# 2D numerical model for gas mass transfers in modified atmosphere packaging systems

Anh Khoa Nguyen, Fanny Coffigniez, Valérie Guillard, Sébastien Gaucel

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Anh Khoa Nguyen, Fanny Coffigniez, Valérie Guillard, Sébastien Gaucel. 2D numerical model for gas mass transfers in modified atmosphere packaging systems. Journal of Food Engineering, 2026, 402, pp.112701. 10.1016/j.jfoodeng.2025.112701. hal-05283442

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# 2D numerical model for gas mass transfers in modified atmosphere packaging systems

Anh Khoa Nguyen, Fanny Coffigniez*, Valérie Guillard, Sébastien Gaucel

JRU IATE, University of Montpellier, INRAE, Institut Agro, 2 Place Pierre Viala, Montpellier, 34060, France

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

### Keywords:

- Food-packaging system
- Modified atmosphere packaging
- Mathematical modelling
- Mass transfers
- Sensitivity analysis

## ABSTRACT

The complexity of some food-packaging systems requires mathematical models of mass transfers to accurately predict the evolution of headspace gas composition during storage and better anticipate food shelf-life. In this study, a numerical 2D model of O₂/CO₂ transfer in a cheese packaging system was proposed. The results showed significant involvement of gas sorption/desorption in food and permeation through the tray in evolution of gases in headspace. In this context, only a 2D model provided a good fit to experimental results, as gas permeation at the food/tray interface was non-negligible, limiting 1D approaches. This validated 2D model was then used in a global sensitivity analysis employing the Morris method to quantitatively identify the most influential mass transfer parameters affecting prediction accuracy. The results revealed a significant impact of O₂ and CO₂ permeability of the tray and their activation energies, as well as O₂ and CO₂ solubility and diffusivities in food. Therefore, to obtain a robust model of a food packaging system, it is recommended to integrate gas transfers between food and tray when the food-tray interface area is significant (2D model) and to precisely evaluate the gas permeability of the tray. These transfers can be neglected when this interface area is minimal (1D model).

## 1. Introduction

Over the past decade, efforts have been made to develop well-adapted and "just-necessary" packaging to preserve food quality and extend shelf life. Just necessary means that the properties of the packaging are designed/dimensioned to fulfill the need of the food to mitigate product's alterations, extend shelf life and limit food loss (Verghese et al., 2015; Coffigniez et al., 2021). The quest for extending food shelf life has promoted the widespread use of Modified Atmosphere Packaging (MAP), i.e. replacement of the inner atmosphere of the packaging by flushing a gas mixture, and the use of active compounds (e.g. oxygen scavenger, antimicrobial molecules, etc.) into the packaging materials, to help prevent oxidative degradations, microorganisms growth and enzymatic browning (Mangaraj et al., 2009; Caleb et al., 2012; Di Giuseppe et al., 2022; Bruni et al., 2024). However, to maintain the targeted gas mixture throughout the food's shelf life, it is essential to control gas permeation between the internal and external atmospheres of the packaging. This can be achieved by selecting packaging materials with appropriate gas transfer properties and considering the dissolution and diffusion of gases into the food (Belay et al., 2016; Guillard et al., 2016; Opara et al., 2019).

Due to the complexity of the food-packaging system, the use of mathematical modeling tools is highly recommended to optimize the development of modified atmosphere packaging (Guillard et al., 2017; Coffigniez et al., 2021). So far, such mathematical models have primarily been developed based on gas transfer between the headspace, i.e. inner atmosphere, and the external atmosphere, from permeation through the packaging material, described by Fick's first law combined to Henry's law. The authors of the present paper have performed a lot of previous modelling work, and the current model could be seen as an accomplishment of several preliminary approaches (Chaix et al., 2015a; Guillard et al., 2016; Matar et al., 2018). For respiring products, models generally couple permeation with equations for food respiration, the most classical being Michaelis-Menten models (Song et al., 2002; Matar et al., 2018; Joshi et al., 2019). For non-respiring products, few studies in the literature including co-authors previous work have included gas solubilization and diffusion in food in their modelling approaches, mainly to consider gas sorption/desorption at the food-headspace interface, based on Henry's law and Fick's second law, the gases solubilization having a high impact on the modified atmosphere, especially for carbon dioxide (Fabiano et al., 2000; Simpson et al., 2004; Coutelieris and Kanavouras, 2006; Guillard et al., 2016). However, in these

<footnote>* Corresponding author. IATE, 2 place Pierre Viala, Bât 31, 34060, Montpellier, France.
E-mail address: fanny.coffigniez@umontpellier.fr (F. Coffigniez).</footnote>
works, some assumptions were made regarding the food geometry (reduced to 1D essentially) and the transfers at the food-packaging interface, to reduce model complexity. This allowed the diffusion in food to be simplified to one-dimensional diffusion, along the thickness of the food. For instance, in previous work, the authors of the present paper considered several food commodities (poultry, salmon and processed cheese) packed in sealed trays Chaix et al. (2015a) and Guillard et al. (2016) where trays were assumed to have a much higher barrier to gases than the sealing films so the transfer at the food-tray interface, and more generally through the tray, was neglected. This simplification therefore reduces the range of applications to conditions where mass transfer through the tray could be neglected. Even though this approach was validated for some food product with negligible contact between food and packaging (poultry, salmon), it cannot be extrapolated to food products where the food-packaging interface is predominant, such as liquid or injected food (Bacigalupi et al., 2013). As far as we know, these are the most advanced modelling approaches for food and packaging systems that can be used to dimension modified atmospheres. However, the lack of models that consider transfers at the food-packaging interface makes it difficult to estimate the error introduced by the aforementioned assumptions. This can affect the quality of the predictions and is therefore worth studying.

Apart from the geometry of the food and packaging, many mass transfer parameters are required for the development of these models, such as the gas (O₂, CO₂) permeability of the packaging material, gas solubility and apparent diffusivity in food, and activation energy (to account for the temperature dependency of parameters) (Chaix et al., 2015a; Guillard et al., 2016; Matar et al., 2018). However, the accuracy of the parameter values should be considered with caution due to frequently observed variability (e.g. O₂ packaging permeability), complex measurement (e.g. gas diffusivities) and, in some cases, a lack of data (e.g. CO₂ packaging permeability, activation energy) which

necessitates making assumptions. Therefore, it is essential to understand the impact of the variability of these parameter values on the accuracy of the model, which has never been done in the literature.

In this context, the objective of this work was to propose for the first time a 2D model for modified atmosphere packaging system -injected food packed in a sealed tray-where the tray significantly contributes to the mass transfer in the axisymmetric food packaging system composed. The model is based on gas permeation between the headspace and external atmosphere through the packaging components (tray and lid film), gas sorption and desorption at food-headspace interface, gas diffusion in food and transfers between food and external atmosphere through the tray material. To assess the predictability of the model and the importance of the mass transfers at food-packaging interface, the model validation and comparison with simplified models were performed on a fresh cheese case study. This enabled us to recommend when a 2D geometry is necessary, and when it can be simplified to a 1D geometry. Even when computational resources are no longer limited, simplified geometry could be a good option for optimizing parameters to save running time and limit the environmental cost of the simulation. It could also be useful for proposing an open-source model for teaching, training or R&D teams. Finally, a global sensitivity analysis was conducted to identify the most influential inputs, specifically among mass transfer parameters, on the prediction of the headspace gas composition.

## 2. Material & methods

### 2.1. Material

#### 2.1.1. Packaging

An injected polypropylene cup was considered. The cup has a thickness of 55 µm and an overall truncated cone shape (see Fig. 1A) with inner upper and lower diameters of 86.9 mm and 56.46 mm

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** 3D representation of the actual cheese packaging system, without sealing film (A). Scheme of an equivalent 2D model exploiting the axis symmetry of the food-packaging system, and presenting the main mass transfers (B). Adaptive triangular mesh (min and max element sizes of the triangular element of the adaptive mesh are given) for the food-tray system (C) and visualization of the simulated O₂ concentration (mol · m⁻³) at 1 day and 60 days (D).</figcaption></figure>
respectively, and an inner height of 47.8 mm. Dimensions may vary with a tolerance of ±5 %. The cup is given for an inner volume of 199 cm³ and a total inner surface area of 146 cm².

The top film considered for sealing the polypropylene cup was a multilayer material provided by Soplaril, Arras, France. The film is composed of a metalized polyethylene tereptalate layer and a poly-propylene layer with a total thickness of 63 µm. Based on the geometry of the cup, the film surface available for transfer is a disk of diameter 86.9 mm, resulting in an area of 0.0059 m².

### 2.1.2. Food

The food considered was commercial, spreadable cheese, with a density of approximately 1000 kg.m⁻³, a fat content of 34.5 g per 100 g of product and a dry matter content between 45 % and 48 %.

