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Simultaneous determination of oxygen diffusivity and respiration in pear skin and tissue
Postharvest Biology and Technology 23 (2001) 93 – 104
www.elsevier.com/locate/postharvbio
Simultaneous determination of oxygen diffusivity and respiration in pear skin and tissue J. Lammertyn *, N. Scheerlinck, B.E. Verlinden, W. Schotsmans, B. M. Nicolaı¨ Flanders Centre/Laboratory of Posthar6est Technology, Katholieke Uni6ersiteit Leu6en, Willem de Croylaan 42, 3001 Leu6en, Belgium Received 1 December 2000; accepted 2 April 2001
Abstract A novel method to measure gas diffusion properties of pear tissue and skin was developed. In a temperature and pressure controlled system, a diffusion cell was attached to a polarographic oxygen electrode to measure gas diffusivity. A dynamic finite element model based on simultaneous gas diffusion and respiration for pear slices and pear skin was developed. The average experimentally determined values for oxygen diffusivity in tissue and skin were, respectively, 1.71 ×10 − 9 m2 s − 1 and 2.84 ×10 − 10 m2 s − 1. The mean estimated tissue and skin respiration values did not differ statistically, at a 5% level of significance, from the mean respiration values obtained in an independent traditional respiration experiment. If the oxygen consumption and the oxygen diffusivity needed to be estimated accurately and simultaneously from one experiment, the experiment typically lasted 15 h. The duration of the diffusion measurement could be reduced to 3 h, when the respiration characteristics were measured in a separate experiment. Monte-Carlo simulations were performed to calculate the variability due to biological variation of oxygen transport and consumption in slices of pear tissue covered with skin. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Modelling; Pyrus communis; Finite elements; Diffusion; Gas transport
1. Introduction The knowledge of the internal atmosphere of the fruit can be of great importance for the development of controlled atmosphere treatments and of controlled atmosphere (CA) packaging for extended product life (Dadzie et al., 1996; Yearsley * Corresponding author. Tel.: +32-16-322376; fax: + 32-16322955. E-mail address: [email protected] (J. Lammertyn).
et al., 1996). It can also explain the variability in responses of fruits to their storage atmospheres, as well as improve the understanding of the development of storage related disorders, such as core breakdown in ‘Conference’ pears (Pyrus communis L.; Lammertyn et al., 2000). Several methods have been used in the past to measure or calculate the internal concentration of gases in various horticultural commodities (Burg and Burg, 1965; Solomos, 1987; Banks and Kays, 1988; Zhang and Bunn, 2000). Diffusion of gases in fruits and
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vegetables occurs through the gaseous (e.g. the lenticels and the stomata) and/or aqueous pores and the wax layer of the epidermis. The gas exchange in fruits can be approximated by Fick’s First Law of Diffusion, which states that the flux of a gas diffusing through a barrier, is proportional to the diffusion coefficient and the concentration gradient (Incropera and Dewitt, 1990): J= D
(C (x
(1)
where J is the molar flux (mol m − 2 s), D the diffusion coefficient (m2 s − 1), and C the gas concentration (mol m − 3). The application of this steady-state equation to estimate and describe the diffusion properties of fruit tissue and skin has some limitations. First, the concentration gradient #C/#x is often replaced by DC/Dx, which is only permitted if the concentration gradient in the skin or tissue is linear. Secondly, the fruit internal gas concentration is assumed constant and the skin is taken as the only barrier to gas diffusion. This assumption has been made for all kinds of horticultural products (Cameron and Yang, 1982; Knee, 1991; Banks et al., 1993). However, this does not hold for all types of commodities, and especially not for fruits with a high tissue density (Solomos, 1987; Banks and Nicholson, 2000). Banks and Kays (1988) demonstrated that potato tuber flesh has a significant resistance to gas diffusion, which can give rise to measurable oxygen gradients within the tissue. Similarly, Solomos (1987) has shown that the diffusivity of apple flesh is only ten- to twenty-fold higher than that of the skin, which, under non-steady-state conditions, may create appreciable concentration gradients within the flesh. Finally, Fick’s First Law of Diffusion does not take into account the respiratory activity of the fruit. When diffusivities are determined on relatively thick slices of tissue, respiration cannot be neglected. Most of these limitations can be dealt with by solving the onedimensional version of Fick’s Second Law of Diffusion: (C =D92C+q (t
(2)
were q is the gas production or consumption term (mol m − 3 s − 1), D the diffusion coefficient (m2 s − 1), C the gas concentration (mol m − 3), 92 is the Laplace operator (1 m − 2) and t is the time (s). This law describes the non-steady-state gas transport through the fruit skin or tissue, including the consumption and production of metabolically active gases. In the literature, often the shape of the product is approximated by a sphere or a cylinder, for which a steady-state solution can easily be calculated. Mannapperuma et al. (1991) proposed a steady-state spherical mathematical model based on simultaneous gas diffusion and chemical reaction to represent the gas exchange of ‘Golden Delicious’ apples stored in controlled atmosphere. Abdul-Baki and Solomos (1994) solved the second law of diffusion under steady-state conditions, approximating the skin and the tissue of a potato tuber, respectively by a hollow metabolically inert cylindrical shell and a metabolically active solid cylinder. Banks et al. (1993) constructed a steady-state model to describe the effect of surface coatings on the gas exchange of fruits. This model combined gas diffusion and respiration, but did not take the heterogeneity in internal atmosphere composition into account. Banks and Nicholson (2000) offered an alternative theoretical basis by which fruit skin permeance could be calculated from other gas exchange variables. They found that removal of the cuticle dramatically increased the permeance of the sweet pepper (Capsicum annum cv. Reflex) surface. All models above were based on Fick’s First and Second laws of diffusion. In most cases, reasonable assumptions concerning shape, and presence or absence of internal gradients, were made, and reliable skin diffusivities and permeances were obtained. However, when the shape of the product is irregular (e.g. pears), and both the skin and the tissue cause considerable gas gradients, the solution of the reaction-diffusion equation becomes more complex and numerical techniques are needed to solve the problem. To simplify this problem, we chose to uncouple the diffusion measurements for skin and tissue, and then combine them again accounting for the biological variability among pears.
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The specific objectives of this study were (1) to construct a system to measure the oxygen diffusivity and consumption of skin and tissue simultaneously and accurately, (2) to design the diffusion experiment in such a way that the role of the initial internal gas concentration is of minor importance, (3) to obtain a non-steady-state numerical solution of Fick’s Second Law of Diffusion, including respiration, by means of a finite element technique, and (4) to quantify the biological variability in the measured diffusivities and respiration values for simulation and validation purposes. The reliable parameter estimates, obtained in this way, will be used in further research to construct a three dimensional finite element reaction-diffusion model of intact fruit with an irregular shape, such as, e.g. pears.
2. Materials and methods
2.1. Fruit material Pears (Pyrus communis cv. Conference) were harvested in September 1999 at a pre-climacteric stage at the Nationale Proeftuin voor Grootfruit in Velm (Belgium), and cooled and stored according to commercial protocols for a period of 21 days at − 0.5°C preceeding CA storage (2 kPa O2, 0.7 kPa CO2 at − 0.5°C) until they were used for the experiments.
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diffusion cell, to monitor the pressure changes in the cell during the measurements. A second pressure sensor was inserted in the outer chamber, to estimate the pressure difference between the two chambers. Too large a difference (\ 0.5 kPa) would cause forced diffusion (permeation) and would falsify the results. For all experiments the pressure drop over the sample never exceeded 0.5 kPa and the temperature of the diffusion cell was kept constant at 209 0.3°C during the measurements using a water jacket, surrounding the outer chamber, and a temperature controlled waterbath (F-26, Haake, Germany). Longitudinal pear sections of varying thickness (1.5–3 mm) were cut with a professional slice cutter (EH 158-L, Graef, Germany), from which cylinders with a diameter of 2.5 cm were cut with a cork borer. The exact thickness of the sample was measured with a digital caliper (Mitutoyo Ltd, UK). The sample was rinsed with deionised water to remove free starch and cell wall debris and immediately dried with tissue paper to prevent the water from penetrating the intracellular airspace, where it could disturb the diffusion properties of the sample. The sample was glued to the bottom of the diffusion cell with cyanoacrylate glue. The glue was only applied to the very edge of the sample to avoid a toxic effect on the cells through which the diffusion was measured.
