# Modelling CO 2 transfer in foil ripened semi-hard Swiss-type cheese

F. Acerbi, Valérie V. Guillard, M. Saubanère, C. Guillaume, Nathalie Gontard

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F. Acerbi, Valérie V. Guillard, M. Saubanère, C. Guillaume, Nathalie Gontard. Modelling CO 2 transfer in foil ripened semi-hard Swiss-type cheese. Journal of Food Engineering, 2018, 222, pp.73 - 83. 10.1016/j.jfoodeng.2017.10.025. hal-01645481

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Modelling CO₂ transfer in foil ripened semi-hard Swiss-type cheese

F. Acerbi, V. Guillard, M. Saabanere, C. Guillaume, N. Gontard

PII: S0260-8774(17)30458-2

DOI: 10.1016/j.jfoodeng.2017.10.025

Reference: JFOE 9058

To appear in: *Journal of Food Engineering*

Received Date: 30 May 2017

Revised Date: 27 October 2017

Accepted Date: 28 October 2017

Please cite this article as: Acerbi, F., Guillard, V., Saabanere, M., Guillaume, C., Gontard, N., Modelling CO₂ transfer in foil ripened semi-hard Swiss-type cheese, *Journal of Food Engineering* (2017), doi: 10.1016/j.jfoodeng.2017.10.025.

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1 Modelling CO₂ transfer in foil ripened cheese with CO₂ production

2 Modelling CO₂ transfer in foil ripened semi-hard Swiss-type cheese

3

4 F. Acerbi<sup>a</sup>, V. Guillard<sup>a*</sup>, M. Saabanere<sup>b</sup>, C. Guillaume<sup>a</sup>, N. Gontard<sup>a</sup>

5 <sup>a</sup>UMR 1208 IATE Agro-polymer Engineering and Emerging Technologies, Montpellier (France),

6 University of Montpellier, INRA - 2 place Viala - bât. 31, F-34060 Montpellier Cedex 01

7 <sup>b</sup>UMR 5253 - Institut Charles Gerhardt, Montpellier (France), Place Eugène Bataillon, 34090 Montpellier

8 * corresponding author: valerie.guillard@umontpellier.fr

9
**Abstract**

Eye growth in cheese with intense CO₂ production during ripening mainly depends on CO₂ production and transfer properties. Despite CO₂ production and diffusion during ripening of semi-hard Swiss-type cheese eyes in such cheeses are considered as important quality parameters, the research concerning key gas production and transfer in cheese remains widely overlooked. In this study, experimentally assessed CO₂ production was coupled with transfer coefficients in a mathematical model in order to predict CO₂ gradients formed inside the cheese during ripening. The permeability coefficient of CO₂ through the multilayer barrier packaging which wraps the cheese during ripening was also included in the model. The presented model was validated by assessing the CO₂ concentration in the cheese and its partial pressure in the packaging headspace. CO₂ production rate was found to be the most important input parameter affecting CO₂ gradients formed in cheese during ripening whereas the other input parameters (solubility, diffusivity, permeability) had little effect on the total CO₂ gradient.