### 2.1.3. Modified atmosphere - headspace

A 30 %/70 % mixture of CO₂/N₂ was considered to preserve the food product. This modified atmosphere was selected for its bacteriostatic properties and absence of defects on the products, aiming to achieve a target shelf-life of 60 days.

## 2.2. Method

### 2.2.1. Thermosealing of packaging

For each cup, 150 g of cheese, or equivalently 150 cm³, was poured into the cup at a temperature approximately 10 °C. The headspace was replaced by a modified atmosphere, 30 % CO₂/70 % N₂ at atmospheric pressure, and then sealed with the metalized PET/PP film using a Reepack chamber machine Reetray 20 VGT (Reepack, Seriate (BG), Italy).

Alternatively, empty cups were sealed under the same conditions to monitor the overall gas transfer through the entire packaging.

### 2.2.2. Characterization of oxygen transmission rate

The oxygen transmission rate (OTR) of the PP cups was assessed using an oxygen permeation analyzer, model 8001, from Systech Instruments, UK. The polypropylene cup was sealed on an instrumented airtight support dedicated to measuring the OTR of packaging, provided by Systech Instruments. To remove the oxygen from the headspace, the cup was first flushed with pure nitrogen for 30 min. After this flushing step, while the system was kept under nitrogen flux, a pressure gradient of oxygen was established between the external atmosphere, i.e. ambient air with 20.9 % of oxygen, and the headspace, i.e. 0 % of oxygen. This gradient imposed a flux of O₂ molecules from the external atmosphere to the headspace. The O₂ molecules that transferred through the cup were transported by the N₂ flux, to be measured by a Coulometric sensor. The linear regression of the O₂ flux allowed to calculate the OTR for the cup, given in cm³.pack⁻¹.day⁻¹. The OTR can be converted into oxygen permeability of the cup P, in mol.m.m⁻².s⁻¹.Pa⁻¹, by Equation (1), where l_cup (m) and A_cup (m²) are the thickness and surface area of the cup respectively. The oxygen partial pressure gradient is 0.209 P_atm, where P_atm (Pa) is the atmospheric pressure (1.013 × 10⁵ Pa).

$$P = \frac{\text{OTR}}{0.209 P_{\text{atm}} A_{\text{cup}}} l_{\text{cup}} \quad (1)$$

where OTR was converted to mol.pack⁻¹.s⁻¹ by considering a molar volume of 23.2 × 10³ cm³.mol⁻¹ at 10 °C.

Four measurements were performed under lab conditions, at a temperature of 23 °C and without regulation of relative humidity, which was then considered to be 50 %.

For the top film, the OTR was given by the technical data sheet from by the supplier, i.e. 1.2 × 10⁻¹⁸ mol.m.m⁻².s⁻¹.Pa⁻¹ following standard ASTM D 3985-81, at 23 °C and 50 % of relative humidity.

### 2.2.3. Experimental assessment of the headspace gas composition

The sealed cups, with or without food, were placed in a thermo-

regulated oven at 10 °C. The evolution of headspace gases (O₂ and CO₂) concentrations was characterized using an headspace gas analyser Oxybaby WITT, Germany. A needle was inserted into the headspace through a septum attached to the surface of the film and a small volume of gas was analyzed by an electrochemical sensor for O₂ and an infrared optical sensor for CO₂. Each experiment was conducted in triplicate, using three different cups. The measuring range of the Oxybaby is 0–100 % for both O₂ and CO₂, resolution is 0.01 % for O₂ and 0.1 % for CO₂ and accuracy is ±0.1 % absolute for concentrations O₂ ≤ 10 % and ±1 % relative for CO₂ concentrations between 10 and 100 %.

## 3. Model description

### 3.1. System description and modelling assumptions

The food-packaging system consists in five different domains: the food product, the tray, the lid film, the headspace, and the external atmosphere. The cup was simplified for the modeling purpose and was assumed to have a perfect truncated cone shape with inner upper diameter, inner lower diameter, and inner height corresponding to those of the actual cup, i.e. of 86.9 mm, 56.46 mm and 47.8 mm respectively, leading to a volume of 196 cm³ close to the actual volume (199 cm³). The total inner surface area of the truncated cone tray was 138 cm² (compared to 146 cm² for the real cup) which decomposed into the lateral surface area (113 cm²) and the bottom surface area (25 cm²).

The injected food was assumed to adapt its shape to the cup as shown in Fig. 1A. Additionally, for modeling purpose, the food-headspace interface was assumed to be perfectly flat and horizontal. Thus, the food compartment also presented a truncated cone shape. Assuming that the volume of the food was known, the height of the food and the area of the food-headspace interface could be calculated using algebraic formulas for the volume and surface area of a truncated cone. The dimensions of the packaging and food product used for the present work are gathered in Table 1.

Due to the rotational axis symmetry of the food-packaging system, and under the assumptions of homogeneous food and packaging materials, initial and boundary conditions, i.e. initial and boundary gas concentrations, the modelling was reduced to a 2D description, as presented in Fig. 1B.

Building the mathematical model involved the following additional assumptions.

* The temperature of the food-packaging system was assumed to be homogeneous in space, equal to the storage temperature;

* Gas concentration in the headspace was homogeneous; in other words gas diffusion is considered instantaneous in the headspace;

* The total pressure in the headspace was constant, equal to its initial value, i.e. atmospheric pressure;

* The solubilization of nitrogen in food was negligible;

* The resistance to transfer at the interfaces were neglected, by considering an arbitrary high mass Biot number, i.e. 10⁵.

* The bottom side of the tray was assumed to be in contact with a plane surface, so mass transfers were neglected at this interface;

* The food is considered non-porous and homogeneous.

### 3.2. Building of the 2D model

The mathematical model was derived from the work of Chaix et al. (2015a) and relied on the mass transfer of gases occurring in and between the different compartments of the food packaging system: headspace, food, tray, film and external atmosphere. Direct transfers between headspace and external atmosphere occurred by permeation through the film and the part of the tray in contact with the headspace. Transfers occurring in the food compartment, the tray in contact with food, and at the interfaces between these compartments and others required a more
<caption>**Table 1**</caption>

Summary of input parameters for the numerical model. Uncertainty ranges are given for the 21 uncertain parameters considered in the global sensitivity analysis.