2.2. Experimental system and procedure The system used to measure oxygen diffusivity of fruit skin and tissue is shown in Fig. 1. It consisted of a large glass outer chamber (1.2 litres) and a small inner polyvinylchloride diffusion cell (4.5 ml), attached to the end of a polarographic oxygen electrode (T-243, Consort, Belgium) which was mounted on the lid of the outer chamber. The connections between the different materials were made airtight with rubber o-rings. The inner diameter of the diffusion cell was 2 cm. An inlet and outlet gas channel, including needle valves, was used to flush the gas atmosphere in the diffusion cell. A pressure sensor (PTX 1400, Druck, Germany) was attached to the
Fig. 1. System for measuring oxygen diffusivity of fruit skin and tissue.
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The edge of the cylinder was covered with silicon grease to prevent diffusion from the edge of the tissue. The skin samples were prepared from the top of a cylindrical plug (2.5 cm diameter) of pear tissue taken in the radial direction (perpendicular to the long axis of the pear). A razor blade was used to remove the flesh from the skin slice. The thickness of each slice was measured with a caliper. The samples were attached in a similar way to the diffusion cell as for the tissue samples. Once the sample had been attached to the diffusion cell, the outer and the inner chambers were flushed with, respectively, a gas mixture of 10 kPa O2, 0.03 kPa CO2, balanced with N2, at 10 l/h and 20.8 kPa O2, 0.03 kPa CO2, balanced with N2 at a flow preventing a pressure build up in the diffusion cell. This flow was controlled by a manual needle valve. The gas mixtures were prepared from the gases by means of mass flow controllers (5850 S, Brooks Instruments, The Netherlands) and passed through a heat exchanger and humidifier to prevent the sample from drying and cooling down while flushing the two chambers. After 1 h, the in- and outlet valves of the diffusion cell were closed and the steady decrease in oxygen partial pressure was monitored for at least 13 h until an equilibrium had been reached. The oxygen electrode, the temperature and the two pressure sensor signals were logged at 4 s intervals during measurement using a Fieldpoint I/O module (National Instruments, Zaventem, Belgium) connected to a computer. The graphical user interface was programmed in Labview 5.0 (National Instruments, Zaventem, Belgium). The oxygen electrode was calibrated using a high O2 gas mixture (20.8 kPa O2, and 0.03 kPa CO2, balanced with N2) and an O2-free gas (pure N2). The diffusion experiments were repeated ten and six times, respectively, for tissue and skin.
2.3. Respiration measurements Respiration measurements on pear tissue and skin were carried out at 20°C as described by de Wild et al. (1999). Either ten cylindrical tissue or skin samples (2 mm height and 2.5 cm diameter) were placed in each of six 1.2 litre glass jars. After 1 h, during which time the jar headspace was
flushed at 20 l/h with the applied humidified gas mixture (10 kPa and 20.8 kPa O2 balanced with N2), the jars were closed and the initial headspace gas (O2, N2 and CO2) measured using a gas chromatograph (Chrompack CP 2002, Bergen op Zoom, The Netherlands). After 6, 21 and 27 h a gas sample was taken from the jar headspace and analysed. The total pressure was recorded before and after each measurement (PTX 1400, Druck, Germany), and the pressure drop caused by the sampling technique was corrected for. The difference in gas partial pressure was converted to moles according to the ideal gas law. The density of the tissue was measured based on the water displacement method. The oxygen consumption rate was expressed in mol per unit volume and per unit time (mol/m3 s).
2.4. Model construction and parameter estimation Transient linear mass transfer in solid foods subjected to convection boundary conditions Eq. (4) is governed by Fick’s Law of Diffusion Eq. (3). ([O2] = D92[O2]+ q (t −D
)
([O2] = h([O2]r − [O2] ) (x x = r
(3) (4)
were [O2] is the oxygen concentration (mol m − 3), D is the diffusion coefficient (m2 s − 1), q is the oxygen consumption (mol m − 3 s − 1), 92 is the Laplace operator (1 m − 2), h is the convective mass transfer coefficient (m s − 1) and t is the time (s). Indices and r refer, respectively, to the ambient atmosphere in the outer chamber and the length of the modelled system (Fig. 2). For many realistic mass transfer problems no analytical solutions of Eq. (3) subjected to Eq. (4) are known. In this case numerical discretization techniques such as the finite difference or finite element method can be used to obtain an approximate solution. The finite element method in particular is a very flexible and accurate method for solving partial differential equations such as Fick’s Law of Diffusion. In the framework of the finite element method, the continuum is subdivided in
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Fig. 2. Schematic overview of the two-material finite element model. Both materials consist of 20 elements, each element has eight nodes. Dair, Dsample, Rsample, h, [O2]r, [O2] , and r are, respectively, the diffusion coefficient of oxygen in air, the diffusion coefficient of oxygen in the sample, the respiration rate of the sample, the convective oxygen transfer coefficient, the oxygen partial pressure at position r, the ambient oxygen partial pressure in the outer chamber.
elements of variable size and shape, which are interconnected in a finite number, N, of nodal points. In every element the unknown oxygen concentration is approximated by a low order interpolating polynomial. The application of a suitable spatial discretization technique such as the Galerkin weighted residual method to Eq. (3) subjected to Eq. (4) results in the following differential system (Segerlind, 1984). d C u +Ku =f dt
(5) (6)
u(t =0)= u0 T
with u= [u1 u2 …uN ] the overall nodal oxygen concentration vector, C the capacitance matrix and K the conductance matrix, both N ×N matrices, and f a N ×1 vector. The system Eq. (5) is solved by finite differences in the time domain. In this work the Chamspack code (Scheerlinck et al., 2000) has been used. In the two material approach, the finite element model was composed of two materials, attached to each other (Fig. 2). The internal volume of the diffusion cell, was taken as the first material. The diffusion coefficient of oxygen molecules in air at 20°C was set equal to 6×10 − 5 m2 s − 1, four orders of magnitude higher than the diffusivity in fruit tissue (Lide, 1999). The tissue was modelled as the second material, for which the diffusion coefficient was to be estimated. The gas transfer
from the outer chamber to the tissue or the skin was expressed by Eq. (4) as a convective boundary condition. The convective mass transfer coefficient was taken as very high (1× 106 m s − 1), such that no resistance to gas transport occurred at the interface. Eq. (4) could then be simplified to [O2](r,t) = [O2]
(7)
The finite element mesh consisted of 20 quadratic quadrilateral elements for each material, resulting in a total of 206 nodes (Fig. 2). An iterative least squares estimation procedure was written in MATLAB 5 (The Mathworks, Inc., Natick, USA) to estimate the tissue respiration and the oxygen diffusivity from the measured oxygen partial pressure versus time profiles measured in the diffusion cell with the oxygen electrode. A similar two material model was constructed for the skin. In the three material approach, a finite element model was constructed, consisting of three materials: the internal volume of the diffusion cell, the tissue and the skin. This model was not used for estimation purposes, since this would have required the simultaneous estimation of at least three parameters, which undoubtedly would have resulted in badly identified parameters. However, this model was useful in validation and simulation experiments. In practice, skin and tissue occur as a coupled system and, therefore, a validation ex-
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periment, on a coupled system of skin and tissue, was performed and compared with the simulation result of the model accounting for the biological variability on the estimated parameters. To simplify the complexity of the problem, three assumption were made. First, it was assumed that the oxygen diffusivity for the skin (Ds) and the tissue (Dt) and the respiration of the skin (Rs) and the tissue (Rt) were normally distributed. Second, the covariance between Ds and Dt, and between Ds and Rt, was considered negligible. Third, it was chosen to set the skin respiration equal to that of the tissue, since the former was most probably caused by wound respiration and it would be inappropriate to include this value in a model that describes a coupled system without this wound respiration. Besides, the contribution of skin respiration is only minor compared to the amount of tissue respiration. The variance–covariance structure of the tridimensional space based on Dt, Rt and Ds was used to calculate the trivariate normal distribution. Monte-Carlo simulations (n =1000) (Rubinstein, 1981), based on this distribution, were performed, resulting in 1000 oxygen partial pressure versus time profiles. To validate the model on samples with different geometrical shapes, Monte-Carlo simulations were performed to determine the area of biological variation in which the measured curve should be situated. A hollow cylinder with an inner and outer diameter of, respectively, 28 and 35 mm was cut in the longitudinal direction of the pear by means of two concentric steel knives. It was attached to the diffusion cell in a similar way to the other samples and closed at the bottom with an airtight plastic disk. The measurements were carried out in exactly the same way as described above. Note that the inner volume of the cylinder plays a role as a diffusion cell on itself. When, in the three material approach, the thickness of the skin was considered negligible, its resistance to diffusion could be treated like the resistance of a boundary layer (Mannapperuma et al., 1991). This resulted in a two material model with a non-negligible convective mass transfer coefficient at the boundary between the tissue and the outer chamber. The convective mass transfer coefficient was calculated as the ratio of the skin
diffusion coefficient and the skin thickness. The reciprocal ratio corresponded to skin resistance to oxygen transport.