**Keywords:** solubility, diffusivity, permeability, CO₂, cheese ripening
Nomenclature

<table><tr><td>α</td><td>Level of statistical significance</td></tr><tr><td>A</td><td>Surface (m²)</td></tr><tr><td>Bi</td><td>Biot number (non-dimensional)</td></tr><tr><td>br</td><td>Bottom rind position of the cheese</td></tr><tr><td>C</td><td>Concentration (kg/m³)</td></tr><tr><td>ch</td><td>Relative to cheese</td></tr><tr><td>D</td><td>Effective diffusivity coefficient of CO₂ in cheese (m² s⁻¹)</td></tr><tr><td>exp</td><td>Experimental values</td></tr><tr><td>HS</td><td>Packaging headspace</td></tr><tr><td>I</td><td>Relative to the interface</td></tr><tr><td>j</td><td>Relative to a gas specie</td></tr><tr><td>k</td><td>External mass transfer coefficient (m/s)</td></tr><tr><td>K</td><td>Degree in Kelvin, with K = 273.15 + degree in Celsius</td></tr><tr><td>KP</td><td>Gas permeability through the packaging film (cm³ µm⁻¹ m⁻² d⁻¹ bar⁻¹)</td></tr><tr><td>l</td><td>Cheese thickness (m)</td></tr><tr><td>M</td><td>Molar mass (kg/mol)</td></tr><tr><td>p</td><td>Partial pressure (Pa)</td></tr><tr><td>P<sub>T</sub></td><td>Total pressure (atm)</td></tr><tr><td>Pe</td><td>Gas permeability through the packaging film (mol m⁻¹ s⁻¹ Pa⁻¹)</td></tr><tr><td>PAB</td><td>Propionic Acid Bacteria</td></tr><tr><td>pred</td><td>Predicted values</td></tr><tr><td>PTATN</td><td>ratio of phosphotungstic acid soluble nitrogen on total nitrogen (g/100g)</td></tr><tr><td>S or SCO2</td><td>Solubility coefficient of CO₂ in cheese (mol m⁻³ Pa⁻¹ or mmol kg⁻¹ atm⁻¹) or CO₂ solubility</td></tr><tr><td>SM</td><td>Salt in moisture ratio (g/100g)</td></tr><tr><td>t</td><td>Time (s)</td></tr><tr><td>T</td><td>Temperature (K)</td></tr><tr><td>R</td><td>Universal gas constant: 8.314 (J mol⁻¹ K⁻¹)</td></tr><tr><td>CVRMSD</td><td>Coefficient of variation of the root mean squared deviation</td></tr><tr><td>V</td><td>Volume (m³)</td></tr><tr><td>x</td><td>Position in the cheese (m)</td></tr><tr><td>y</td><td>Experimental or predicted variable</td></tr><tr><td>μ</td><td>average</td></tr><tr><td>φ</td><td>Mass flow (kg s⁻¹)</td></tr><tr><td>ν</td><td>Production rate (kg m⁻³ s⁻¹)</td></tr><tr><td>∞</td><td>Relative to the surrounding atmosphere</td></tr></table>
## 1. Introduction

In food engineering, mathematical modelling of physical mechanisms such as heat or mass transfer was used for many years to simulate and optimize/control operation units such as drying or salting (Bona et al., 2007; Mayor and Sereno, 2004; Payne and Morison, 1999; Santapaola et al., 2013). More recently these models were coupled to biological ones such as Mickaëlis Menten equation that formalize the aerobic respiration (Ho et al., 2011; Guillard et al., 2012), chemical reaction for oxidation (Bacigalupi et al., 2013) or predictive microbiology models in order to better represent the evolution of the packed food during storage (Chaix et al., 2015).

This modelling approach coupling equations coming from different disciplinary fields was recently applied to semi-hard Swiss-type cheese in order to model the eyes' growth during ripening (Laridon, 2014; Laridon et al., 2016). CO₂-source in this type of cheese is propionic acid bacteria (PAB) which means that CO₂-production rate and amount in such cheese is much higher compared to that of lactic acid bacteria. The ripening of these cheeses includes a step during which cheeses are stored at warm temperature (about 1-2 weeks at 20-25°C) for stimulating the PAB fermentation. During this step, CO₂ is intensively produced, leading to the growth of eyes (Fröhlich-Wyder and Bachmann, 2004). Cheeses are often foil ripened and a plastic packaging is used to wrap the cheeses during the whole ripening period for avoiding excessive loss of gaseous compounds. The modelling approach proposed by Laridon (2014) therefore concomitantly took into account mass transfer of gas (CO₂ produced by bacteria and responsible of eyes' growth), production of the CO₂ and mechanical constraint imposed to cheese paste by this production. This model was based on experimentally assessed rheological parameters (stress) and CO₂ production rate in simplified condition, but some input parameters such as CO₂ diffusivity and solubility into the cheese were estimated from the literature and assumed constant throughout the cheese paste and the ripening age. The sensitivity analysis performed by these authors revealed
afterwards the higher influence of CO₂ production and diffusion parameters compared to the rheological ones in semi-hard cheese (Laridon, 2014).