<table><thead><tr><th>Parameter</th><th>Value</th><th>Uncertainty range</th><th>Unit</th><th>Reference</th></tr></thead><tbody><tr><td colspan="5"><strong>Food characteristics</strong></td></tr><tr><td>Food mass (m<sub>F</sub>)</td><td>0.150 ± 0.008</td><td>-</td><td>kg</td><td>a</td></tr><tr><td>Food thickness (e<sub>F</sub>)</td><td>0.0415 ± 0.0021</td><td>-</td><td>m</td><td>b</td></tr><tr><td>Surface area of headspace-food interface (A<sub>F</sub>)</td><td>53.974 · 10<sup>-4</sup>±2.699 · 10<sup>-4</sup></td><td>-</td><td>m<sup>2</sup></td><td>b</td></tr><tr><td>Food volume (V<sub>F</sub>)</td><td>1.50 · 10<sup>-4</sup>±0.08 · 10<sup>-4</sup></td><td>-</td><td>m<sup>3</sup></td><td>a</td></tr><tr><td colspan="5"><strong>Packaging container characteristics</strong></td></tr><tr><td>Film thickness (e<sub>L</sub>)</td><td>63 · 10<sup>-6</sup></td><td>-</td><td>m</td><td>a</td></tr><tr><td>Film area (A<sub>L</sub>)</td><td>0.0059 ± 0.0003</td><td>-</td><td>m<sup>2</sup></td><td>a</td></tr><tr><td>Tray thickness (e<sub>T</sub>)</td><td>550 · 10<sup>-6</sup>±28 · 10<sup>-6</sup></td><td>-</td><td>m</td><td>a</td></tr><tr><td>Surface area of headspace-tray interface (A<sub>T</sub>)</td><td>0.0018 ± 0.0001</td><td>-</td><td>m<sup>2</sup></td><td>b</td></tr><tr><td>Headspace volume (V<sub>HS</sub>)</td><td>4.58 · 10<sup>-5</sup>±0.23 · 10<sup>-5</sup></td><td>-</td><td>m<sup>3</sup></td><td>b</td></tr><tr><td colspan="5"><strong>Mass transfer parameters of the food at 23°C<sup>d</sup></strong></td></tr><tr><td>O<sub>2</sub> diffusivity in food (D<sub>0, O<sub>2</sub>, F</sub>)</td><td>1.24 · 10<sup>-9</sup></td><td>[1.24 · 10<sup>-10</sup>, 1.24 · 10<sup>-8</sup>]</td><td>m<sup>2</sup>s<sup>-1</sup></td><td>(Chaix et al., 2016) (20 °C)</td></tr><tr><td>CO<sub>2</sub> diffusivity in food (D<sub>0, CO<sub>2</sub>, F</sub>)</td><td>9.35 · 10<sup>-10</sup> ±1.49 · 10<sup>-10</sup></td><td>[9.35 · 10<sup>-11</sup>, 9.35 · 10<sup>-9</sup>]</td><td>m<sup>2</sup>s<sup>-1</sup></td><td>(Chaix et al., 2015b) (22 °C)</td></tr><tr><td>O<sub>2</sub> solubility in food (S<sub>0, O<sub>2</sub>, F</sub>)</td><td>1.26 · 10<sup>-5</sup></td><td>[4.197 · 10<sup>-6</sup>, 3.78 · 10<sup>-5</sup>]</td><td>mol.m<sup>-3</sup>Pa<sup>-1</sup></td><td>(Battino and Clever, 1966) (23 °C)</td></tr><tr><td>CO<sub>2</sub> solubility in food (S<sub>0, CO<sub>2</sub>, F</sub>)</td><td>3.24 · 10<sup>-4</sup> ±0.14 · 10<sup>-4</sup></td><td>[1.08 · 10<sup>-4</sup>, 9.73 · 10<sup>-4</sup>]</td><td>mol.m<sup>-3</sup>Pa<sup>-1</sup></td><td>(Chaix et al., 2015b) (22 °C)</td></tr><tr><td colspan="5"><strong>Mass transfer parameters of the tray compartment at 23°C<sup>d</sup></strong></td></tr><tr><td>O<sub>2</sub> diffusivity in the tray (D<sub>0, O<sub>2</sub>, T</sub>)</td><td>2.14 · 10<sup>-11</sup></td><td>-</td><td>m<sup>2</sup>s<sup>-1</sup></td><td>(Beltrame et al., 1999) (23 °C)</td></tr><tr><td>CO<sub>2</sub> diffusivity in the tray (D<sub>0, CO<sub>2</sub>, T</sub>)</td><td>1.2 · 10<sup>-11</sup></td><td>-</td><td>m<sup>2</sup>s<sup>-1</sup></td><td>(Siracusa and Ingrao, 2017) (23 °C)</td></tr><tr><td>O<sub>2</sub> permeability in the tray (P<sub>0, O<sub>2</sub>, T</sub>)</td><td>3.44 · 10<sup>-16</sup> ±0.24 · 10<sup>-16</sup></td><td>[3.44 · 10<sup>-17</sup>, 3.44 · 10<sup>-15</sup>]</td><td>mol.m. m<sup>-2</sup>s<sup>-1</sup>Pa<sup>-1</sup></td><td>a</td></tr><tr><td>CO<sub>2</sub> permeability in the tray (P<sub>0, CO<sub>2</sub>, T</sub>)</td><td>1.09 · 10<sup>-15</sup> ±0.08 · 10<sup>-15</sup></td><td>[1.09 · 10<sup>-16</sup>, 1.09 · 10<sup>-14</sup>]</td><td>mol.m. m<sup>-2</sup>s<sup>-1</sup>Pa<sup>-1</sup></td><td>Siracusa (2012)</td></tr><tr><td>N<sub>2</sub> permeability in the tray (P<sub>0, N<sub>2</sub>, T</sub>)</td><td>8.00 · 10<sup>-17</sup> ±0.56 · 10<sup>-17</sup></td><td>-</td><td>mol.m. m<sup>-2</sup>s<sup>-1</sup>Pa<sup>-1</sup></td><td>Siracusa (2012)</td></tr><tr><td>O<sub>2</sub> solubility in the tray (S<sub>0, O<sub>2</sub>, T</sub>)</td><td>1.61 · 10<sup>-5</sup></td><td>[1.61 · 10<sup>-6</sup>, 1.61 · 10<sup>-4</sup>]</td><td>mol.m<sup>-3</sup>Pa<sup>-1</sup></td><td>c</td></tr><tr><td>CO<sub>2</sub> solubility in the tray (S<sub>0, CO<sub>2</sub>, T</sub>)</td><td>8.2 · 10<sup>-5</sup></td><td>[8.2 · 10<sup>-6</sup>, 8.2 · 10<sup>-4</sup>]</td><td>mol.m<sup>-3</sup>Pa<sup>-1</sup></td><td>c</td></tr></tbody></table>

Mass transfer parameters of the lid film compartment at 23°C<sup>d</sup>

<table><caption>**Table 1 (continued)**</caption><thead><tr><th>Parameter</th><th>Value</th><th>Uncertainty range</th><th>Unit</th><th>Reference</th></tr></thead><tbody><tr><td>O<sub>2</sub> permeability in the lid film (P<sub>0, O<sub>2</sub>, L</sub>)</td><td>1.2 · 10<sup>-18</sup></td><td>[1.2 · 10<sup>-19</sup>, 1.2 · 10<sup>-17</sup>]</td><td>mol.m. m<sup>-2</sup>s.<sup>-1</sup>Pa<sup>-1</sup></td><td>a</td></tr><tr><td>CO<sub>2</sub> permeability in the lid film (P<sub>0, CO<sub>2</sub>, L</sub>)</td><td>2.4 · 10<sup>-18</sup></td><td>[2.4 · 10<sup>-19</sup>, 2.4 · 10<sup>-17</sup>]</td><td>mol.m. m<sup>-2</sup>s.<sup>-1</sup>Pa<sup>-1</sup></td><td>a</td></tr><tr><td>N<sub>2</sub> permeability in the lid film (P<sub>0, N<sub>2</sub>, L</sub>)</td><td>4 · 10<sup>-19</sup></td><td>-</td><td>mol.m. m<sup>-2</sup>s.<sup>-1</sup>Pa<sup>-1</sup></td><td>Siracusa (2012)</td></tr><tr><td colspan="5"><strong>Activation energies (in food, tray and lid film)</strong></td></tr><tr><td>Activation energy of the O<sub>2</sub> diffusivity in food (E<sub>a, D<sub>0</sub>, F</sub>)</td><td>20.3</td><td>[10.15, 40.6]</td><td>kJ.mol<sup>-1</sup></td><td>(Simpson et al., 2004)</td></tr><tr><td>Activation energy of the CO<sub>2</sub> diffusivity in food (E<sub>a, D<sub>0</sub>, F</sub>)</td><td>14.5</td><td>[7.25, 29]</td><td>kJ.mol<sup>-1</sup></td><td>(Simpson et al., 2004)</td></tr><tr><td>Activation energy of the O<sub>2</sub> solubility in food (E<sub>a, S<sub>0</sub>, F</sub>)</td><td>-13.2</td><td>[-26.4, -6.6]</td><td>kJ.mol<sup>-1</sup></td><td>Simpson et al. (2004)</td></tr><tr><td>Activation energy of the CO<sub>2</sub> solubility in food (E<sub>a, S<sub>0</sub>, F</sub>)</td><td>-19.7</td><td>[-39.4, -9.85]</td><td>kJ.mol<sup>-1</sup></td><td>(Simpson et al., 2004)</td></tr><tr><td>Activation energy of the O<sub>2</sub> diffusivity in the tray (E<sub>a, D<sub>0</sub>, T</sub>)</td><td>25.5</td><td>-</td><td>kJ.mol<sup>-1</sup></td><td>(Kurek et al., 2011)</td></tr><tr><td>Activation energy of the CO<sub>2</sub> diffusivity in the tray (E<sub>a, D<sub>0</sub>, T</sub>)</td><td>28.2</td><td>-</td><td>kJ.mol<sup>-1</sup></td><td>Kurek et al. (2011)</td></tr><tr><td>Activation energy of the O<sub>2</sub> solubility in the tray (E<sub>a, S<sub>0</sub>, T</sub>)</td><td>24.4</td><td>[12.2, 48.8]</td><td>kJ.mol<sup>-1</sup></td><td>Kurek et al. (2011)</td></tr><tr><td>Activation energy of CO<sub>2</sub> solubility in the tray (E<sub>a, S<sub>0</sub>, T</sub>)</td><td>21.1</td><td>[10.55, 42.2]</td><td>kJ.mol<sup>-1</sup></td><td>Kurek et al. (2011)</td></tr><tr><td>Activation energy of the CO<sub>2</sub> permeability in the tray (E<sub>a, P<sub>0</sub>, T</sub>)</td><td>49.9</td><td>[24.95, 99.8]</td><td>kJ.mol<sup>-1</sup></td><td>Kurek et al. (2011)</td></tr><tr><td>Activation energy of the N<sub>2</sub> permeability in the tray (E<sub>a, P<sub>0</sub>, T</sub>)</td><td>49.4</td><td>[24.7, 98.8]</td><td>kJ.mol<sup>-1</sup></td><td>Kurek et al. (2011)</td></tr><tr><td>Activation energy of the N<sub>2</sub> permeability in the tray (E<sub>a, P<sub>N</sub>, T</sub>)</td><td>51.2</td><td>-</td><td>kJ.mol<sup>-1</sup></td><td>Kurek et al. (2011)</td></tr></tbody></table>

