through the application of the gas ideal law, the mass of the tissue mti in terms of the tissue density rti in kg m3 and the element volume V in m3, and Eqn (3) is

199

substituted in Eqn (2), then the result is Eqn (4), where T in 1C is temperature and R is the ideal gas constant, equal to 8314 Pa m3 kg1[mol] K1. This equation describes the rate of change of the partial pressure of oxygen in a spherical band of internal tissue:  2 i
 qpiO2 q pO2 2 qpiO2 ¼  DO2 þ r qr qt qr2
max i 
 gO2 pO2 rti RðT þ 273
15Þ  ð4Þ M O2 K p þ piO2 2.1.2. Oxygen transport mechanism in the fruit skin Equation (4) applies to internal tissue regions, but not to the fruit skin region. Due to the layers of cutine and waxes deposited on the external surface of the fruit, the O2 exchange through the skin occurs by a permeation phenomena, in a similar way as it takes place in polymeric films (Kertiens, 1996; Amarante & Banks, 2001; Maguire et al., 2001); hence, the O2 mass balance for the element immediately below the skin cannot catch the diffusion as the unique mechanism for O2 transport. Under this consideration the O2 gas exchange rO2 in kg ½gas kg1 ½fruit s1 through the fruit skin, referred to as a permeation process, was considered in the terms of Eqn (5) (Banks & Nicholson, 2000):   (5) rO2 ¼ P0sk Ask peO2  piO2 where: P0sk is the permeance to O2 in kg½gas s1 m2 Pa1 of the fruit skin; Ask is its superficial area in m2, and peO2 and piO2 are O2 partial pressures in Pa outside and below the fruit skin, respectively. Thus, the expression of the O2 mass balance results in Eqn (6), where the terms of entrance and exit correspond to permeation and diffusional processes, respectively.  

 P0sk Ask peO2  piO2 qcO2 qcO2 ¼  DO2 Aus þV M O2 qr qt
max 
 gO2 cO2 mti þ ð6Þ K M þ cO2 M O2 In this equation, Aus is the surface area in m2 of a tissue element located immediately below the skin. The region below the peel constitutes the most external zone where O2 consumption phenomena by tissue occurs and the diffusive flow of the gas towards the interior may be identified, in addition to its infiltration through the skin by a permeation process. This region is normally considered (Holman, 1986) as an element of thickness equal to half the differential element analysed (dr in this case), hence the fruit boundary region may be considered to have a thickness of dr/2 or Dr/2, and, from here, Ask and Aus are spherical areas defined as (4pr2)

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2.2. Model solution The fruit was considered as a spherical nodal system; thus, each partial derivative was expressed in terms of finite differences (Burden & Faires, 1985; Holman, 1986) to produce, after reorganisation, work equations for the description of the change of the O2 partial pressure with time in a boundary node Eqn (8), and in an internal node Eqn (9). 
 Zr2 P0sk RðT þ 273
15Þ ZDO2 t ð2r  DrÞ2 ptþ1 ¼ p 1   r r M O2 4Dr  max  gO2 rti RðT þ 273
15ÞDt  ðK p þ ptr ÞM O2  2 0  Zr Psk RðT þ 273
15Þ e þ pO2 M O2 
  ZDO2 þ ptr1 ð8Þ ð2r  DrÞ2 4Dr   max 

 gO2 rti RðT þ 273
15ÞDt 2DO2 Dt Dr t  ptþ1 ¼ p 1  1  r r r ðK p þ ptr ÞM O2 ðDrÞ2 

 
DO2 Dt DO2 Dt 2Dr t ð9Þ þ p 1  þ ptrþ1 r1 r ðDrÞ2 ðDrÞ2

In these expressions the subscript r identifies the position of the analysis spherical node, given as the distance in m from the centre of the fruit; subscripts (r+1) and (r1) represent the positions of the immediate nodes, external and internal, with regard to the analysis node, respectively; superscripts t and (t+1) refer to the previous and subsequent conditions, in time, respectively; Dr and Dt are the distance in m between nodes and the time interval in s between a previous and a subsequent condition, respectively. Partial pressure coefficients in Eqns (8) and (9) are dimensionless and the parameter Z in s m3 is defined by

Z¼

24Dt 8r3  ð2r  DrÞ3

(10)