3. Results and discussion
3.1. Respiration measurements on skin and tissue The oxygen consumption rate of intact pears and pear disks strongly depends on the oxygen partial pressure in the gas atmosphere (de Wild et al., 1999). At low oxygen partial pressures (0–2 kPa), the oxygen consumption rate decreases almost linearly with the partial pressure. For high oxygen partial pressures (\8 kPa), the consumption rate is independent of the partial pressure. To check whether the consumption rate was really constant between the two oxygen partial pressures used for the determination of the diffusion constant, the oxygen consumption rate was measured six times at both 10 kPa and 20.8 kPa O2. A statistical t-test indicated no difference in mean respiration rate between 10 kPa or 20.8 kPa O2, at a 5% level of significance. An average oxygen consumption rate of 2.69× 10 − 4 mol m − 3 s − 1 (standard deviation 0.371× 10 − 4 mol m − 3 s − 1) in tissue was observed. Similarly, no significant difference was found between skin respiration rates at 10 kPa and 20.8 kPa O2. The average oxygen consumption rate of skin equalled 21.14× 10 − 4 mol m − 3 s − 1 (standard deviation 4.46× 10 − 4 mol m − 3 s − 1). Extensive wound respiration, induced by scratching the remaining tissue from the skin, may explain why the respiration rate of skin is eight times higher than that for tissue. However, it may also be that skin respiration is higher than tissue respiration. Most probably a combination of both explanations will be true.
3.2. Oxygen diffusi6ity of skin and tissue To solve Eq. (3) properly, the initial oxygen partial pressure profile must be known both in the diffusion cell and in the tissue or skin. The former can be measured easily since the oxygen molecules are uniformly distributed over the internal volume
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and the initial partial pressure is exactly the composition of the gas mixture used to flush the cell. However, it is not trivial to determine the initial oxygen partial pressure profile in the tissue or the skin. To avoid this problem an experimental design was developed that minimizes the influence of the initial conditions on the estimated parameters. A typical oxygen partial pressure versus time profile for skin and tissue is given in Fig. 3. This profile can be divided into two parts. The first part, recorded in the first 3600 s, corresponded to the situation where both the inner and the outer chamber were flushed with, respectively, 20.8 kPa and 10 kPa O2. At the end of this period, a steady-state gas equilibrium was established in the sample, which was then used as the initial oxygen partial pressure profile to solve Eq. (3) for the second part of the curve. This dynamic part of the profile describes the decrease in oxygen partial pressure in the diffusion cell when the flushing has been stopped and the valves were closed, while the
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outer chamber was continually flushed with 10 kPa O2. Since the oxygen diffusion depends on the oxygen gradient over the sample, a fast decrease was observed in the beginning, corresponding to a high gradient. In time, the gradient decreased and, consequently, the gas exchange between the two chambers slowed down, until an equilibrium was reached. Note that the equilibrium was not reached at 10 kPa O2, but at some lower value, because of the oxygen consumption caused by respiration. During the whole experiment the total pressure drop over the sample was constant and very small to non-existent (B 0.5 kPa). This implies that nitrogen molecules diffuse in the opposite direction to the oxygen molecular diffusion. The measured and the modelled oxygen partial pressure versus time profiles for the experimental cell are shown in Fig. 3, for both a tissue slice of 2 mm and a skin of 0.4 mm thickness. The model described the experimentally determined values very well.
Fig. 3. Oxygen partial pressure in the inner chamber of the diffusion cell as a function of time for diffusion of oxygen through pear tissue and skin. Symbols represent measured values, and lines values predicted by the model.
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Table 1 Estimated oxygen diffusivities for skin (Ds) and tissue (Dt) and measured and estimated respiration values for skin (Rs) and tissue (Rt) Source of data for estimates
Rs (×10−4 mol m−3 s−1)
Ds (×10−10 m2 s−1)
Rt (×10−4 mol m−3 s−1)
Dt (×10−9 m2 s−1)
Measured values Full profile Steady state +5000s, Rt = Rs = 0 Steady state +5000s, Rt or Rs
21.149 4.46 25.239 8.56 0 21.14
– 2.849 0.79 3.68 9 0.89 3.03 9 0.81
2.69 90.37 2.64 90.92 0 2.69
– 1.71 90.51 2.33 91.14 1.80 90.51
Values were estimated from three different data structures, 9the standard deviation. Values without standard deviation were not estimated, but kept constant during the estimation algorithm.
The average experimentally determined values for oxygen diffusivity in tissue and skin were, respectively, 1.71×10 − 9 and 2.84× 10 − 10 m2 s − 1. The mean estimated tissue (2.64×10 − 4 mol m − 3 s − 1) and skin (25.2×10 − 4 mol m − 3 s − 1) respiration values did not differ statistically, at a 5% level of significance, from the mean respiration values obtained in the independent traditional respiration experiment. Within one experiment the two estimated parameters were not correlated which indicated that the measured profile contained sufficient information to estimate the respiration and the diffusion simultaneously. The high parameter accuracy was confirmed by the standard deviations, which typically were two orders of magnitude lower than the parameter estimate, and thus, completely negligible compared to the biological variation. Although no literature was found about the coupled effect of gas diffusivity and respiration in pears, it can be useful to compare the obtained results with those found in the literature for other horticultural commodities. Mannapperuma et al. (1991) experimentally determined diffusivities of the flesh for oxygen of 2.67× 10 − 9 m2 s − 1 and for carbon dioxide of 3.28×10 − 9 m2 s − 1 for ‘Golden Delicious’ apples. The conductance (permeance) of the skin to oxygen was 2.9×10 − 7 m s − 1 and to carbon dioxide 2.2×10 − 7 m s − 1. When the skin diffusivities for oxygen obtained in this study for pears, were recalculated as convective mass transfer coefficients, an average value of 4.4× 10 − 7 m s − 1 was found. Abdul-Baki and Solomos, (1994) measured for potato tubers a
carbon dioxide diffusivity for the skin and tissue of, respectively, 7.26× 10 − 11 and 2.65× 10 − 8 m2 s − 1 at 27°C. Zhang and Bunn (2000) measured an oxygen diffusivity for apple flesh of 1.81× 10 − 8 m2 s − 1. Similar to the results in this paper, they found a skin diffusion coefficient which was six times lower (3.1× 10 − 9 m2 s − 1) than that for apple tissue. The resistance to skin gas transport of horticultural products is often expressed as a resistance or permeance. The conversion factors calculated by Banks et al. (1995) were used to convert and compare the literature results with those presented in this study. The resistance of pear skin (2.3× 104 s m − 1) obtained in this study had the same order of magnitude as that of potato tubers (Banks, 1985), but was much larger than that for citrus fruits and apples (Cameron and Yang, 1982; Ben-Yehoshua et al., 1985; Knee, 1991).