(continued on next page)
<table><caption>**Table 1 (continued)**</caption><thead><tr><th>Parameter</th><th>Value</th><th>Uncertainty range</th><th>Unit</th><th>Reference</th></tr></thead><tbody><tr><td>Activation energy of the O<sub>2</sub> permeability in the lid film (E<sub>a,P<sub>O<sub>2</sub></sub>,L</sub>)</td><td>35.97</td><td>[17.99, 71.94]</td><td>kJ.mol<sup>-1</sup></td><td>Charles et al. (2005)</td></tr><tr><td>Activation energy of the CO<sub>2</sub> permeability in the lid film (E<sub>a,P<sub>CO<sub>2</sub></sub>,L</sub>)</td><td>32.12</td><td>[16.06, 64.24]</td><td>kJ.mol<sup>-1</sup></td><td>Charles et al. (2005)</td></tr><tr><td>Activation energy of the N<sub>2</sub> permeability in the lid film (E<sub>a,P<sub>N<sub>2</sub></sub>,L</sub>)</td><td>42.88</td><td>-</td><td>kJ.mol<sup>-1</sup></td><td>Siracusa (2012)</td></tr><tr><td colspan="5"><strong>Initial conditions</strong></td></tr><tr><td>Surrounding atm. total pressure (P<sub>T</sub>)</td><td>1.013 &middot; 10<sup>5</sup></td><td>-</td><td>Pa</td><td>-</td></tr><tr><td>Surrounding atm. O<sub>2</sub> partial pressure (p<sub>O<sub>2</sub>,atm</sub>)</td><td>20.95</td><td>-</td><td>%</td><td>(Widory and Javoy, 2003)</td></tr><tr><td>Surrounding atm. CO<sub>2</sub> partial pressure (p<sub>CO<sub>2</sub>,atm</sub>)</td><td>0.04</td><td>-</td><td>%</td><td>Widory and Javoy (2003)</td></tr><tr><td>O<sub>2</sub> partial pressure in the headspace (p<sub>O<sub>2</sub>,HS</sub>)</td><td>0.9</td><td>-</td><td>%</td><td>a</td></tr><tr><td>CO<sub>2</sub> partial pressure in the headspace (p<sub>CO<sub>2</sub>,HS</sub>)</td><td>30</td><td>-</td><td>%</td><td>a</td></tr><tr><td>Ideal gas constant (R)</td><td>8.314</td><td>-</td><td>J.mol<sup>-1</sup>.K<sup>-1</sup></td><td>-</td></tr><tr><td>Biot number (Bi)</td><td>10<sup>5</sup></td><td>-</td><td>dimensionless</td><td>-</td></tr><tr><td>Storage temperature (T)</td><td>283.15</td><td>[281.15, 285.15]</td><td>K</td><td>a</td></tr><tr><td>Storage time (t)</td><td>83</td><td>-</td><td>day</td><td>a</td></tr></tbody></table>

a value from this study.

b value was calculated based on the truncated-cone shape of the packaging container and.

c value was calculated from the relation $P = D \times S$.

d when not available, values at 23 °C were recalculated from values at different temperature, and activation energy, from the literature, assuming that Arrhenius law depicted the temperature dependency.

complex approach based on the two-dimensional diffusion equation.

The system is described in Fig. 2, with the definition of the compartments and boundaries.

### 3.2.1. Permeation fluxes through packaging

The gas transfers between the headspace and the external atmosphere, through the lid film and the cup, is characterized by simple permeation equations, obtained by coupling the Fick's first law and Henry's law. The molar fluxes, in mol.s⁻¹, of gas species (i ∈ {O₂, CO₂, N₂}), through the lid film $\phi_i^L$ and the tray $\phi_i^T$ are described by the following equations:

$$ \phi_i^L = P_{i,L} \left( \frac{A_L}{e_L} \right) (p_{i,\infty} - p_{i,HS}) \quad (2) $$

<figure><img src="image_5.png" /><figcaption>**Fig. 2.** Representation of the 2D axis symmetric model of packed food with definition of the boundaries.</figcaption></figure>

$$ \phi_i^T = P_{i,T} \left( \frac{A_T}{e_T} \right) (p_{i,\infty} - p_{i,HS}) \quad (3) $$

where $P_{i,L}$ and $P_{i,T}$ (mol.m⁻¹s⁻¹Pa⁻¹) are the permeabilities of gas species i through the lid film (L) and the tray (T) respectively; $A_L$ and $A_T$ (m²) are the lateral surface areas of the lid film and tray respectively; $e_L$ and $e_T$ (m) are the thickness of the lid film and tray, respectively; and $p_{i,\infty}$ and $p_{i,HS}$ (Pa) are the partial pressures in the ambient and in the headspace of gas species i, respectively.

### 3.2.2. Diffusion of gases in the food compartment and tray compartment

The 2D spatial diffusion for the dissolved gas species (O₂ and CO₂ only, N₂ being neglected) in the food compartment and the tray compartment, reduced to its part which is in contact with the food, is described by Fick's second law. Due to the axial symmetry of the food-packaging system, cylindrical coordinates were considered, leading to the following expression for the diffusion of gas species i in the food compartment:

$$ \frac{\partial C_{i,F}}{\partial t}(r, z, t) = D_{i,F} \left( \frac{\partial^2 C_{i,F}}{\partial r^2}(r, z, t) + \frac{1}{r} \frac{\partial C_{i,F}}{\partial r}(r, z, t) + \frac{\partial C_{i,F}}{\partial z}(r, z, t) \right) \quad (4) $$

where $C_{i,F}(r, z, t)$ is the concentration of the dissolved gas species i in the food commodity (mol.m⁻³), at time t and coordinates $(r, z)$ and $D_{i,F}$ is the diffusivity of the gas species i in the food commodity (m².s⁻¹). A similar equation applies to the domain of the tray compartment that is in contact with the food (see Fig. 2).

$$ \frac{\partial C_{i,T}}{\partial t}(r, z, t) = D_{i,T} \left( \frac{\partial^2 C_{i,T}}{\partial r^2}(r, z, t) + \frac{1}{r} \frac{\partial C_{i,T}}{\partial r}(r, z, t) + \frac{\partial C_{i,T}}{\partial z}(r, z, t) \right) \quad (5) $$

### 3.2.3. Gas transfer at food-headspace interface

Sorption or desorption occurs at food-headspace interface. Since the solubilization of nitrogen in food is neglected, only the gas molar flux $\phi_i^L$, in mol.s⁻¹, of gas species i (i ∈ {O₂, CO₂}) at the food-headspace interface is described as follows:
$$q_i^L = k_{i,\partial\Omega_{top}^{food}} \frac{A_L}{RT} (p_{i,HS,i} - p_{i,HS}) \quad (6)$$

with

$$k_{i,\partial\Omega_{top}^{food}} = \frac{BiD_i}{L_c} \quad (7)$$

where $k_{i,\partial\Omega_{top}^{food}}$ is the external mass transfer coefficient (in m.s⁻¹) at the interface $\partial\Omega_{top}^{food}$ for each gas species i, Bi is the Biot number (dimensionless), $D_i$ is the apparent diffusivity of the species i in the food (m².s⁻¹), and $L_c$ is a characteristic length (taken as the half of the food thickness) (m); $A_i$ denotes the food-headspace interfacial area (m²), R is the ideal gas constant, T is the storage temperature (K) and $p_{i,HS,i}$ is the partial pressure of gas species i at the food-headspace interface (Pa). Assuming that equilibrium is reached, this latter term $p_{i,HS,i}$ can be derived from the molar concentration at the food surface using Henry's law:

$$p_{i,HS,i} = \frac{C_{i,F} |_{\partial\Omega_{top}^{food}}}{S_{i,F}} \quad (8)$$

where $C_{i,F} |_{\partial\Omega_{top}^{food}}$ denotes the food concentration of gas species i at the food-headspace interface (mol.m⁻³) and $S_{i,F}$ is the solubility coefficient of gas species i in food (mol.Pa⁻¹.m⁻³).

### 3.2.4. Gas transfer at food-tray interface

At the food-tray interface, denoted as $\partial\Omega_{in}^{tray}$, the transfer of gas species i between the food and the tray is given by the following boundary condition:

$$D_{i,F} \nabla C_{i,F} = k_{i,\partial\Omega_{in}^{tray}} (C_{i,T} - K_i C_{i,F}), \quad \forall (r, z) \in \partial\Omega_{in}^{tray}, \forall t \ge 0 \quad (9)$$

where $k_{i,\partial\Omega_{in}^{tray}}$ is the mass transfer coefficient at this interface, also calculated using Equation (7) with a Biot number, and $K_i$ (dimensionless) is the partition coefficient defined as the ratio of the concentration of gas species i in the tray domain $C_{i,T}$ at equilibrium, to the concentration of gas species i in the food domain $C_{i,F}$ at equilibrium, as described by (Tehrany and Desobry, 2004):

$$K_i = \frac{C_{i,T}}{C_{i,F}} = \frac{S_{i,T}}{S_{i,F}} \quad \text{at equilibrium and for } (r, z) \in \partial\Omega_{in}^{tray} \quad (10)$$

where $S_{i,T}$ and $S_{i,F}$ are the solubilities of gas species i in the tray domain and in the food domain, respectively (mol.m⁻³Pa⁻¹).