2.3. Parameterisation 2.3.1. Evaluation of O2 consumption rate The relationship between the O2 consumption rate gO2 and the O2 availability was expressed in terms of the Michaelis–Menten enzymatic kinetic theory in the form expressed by Eqn (3). The values for gmax O2 and K p were evaluated to be 1
4  108 kg½gas kg1 ½fruit s1 and 4961 Pa, respectively (Fig. 1). These parameters were measured from intact fruits and it was assumed that they could be applied to the tissue level. The procedure used in this evaluation consisted of placing 15 ‘Hass’ avocado fruits, physiologically mature, in controlled atmospheres at 20 1C with oxygen concentrations between 2 and 21% (environmental condition). The fruits were harvested in January 2003; they had an average weight of 217 g, an approximate volume of 2
05  104 m3 and a surface to weight ratio of 0
088 m2 kg1. Gas mixtures were applied to individual fruits in containers of 0
0012 m3, with flows of 75 ml min1, and over a period of 24 h. At the end of the treatment, the O2 consumption was evaluated by means of the gas exchange of the fruit with the environment inside the containers. For this, controlled atmosphere units were hermetically separated from the flow system, counting this as time zero (t ¼ 0). The O2 concentration changes inside the containers were registered on ten occasions, for over an approximate period of 4 h. The data from each run were treated with regression routines and the resulting models were used to estimate the concentrations inside the containers corresponding to 1 h of evaluation. The oxygen consumption

Oxygen consumption, µg[gas] kg−1 [fruit] s−1

and [4p (rDr/2)2], respectively. Likewise, the volume V of the tissue element analysed is [(4/3) p(r3(rDr/2))3]. The substitution of these definitions in Eqn (9) and the expression of cO2 and K M in terms of partial pressures resulted in Eqn (7), which explains the rate of change of the internal partial pressure of O2 piO2 in the region immediately below the skin: "  e 
 2 0 i r Psk pO2  piO2 RðT þ 273
15Þ qpO2 Dr 2 ¼ þ DO2 r  qt 2 M O2
i  
max i  qpO2 24 gO2 pO2  qr K p þ piO2 8r3  ð2r  DrÞ3
 rti RðT þ 273
15Þ ð7Þ  M O2

12

0

2

4

Oxygen concentration, % 6 8 10 12 14 16

18

20

10 8 Equation (3)

6 4 2 0

0

2

12 10 4 6 8 Oxygen partial pressure, kPa

14

16

Fig. 1. Oxygen internal partial pressure (piO2 ) influence on the oxygen consumption rate (gO2 ), in ‘Hass’ avocado fruits at 20 1C; points represent experimental data and the solid line the adjustment of these to the equality shown in Eqn (3) (determination coefficient equal to 0
8915)

ARTICLE IN PRESS OXYGEN DIFFUSIVITY IN AVOCADO FRUIT TISSUE

rate gO2 in kg½gas kg1 ½fruit s1 for each fruit was determined by using Eqn (11):

 fin yini O2  yO2 V c ptot M O2 (11) gO2 ¼ mfr tc RðT þ 273
15Þ ini where yfin O2 and yO2 are the final (superscript fin) and the initial (superscript ini) O2 concentrations as volume fractions inside the container, respectively; Vc is its free volume in m3; ptot is the total pressure (atmospheric pressure, 77 993 Pa); T is the temperature in 1C; R is the ideal gas constant (8314 Pa m3 kg1 ½mol K1 ); mfr is the fruit mass in kg; tc is the evaluation time in s, MO2 is the molecular weight of O2 (32 kg½mass kg1 ½mol ). Oxygen concentrations applied in the controlled atmosphere were converted to partial pressures (1 kPa of partial pressure is equivalent to a concentration of 1
286% at the mentioned atmospheric pressure). The data were fitted to Eqn (3) with the Sigma Plots software (SPSS Incorporation, 2000). Concentrations were determined with a gas chromatograph (Varian model 3400CX, USA) with a poraplot Q type Chrompacks capillary column, thermal conductivity detector (TCD) and flame ionising detector (FID). Temperatures of 80, 150 and 170 1C were used in the column, injector and detectors, respectively, and a column manometric pressure of 158
5 kPa. Oxygen quantification was assisted by calibration standards (Praxair of Mexico S.A. of C.V.) in concentrations of (2
01% O2–97
99% N2) and (4
98% O2–4
98% CO2–90
04% N2). In all cases, chromatograph samples were injected using glass syringes for gases (Hamiltons) in aliquots of 100 ml.

