The comprehensive mass balances of differential equations involving gas diffusion and hydraulic convection through package perforation, gas permeation through polymeric film, and produce respiration have commonly been used to predict the atmosphere of perforated fresh produce packages. However, the predictions often suffer from instability, and to circumvent this problem, a simplified diffusion model that omits the convective gas transfer and empirical models based on experimental mass transfer data have been developed and investigated previously by several researchers. This study investigated the potential and limitations of the simplified diffusion model and two empirical models for predicting the atmosphere in perforated produce packages. The simplified diffusion model satisfactorily estimated the atmosphere inside the perforated packages of fresh produce under the aerobic conditions examined. Published empirical models of the mass transfer coefficients of the perforation seem to be valid only for the measured conditions and thus should be used carefully for that specific purpose.

Perforation or diffusion channel has been made or attached to plastic packages of fresh produce to maintain the desired modified atmosphere (MA) for a variety of commodities and storage conditions. The perforation or diffusion channel can partially overcome the gas permeability limitations of the available plastic packaging layers. Plastic packaging materials have narrow ranges of CO_{2} and O_{2} permeabilities, with the relative ratio of the former to the latter in the range of 3–7, which is suitable for creating an MA of low-to-high O_{2} concentration and low CO_{2} concentration [_{2} permeation to O_{2} permeation, allowing an MA with low O_{2} and high CO_{2} concentrations [

The perforated packages for a desired MA can be designed to balance the produce respiration and the package gas permeation by adjusting the number, diameter, and length of the perforations. In order to expedite the design process, mathematical models for estimating the package atmosphere have been developed and proposed by several researchers [_{2} and CO_{2} gases as a function of perforation diameter, length, and/or temperature [_{2} solubility of the produce, gas concentration distribution in the container, transpiration and water vapor balance, and external air velocity have been proposed [

Although any model has specific targets for application, with associated advantages and disadvantages, the models should be evaluated by their prediction accuracy and convenience. By comparing the mathematical models based on prediction outcomes for different commodities and package configurations, the models can be used to successfully design perforated produce packages.

Therefore, this study aims to evaluate the different mathematical models of perforated fresh produce packages with respect to their prediction accuracy and practical usefulness. The results of the simulations of each model were compared to the literature data first and then the effects of omitting the convective gas transfer were examined for a model package system of two commodities.

Four different mathematical models used for estimating the atmosphere of perforated fresh produce packages were compared. The first model examined was the comprehensive theoretical mass balance model, which includes diffusive gas fluxes through the plastic wall and the perforations, convective hydraulic gas flow through the perforation, and respiration rates of O_{2} consumption and CO_{2} production [_{2}, CO_{2}, and N_{2} gases at time _{2}, CO_{2}, and N_{2} gases in air (m^{2} h^{−1}); ^{2}); ^{2}) of the plastic layer, respectively; _{2}, CO_{2}, and N_{2} (mol mm m^{−2} h^{−1} Pa^{−1}); ^{−1} mol^{−1}); ^{−1} h^{−1}); _{2} consumption (mol kg^{−1} h^{−1}); _{2} production (mol kg^{−1} h^{−1}); and _{2}, CO_{2}, and N_{2} in the ambient air when ^{5} Pa) and are the partial pressures in the container when

Although (_{2} and CO_{2}, (

Two empirical models correlating the gas exchange flow of the perforation to its physical dimensions were also tested for their ability to predict the atmosphere of the perforated packages. These models, which were built using experimental gas transfer data for various physical perforation dimensions, lump the first and last terms in (^{3} h^{−1}). Nitrogen transfer across the package layer is neglected. Hence, the empirical models can be described as follows:

The model of Fonseca et al. [_{2} and CO_{2} mass transfer coefficients (

The model of Emond et al. [_{2} and CO_{2} mass transfer coefficients to the perforation dimensions and temperature using polynomial functions:

The respiration rates (_{2} and CO_{2} mass balance equations ((_{2} and CO_{2} concentrations can be conveniently used in the simulation [

The solution of the equations obtained for the moles of gas in the container for any time _{2}, CO_{2}, and N_{2} (

Experimental data for perforated produce packages in the literature that contained all the information on package dimensions, gas permeability, perforation conditions, storage temperature, and produce respiration were collected. Simulations using the four models examined were performed for each specific package and storage condition. The results of the simulations were compared to the literature data for the experimental package atmosphere.