### 3.2.5. Other boundary conditions

At the interface between the tray and the external atmosphere, denoted as $\partial\Omega_{ext}^{tray}$, similarly to the flux at the food-headspace interface (Equations (6)–(8)) (Chaix et al., 2015a), the flux is given by:

$$D_{i,T} \nabla C_{i,T} = k_{i,\partial\Omega_{ext}^{tray}} (C_i^{ext} - C_{i,T}) \quad \text{for } (r, z) \in \partial\Omega_{ext}^{tray}, \forall t \ge 0 \quad (11)$$

where $k_{i,\partial\Omega_{ext}^{tray}}$ is the mass transfer coefficient at this interface, also calculated using Equation (7). This calculation considers the diffusivity $D_{i,T}$ in the tray, a Biot number of 10⁵ and a characteristic length taken as the half of the tray thickness. $C_i^{ext}$ represents the concentration of species i (O₂ or CO₂) in the external atmosphere (mol.m³).

The tray material in contact with the food is assumed to have no exchange with the tray part in contact with the headspace, i.e. interface $\partial\Omega_{top}^{tray}$, and with the bottom part of the tray, i.e. interface $\partial\Omega_{bottom}^{tray}$. The fluxes at these interfaces are then expressed by the null flux condition:

$$D_{i,T} \nabla C_{i,T} = 0 \quad \text{for } (r, z) \in \partial\Omega_{top}^{tray} \cup \partial\Omega_{bottom}^{tray}, \forall t \ge 0 \quad (12)$$

The mass transfers at the interface between the bottom of the cup and

the storage surface, e.g. shelf, are neglected. Solubilization of gases from food to the bottom part of the cup is also neglected, based on the small volume of this part and the low gas solubilities. Consequently, the model considers a null flux condition at the interface between the food and the bottom of the tray.

$$D_{i,F} \nabla C_{i,F} = 0 \quad \text{for } (r, z) \in \partial\Omega_{bottom}^{food}, \forall t \ge 0 \quad (13)$$

At boundary corresponding to the axis of symmetry, $\partial\Omega_{axis}^{food}$, a null flux is also applied.

$$D_{i,F} \nabla C_{i,F} = 0 \quad \text{for } (r, z) \in \partial\Omega_{axis}^{food}, \forall t \ge 0 \quad (14)$$

### 3.2.6. Mass balance in the headspace

For gas species i (O₂, CO₂, N₂) involved in the mass balance in headspace, the variation of gas concentration depends on the permeation through the lid film and the tray (equations (2) and (3)), the solubilization at the food-headspace interface (equations (6)–(8)) and the dilution effect due to the variation of the headspace volume. Assuming that temperature and pressure remain constant, the instantaneous volume variation was obtained from the ideal gas law, leading to the following equation:

$$\frac{dV_{HS}}{dt} = \frac{RT}{P_T} \left( \varphi_{O_2}^L + \varphi_{O_2}^T + \varphi_{O_2}^I + \varphi_{CO_2}^L + \varphi_{CO_2}^T + \varphi_{CO_2}^I + \varphi_{N_2}^L + \varphi_{N_2}^T \right) \quad (15)$$

where $V_{HS}$ is the headspace volume (m³) and $P_T$ is the total pressure in the headspace (Pa).

Consequently, the mass balance of gas species i (O₂, CO₂, N₂) is written as:

$$\frac{dC_{i,HS}}{dt} = \frac{1}{V_{HS}} \left( \varphi_i^L + \varphi_i^T + \varphi_i^I - C_{i,HS} \frac{dV_{HS}}{dt} \right) \quad (16)$$

where $C_{i,HS}$ is the headspace concentration (mol.m⁻³) of gas species i (O₂, CO₂, N₂).

### 3.2.7. Initial conditions

The initial partial pressure for gas species i (O₂, CO₂, N₂) in headspace is calculated using the following relationship:

$$p_{i,HS}(t=0) = P_T \frac{x_{ini,i}}{100} \quad (17)$$

where $x_{ini,i}$ (%) is the initial percentage of gas species i in the headspace after the flush for setting the modified atmosphere.

The initial concentration of gas species i (O₂ and CO₂) in the food (mol.m⁻³) is assumed to be homogeneous, meaning that the food material is at equilibrium with the external atmosphere. The effect of the gas flush for establishing the modified atmosphere is neglected due to the usually very short duration of this step. The initial concentration of gas species i (O₂ and CO₂) in the food is then expressed according to Henry's law as:

$$C_{i,F}(t=0, r, z) = p_i^{ext} S_{i,F}(T) \quad \text{at } t=0, \forall (r, z) \in \text{Food} \quad (18)$$

where $p_i^{ext}$ is the partial pressure of gas species i in the external atmosphere (Pa) and $S_{i,F}(T)$ is the solubility of gas species i in the food (mol.m⁻³.Pa⁻¹) at the considered storage temperature T (K).

Similarly, the initial concentration of gas species i (O₂ and CO₂) in the tray compartment (mol.m⁻³) is assumed to be homogeneous and is written as:

$$C_{i,T}(t=0, r, z) = p_i^{ext} S_{i,F}(T) \quad \text{at } t=0, \forall (r, z) \in \text{Tray} \quad (19)$$

where $S_{i,F}(T)$ is the solubility of gas species i in the tray compartment (mol.m⁻³.Pa⁻¹) at the considered storage temperature T (K).

### 3.2.8. Temperature dependency – Arrhenius equation

Generally, the transfer parameters are determined at standard con-
ditions, and especially at standard temperature, e.g. 23 °C for the gas permeabilities. To be able to simulate any storage temperature, the diffusivities, solubilities and permeabilities are assumed to follow the Arrhenius law, expressed hereafter for a generic mass transfer parameter $X_i$ for gas species i, at temperature T (K)

$$X_i(T) = X_{0,i} \exp \left( - \frac{E_{a,X_i}}{R} \left( \frac{1}{T} - \frac{1}{T_{\text{ref},X_i}} \right) \right) \quad (20)$$

where $X_{0,i}$ is the value of the parameter at its reference temperature $T_{\text{ref},X_i}$ (K), $E_{a,X_i}$ is the activation energy (kJ.mol⁻¹) and R is the ideal gas constant (J.mol⁻¹.K⁻¹). All the parameters are gathered in Table 1.

### 3.3. Simplified models

#### 3.3.1. Model for empty packaging

For simulation purpose of an empty packaging, the model relied on simple permeation equations between headspace compartment and the external atmosphere through the lid film and the cup (Equations (2) and (3)), considering the full inner lateral surface area, the mass balance and initial condition in the headspace compartment (Equations (15)–(17)) and the Arrhenius law for the temperature dependency (Equation (20)).

#### 3.3.2. 2D insulated model

To investigate the contribution of food-tray gas transfers, an alternative model, named the 2D insulated model, was considered. This model is strictly similar to the one described in part 2.2, apart from a difference to the food/tray interface where the null flux was applied, so that Eq. (9) was replaced by Eq. (21).

$$D_{i,P} \nabla C_{i,P} = 0, \text{ for } (r, z) \in \partial \Omega_{\text{in}}^{\text{tray}}, \forall t \ge 0 \quad (21)$$

#### 3.3.3. 1D model

For comparison, a 1D model (Chaix et al., 2015a; Guillard et al., 2016) was also considered. This model was similar in terms of the mathematical formulation of gas permeation through packaging, gas solubilization at the food-headspace interface, and gas diffusion in the food compartment. Like the 2D insulated model, this 1D model did not consider any flux at the food-tray vertical interface or at the food bottom interface. Finally, this model, coupled with the no-flux conditions, reduced the diffusion to a mono dimensional one, along the thickness of the food material.

### 3.4. Numerical simulations

Simulation with the 1D model were conducted using Matlab (version 2017b) based on the finite difference method and the built-in ODE15s solver.

Both the 2D model and the 2D insulated model were implemented in COMSOL Multiphysics (version 5.3a) based on the finite element method. An adaptive-size triangular mesh was considered to refine the mesh near the interfaces (Fig. 1C). For gas species i (O₂ and CO₂), the spatial mesh size $\Delta l$ of a single mesh element, and the time increment $\Delta t$ followed the criterion $\Delta t \le \sqrt{\Delta t \times D_i/2}$ (Thomas and Zhou, 1997), to avoid spurious oscillations at the first time steps (Vermeer and Verruijt, 1981). Numerical solving was based on the MUMPS time-dependent solver (Multifrontal Massively Parallel Solver) which implements a parallel distributed LU factorization of large sparse matrices.