2.3.2. Analysis constants Spherical nodes were spaced by a thickness Dr of 0
0005 m; time intervals Dt were fixed to have a value of 10 s; environmental O2 partial pressure peO2 was considered to remain at a constant value of 15 988 Pa (environmental concentration at the atmospheric pressure condition) and the O2 skin permeance P0sk was taken from Valle-Guadarrama et al. (2002) to be 1
338  1010 kg s1 m2 Pa1. 2.4. Oxygen mass diffusivity coefficient estimation A procedure based on simulation trials was used: a set of values in the range of 1
0  1010–1
2  108 m2 s1 was tested for DO2 (O2-MDC). Each value was proved in the following manner: the fruit was considered as a mass of tissue without skin and with a homogeneous O2 partial pressure of 15
988 kPa (p0r ¼ 15
988) for all nodes; at time zero (t ¼ 0) the placement of a layer equivalent to the skin and with similar properties was

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declared, and the variation of the partial pressure of O2 for different tissue points (nodes) was simulated during 24 h with the use of Eqns (8) and (9). The value of the O2 mass diffusivity coefficient (O2MDC) was determined through the comparison of the O2 partial pressure data, obtained in each trial for the region immediately below the skin, with an experimental value taken as a reference. Since that boundary region was defined as a spherical element of thickness equal to Dr/2, the corresponding O2 partial pressure was determined as the average of the value acquired for the two most external nodes. In accordance with ValleGuadarrama et al. (2002) the O2 partial pressure in that region must be 14
896 kPa for ‘Hass’ avocado fruits in pre-climacteric stage at 20 1C; hence, the correct value for O2-MDC was the one that produced an O2 partial pressure that agreed with that reference. 2.5. Model stability requirement When a partial differential equation is solved by the finite difference method, a stability criterion must be defined for its solution (Burden & Faires, 1985; Holman, 1986) in order to avoid the use of work conditions that violate the second law of thermodynamics. In this work, the stability criterion derived from the solution of nodal Eqns (8) and (9) was defined through Eqn (12):   DO2 Dt o0
484 (12) ðDrÞ2

3. Results and discussion 3.1. Oxygen mass diffusivity coefficient in tissue The set of simulation trials done with the use of Eqns (8) and (9) produced different values for piO2 underneath the skin, each one corresponding to one of the DO2 values. Figure 2 shows that when the diffusion coefficient adopts a value of 2
2  109 m2 s1 the average oxygen partial pressure reaches 14
89 kPa, which agreed with the experimental reference established; hence, the O2 mass diffusivity coefficient (O2MDC) for ‘Hass’ avocado fruits in pre-climacteric stage has the value pointed out. This result was of similar order of magnitude of several fruits reported in the literature, such as pear (1
71  109 m2 s1, Lammertyn et al., 2001) and apple (2
67  109 m2 s1, Mannapperuma et al. 1991). Rajapakse et al. (1990) reported higher values for apple (1
7  107 m2 s1), pear (3
0  108 m2 s1) and nectarines (1
7  107 m2 s1), although the authors recognised

ARTICLE IN PRESS S. VALLE-GUADARRAMA ET AL.

16

21

a

20 15

19 b

14

18 17

c

13

16 12 11

15 0

2

4

6

8

10

Oxygen internal concentration, %

Oxygen internal partial pressure, Kpa

202

12

Mass diffusivity, nm2 s−1 Fig. 2. Oxygen mass diffusivity coefficient (DO2 ) influence on the oxygen partial pressure (piO2 ) in ‘Hass’ avocado fruits tissue, at the region immediately below the skin: line (a) represents the behaviour of an exact point below the skin; (c) of one point located at 0
5 mm below the skin; and (b) corresponds to the average of both; the arrow describes the method followed to determine the O2 mass diffusivity coefficient in the fruit tissue