3.3. Relationship between full and partial use of the oxygen-time profile Until now the full oxygen partial pressure versus time profile was used to identify the oxygen diffusivity for the skin (Ds) and the tissue (Dt) and the respiration of the skin (Rs) and the tissue (Rt). However, such experiments typically lasted 15 h or even longer before an equilibrium was established. Parameter estimation based on a partial profile would reduce the measurement time considerably. In Table 1 an overview is given of the diffusivity and respiration parameters, estimated from three different sources of information. In the
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first case the full profile was used to estimate the diffusivity and the respiration simultaneously. The correlation between both estimated parameters was relatively low (typically ranging between 0.5 and 0.8), indicating that the profile contains sufficient information about both respiration and diffusion. In the second case, only the diffusivity was estimated based on the steady state part and the first 5000 s of the dynamic part of the profile. The respiration of the skin or the tissue was set to zero, assuming that the decrease in oxygen partial pressure caused by respiration during this 5000 s was negligible compared to the decrease due to diffusion. In the third case, only the diffusivity was estimated based on the steady state part and the first 5000 s of the dynamic part, but the average measured respiration rate for the skin or the tissue was kept constant during the estimation algorithm. No statistical difference was observed at the 5% significance level between the mean estimated diffusivities. However, when the correlation coefficients were calculated between oxygen diffusivities, both for the skin and the tissue, high correlations existed between the diffusion coefficients estimated based on the full profile and those estimated based on the partial profile with the average respiration rate. For skin and tissue, the coefficients of determination (R 2) were equal to 0.90 and 0.94, respectively. The other coefficients of determination were lower than 0.5.
3.4. Biological 6ariation in respiration and diffusi6ity In biological experimentation, the ideal of creating multiple sets of circumstances in which only one relevant factor varies is unrealistic. Inevitably, there will be what is called ‘biological variation’, which refers to variation in the set of conditions that produces the effect (Greenland and Rothman, 1998). Although the pears were picked on the same picking date and stored under the same conditions, they were of different maturities, resulting in slightly different diffusivities to oxygen and respiration characteristics. In-
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stead of trying to control all those factors which cause this biological variation, it is better to take them into account in the calculation, simulation and validation experiments. Until now, both experiments on skin and tissue were carried out separately, to obtain accurate diffusivity and respiratory parameter estimates. In practice, skin and tissue occur as a coupled system and, therefore, a coupled validation experiment was performed and compared with the Monte-Carlo simulation result of the model accounting for the biological variability on the estimated parameters. The upper and lower line in Fig. 4A represent, respectively, the 97.5th upper and 2.5th lower percentiles of the 1000 simulated curves, illustrating the inherent large biological variability. The full line represents the measured oxygen partial pressure versus time profile of the coupled system composed of 2 mm tissue covered by 0.4 mm skin. The measured curve perfectly lies between the two percentile curves indicating that the model, accounting for biological variation, describes reality well. The present model allows simulations of oxygen diffusion and respiration in tissue and skin slices of different thicknesses. With some small geometrical adaptations in the finite element model, it can even be used to simulate what happens in samples with different geometrical shapes. Monte-Carlo simulations (1000) were performed to determine the area of biological variation in which the measured curve for a hollow tissue cylinder should be situated. Fig. 4B indicates that the measured curve fits perfectly within the simulated range of biological variation. This indicates strongly that the estimated diffusivities and respiration parameters are shape independent and transferable to other geometries. The latter is of major importance in the construction of a three dimensional reaction-diffusion finite element model for intact pears.
3.5. Time dependent oxygen partial pressure profiles The developed finite element model allows simulation of the oxygen partial pressures at ev-
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ery spot in the tissue and skin at any time. In Fig. 5 the spatial oxygen partial pressure profiles of the coupled skin (0.4 mm)-tissue (2 mm) system are plotted at different time intervals. For reasons of clarity, only a part of the diffusion cell is given. The dashed profiles describe the changes in oxygen partial pressure profiles from the start of the experiment until a steady-state was reached around 3600 s. The diffusion cell oxygen partial pressure was kept constant at 20.8 kPa, by flushing. After 3600 s the flushing was stopped and the oxygen partial pressure started to decrease in the cell. The full lines display the changes in oxygen partial pressure in the cell, skin and tissue due to diffusion and respiration. Around 70 000 s, a steady-state was obtained and the oxygen partial pressure reached a value in the tissue lower than that in the diffusion cell and outer chamber, indicating that respiration plays an important role, and that it can not be neglected.
4. Conclusion A new system for the simultaneous measurement of the respiration and the oxygen diffusivity for pear skin and tissue was proposed. The use of a polarographic oxygen electrode resulted in informative oxygen partial pressure versus times profiles, which in turn resulted in good model parameter estimates. A dynamic finite element solution for the model, based on simultaneous gas diffusion and respiration for pear slices and pear skin, was developed. An experimental design was applied to minimize the effect of the initial oxygen conditions on the parameter identification. The average experimentally determined oxygen diffusivities for tissue and skin correspond to those found in literature for similar horticultural products and the oxygen diffusivity for the skin is an order of magnitude lower than that for tissue. The respiration values, measured in an independent respiration experiment, correspond very well to
Fig. 4. Simulated oxygen partial pressures as a function of time for (A) a skin (0.4 mm)-tissue (2 mm) coupled system and (B) a validation experiment on a hollow tissue cylinder. The upper (dotted line) and lower (dashed line) line indicate respectively the 97.5th and the 2.5th percentile of the Monte-Carlo simulations which account for the biological variation on the diffusivities for skin and tissue.
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Fig. 5. Predicted spatial oxygen partial pressure profiles of the coupled skin (0.4 mm)-tissue (2 mm) system as a function of different time intervals. The dashed profiles describe the changes in oxygen partial pressures from the start until a steady-state was reached (3600 s). The full lines display the changes in oxygen partial pressure in the cell, skin. and tissue due to diffusion and respiration.
those obtained with the diffusion model. It was shown that with this technique, the duration of the diffusion measurement could be reduced from fifteen to three hours, if the respiration characteristics of tissue were measured in a separate experiment. The diffusivity values estimated from the full oxygen partial pressure versus time profile were highly correlated to those obtained based on the first section of the profile. A technique, based on Monte-Carlo simulations, was used to quantify the effect of biological variation on the measured diffusivities and respiration values. Finally, simulations and validation measurements on a hollow cylindrical sample indicated that the estimated diffusivities and respiration parameters were shape independent and transferable to other geometries. The latter is of major importance in the construction of a three dimensional reaction-diffusion model for intact pears.
Acknowledgements The Belgian Ministry of Small Enterprises, Traders and Agriculture, the Flemish Government (project S-5901) and the European Communities (FAIR-project CT 96-1803) are gratefully acknowledged for financial support. Jeroen Lammertyn is Research Assistant of the Fund for Scientific Research-Flanders (Belgium) (F.W.OVlaanderen).
References Abdul-Baki, A.A., Solomos, T., 1994. Diffusivity of carbon dioxide through the skin and flesh of ‘Russet Burbank’ potato tubers. J. Am. Soc. Hortic. Sci. 119, 742 – 746. Banks, N.H., 1985. Estimating skin resistance to gas diffusion in apples and potatoes. J. Exp. Bot. 36, 1842 – 1850. Banks, N.H., Kays, S.J., 1988. Measuring internal gases and
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lenticel resistance to gas diffusion in potato tubers. J. Am. Soc. Hortic. Sci. 113, 577 –580. Banks, N.H., Nicholson, S.E., 2000. Internal atmosphere composition and skin permeance of pepper fruit. Postharvest Biol. Technol. 18, 33 –41. Banks, N.H., Dadzie, B.K., Cleland, D.J., 1993. Reducing gas exchange of fruits with surface coatings. Postharvest Biol. Technol. 3, 269 – 284. Banks, N.H., Cleland, D.J., Cameron, A.C., Beaudry, R.M., Kader, A.A., 1995. Proposal for a rationalized system of units for postharvest research in gas exchange. HortScience 30, 1129 – 1131. Ben-Yehoshua, S., Burg, S.P., Young, R., 1985. Resistance of citrus fruit to mass transport of water vapor and other gases. Plant Physiol. 79, 1048 –1053. Burg, S.P., Burg, E.A., 1965. Molecular requirements for the biological activity of ethylene. Plant Physiol. 42, 144 –152. Cameron, A.C., Yang, S., 1982. A simple method for the determination of resistance to gas diffusion in plant organs. Plant Physiol. 70, 21 –23. de Wild, H.P., Woltering, E.J., Peppelenbos, H.W., 1999. Carbon dioxide and 1-MCP inhibit ethylene production and respiration of pear fruit by different mechanisms. J. Exp. Bot. 50, 837 – 844. Dadzie, B.K., Banks, N.H., Cleland Donald, J., Hewett, E.W., 1996. Changes in respiration and ethylene production of apples in response to internal and external oxygen partial pressures. Postharvest Biol. Technol. 9, 297 –309. Greenland, S., Rothman, K.J., 1998. Modern Epidemiology. Lippincott Williams & Wilkins, Philadelphia.