Faced to the importance of CO₂ diffusion and CO₂ production rate in the ripening of semi-hard Swiss-type cheese, this paper aimed at deepening these two phenomena by coupling them in a mathematical model in order to simulate and predict evolution with time of CO₂ gradients in the cheese paste and in the packed cheese (including CO₂ permeation through the ripening foil). In this purpose, the aim of this study was to develop and experimentally validate in various conditions, mimicking the ripening conditions of Swiss-type cheese, a mathematical diffusion-reaction model to predict CO₂ gradients in packed cheeses during ripening.

## 2. Materials and methods

### 2.1 Cheese and packaging

Semi-hard cheese blocks of 1 kg were kindly supplied by a cheese company. The cheeses included about 42% w/w moisture and 26.5% w/w fat, with pH varying from 5.50 to 5.70 from 14 to 28 days from renneting. The cheeses used for the model validation without CO₂ production (same batch) did not include any intentionally added CO₂ producing bacteria (PAB and hetero-fermentative lactic acid bacteria or LAB) and was produced from pasteurized milk to avoid any sources of hetero-fermentative LAB or PAB. The cheeses used for the validation with CO₂ production included PAB (10⁶ CFU ml⁻¹ cheese milk) as main source of CO₂ production. They were produced from a second, dedicated batch from pasteurized milk. After brining, all cheeses were packed in a multilayer barrier packaging film made of PE/PVDC/PE. Indicatively, the permeability to CO₂ at 19°C equalled 40000 cm³ µm m⁻² d⁻¹ bar⁻¹ (Acerbi et al., 2016c). The ripening schedule of all ripened cheeses was as follows: 13°C until 14 days from renneting, 19°C
74 for 14 days, 2–3 days at 13°C, longer storage at 6°C. The target chemical composition of cheeses
75 used in this study is described in Table 1.

76 Cheese blocks were about 15 to 8 cm square in shape (see Supplementary Material A) and were
77 sampled in their core, orthogonally to the interface exposed to headspace/surrounding atmosphere
78 (Figure 1), at least 3 cm away from side rinds, resulting in a cylinder of 8 cm height and about 2
79 cm of diameter. The sampling region was then cut in thin slices of minimum 0.5 cm of thickness
80 for assessment of chemical composition gradient or CO₂ gradient.

81

82

83

84 **2.2 Chemical analyses**

85 The chemical composition (fat, moisture, pH, total nitrogen (TN) and fractions of TN, salt,
86 organic acids) of three cheeses per production was measured, in order to verify that the cheese
87 production was on target. Dry matter and sodium chloride were measured according to the ISO
88 5534 (Anonymous, 2004a) and ISO 5943 (Anonymous, 2006) standards. Total nitrogen (TN),
89 water soluble nitrogen (SN) and phosphotungstic acid soluble nitrogen (PTA-N) were measured
90 using Kjeldahl based methods according to ISO 8968-3 standard (Anonymous, 2004b; Bütikofer
91 et al., 1993). The ratio of PTA-N on TN (g/100g) was considered as a good proteolytic indicator.
92 pH and fat content were characterized according to respectively, FD V04-035 (Anonymous,
93 2009) and NF V04-287 (Anonymous, 2002) French standards. Organics acids were assayed using
94 internal method based on high-performance liquid chromatography.

95 CO₂ determination in cheese was carried out with the protocol described in Acerbi et al. (2016b).

96 The protocol included a first step where CO₂ was extracted from the cheese sample by immersing
97 it in acidic solution. The extracted CO₂ was then scavenged by a soda solution of known
molarity, which was finally titrated with hydrochloric acid. The difference between the initial and
final soda molarity related to the moles which reacted with CO2.

## 2.3 Microbiology analyses

Propionic acid bacteria were enumerated with a method based on the count of the diluted colonies grown on agar plates enriched mainly with sodium lactate and yeast extract after 1 week of incubation at 30°C. The results were expressed in colony-forming unit per gram of cheese (CFU g⁻¹).