certain overestimation due to the fact that the fruit conformation did not fit into a regular geometrical shape. The strategy used in the present work to determine the O2-MDC was similar to the method followed by Lammertyn et al. (2001): a diffusion–respiration model was developed and solved by using experimental information that was taken as reference. A set of simulation trials were needed to be done until the concentration gradient predicted corresponded to the experimental information. Under this consideration, the set formed by the model plus the experimental information has to be taken as a tool to estimate the O2MDC and the simulation trials in experiments which allow estimation of the parameter. Nevertheless, it is still necessary to work with satisfactory forms of validation of this kind of models. As the model was solved numerically, a way of validating it is the use of an analytical solution for the theoretical equations (Abbaspour-Fard, 2004), in this case Eqns (4) and (7); unfortunately, in the open literature such a solution is not available. Another option is comparing simulated data with experimental data. The model developed in the present work was structured by two kinds of equations, one of them representing the internal tissue elements and the other, the one located immediately below the pericarp. If experimental data corresponding to the most external region or that relative to the gas exchange with the environment are reproduced with the model, then the second equation would be validated but, for the first one, the strategy might not be enough and experimental information relative to internal concentration gradients would be required. Due to the difficulty of

achieving this and as the main application objective of a diffusion–respiration model is the prediction of the internal atmosphere of the fruits, the experimental characterisation of internal concentration gradients would have to be done indirectly through other physiological gradients.

3.2. Relation between respiration and diffusion parameters There are some differences between the method followed in this work and the method used by Lammertyn et al. (2001). One of them is that while the proposal of these authors requires extracting samples from the fruit (destructive procedure) the method followed in the present work uses the information in a natural condition of the item (non-destructive procedure). Another difference is the way of considering the respiration parameters. Lammertyn et al. (2001) assayed respiration measurements with thin tissue elements; based on this strategy, they assumed that inside them the O2 concentration gradients were negligible or zero and because of this the diffusion transport within an element could be declared null. Due to this consideration, the respiration parameter obtained was considered free of diffusional effects and, after the substitution of this in the model, the O2 mass diffusion coefficient determined was qualified as free of respiration effects, since an uncoupled process was employed. In the present work the respiration parameters were evaluated with intact fruits; this implies that no method for uncoupling the diffusion and respiration effects was used and, because of this, the K p obtained is considered an apparent Michaelis–Menten constant K app p : Lammertyn et al. (2003) showed, through experiments with pear fruits, that the Michaelis–Menten constant corresponding to intact fruits is higher than the one matching at tissue level and this is higher than that relative to cell suspensions. These differences have suggested that, at the tissue level, even when working with thin elements, if there are no gas convective movements, the availability of oxygen to internal zones is restricted by a diffusional effect still present. In an intact fruit, the bigger tissue thickness and the presence of both the skin and the cuticle formed on this induce a higher reduction in O2 availability, mainly to deep zones and, therefore, the Kp (or KM), it means the O2 concentration at which the O2 consumption rate is reduced to half the maximum, occurs at higher levels. These aspects have been widely studied in the field of designing catalytic chemical reactors with heterogeneous solid–fluid reactions (Masel, 2001); it has been observed that the rate at which a reaction occurs depends, among

ARTICLE IN PRESS OXYGEN DIFFUSIVITY IN AVOCADO FRUIT TISSUE

other factors, on the resistance that diffusion establishes to the transport of reactants until the active site. If a reaction occurs slowly, then the diffusion does not affect the rate, but if it occurs rapidly, then the diffusion becomes the controlling stage. In order to quantify this phenomenon, an effectiveness factor e (dimensionless) is used; this is defined as the ratio of the reaction rate affected by diffusion to the reaction rate not affected by diffusion, and it is evaluated as a function of the mass diffusivity coefficient and a characteristic length. The results obtained by Lammertyn et al. (2003) with pear fruits suggest that, when the respiration rate is being modelled through the enzymatic kinetic theory of Michaelis–Menten, the process as a whole has to be considered as a diffusion-dependant phenomenon. Since the process occurring inside a fruit is similar to that in a catalytic chemical reactor with a heterogeneous solid–fluid reaction, it will be convenient to include, in Eqns (3), (4) and (7), an effectiveness factor e in order to be able to use an apparent Michaelis–Menten constant. The Michaelis–Menten theory establishes that the enzymatic kinetic consists of two reactions that occur in sequence. In the first one, a substrate joins to the active site forming a complex enzyme–substrate. In the second, the complex is separated to form a free enzyme and a product (Lehninger, 1998). As the diffusion restricts the availability of the substrate to the active sites, its effect acts on the first stage of the kinetic; therefore, the adjustment of the reaction rate for this stage, due to the diffusion effect, can be done if the effectiveness factor e multiplies its reaction rate constant. Under this consideration, it can be shown that the Kp (or KM) intrinsic to the tissue results as a product of the effectiveness factor e and the apparent app Michaelis-Menten constant K app p (or K M ). The fact that in app the present work a K p was used implies that the effectiveness factor e adopted a value equal to 1
0; nevertheless, further experiments will be necessary to verify this assumption or to determine the adequate value for e.