Incropera, F.P., Dewitt, D.P., 1990. Fundamentals of Heat and Mass Transfer. Wiley, New York. Knee, M., 1991. Rapid measurement of diffusion of gas through the skin of apple fruits. HortScience 26, 885 – 887. Lammertyn, J., Aerts, M., Verlinden, B.E., Schotmans, W., Nicolaı¨, B.M., 2000. Logistic regression analysis of factors influencing core breakdown in ‘Conference’ pears. Postharvest Biol. Technol. 20, 25 – 37. Lide, D.R., 1999. Handbook of Chemistry and Physics. CRC Press, New York. Mannapperuma, J.D., Singh, R.P., Montero, M.E., 1991. Simultaneous gas diffusion and chemical reaction in foods stored in modified atmospheres. J. Food Eng. 14, 167 – 183. Rubinstein, R.Y., 1981. Simulation and the Monte Carlo Method. Wiley, New York. Scheerlinck, N., Verboven, P., Stigter, J.D., De Baerdemaeker, J., Van Impe, J.F., Nicolaı¨, B.M., 2000. Stochastic finite element analysis of coupled heat and mass transfer problems with random field parameters. Numer. Heat Transfer, Part B Fundament. 37, 309 – 330. Segerlind, L., 1984. Applied Finite Element Analysis. Wiley, New York. Solomos, T., 1987. Principles of gas exchange in bulky plant tissues. HortScience 22, 766 – 771. Yearsley, C.W., Banks, N.H., Ganesh, S., Cleland, D.J., 1996. Determination of lower oxygen limits for apple fruit. Postharvest Biol. Technol. 8, 95 – 109. Zhang, J., Bunn, J.M., 2000. Oxygen diffusivity of apple flesh and skin. Trans. ASAE 43, 359 – 363.
www.elsevier.com/locate/postharvbio
Simultaneous determination of oxygen diffusivity and respiration in pear skin and tissue J. Lammertyn *, N. Scheerlinck, B.E. Verlinden, W. Schotsmans, B. M. Nicolaı¨ Flanders Centre/Laboratory of Posthar6est Technology, Katholieke Uni6ersiteit Leu6en, Willem de Croylaan 42, 3001 Leu6en, Belgium Received 1 December 2000; accepted 2 April 2001
Abstract A novel method to measure gas diffusion properties of pear tissue and skin was developed. In a temperature and pressure controlled system, a diffusion cell was attached to a polarographic oxygen electrode to measure gas diffusivity. A dynamic finite element model based on simultaneous gas diffusion and respiration for pear slices and pear skin was developed. The average experimentally determined values for oxygen diffusivity in tissue and skin were, respectively, 1.71 ×10 − 9 m2 s − 1 and 2.84 ×10 − 10 m2 s − 1. The mean estimated tissue and skin respiration values did not differ statistically, at a 5% level of significance, from the mean respiration values obtained in an independent traditional respiration experiment. If the oxygen consumption and the oxygen diffusivity needed to be estimated accurately and simultaneously from one experiment, the experiment typically lasted 15 h. The duration of the diffusion measurement could be reduced to 3 h, when the respiration characteristics were measured in a separate experiment. Monte-Carlo simulations were performed to calculate the variability due to biological variation of oxygen transport and consumption in slices of pear tissue covered with skin. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Modelling; Pyrus communis; Finite elements; Diffusion; Gas transport
1. Introduction The knowledge of the internal atmosphere of the fruit can be of great importance for the development of controlled atmosphere treatments and of controlled atmosphere (CA) packaging for extended product life (Dadzie et al., 1996; Yearsley * Corresponding author. Tel.: +32-16-322376; fax: + 32-16322955. E-mail address: [email protected] (J. Lammertyn).
et al., 1996). It can also explain the variability in responses of fruits to their storage atmospheres, as well as improve the understanding of the development of storage related disorders, such as core breakdown in ‘Conference’ pears (Pyrus communis L.; Lammertyn et al., 2000). Several methods have been used in the past to measure or calculate the internal concentration of gases in various horticultural commodities (Burg and Burg, 1965; Solomos, 1987; Banks and Kays, 1988; Zhang and Bunn, 2000). Diffusion of gases in fruits and
0925-5214/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 5 - 5 2 1 4 ( 0 1 ) 0 0 1 1 3 - 2
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vegetables occurs through the gaseous (e.g. the lenticels and the stomata) and/or aqueous pores and the wax layer of the epidermis. The gas exchange in fruits can be approximated by Fick’s First Law of Diffusion, which states that the flux of a gas diffusing through a barrier, is proportional to the diffusion coefficient and the concentration gradient (Incropera and Dewitt, 1990): J= D
(C (x
(1)
where J is the molar flux (mol m − 2 s), D the diffusion coefficient (m2 s − 1), and C the gas concentration (mol m − 3). The application of this steady-state equation to estimate and describe the diffusion properties of fruit tissue and skin has some limitations. First, the concentration gradient #C/#x is often replaced by DC/Dx, which is only permitted if the concentration gradient in the skin or tissue is linear. Secondly, the fruit internal gas concentration is assumed constant and the skin is taken as the only barrier to gas diffusion. This assumption has been made for all kinds of horticultural products (Cameron and Yang, 1982; Knee, 1991; Banks et al., 1993). However, this does not hold for all types of commodities, and especially not for fruits with a high tissue density (Solomos, 1987; Banks and Nicholson, 2000). Banks and Kays (1988) demonstrated that potato tuber flesh has a significant resistance to gas diffusion, which can give rise to measurable oxygen gradients within the tissue. Similarly, Solomos (1987) has shown that the diffusivity of apple flesh is only ten- to twenty-fold higher than that of the skin, which, under non-steady-state conditions, may create appreciable concentration gradients within the flesh. Finally, Fick’s First Law of Diffusion does not take into account the respiratory activity of the fruit. When diffusivities are determined on relatively thick slices of tissue, respiration cannot be neglected. Most of these limitations can be dealt with by solving the onedimensional version of Fick’s Second Law of Diffusion: (C =D92C+q (t
(2)
were q is the gas production or consumption term (mol m − 3 s − 1), D the diffusion coefficient (m2 s − 1), C the gas concentration (mol m − 3), 92 is the Laplace operator (1 m − 2) and t is the time (s). This law describes the non-steady-state gas transport through the fruit skin or tissue, including the consumption and production of metabolically active gases. In the literature, often the shape of the product is approximated by a sphere or a cylinder, for which a steady-state solution can easily be calculated. Mannapperuma et al. (1991) proposed a steady-state spherical mathematical model based on simultaneous gas diffusion and chemical reaction to represent the gas exchange of ‘Golden Delicious’ apples stored in controlled atmosphere. Abdul-Baki and Solomos (1994) solved the second law of diffusion under steady-state conditions, approximating the skin and the tissue of a potato tuber, respectively by a hollow metabolically inert cylindrical shell and a metabolically active solid cylinder. Banks et al. (1993) constructed a steady-state model to describe the effect of surface coatings on the gas exchange of fruits. This model combined gas diffusion and respiration, but did not take the heterogeneity in internal atmosphere composition into account. Banks and Nicholson (2000) offered an alternative theoretical basis by which fruit skin permeance could be calculated from other gas exchange variables. They found that removal of the cuticle dramatically increased the permeance of the sweet pepper (Capsicum annum cv. Reflex) surface. All models above were based on Fick’s First and Second laws of diffusion. In most cases, reasonable assumptions concerning shape, and presence or absence of internal gradients, were made, and reliable skin diffusivities and permeances were obtained. However, when the shape of the product is irregular (e.g. pears), and both the skin and the tissue cause considerable gas gradients, the solution of the reaction-diffusion equation becomes more complex and numerical techniques are needed to solve the problem. To simplify this problem, we chose to uncouple the diffusion measurements for skin and tissue, and then combine them again accounting for the biological variability among pears.