The solution outcomes of two mechanistic models, the comprehensive model and the simplified diffusion model, were investigated for a model package with a variety of perforation dimensions. The model package for the study, a plastic box made of 2 mm thick polypropylene with dimensions of 0.32 × 0.23 × 0.18 m, had a perforation, and it could hold 0.35 kg of spinach or 0.9 kg of king oyster mushrooms; storage was at 10°C [^{−10}, 1.63 × 10^{−9}, and 1.14 × 10^{−10} mol mm m^{−2} h^{−1} Pa^{−1} for O_{2}, CO_{2}, and N_{2} at 10°C [^{−3} mol kg^{−1} h^{−1}, ^{−3} mol kg^{−1} h^{−1}, ^{−3} mol kg^{−1} h^{−1}, ^{−3} mol kg^{−1} h^{−1},

Figures

Comparison of the different simulation models used to estimate the gas composition of a strawberry package used by Renault et al. [^{−3} m^{3} and 8 perforations (diameter: 8.0 × 10^{−5} m, length: 5.0 × 10^{−5} m) over a surface area of 0.1 m^{2} contained 0.5 kg of strawberries at 10°C. (a) Comprehensive model, (b) simplified diffusion model, (c) Fonseca’s empirical model, and (d) Emond’s empirical model. Solid lines are the estimated gas compositions.

Comparison of the different simulation models used to estimate the gas composition of a cut onion package used by Lee and Renault [^{−4} m^{3} and a single perforation (diameter: 3.0 × 10^{−4} m, length: 2.1 × 10^{−2} m) contained 0.161 kg of cut onions at 10°C. (a) Comprehensive model, (b) simplified diffusion model, (c) Fonseca’s empirical model, and (d) Emond’s empirical model. Solid lines are the estimated gas compositions.

Comparison of the different simulation models used to estimate the gas composition of a peeled garlic package used by Lee et al. [^{−4} m^{3} and 11 perforations (diameter: 4.0 × 10^{−4} m, length: 3.0 × 10^{−5} m) over a surface area of 0.113 m^{2} and contained 0.117 kg of garlic cloves at 20°C. (a) Comprehensive model, (b) simplified diffusion model, (c) Fonseca’s empirical model, and (d) Emond’s empirical model. Solid lines are the estimated gas compositions.

The data in Table _{2}, CO_{2}, and/or N_{2}), and

Mean relative deviation modulus (%) of the package atmosphere simulated by the different models.

Data set | Comprehensive model | Simplified diffusion model | Fonseca’s empirical model | Emond’s empirical model |
---|---|---|---|---|

Figure |
20.0 | 21.7 | 35.4 | 74.6 |

Figure |
11.4 | 12.1 | 16.4 | 123.3 |

Figure |
12.0 | 11.0 | 45.6 | 59.2 |

The difference in estimation ability between the comprehensive and simplified diffusion models was minimal, and thus, the two models are comparable.

The empirical models cannot simulate the behavior of the package atmosphere because of their limitations in predicting accurate mass transfer coefficients for different perforation conditions, even though the empirical models can provide the relevant coefficient values and produce an estimation for their range of experimental conditions. There was a better agreement between the estimation of Fonseca’s model and the experimental literature data from Figure ^{−3}−1.7 × 10^{−2} m and length of 6.0 × 10^{−3}−3.0 × 10^{−2} m) used to build the model [

Considering that a wide variety of perforation conditions are being adopted for the packaging of many types of fresh produce, it is difficult to find any empirical gas transfer model that is universally applicable to all packaging situations. In terms of general applicability, the comprehensive and simplified diffusion models are more effective for designing a package with optimal perforation conditions for different commodities.