3.3. Additional considerations to the oxygen consumption rate The relationship between the O2 consumption rate and the O2 availability evaluated in the present work showed the typical behaviour described by the theory of Michaelis–Menten (Fig. 1). However, the fact that a model without competitive effects was used means it was assumed that the internal CO2 accumulation did not affect the O2 internal consumption rate. The evaluation of the O2 exchange rate in ‘Hass’ avocado fruits based on the Michaelis–Menten theory has also been assayed by Hertog et al. (2003) with experiments conducted at 7 1C by using intact fruits. They found that the rate of

203

O2 consumption increases with increasing levels of O2, which is in agreement with the behaviour observed in this work. Also, they pointed out that the rate of consumption decreases with increasing levels of CO2 and determined that the relationship can be described by an uncompetitive model (Peppelenbos & van’t Leven, 1996; Hertog et al., 1998), where the value for Kp was 10
5 kPa. This result is higher than the value used in the present work, which confirms that the Kp parameter increases when temperature decreases (Lakakul et al., 1999). Also, the parameters used in this work corresponded to the pre-climacteric phase of the fruits, where CO2 accumulation is small and, because of this, a small inhibitory effect of this gas is foreseen. To this respect, Hertog et al. (2003) found a value for the Michaelis–Menten constant in an uncompetitive inhibition of respiration by CO2 of 1
96 kPa, which supports the small effect of this gas; nevertheless, if the model application is extended to more advanced maturity stages, it will be necessary to consider these aspects.

3.4. Geometrical considerations It is still necessary to take into account the geometrical shape of the fruit. The spherical condition assumed implies that gas gradients occur only in the radial direction of the fruit, and those in the latitude and azimuth directions [see Eqn (1)] are negligible or zero; thus, points located inside the pulp at similar distances to the skin have similar O2 partial pressure values. The sphericity is defined as the ratio of the external surface of a sphere to the external surface of the system evaluated, both referring to the same volume (Foust et al., 1980). The fruits used in the present work had an average weight of 217 g, an approximate volume of 2
05  104 m3 and a surface to weight ratio of 0
088 m2 kg1; hence, the external surface was 0
0191 m2. On the other hand, a sphere of the same volume has an external surface of 0
0168 m2; therefore, a sphericity of 0
88 was obtained for ‘Hass’ avocado fruits of the dimensions mentioned. In order to compensate the lack of sphericity, the simulation routines were conducted beginning from the external layers of the fruit to the internals. In spite of this, it must be considered that, as sphericity is faraway from a value of 1
0, the concentration gradients become different as a function of the direction (Wu et al., 2004). Likewise, if an item having an ellipsoidal shape is modelled as a sphere, then the mass diffusivity coefficient can be misestimated, as shown by Gasto´n et al. (2004), who found in grains of wheat that the ratio of water diffusivities for ellipsoids to those for spheres was approximately equal to the sphericity square.