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The specific objectives of this study were (1) to construct a system to measure the oxygen diffusivity and consumption of skin and tissue simultaneously and accurately, (2) to design the diffusion experiment in such a way that the role of the initial internal gas concentration is of minor importance, (3) to obtain a non-steady-state numerical solution of Fick’s Second Law of Diffusion, including respiration, by means of a finite element technique, and (4) to quantify the biological variability in the measured diffusivities and respiration values for simulation and validation purposes. The reliable parameter estimates, obtained in this way, will be used in further research to construct a three dimensional finite element reaction-diffusion model of intact fruit with an irregular shape, such as, e.g. pears.
2. Materials and methods
2.1. Fruit material Pears (Pyrus communis cv. Conference) were harvested in September 1999 at a pre-climacteric stage at the Nationale Proeftuin voor Grootfruit in Velm (Belgium), and cooled and stored according to commercial protocols for a period of 21 days at − 0.5°C preceeding CA storage (2 kPa O2, 0.7 kPa CO2 at − 0.5°C) until they were used for the experiments.
95
diffusion cell, to monitor the pressure changes in the cell during the measurements. A second pressure sensor was inserted in the outer chamber, to estimate the pressure difference between the two chambers. Too large a difference (\ 0.5 kPa) would cause forced diffusion (permeation) and would falsify the results. For all experiments the pressure drop over the sample never exceeded 0.5 kPa and the temperature of the diffusion cell was kept constant at 209 0.3°C during the measurements using a water jacket, surrounding the outer chamber, and a temperature controlled waterbath (F-26, Haake, Germany). Longitudinal pear sections of varying thickness (1.5–3 mm) were cut with a professional slice cutter (EH 158-L, Graef, Germany), from which cylinders with a diameter of 2.5 cm were cut with a cork borer. The exact thickness of the sample was measured with a digital caliper (Mitutoyo Ltd, UK). The sample was rinsed with deionised water to remove free starch and cell wall debris and immediately dried with tissue paper to prevent the water from penetrating the intracellular airspace, where it could disturb the diffusion properties of the sample. The sample was glued to the bottom of the diffusion cell with cyanoacrylate glue. The glue was only applied to the very edge of the sample to avoid a toxic effect on the cells through which the diffusion was measured.
2.2. Experimental system and procedure The system used to measure oxygen diffusivity of fruit skin and tissue is shown in Fig. 1. It consisted of a large glass outer chamber (1.2 litres) and a small inner polyvinylchloride diffusion cell (4.5 ml), attached to the end of a polarographic oxygen electrode (T-243, Consort, Belgium) which was mounted on the lid of the outer chamber. The connections between the different materials were made airtight with rubber o-rings. The inner diameter of the diffusion cell was 2 cm. An inlet and outlet gas channel, including needle valves, was used to flush the gas atmosphere in the diffusion cell. A pressure sensor (PTX 1400, Druck, Germany) was attached to the
Fig. 1. System for measuring oxygen diffusivity of fruit skin and tissue.
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The edge of the cylinder was covered with silicon grease to prevent diffusion from the edge of the tissue. The skin samples were prepared from the top of a cylindrical plug (2.5 cm diameter) of pear tissue taken in the radial direction (perpendicular to the long axis of the pear). A razor blade was used to remove the flesh from the skin slice. The thickness of each slice was measured with a caliper. The samples were attached in a similar way to the diffusion cell as for the tissue samples. Once the sample had been attached to the diffusion cell, the outer and the inner chambers were flushed with, respectively, a gas mixture of 10 kPa O2, 0.03 kPa CO2, balanced with N2, at 10 l/h and 20.8 kPa O2, 0.03 kPa CO2, balanced with N2 at a flow preventing a pressure build up in the diffusion cell. This flow was controlled by a manual needle valve. The gas mixtures were prepared from the gases by means of mass flow controllers (5850 S, Brooks Instruments, The Netherlands) and passed through a heat exchanger and humidifier to prevent the sample from drying and cooling down while flushing the two chambers. After 1 h, the in- and outlet valves of the diffusion cell were closed and the steady decrease in oxygen partial pressure was monitored for at least 13 h until an equilibrium had been reached. The oxygen electrode, the temperature and the two pressure sensor signals were logged at 4 s intervals during measurement using a Fieldpoint I/O module (National Instruments, Zaventem, Belgium) connected to a computer. The graphical user interface was programmed in Labview 5.0 (National Instruments, Zaventem, Belgium). The oxygen electrode was calibrated using a high O2 gas mixture (20.8 kPa O2, and 0.03 kPa CO2, balanced with N2) and an O2-free gas (pure N2). The diffusion experiments were repeated ten and six times, respectively, for tissue and skin.
2.3. Respiration measurements Respiration measurements on pear tissue and skin were carried out at 20°C as described by de Wild et al. (1999). Either ten cylindrical tissue or skin samples (2 mm height and 2.5 cm diameter) were placed in each of six 1.2 litre glass jars. After 1 h, during which time the jar headspace was
flushed at 20 l/h with the applied humidified gas mixture (10 kPa and 20.8 kPa O2 balanced with N2), the jars were closed and the initial headspace gas (O2, N2 and CO2) measured using a gas chromatograph (Chrompack CP 2002, Bergen op Zoom, The Netherlands). After 6, 21 and 27 h a gas sample was taken from the jar headspace and analysed. The total pressure was recorded before and after each measurement (PTX 1400, Druck, Germany), and the pressure drop caused by the sampling technique was corrected for. The difference in gas partial pressure was converted to moles according to the ideal gas law. The density of the tissue was measured based on the water displacement method. The oxygen consumption rate was expressed in mol per unit volume and per unit time (mol/m3 s).
2.4. Model construction and parameter estimation Transient linear mass transfer in solid foods subjected to convection boundary conditions Eq. (4) is governed by Fick’s Law of Diffusion Eq. (3). ([O2] = D92[O2]+ q (t −D
)
([O2] = h([O2]r − [O2] ) (x x = r
(3) (4)
were [O2] is the oxygen concentration (mol m − 3), D is the diffusion coefficient (m2 s − 1), q is the oxygen consumption (mol m − 3 s − 1), 92 is the Laplace operator (1 m − 2), h is the convective mass transfer coefficient (m s − 1) and t is the time (s). Indices and r refer, respectively, to the ambient atmosphere in the outer chamber and the length of the modelled system (Fig. 2). For many realistic mass transfer problems no analytical solutions of Eq. (3) subjected to Eq. (4) are known. In this case numerical discretization techniques such as the finite difference or finite element method can be used to obtain an approximate solution. The finite element method in particular is a very flexible and accurate method for solving partial differential equations such as Fick’s Law of Diffusion. In the framework of the finite element method, the continuum is subdivided in
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Fig. 2. Schematic overview of the two-material finite element model. Both materials consist of 20 elements, each element has eight nodes. Dair, Dsample, Rsample, h, [O2]r, [O2] , and r are, respectively, the diffusion coefficient of oxygen in air, the diffusion coefficient of oxygen in the sample, the respiration rate of the sample, the convective oxygen transfer coefficient, the oxygen partial pressure at position r, the ambient oxygen partial pressure in the outer chamber.