The simulations were run on the same PC having 2.60 GHz, x64-based processor with 32 GB of RAM. The typical simulation duration was less than 2 min for the 2D model on COMSOL and less than 30 s for 1D model on Matlab.

### 3.5. Global sensitivity analysis

The Elementary Effects Method, also known as the Morris Method

(Morris, 1991), was selected to perform the global sensitivity analysis on the 2D model. The Morris method belongs to the OAT (“One At a Time”) group, meaning it aims to characterize locally the effect of the variation of a single input on the variation of the output. However, the method is designed to explore the parameter space.

For a generic model $Y = f(X) = f(X_1, X_2, ..., X_K)$ where $Y \in R^M$ is the output vector and $X_1, X_2, ..., X_K$ are K input parameters, the Morris method assumes independence of the input parameters and requires a defined range for each input parameter. The input space is transformed into a K-dimensional grid by defining a constant number of levels for each range. The Morris method explores the parameter space by defining N trajectories. A single trajectory starts from an initial input point, randomly drawn in the grid. Then, the inputs were sequentially modified, one at a time, to build a $(K+1)$ points trajectory. Thus, for each input, $X_k$ ($1 \le k \le K$), there is one and only one variation of this input for each trajectory. Once the outputs are calculated for the entire set of points, i.e. $N(K+1)$ runs of the model, the Morris method computes the elementary effects $EE_{n,k}$, where $n \in [1, N]$ is the trajectory index and $k \in [1, K]$ is the input index, expressed as:

$$EE_{n,k} = \frac{Y(X + e_k \Delta_k) - Y(X)}{\Delta_k} \quad (22)$$

where $e_k$ is the standard unit vector with the k-th element equal to 1, $\Delta_k$ is the variation of the k-th input for the n-th trajectory and X is the base point for this variation.

Two sensitivity measures are proposed to assess the overall importance of the input and the interaction and nonlinear effect (Morris, 1991): the mean and the standard deviation of the elementary effects. The mean of absolutes effects $\mu^*$ (Equation (23)) was proposed by Campolongo et al. (2007) as a replacement for the standard mean, to avoid compensation of the effects. This measure $\mu^*$ allows the identification of non-influential and influential inputs, and the ranking of these inputs, i.e. the highest the measure $\mu^*(k)$, the more important the input $X_k$.

$$\mu^*(k) = \frac{1}{N} \sum_{n=1}^{N} |EE_{n,k}| \quad (23)$$

The second sensitivity measure is the standard deviation of the elementary effects $\sigma$ (Equation (24)) which provides information regarding the linearity of the effect and the potential presence of interactions.

$$\sigma(k) = \sqrt{\frac{1}{N-1} \sum_{n=1}^{N} \left( EE_{n,k} - \frac{1}{N} \sum_{n=1}^{N} EE_{n,k} \right)^2} \quad (24)$$

The ratio $(\sigma/\mu^*)$ allows for the detection of the qualitative behavior of the effects, including parameter interactions (Sanchez et al., 2014): almost linear elementary effect of the input $(\sigma/\mu^* \in [0, 0.1])$, a model response monotonic with respect to the input $(\sigma/\mu^* \in [0.1, 0.5])$, a model response almost monotonic $(\sigma/\mu^* \in [0.5, 1])$ and inputs that lead to a non-monotonic or non-linear model response or presented interactions with other inputs $(\sigma/\mu^* \in [1, +\infty])$.

In this work, SAFE (Sensitivity Analysis For Everyone), an open-source Matlab Toolbox for Global Sensitivity Analysis (Pianosi et al., 2015), was employed to implement the Morris method.

## 4. Results and discussions

### 4.1. Experimental gas evolution into headspace

After an initial gas flush (30 %–70 % CO₂–N₂) to establish the modified atmosphere, the evolution of the headspace gas composition (O₂ and CO₂) was experimentally assessed at 10 °C, for both the packed food and the empty closed packaging, see Fig. 3. At initial time, the
<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Experimental evolution at 10 °C of mean headspace percentage of O₂ (blue solid circles) and CO₂ (black empty circles) after an initial flush with 30%–70% CO₂–N₂ for an empty packaging (left) and a packed food (right). Error bars represent the experimental standard deviation obtained from triplicates. Error bars are included in the symbol for O₂ results (blue solid circles).</figcaption></figure>

<figure><img src="image_7.png" /></figure>

headspace gas composition did not precisely match the target mixture, with a deviation for O₂ (0.9 % instead of 0 %) in the packed food headspace, and a deviation for CO₂ (29.4 % instead of 30 %) in the empty packaging headspace, indicating that a total flush could not be achieved under these conditions.

For the empty packaging (Fig. 3), the combined permeation through the cup material and the lid film led to a decrease of the CO₂, from 29.4 % to 17.3 % after 49 days, and an increase of the O₂, from 0.1 % to around 4.1 % after 49 days, both showing a nearly linear trend confirming that permeation governs the transfer in that case. For the packed food (Fig. 3), an increase of O₂, from 0.9 % to 11.2 % after 83 days, and a decrease of CO₂, from 30.0 % to 3.2 % after 83 days, were observed, presenting a more complex behavior that cannot be explained by permeation alone. Although permeation occurred throughout the experiment, the rapid changes observed during the first 4 days were driven by the sorption of CO₂ into the food and the desorption of solubilized O₂ from the food, followed by a slower and linear dynamic where permeation became predominant. Assuming the food was at equilibrium with the standard atmosphere, i.e. 20.95 % of O₂ and 0.04 % of CO₂ (Widory and Javoy, 2003), and neglecting the perturbation due to the flushing step, a gradient of O₂ and CO₂ between the food and the headspace was present at the initial time, i.e. after the packaging closing by thermosealing, inducing a release of O₂ from the food to the headspace and a sorption of CO₂ from the headspace to the food. Due to the high solubility of CO₂, and in a lesser extent of O₂, in food, the food compartment significantly impacted the headspace composition.

This observation was previously highlighted by Chaix et al. (2015a); Guillard et al. (2016) and Guillard et al. (2017) who studied headspace gas evolution in modified atmosphere packaging (MAP) of fresh products, including processed cheese. The authors showed that the sorption and desorption of gases in food strongly depended on its composition, particularly on water and fatty phases (Sivertsvik et al., 2004; Jakobsen and Bertelsen, 2006; Chaix et al., 2015b; Münch et al., 2023).

## 4.2. Validation of the 2D predicting model

First, the oxygen transmission rate (OTR) of the tray was experimentally assessed at 23 °C and room relative humidity, resulting in a value of 0.401 ± 0.028 cm³.pack⁻¹.day⁻¹. Using Equation (1) and the geometrical parameters from Table 1, the OTR was converted into an oxygen permeability, yielding a value of 3.44 · 10⁻¹⁶ ± 0.24 · 10⁻¹⁶ mol.m.m⁻².Pa⁻¹.s⁻¹. The ratio between carbon dioxide permeability

and oxygen permeability, known as CO₂/O₂ permeability selectivity, for polypropylene (the material of the tray) was taken equal to 3.16 (Siracusa, 2012). Therefore, the carbon dioxide permeability of the tray was calculated as 1.09 · 10⁻¹⁵ ± 0.08 · 10⁻¹⁵ mol.m.m⁻².Pa⁻¹.s⁻¹. Similarly, nitrogen permeability was estimated by considering the N₂/O₂ permeability selectivity for polypropylene, which is 4.3 (Siracusa, 2012), resulting in a nitrogen permeability value of 8.00 · 10⁻¹⁷ ± 0.56 · 10⁻¹⁷ mol.m.m⁻².Pa⁻¹.s⁻¹.

All other parameters used in the model were sourced from the literature (see Table 1). For the food compartment, O₂ and CO₂ diffusivities and CO₂ solubility in cheese were taken from works of Chaix et al., 2015b, 2016; the O₂ solubility in cheese was estimated from its value in water (Battino and Clever, 1966; Chaix et al., 2015b). For the tray compartment, O₂ and CO₂ diffusivities for polypropylene were taken from Beltrame et al. (1999) and Siracusa and Ingrao (2017). Finally, activation energies were extracted from the literature: for O₂ and CO₂ diffusivities and solubility in food (Simpson et al., 2004), O₂, CO₂ diffusivities, solubility and permeability, as well as N₂ permeability in the tray (Kurek et al., 2011), and O₂, CO₂ and N₂ permeability in the film (Charles et al., 2005; Siracusa, 2012).

**Simulation of headspace gas composition.** Numerical simulations were performed with the 2D model, corresponding to Equations (2)–(20), testing the two hypotheses of variable and constant headspace volume. The simulated results of O₂ and CO₂ in headspace were compared to the experimental results in Fig. 4.