ARTICLE IN PRESS 3.5. Simulation of a modified atmosphere system If the fruit skin plus a polymeric film or coating placed on it is defined as a ‘shell’, the characteristics of the plant material, in a modified atmosphere system, can be analysed. Figure 3 shows the O2 concentration profiles throughout the radius, with a value for DO2 of 2
2  109 m2 s1 (which was assumed constant), and distinct permeance values for the shell. From Fig. 3, we can note that the bigger the shell permeance, the bigger the O2 internal partial pressure and, for a specific shell permeance value, the minor the radius, the minor the O2 concentration in a logarithmic tendency. Valle-Guadarrama et al. (2004) determined that the anaerobic compensation point (ACP), that is the O2 concentration at which the transition between aerobic and anaerobic metabolisms occurs, has a value of 1
81% (O2 partial pressure of 1
41 kPa) for pre-climacteric ‘Hass’ avocado fruits stored at 20 1C. If it is assumed that this parameter remains constant even at tissue level, then an adequate shell permeance must produce an O2 internal concentration profile with values greater than ACP for the fermentative metabolism to be avoided. Lines (f), (g) and (h) in Fig. 3 show behaviours where a fruit would have, in the external layers, an equivalent to an environmentally controlled atmosphere with O2 between 5
0 and 3
2 kPa (between 6
4 and 4
1%), but in the internal zone of the equatorial region, partial pressures between 1
2 and 0
7 kPa (between 1
5 and 0
9%) would be developed and, in points of greater depth, as those found in the seed proximity, on the side of the peduncle scar, the concentrations would be even lower. Thus, internal values would be lower than ACP and the fruit indeed would have sections under both aerobic and anaerobic conditions. In contrast, line (c) shows a behaviour where the fruit does not contain zones with O2 below ACP, but the external layers conditions would be equivalent to have O2 in concentration of 14
4% approximately, which is faraway from commercial recommendations (Kader, 1997). These facts suggest that the determination of an adequate shell permeance implies a compromise between the search for longer fruit shelf-life and storage conditions with fewer injuries caused by an excessive reduction in O2 concentration. From Fig. 3, when the shell permeance acquires a value of 6
0  1012 kg s1 m2 Pa1 [line (e)], the region of greatest tissue depth in the equatorial zone of the fruit registers an O2 partial pressure of 1
561 kPa (approximately 2%), and a partial pressure of 5
8 kPa (7
4%) in the most external regions, which would result in a value slightly greater than that recommended for a commercial application (5%; Kader, 1997), hence, the referred gas exchange capacity value could be defined as a minimal permissible shell permeance P0shmin in kg s1 m2 Pa1 (MPSP).

16 14

a b

DO2 = constant = 2.2 nm2 s−1 k

12

c

10 8 ACP = 1. 41 kpa (1.81%) 6

d e f g h i j

4 2 0

5

10

15

20 25 30 Fruit radius, mm

35

20 18 16 14 12 10 8 6 4 2 0 40

Oxygen internal concentration, %

S. VALLE-GUADARRAMA ET AL.

Oxygen internal partial pressure, Kpa

204

Fig. 3. Oxygen concentration profiles along fruit radius considering an oxygen mass diffusivity coefficient DO2 in tissue of 2.2 nm2 s1, for different shell permeance values of: (a) 133
8 ng s1m2 Pa1 (natural condition; ValleGuadarrama et al., 2002); (b) 50
0 ng s1m2 Pa1; (c) 20
0 ng s1m2 Pa1 (d) 8
0 ng s1m2 Pa1; (e) 6
0 ng s1m2 Pa1; (f) 5
0 ng s1m2 Pa1; (g) 4
0 ng s1 m2 Pa1; (h) 3
0 ng s1m2 Pa1; (i) 2
0 ng s1m2 Pa1 and (j) 1
0 ng s1m2 Pa1. In condition (a) the shell is formed only by skin and in conditions (b) to (j), the shell is formed by the skin and a polymeric coating placed over it; line (k) indicates the position of greatest tissue depth in the equatorial region of a fruit of 200 g approximately; the zone found at the right of line (k) only exists physically in the region next to the seed side of the peduncle scar; ACP, anaerobic compensation point of the fruit

Figure 4 shows the development of a modified atmosphere inside the fruit as a function of time. Starting with an O2 concentration profile corresponding to a natural condition (shell permeance equal to 1
338  1010 kg s1 m2 Pa1 ; line a–b), it was seen that, when shell permeance is modified to the value 6
0  1012 kg s1 m2 Pa1 by placing a polymeric material on the skin, the most external zones of the fruit would experience an immediate displacement in their O2 concentration; in contrast, that effect would be observable only after 2 h in the internal zones. However, the new steady-state condition would be reached within a period of 10–15 h in the entire fruit. Berrios (2002) evaluated the behaviour of ‘Hass’ avocado fruits stored in modified atmosphere systems at room temperature using, as wrapping, bags of low-density polyethylene (LDP) 1
15 mm thick. He reported the development of O2 concentrations inside the package between 2 and 4% and the presence of symptoms of a fermentative metabolism. The concentration levels found by Berrios (2002) correspond to behaviour similar to that described by line (h) in Fig. 3, which allows realising that anaerobic conditions would be present in most part of the fruit.