elements of variable size and shape, which are interconnected in a finite number, N, of nodal points. In every element the unknown oxygen concentration is approximated by a low order interpolating polynomial. The application of a suitable spatial discretization technique such as the Galerkin weighted residual method to Eq. (3) subjected to Eq. (4) results in the following differential system (Segerlind, 1984). d C u +Ku =f dt
(5) (6)
u(t =0)= u0 T
with u= [u1 u2 …uN ] the overall nodal oxygen concentration vector, C the capacitance matrix and K the conductance matrix, both N ×N matrices, and f a N ×1 vector. The system Eq. (5) is solved by finite differences in the time domain. In this work the Chamspack code (Scheerlinck et al., 2000) has been used. In the two material approach, the finite element model was composed of two materials, attached to each other (Fig. 2). The internal volume of the diffusion cell, was taken as the first material. The diffusion coefficient of oxygen molecules in air at 20°C was set equal to 6×10 − 5 m2 s − 1, four orders of magnitude higher than the diffusivity in fruit tissue (Lide, 1999). The tissue was modelled as the second material, for which the diffusion coefficient was to be estimated. The gas transfer
from the outer chamber to the tissue or the skin was expressed by Eq. (4) as a convective boundary condition. The convective mass transfer coefficient was taken as very high (1× 106 m s − 1), such that no resistance to gas transport occurred at the interface. Eq. (4) could then be simplified to [O2](r,t) = [O2]
(7)
The finite element mesh consisted of 20 quadratic quadrilateral elements for each material, resulting in a total of 206 nodes (Fig. 2). An iterative least squares estimation procedure was written in MATLAB 5 (The Mathworks, Inc., Natick, USA) to estimate the tissue respiration and the oxygen diffusivity from the measured oxygen partial pressure versus time profiles measured in the diffusion cell with the oxygen electrode. A similar two material model was constructed for the skin. In the three material approach, a finite element model was constructed, consisting of three materials: the internal volume of the diffusion cell, the tissue and the skin. This model was not used for estimation purposes, since this would have required the simultaneous estimation of at least three parameters, which undoubtedly would have resulted in badly identified parameters. However, this model was useful in validation and simulation experiments. In practice, skin and tissue occur as a coupled system and, therefore, a validation ex-
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periment, on a coupled system of skin and tissue, was performed and compared with the simulation result of the model accounting for the biological variability on the estimated parameters. To simplify the complexity of the problem, three assumption were made. First, it was assumed that the oxygen diffusivity for the skin (Ds) and the tissue (Dt) and the respiration of the skin (Rs) and the tissue (Rt) were normally distributed. Second, the covariance between Ds and Dt, and between Ds and Rt, was considered negligible. Third, it was chosen to set the skin respiration equal to that of the tissue, since the former was most probably caused by wound respiration and it would be inappropriate to include this value in a model that describes a coupled system without this wound respiration. Besides, the contribution of skin respiration is only minor compared to the amount of tissue respiration. The variance–covariance structure of the tridimensional space based on Dt, Rt and Ds was used to calculate the trivariate normal distribution. Monte-Carlo simulations (n =1000) (Rubinstein, 1981), based on this distribution, were performed, resulting in 1000 oxygen partial pressure versus time profiles. To validate the model on samples with different geometrical shapes, Monte-Carlo simulations were performed to determine the area of biological variation in which the measured curve should be situated. A hollow cylinder with an inner and outer diameter of, respectively, 28 and 35 mm was cut in the longitudinal direction of the pear by means of two concentric steel knives. It was attached to the diffusion cell in a similar way to the other samples and closed at the bottom with an airtight plastic disk. The measurements were carried out in exactly the same way as described above. Note that the inner volume of the cylinder plays a role as a diffusion cell on itself. When, in the three material approach, the thickness of the skin was considered negligible, its resistance to diffusion could be treated like the resistance of a boundary layer (Mannapperuma et al., 1991). This resulted in a two material model with a non-negligible convective mass transfer coefficient at the boundary between the tissue and the outer chamber. The convective mass transfer coefficient was calculated as the ratio of the skin
diffusion coefficient and the skin thickness. The reciprocal ratio corresponded to skin resistance to oxygen transport.
3. Results and discussion
3.1. Respiration measurements on skin and tissue The oxygen consumption rate of intact pears and pear disks strongly depends on the oxygen partial pressure in the gas atmosphere (de Wild et al., 1999). At low oxygen partial pressures (0–2 kPa), the oxygen consumption rate decreases almost linearly with the partial pressure. For high oxygen partial pressures (\8 kPa), the consumption rate is independent of the partial pressure. To check whether the consumption rate was really constant between the two oxygen partial pressures used for the determination of the diffusion constant, the oxygen consumption rate was measured six times at both 10 kPa and 20.8 kPa O2. A statistical t-test indicated no difference in mean respiration rate between 10 kPa or 20.8 kPa O2, at a 5% level of significance. An average oxygen consumption rate of 2.69× 10 − 4 mol m − 3 s − 1 (standard deviation 0.371× 10 − 4 mol m − 3 s − 1) in tissue was observed. Similarly, no significant difference was found between skin respiration rates at 10 kPa and 20.8 kPa O2. The average oxygen consumption rate of skin equalled 21.14× 10 − 4 mol m − 3 s − 1 (standard deviation 4.46× 10 − 4 mol m − 3 s − 1). Extensive wound respiration, induced by scratching the remaining tissue from the skin, may explain why the respiration rate of skin is eight times higher than that for tissue. However, it may also be that skin respiration is higher than tissue respiration. Most probably a combination of both explanations will be true.
3.2. Oxygen diffusi6ity of skin and tissue To solve Eq. (3) properly, the initial oxygen partial pressure profile must be known both in the diffusion cell and in the tissue or skin. The former can be measured easily since the oxygen molecules are uniformly distributed over the internal volume
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and the initial partial pressure is exactly the composition of the gas mixture used to flush the cell. However, it is not trivial to determine the initial oxygen partial pressure profile in the tissue or the skin. To avoid this problem an experimental design was developed that minimizes the influence of the initial conditions on the estimated parameters. A typical oxygen partial pressure versus time profile for skin and tissue is given in Fig. 3. This profile can be divided into two parts. The first part, recorded in the first 3600 s, corresponded to the situation where both the inner and the outer chamber were flushed with, respectively, 20.8 kPa and 10 kPa O2. At the end of this period, a steady-state gas equilibrium was established in the sample, which was then used as the initial oxygen partial pressure profile to solve Eq. (3) for the second part of the curve. This dynamic part of the profile describes the decrease in oxygen partial pressure in the diffusion cell when the flushing has been stopped and the valves were closed, while the
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outer chamber was continually flushed with 10 kPa O2. Since the oxygen diffusion depends on the oxygen gradient over the sample, a fast decrease was observed in the beginning, corresponding to a high gradient. In time, the gradient decreased and, consequently, the gas exchange between the two chambers slowed down, until an equilibrium was reached. Note that the equilibrium was not reached at 10 kPa O2, but at some lower value, because of the oxygen consumption caused by respiration. During the whole experiment the total pressure drop over the sample was constant and very small to non-existent (B 0.5 kPa). This implies that nitrogen molecules diffuse in the opposite direction to the oxygen molecular diffusion. The measured and the modelled oxygen partial pressure versus time profiles for the experimental cell are shown in Fig. 3, for both a tissue slice of 2 mm and a skin of 0.4 mm thickness. The model described the experimentally determined values very well.
Fig. 3. Oxygen partial pressure in the inner chamber of the diffusion cell as a function of time for diffusion of oxygen through pear tissue and skin. Symbols represent measured values, and lines values predicted by the model.
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Table 1 Estimated oxygen diffusivities for skin (Ds) and tissue (Dt) and measured and estimated respiration values for skin (Rs) and tissue (Rt) Source of data for estimates
Rs (×10−4 mol m−3 s−1)
Ds (×10−10 m2 s−1)
Rt (×10−4 mol m−3 s−1)
Dt (×10−9 m2 s−1)
Measured values Full profile Steady state +5000s, Rt = Rs = 0 Steady state +5000s, Rt or Rs
21.149 4.46 25.239 8.56 0 21.14
– 2.849 0.79 3.68 9 0.89 3.03 9 0.81
2.69 90.37 2.64 90.92 0 2.69
– 1.71 90.51 2.33 91.14 1.80 90.51
Values were estimated from three different data structures, 9the standard deviation. Values without standard deviation were not estimated, but kept constant during the estimation algorithm.