Whereas the empirical gas transfer models have the limitation that they can only be used with very similar conditions of their sources, the comprehensive and simplified diffusion models are generally applicable to a variety of conditions, and thus, their properties under different perforation dimensions were investigated for packages of two typical commodities, spinach and king oyster mushrooms.

The comprehensive model of (

Numerical solution of (^{−3} m in diameter and 2.0 × 10^{−3} m in length that contained 0.35 kg of spinach at 10°C was used. The value on the upper right part of each panel is the maximum time step.

For most of the perforation dimensions examined (diameter 3.0 × 10^{−3}−2.0 × 10^{−2} m and length 2.0 × 10^{−3}−5.0 × 10^{−2} m for the spinach packages; diameter 5.0 × 10^{−3}−2.0 × 10^{−2} m and length 2.0 × 10^{−3}−1.5 × 10^{−2} m for the king oyster mushroom packages), the O_{2} and CO_{2} concentrations of the packages in the aerobic atmosphere range were not appreciably different between the two models when the time step size was chosen properly for the comprehensive model (as shown in Figure _{2} concentration behaved slightly differently between the two models: it was maintained at a constant level of 78% for the simplified diffusion model, whereas it showed slight fluctuation around 78% for the comprehensive model. Figure _{2} and CO_{2} concentrations of the aerobic atmosphere range (

A typical solution set from the comprehensive model (a) and the simplified diffusion model (b) for a perforated model package (perforation with a diameter of 1.0 × 10^{−2} m and length of 2.0 × 10^{−3} m) containing 0.9 kg of king oyster mushrooms at 10°C.

Comparison of the estimated gas composition between the comprehensive and simplified diffusion models for the perforated model package containing 0.35 kg spinach (a) or 0.9 kg king oyster mushrooms (b). The adopted perforations were 3.0 × 10^{−3}–2.0 × 10^{−2} m in diameter and 2.0 × 10^{−3}–5.0 × 10^{−2} m in length for spinach and 5.0 × 10^{−3}−2.0 × 10^{−2} m in diameter and 2.0 × 10^{−3}−1.5 × 10^{−2} m in length for king oyster mushrooms.

Omitting the convection terms from the comprehensive model did not result in significant changes in the package O_{2} and CO_{2} concentrations when the N_{2} concentration was maintained at a relatively constant level of approximately 78% with an aerobic MA. The vanishing of the convection terms under the constant N_{2} concentration has been reported by Paul and Clarke [_{2} concentration increasing above 23% could not sustain the N_{2} concentration at 78% (specific data not shown) and caused the discrepancy in the estimated gas concentrations between the two models (Figure _{2} concentration and high CO_{2} concentration in the aerobic atmospheric range [_{2} concentration of approximately 78%. Thus, a perforated fresh produce package with the desired MA can be designed using the simplified diffusion model in practical terms.

By accounting for gas diffusion through package perforation, gas permeation through the polymeric film, and produce respiration, the simplified diffusion model can estimate the atmosphere within the aerobic range inside perforated packages of fresh produce. The omission of the hydraulic convection terms in the mass balance differential equations led to stability in the solutions without affecting the accuracy of the estimated package atmosphere. Published empirical models of the mass transfer coefficients of the perforation are valid only for conditions similar to those used in the measurement setup.

Mention of trade name in this paper is solely for the purpose of providing specific information and does not imply recommendation or preference by the authors. The authors of this paper do not have any conflict of interests with the trade name.

This study was supported by the Agriculture Research Center Program of the Ministry for Food, Agriculture, Forestry and Fisheries, Republic of Korea (Project no. 710003).

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_{2}transmission rates through microperforated films for modified atmosphere packaging of fresh fruits and vegetables

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_{2}exchange for development of perforation-mediated modified atmosphere packaging