ARTICLE IN PRESS OXYGEN DIFFUSIVITY IN AVOCADO FRUIT TISSUE

If the MPSP is disaggregated into two parts, the skin permeance to O2 P0sk in kg s1 m2 Pa1 under a natural condition (without covering), and the permeance of a polymeric coating placed over it, then a minimal permissible coating permeance P0ctmin in kg s1 m2 Pa1 (MPCP) can be evaluated, as described by Eqn (13) (expressed in analogue form to the proposal by Cameron et al., 1995; Cooksey et al., 1999 and Amarante & Banks, 2001): 1 1 1 ¼ þ P0shmin P0sk P0ctmin

(13)

With values of 6
0  1012 and 1
338  1010 kg s1 m2 Pa1 for P0shmin and P0sk (Valle-Guadarrama et al., 2002), respectively, the MPCP (P0ctmin ) would have a value of 6
28  1012 kg s1 m2 Pa1. Data from Cameron et al. (1995), and the definition given by Banks et al. (1995) (permeance is equal to film permeability divided by film thickness), allow knowing that 10 mm films of LDP, polypropylene (PP), polyvinyl chloride (PVC) and cellulose acetate (CA) have permeance values

a

16 Oxygen internal partial pressure, kpa

14 12 c

10 8 6 b

30 25

of 8
64  1012, 2
88  1012, 0
06  1012 and 3
20  1012 kg s1 m2 Pa1, respectively; hence, only LDP could be used to store ‘Hass’ avocado fruits without significant damage originated by an excessive decrement in O2 internal concentration. Cameron et al. (1995) determined, with the use of a model for a steadystate condition, that lettuce could be stored under a modified atmosphere system by using LDP with 25 mm thickness. Also, Lakakul et al. (1999) determined that an LDP of 14
1 mm could be used to store apple slices under modified atmosphere at 0 1C. The lesser thickness required in the case of avocado fruits responds to its greater respiratory rate at storage conditions.

4. Conclusions A respiration–diffusion model was developed and it was used as a tool to evaluate an O2 mass diffusivity coefficient in ‘Hass’ avocado fruits tissue, which resulted in a value of 2
2  109 m2 s1. The analysis allowed realising that further studies are necessary to clarify the effect of the O2 diffusion transport within the tissue on the respiration parameters. The model permitted simulation analyses about the behaviour of the fruit under a modified atmosphere storage and it was estimated that the minimal permissible permeance a coating must have is 6
28  1012 kg s1 m2 Pa1, and that the use of a film of low density polyethylene with 10 mm of thickness could comply with this requirement. Since this study covered only the pre-climacteric phase of the fruit and, because physical changes occur during ripening, a set of subsequent studies must be done in order to gather further information with a modified atmosphere system.

,m

m

2 ACP = 1. 41 kpa (1.81%)

,h

15

20

15 10

Acknowledgements

ad

d

10 Time

it r

5

20 Fru

0

ius

4

205

Fig. 4. Development of a modified atmosphere system based on the placement of a covering on an intact fruit, causing a fruit wrapping permeance (skin and covering) of 6
0  1012 kg s1 m2 Pa1. The mesh is formed with time intervals of 0
5 h and radius intervals of 0
0005 m. Lines (a-b) represent the concentration profile under natural conditions (wrapping permeance of 1
338  1010 kg s1 m2 Pa1), and lines (c-d), the new concentration profile developed with the placement of the covering. The diffusivity coefficient considered was of 2
2  109 m2 s1. The pulp region with radii from 0 to 20 mm only physically exists in the region near the seed, on the side of the peduncle scar region. All mesh regions located above the reference plane correspond to an aerobic condition and those located below it, are in anaerobic conditions; ACP, anaerobic compensation point of the fruit

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