The average experimentally determined values for oxygen diffusivity in tissue and skin were, respectively, 1.71×10 − 9 and 2.84× 10 − 10 m2 s − 1. The mean estimated tissue (2.64×10 − 4 mol m − 3 s − 1) and skin (25.2×10 − 4 mol m − 3 s − 1) respiration values did not differ statistically, at a 5% level of significance, from the mean respiration values obtained in the independent traditional respiration experiment. Within one experiment the two estimated parameters were not correlated which indicated that the measured profile contained sufficient information to estimate the respiration and the diffusion simultaneously. The high parameter accuracy was confirmed by the standard deviations, which typically were two orders of magnitude lower than the parameter estimate, and thus, completely negligible compared to the biological variation. Although no literature was found about the coupled effect of gas diffusivity and respiration in pears, it can be useful to compare the obtained results with those found in the literature for other horticultural commodities. Mannapperuma et al. (1991) experimentally determined diffusivities of the flesh for oxygen of 2.67× 10 − 9 m2 s − 1 and for carbon dioxide of 3.28×10 − 9 m2 s − 1 for ‘Golden Delicious’ apples. The conductance (permeance) of the skin to oxygen was 2.9×10 − 7 m s − 1 and to carbon dioxide 2.2×10 − 7 m s − 1. When the skin diffusivities for oxygen obtained in this study for pears, were recalculated as convective mass transfer coefficients, an average value of 4.4× 10 − 7 m s − 1 was found. Abdul-Baki and Solomos, (1994) measured for potato tubers a
carbon dioxide diffusivity for the skin and tissue of, respectively, 7.26× 10 − 11 and 2.65× 10 − 8 m2 s − 1 at 27°C. Zhang and Bunn (2000) measured an oxygen diffusivity for apple flesh of 1.81× 10 − 8 m2 s − 1. Similar to the results in this paper, they found a skin diffusion coefficient which was six times lower (3.1× 10 − 9 m2 s − 1) than that for apple tissue. The resistance to skin gas transport of horticultural products is often expressed as a resistance or permeance. The conversion factors calculated by Banks et al. (1995) were used to convert and compare the literature results with those presented in this study. The resistance of pear skin (2.3× 104 s m − 1) obtained in this study had the same order of magnitude as that of potato tubers (Banks, 1985), but was much larger than that for citrus fruits and apples (Cameron and Yang, 1982; Ben-Yehoshua et al., 1985; Knee, 1991).
3.3. Relationship between full and partial use of the oxygen-time profile Until now the full oxygen partial pressure versus time profile was used to identify the oxygen diffusivity for the skin (Ds) and the tissue (Dt) and the respiration of the skin (Rs) and the tissue (Rt). However, such experiments typically lasted 15 h or even longer before an equilibrium was established. Parameter estimation based on a partial profile would reduce the measurement time considerably. In Table 1 an overview is given of the diffusivity and respiration parameters, estimated from three different sources of information. In the
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first case the full profile was used to estimate the diffusivity and the respiration simultaneously. The correlation between both estimated parameters was relatively low (typically ranging between 0.5 and 0.8), indicating that the profile contains sufficient information about both respiration and diffusion. In the second case, only the diffusivity was estimated based on the steady state part and the first 5000 s of the dynamic part of the profile. The respiration of the skin or the tissue was set to zero, assuming that the decrease in oxygen partial pressure caused by respiration during this 5000 s was negligible compared to the decrease due to diffusion. In the third case, only the diffusivity was estimated based on the steady state part and the first 5000 s of the dynamic part, but the average measured respiration rate for the skin or the tissue was kept constant during the estimation algorithm. No statistical difference was observed at the 5% significance level between the mean estimated diffusivities. However, when the correlation coefficients were calculated between oxygen diffusivities, both for the skin and the tissue, high correlations existed between the diffusion coefficients estimated based on the full profile and those estimated based on the partial profile with the average respiration rate. For skin and tissue, the coefficients of determination (R 2) were equal to 0.90 and 0.94, respectively. The other coefficients of determination were lower than 0.5.
3.4. Biological 6ariation in respiration and diffusi6ity In biological experimentation, the ideal of creating multiple sets of circumstances in which only one relevant factor varies is unrealistic. Inevitably, there will be what is called ‘biological variation’, which refers to variation in the set of conditions that produces the effect (Greenland and Rothman, 1998). Although the pears were picked on the same picking date and stored under the same conditions, they were of different maturities, resulting in slightly different diffusivities to oxygen and respiration characteristics. In-
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stead of trying to control all those factors which cause this biological variation, it is better to take them into account in the calculation, simulation and validation experiments. Until now, both experiments on skin and tissue were carried out separately, to obtain accurate diffusivity and respiratory parameter estimates. In practice, skin and tissue occur as a coupled system and, therefore, a coupled validation experiment was performed and compared with the Monte-Carlo simulation result of the model accounting for the biological variability on the estimated parameters. The upper and lower line in Fig. 4A represent, respectively, the 97.5th upper and 2.5th lower percentiles of the 1000 simulated curves, illustrating the inherent large biological variability. The full line represents the measured oxygen partial pressure versus time profile of the coupled system composed of 2 mm tissue covered by 0.4 mm skin. The measured curve perfectly lies between the two percentile curves indicating that the model, accounting for biological variation, describes reality well. The present model allows simulations of oxygen diffusion and respiration in tissue and skin slices of different thicknesses. With some small geometrical adaptations in the finite element model, it can even be used to simulate what happens in samples with different geometrical shapes. Monte-Carlo simulations (1000) were performed to determine the area of biological variation in which the measured curve for a hollow tissue cylinder should be situated. Fig. 4B indicates that the measured curve fits perfectly within the simulated range of biological variation. This indicates strongly that the estimated diffusivities and respiration parameters are shape independent and transferable to other geometries. The latter is of major importance in the construction of a three dimensional reaction-diffusion finite element model for intact pears.
3.5. Time dependent oxygen partial pressure profiles The developed finite element model allows simulation of the oxygen partial pressures at ev-
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ery spot in the tissue and skin at any time. In Fig. 5 the spatial oxygen partial pressure profiles of the coupled skin (0.4 mm)-tissue (2 mm) system are plotted at different time intervals. For reasons of clarity, only a part of the diffusion cell is given. The dashed profiles describe the changes in oxygen partial pressure profiles from the start of the experiment until a steady-state was reached around 3600 s. The diffusion cell oxygen partial pressure was kept constant at 20.8 kPa, by flushing. After 3600 s the flushing was stopped and the oxygen partial pressure started to decrease in the cell. The full lines display the changes in oxygen partial pressure in the cell, skin and tissue due to diffusion and respiration. Around 70 000 s, a steady-state was obtained and the oxygen partial pressure reached a value in the tissue lower than that in the diffusion cell and outer chamber, indicating that respiration plays an important role, and that it can not be neglected.
4. Conclusion A new system for the simultaneous measurement of the respiration and the oxygen diffusivity for pear skin and tissue was proposed. The use of a polarographic oxygen electrode resulted in informative oxygen partial pressure versus times profiles, which in turn resulted in good model parameter estimates. A dynamic finite element solution for the model, based on simultaneous gas diffusion and respiration for pear slices and pear skin, was developed. An experimental design was applied to minimize the effect of the initial oxygen conditions on the parameter identification. The average experimentally determined oxygen diffusivities for tissue and skin correspond to those found in literature for similar horticultural products and the oxygen diffusivity for the skin is an order of magnitude lower than that for tissue. The respiration values, measured in an independent respiration experiment, correspond very well to
Fig. 4. Simulated oxygen partial pressures as a function of time for (A) a skin (0.4 mm)-tissue (2 mm) coupled system and (B) a validation experiment on a hollow tissue cylinder. The upper (dotted line) and lower (dashed line) line indicate respectively the 97.5th and the 2.5th percentile of the Monte-Carlo simulations which account for the biological variation on the diffusivities for skin and tissue.
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Fig. 5. Predicted spatial oxygen partial pressure profiles of the coupled skin (0.4 mm)-tissue (2 mm) system as a function of different time intervals. The dashed profiles describe the changes in oxygen partial pressures from the start until a steady-state was reached (3600 s). The full lines display the changes in oxygen partial pressure in the cell, skin. and tissue due to diffusion and respiration.
those obtained with the diffusion model. It was shown that with this technique, the duration of the diffusion measurement could be reduced from fifteen to three hours, if the respiration characteristics of tissue were measured in a separate experiment. The diffusivity values estimated from the full oxygen partial pressure versus time profile were highly correlated to those obtained based on the first section of the profile. A technique, based on Monte-Carlo simulations, was used to quantify the effect of biological variation on the measured diffusivities and respiration values. Finally, simulations and validation measurements on a hollow cylindrical sample indicated that the estimated diffusivities and respiration parameters were shape independent and transferable to other geometries. The latter is of major importance in the construction of a three dimensional reaction-diffusion model for intact pears.
Acknowledgements The Belgian Ministry of Small Enterprises, Traders and Agriculture, the Flemish Government (project S-5901) and the European Communities (FAIR-project CT 96-1803) are gratefully acknowledged for financial support. Jeroen Lammertyn is Research Assistant of the Fund for Scientific Research-Flanders (Belgium) (F.W.OVlaanderen).
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