First, the 2D model accurately reproduced the experimental trend for both empty tray and packed cheese, regardless the assumption made for the headspace volume, i.e. constant or variable. It is worth noting that no parameter estimation was made to increase the fit between the simulations and the experimental data. However, results showed a better fit of the 2D model using a constant headspace volume to the experimental data for the percentage of O₂ in headspace of the packed food, with an RMSE of 0.62, compared to the 2D model using a variable headspace volume, which had an RMSE of 1.29. In contrast to O₂, the models using constant and variable headspace volume showed lower difference in the fitting to the experimental data for the percentage of CO₂ in headspace, with an RMSE of 0.99 and 0.66 respectively.

Under the hypothesis of variable headspace volume, the simulated headspace volume was reduced from 45.6 mL to about 37.2 mL, an 18.4 % reduction, in the first 5 days for packed cheese (data not shown). This decrease in headspace volume for packed food results from the high sorption of CO₂ in the food compartment, which was not
<figure><img src="image_8.png" /><figcaption>**Fig. 4.** Comparison of experimental headspace gas composition (solid blue circles for O₂ and empty black circles for CO₂) and simulated headspace gas composition obtained by using the 2D model with hypothesis of a variable headspace volume (black lines) and a constant headspace volume (gray lines) for an empty packaging (left column) and a packed food (right column) stored at 10 °C. Error bars represent the experimental standard deviation obtained on triplicates.</figcaption></figure>

counterbalanced by the desorption of O₂ from the food, and the permeation fluxes. Nevertheless, as no variation of the headspace volume was experimentally observed and since the hypothesis of constant headspace volume led to a better fit for O₂, it was decided to consider a constant headspace volume hypothesis in all calculations as the most realistic hypothesis for the rest of this study.

**Simulation of solubilized gas concentration in food.** Fig. 5 shows the O₂ and CO₂ concentrations in the food compartment, which were highly impacted by CO₂ sorption and O₂ desorption in the first 10–20 days, and then by the permeation through the tray after 10–20 days. Indeed, during the first days of storage, a concentration gradient appeared, orthogonal to the interface, due to the diffusion of gases into the cheese and the sorption/desorption at the food-headspace interface. After 1 day, the desorption of O₂ led to a high gradient of O₂ concentration in the cheese with the highest concentration of O₂, around 0.34 mol.m⁻³ at the bottom and the lowest concentration of O₂, around 0.03 mol.m⁻³ at the food-headspace interface. Similarly, due to the sorption of CO₂, its concentration in the food after 1 day was maximal near the food-headspace interface, with a value of 6.3 mol.m⁻³, and minimal at the bottom, with a value of 0.02 mol.m⁻³. Additionally, the isoclines of concentration were close to horizontal lines, indicating no major contribution of the gas transfer at the food-tray interface.

After approximately 10 days for O₂, and 20 days for CO₂ (Fig. 5), a significant gradient appeared near the food-tray interface, with the highest O₂ concentrations and lowest CO₂ concentrations occurring close to the tray. As the external atmosphere was poorer in CO₂ and

richer in O₂ than the food, gas transfer between the food and the tray, and through the tray material, led to the sorption of O₂ and desorption of CO₂ in the food at this food-tray interface. The results showed a heterogeneity of O₂ and CO₂ concentrations in the food compartment.

For CO₂, even though a concentration gradient was present throughout the shelf-life of the product, it is worth noting that after 40 days, the concentration was quite homogeneous, with a span of about 0.1 mol.m⁻³ but was still decreasing, due to the CO₂ outgoing flux from the food to the tray and external atmosphere. This indicates that the decrease in headspace %CO₂ observed experimentally was not purely the consequence of sorption in the food. This suggests that, due to the transfer through the food-tray interface and the tray, the CO₂ solubilized in the food will exit the packed food system, reducing its bacteriostatic effect. Consequently, the dimensioning of packaging, i.e. the selection of materials with adapted mass transfer parameters, for such packed injected food, should consider the transfers at food-packaging interfaces.

### 4.3. Comparison of 2D model and simplified models

To assess the importance of finely modeling the food/packaging system, especially the transfer at food-tray interface, a comparison with a simplified model was made. For this purpose, the flux at the food-tray interface was assumed to be null in the streamlined version of the 2D model, namely the 2D insulated model, corresponding to equations (2)–(8) and 10 to 21. A 1D model adapted from Chaix et al. (2015a) was also implemented for comparison (see section 2.3.2). The evolution of O₂ and CO₂ in the headspace was simulated for these three models and the
<figure><img src="image_9.png" /><figcaption>**Fig. 5.** 2D visualizations of the simulated gas concentration (mol.m⁻³) in food for O₂ (left) and CO₂ (right) at initial time, i.e. after gas flushing, and after 1 day, 10 days, 20 days, 40 days and 60 days of storage. The 2D model considers a constant headspace volume.</figcaption></figure>

results are given in Fig. 6.

Although all models were able to accurately reproduce the experimental trends for the very first days, it was clear that the complete 2D model was the only one able to reproduce the behavior for the full duration of 80 days. These results revealed that the 2D insulated model underestimated the real O₂ ingress and CO₂ output from the system. This gap was particularly noticeable after 10 days, when the gas sorption/desorption at food-headspace interface became minor and evolution of gases in headspace became strongly dependent on permeation through packaging. Indeed, after around 10 days, the discrepancy between the 2D model and the simplified ones, became increasingly apparent, reaching, after 83 days, an O₂ quantity of 11.9 % for the 2D model and 5.8 % for both the 2D insulated model and the 1D model, and a CO₂ quantity of 2.5 % for the 2D model and 5.7 % for both the 2D insulated

model and the 1D model. These results revealed that the gas flux at the food-tray interface could not be neglected in the present case study where contact between food with sides of the tray was significant. This observation confirms previous conclusions from the literature, such as Bacigalupi et al. (2013) who successfully validated a model for packed orange juice that included gases exchanges through the food-packaging interface (high contact between food and packaging in this case too) with experimental data.

A closer look at the results for the two simplified models, i.e. insulated 2D model and 1D model, showed a difference for days 0–20. This was explained by the simplification made on the geometry of the tray. Indeed, the 1D model assumed a rectangular tray. To preserve the headspace volume, headspace-tray interface area, food volume, food-headspace interface area, it was mandatory to move from the
<figure><img src="image_10.png" /><figcaption>**Fig. 6.** Comparison between experimental data for headspace gas composition (solid blue circles for O₂ and empty black circles for CO₂) and simulated data from the 2D model with constant headspace volume (solid black lines), from the 2D insulated model with constant headspace volume (dotted gray lines) and from a 1D model adapted from Guillard et al. (2016) (dashed black lines). Error bars are experimental standard deviation obtained on triplicates.</figcaption></figure>

trapezoidal 2D shape of the food to a rectangular one, by adapting the food thickness (0.0415 m for 2D model and 0.0456 m for 1D model). This change in food thickness was responsible for the small misestimation of sorption/desorption of gases in food during the first 20 days of storage for the 1D model compared to the 2D insulated model.

For injected or molded food product and when gas transfers at food-tray interface are not negligible, the 2D model had shown to accurately reproduce the evolution of gas content in the system. However, when the transfers at food-tray interface can be neglected, e.g. for a tray material with a very high barrier to gases, the 2D insulated model could be relevant. Finally, aside from the injected food type, for food products that have no or negligible contact with the vertical sides of the tray, the 1D model could still give valuable prediction. These conclusions on the usage of simplified models were also confirmed by results observed in literature. Indeed, Chaix et al. (2015b); Guillard et al. (2016); and Simpson et al. (2004) successfully used a 1D model for poultry, meat, or salmon, as these foods have low or no contact with the vertical sides of the packaging. Similarly, Fabiano et al. (2000) successfully modeled the composition of gases in the headspace of the packaging for pesto by neglecting the flux at food-tray interface as the permeability of the tray, being a multilayer composed of polypropylene and polyamide, was very low (and consequently gases fluxes were negligible).

## 4.4. Global sensitivity analysis on the 2D model

The development of the 2D model required a high number of input parameters, such as O₂ and CO₂ permeability, diffusivity and solubility, activation energy, etc., most of which were majority estimated through literature (Table 1). However, due to the complexity of some experimental measurements (as solubility or diffusivity), there is a high variability of such values in literature for the same parameter. In this context, it is essential to understand the impact of this variability of input values on the numerical. The mass transport properties of food, tray, lid film (permeability, solubility and diffusivity of O₂ and CO₂ and activation energy of these parameters) and the storage temperature (T) were investigated to better understand the influence of their variations on the output of the 2D model, i.e. percentages of O₂ and CO₂ in the headspace. The global sensitivity analysis was performed for the current case study, by considering constant and accurate geometrical

parameters, that were excluded from the analysis. The Morris global sensitivity analysis method was employed to perform the parameters screening of the 2D model, as its computational cost is cheap, its reliability in ranking the dominant parameters is high (Menberg et al., 2016) and it can qualitatively infer basic information on the relationship between the input and output (called parameter interaction) (Yang, 2011). As the Morris method requires the independency of the uncertain parameters, a specific treatment was applied to the mass transfer parameters for the tray compartment in contact with the food. Indeed, the 2D model considered both gas permeability *P*, solubility *S* and diffusivity *D*, which are interdependent, following the relationship *P* = *D* × *S*. Consequently, permeability and solubility were considered as uncertain parameters, and the diffusivity was assessed using the relationship *D* = *P*/*S*.

A total of 21 uncertain parameters (on 37 parameters, Table 1) were considered in the Morris method, with 100 random trajectories, to avoid biased estimations (Menberg et al., 2016), representing a total of 2200 simulations of the 2D model. To rank the uncertain parameters regarding their effect on headspace gas composition, statistics were performed on the elementary effects to obtain μ*, the mean of the absolute elementary effects, and σ the standard deviation of the elementary effects. These descriptors μ* and σ were calculated at each storage time, from 1 to 60 days.

After 60 days, i.e. at the end of the expected shelf life, a small group of 5–6 most influential parameters was identified, with elementary effect μ* generally over 1 (Fig. 7). The rest of the inputs can be considered as non-influential since they present very low values for both μ* and σ. At this time, the most influential parameter regarding the headspace %O₂, was the oxygen permeability of the tray, *P*<sub>O₂,T</sub> at reference temperature with μ* > 10. Two clusters of parameters had significant effects with the oxygen diffusivity in food *D*<sub>O₂,F</sub>, and the activation energy of the oxygen tray permeability *E*<sub>a,P<sub>O₂</sub>,T</sub> around μ* = 4, followed by oxygen solubility in food *S*<sub>O₂,F</sub> and storage temperature *T* around μ* = 1.5. For % CO₂, the most influential parameter was, again, the permeability, for CO₂, of the tray, *P*<sub>O₂,CO₂,T</sub> at reference temperature, with a value around μ* = 3.5, followed by the solubility in food *S*<sub>O₂,CO₂,F</sub> (μ* ~ 2.4), and a cluster of three low influential inputs composed of the carbon dioxide diffusivity in food *D*<sub>O₂,CO₂,F</sub>, the temperature *T* and the activation energy
<figure><img src="image_11.png" /><figcaption>**Fig. 7.** Results of the parameter screening with the Morris method for two outputs being the percentages of O₂ (left) and of CO₂ (right) in the headspace after 60 days of storage, i.e. the expected shelf life. The list of considered inputs is given in Table 1. Inputs are plotted according to the sensitivity measures EE μ*, i.e. the mean of absolutes elementary effects, and EE σ, i.e. the standard deviation σ of elementary effects, obtained for 100 trajectories (2200 runs of the 2D model). The labels were only mentioned for the influential parameters. The non-influential parameters are in a gray cluster. Gray lines separate the (μ*, σ) space into domains describing different qualitative behaviors of the input's effects.</figcaption></figure>

of the carbon dioxide solubility in food $E_{a,SCO_2,F}$ at a value around $\mu^* = 0.5$.

The permeability of the tray to O₂ and CO₂ were the most influential parameters for %O₂ and %CO₂ respectively. This is logical since these parameters influence the permeation fluxes between headspace and external atmosphere through the tray and between the food and the external atmosphere through the tray. However, at permeability constant, the variability on gases solubility in the tray (meaning that the variation of solubility is compensated by a variation of diffusivity) does not require precise data, which are more complex to determine compared to permeability. Therefore, only the product of solubility and diffusivity is important to be precisely determined.

On the contrary, the gas permeabilities of the film were non-influential, so they do not need high precision values. For the considered case study, the tray material had a much lower barrier to gases, with permeabilities 30 to 50 times higher compared to those of the film, which could explain the non-influence of film gas permeabilities. However, the thickness ratio between the tray and film is close to 10, reducing the gap between the gas transfer properties of tray and film. For example, the ratio of permeances (permeability * thickness) between tray and film then varied from 3 to 5. By considering the large variations ranges for permeabilities (from *0.1 to *10), the sensitivity analysis actually explored various combinations including cases with higher permeances for the film compared to the film. Finally, this non-influence of gas permeabilities of the film is not only related to the gap in values but probably account for the largest contribution of transfer at food-tray interface on the evolution of the headspace composition and overall dynamic.

Gas solubility and diffusivity in the food had moderate to high influence, since they were related to gas sorption/desorption at the food-headspace interface and to the transfer in food and at the food-tray interface.

Most of the influential parameters had a ratio $\sigma/\mu^*$ between 0.5 and 1, for which the outputs should present almost monotonic response for these parameters. Some parameters presented a ratio $\sigma/\mu^* > 1$, e.g. the carbon dioxide diffusivity and solubility in food $D_{0,CO_2,F}$ and $S_{0,CO_2,F}$, which could be related to non-monotonic, non-linear response or presence of interactions with other inputs. Indeed, the Morris method is not able to quantify the interactions between effects, and presence of some interactions could be expected, such as effects of temperature, permeability and its activation energy, which were used to calculate the permeability at storage temperature, or interactions between effects of the gas solubility and diffusivity in food since they were both involved in the sorption/desorption capacity and kinetic, and they also contributed to the transfer in food and at food-tray interface.

The relatively low effect of storage temperature was explained by the range imposed, i.e. the target temperature ±2 K, that correspond to precision generally observed in the storage rooms or domestic fridges.

Even if the storage duration was not considered as an input of the model, calculations of the elementary effects were performed for each time, e.g. in order to determine the role of a change in the target shelf-life, and the results are presented in Fig. 8 for the influential parameters. The influential parameters observed after 60 days of storage (Fig. 7) remained influential at every time of the product shelf-life (Fig. 8). The activation energy of oxygen solubility in the food appeared to have a low but significant impact on %O₂ for short times, even if it was not influential after 60 days.

The main result was that the ranking of the influential parameters strongly depended on the storage duration. Indeed, for short times, the solubility and diffusivity of O₂ and CO₂ in food were the most influential parameters on the headspace %O₂ and %CO₂ respectively. The duration had to be higher than respectively 7 days for %O₂ and 31 days, half the shelf-life, for %CO₂ to see the effect of gas tray permeability become dominant over the effect of gas solubility in food. This result highlighted the impact of sorption/desorption of O₂ and CO₂ in/from food on the evolution of headspace gas composition, as presented before. For similar food systems, i.e. liquid or injected food, with shorter shelf-life, one should take care to perform a parameter screening, to ensure that the precision of the gas transfer parameters in food (solubility and diffusivity) was high enough to obtain reliable outputs.

## 5. Conclusions

A 2D axis-symmetric model of gas transfers was proposed and successfully validated for an injected food product (fresh cheese). The model successfully reproduced the evolution of O₂ and CO₂ concentrations in the headspace. The results showed a significant role of gas sorption/desorption at the food-headspace interface in the short time, while permeation through the tray became predominant later. For the food model considered (spreadable cheese), the ratio headspace volume on food volume was low, the gas transfers at the food-tray interface
<figure><img src="image_12.png" /><figcaption>**Fig. 8.** Results of the parameter screening with the Morris method for two outputs, the percentages of O₂ (left column) and of CO₂ (right column) in the headspace depending on storage time. Evolution of sensitivity measures EE µ⁺ (upper row), i.e. the mean of absolutes elementary effects, and EE σ (lower row), i.e. the standard deviation σ of elementary effects, is plotted for the influential inputs only.</figcaption></figure>

could not be neglected, and that 1D model failed to predict the experi-
mental results. Consequently, modelling such food type requires a 2D
model, while for food having low contact with tray, the model could be
simplified to a 1D geometry.

Moreover, the global sensitivity analysis conducted using the Morris
method showed a significant impact of O₂ and CO₂ permeability and
their activation energies, as well as O₂ and CO₂ solubility and diffusiv-
ities in food, and storage temperature. A more in-depth study is neces-
sary to better quantify the interactions between these impacts. However,
this analysis emphasizes the importance of having precise values for
these parameters, and especially confirms the importance of gas solu-
bility in food. This study focuses on a realistic case study but for a
specific tray geometry, headspace/food volume ratio, and tray gas
permeability higher than that of the film, etc. The model and its nu-
merical exploration could lead to more generic results that could pave
the way to establishing recommendations for the packaging of injected
food.

CRediT authorship contribution statement

**Anh Khoa Nguyen:** Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Data curation. **Fanny Coffigniez:** Writing – review & editing, Validation, Software, Methodology, Conceptualization. **Valérie Guillard:** Writing – review & editing, Project administration, Funding acquisition. **Sébastien Gaucel:** Writing – review & editing, Visualization, Supervision, Software, Methodology, Formal analysis.

Funding

This project has received funding from the European Union's Hori-
zon 2020 research and innovation program under grant agreement No
773375.

Declaration of competing interest

The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.

Data availability

Data will be made available on request.

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