## 2.4 Gas chromatography for headspace analyses

The partial pressure of N₂, O₂ and CO₂ was measured (resolution: 0.005%) by injecting 10 µl of headspace of cheese packs inside a micro gas chromatography unit including a thermal conductivity detector (MicroGC 3000, SRA Instruments). Homogeneity of the gas composition was ensured by pumping few millilitres of sample gas inside the micro gas chromatography unit before each measurement. The cheese packs were not kept after the analyses because of the too low amount of headspace volume. Therefore, new cheese packs produced from the same batch were used for each analysis in order to follow the kinetics of headspace composition during ripening. The gas chromatography unit was previously calibrated with gas bottles of known compositions. At least two measurements were carried out per each sample.

## 2.5 Description of the model

Evolution of CO₂ partial pressures in the foil packed blind cheese during ripening relies on the interplay of four mechanisms: (1) CO₂ production due to the activity of microorganisms (PAB), (2) gas transport within the cheese paste, (3) gas transfer at the cheese rind/headspace interface and (4) gas transmission through the packaging film. The four described mechanisms are illustrated in figure 1.
121

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138

139

140

141

142

Figure 1

2.5.1 Model assumptions

The following assumptions were made in the present mono-directional modelling study:

- The composition of the atmosphere surrounding the packed cheese during ripening is constant and equals to 78.1% for N₂, 20.9% for O₂ and 0.03% for CO₂ (Widory and Javoy, 2003).

- Temperature in the packed cheese system and surrounding atmosphere is constant, without gradients.

- Total pressure of the system is constant and it equals atmospheric pressure (101325 Pa).

- The solubilization of N₂ inside the cheese was considered negligible compared to CO₂, because of its lower solubility in water (about 50 times less soluble than CO₂ in water at 20°C) (Dean, 1999). O₂ solubility and diffusivity in the cheese were set to constant values of 2.5 10⁻⁹ m² s⁻¹ and 1.3 mmol kg⁻¹ atm⁻¹ respectively and which correspond to solubility and diffusivity of the gas in water at 20°C (Chaix et al., 2014).

- The mechanical resistance to deformation of the cheese packaging is not taken into account in the model. Thus, volume changes according to moles content inside the pack and total pressure of headspace remains equal to external atmospheric pressure. Measuring volume changes during experiment has revealed that they are negligible and that the flexible pouch could easily compensate the small volume change due to its property of elongation.
143 - For the cheese with salt gradient, the initial NaCl gradient was measured in cheeses produced within the same production day and it was considered to remain constant during the experiment. Same assumption was used to quantify the proteolytic indicator (PTATN).

147 - Gradients in chemical composition in the cheese from its upper rind to its core are symmetric to the gradients from lower rind to the core.

## 2.5.2 Gas balance in headspace

150 The variation of concentration of the gas species j in the packaging headspace HS depends on the mass flow of gas permeating the packaging film f from the surrounding atmosphere $\varphi_{jf}$ (kg/s) and the mass flow of the gas species occurring at the interface between the cheese and the headspace $\varphi_{jI}$.

151 Assuming that the packaging film does not oppose any mechanical resistance to deformation, when the amount of gas species in the headspace varies, the headspace volume $V_{HS}$ (m³) changes, whereas the total pressure in the headspace $P_T$ remains constant and equal to atmospheric pressure (101325 Pa). The global mass balance of species j in the headspace can hence be written as follows:

$$V_{HS} \frac{dC_{jHS}}{d_t} + C_{jHS} \frac{dV_{HS}}{d_t} = \varphi_{jf} + \varphi_{jI} \quad (\text{Eq. 1})$$

160 where $C_{jHS}$ (kg/m³) is the concentration of the gas in the headspace and t is time (s).

161 Assuming that the gas mixture in the headspace obeys the ideal gas law, $\frac{dV_{HS}}{d_t}$ can be calculated as a pondered sum of the different aforementioned mass flow $\varphi_{jf}$, $\varphi_{jI}$ as follows:

$$\frac{dV_{HS}}{d_t} = \frac{RT}{P_T} \left( \frac{\varphi_{CO2f} + \varphi_{CO2I}}{M_{CO2}} + \frac{\varphi_{O2f} + \varphi_{O2I}}{M_{O2}} + \frac{\varphi_{N2f}}{M_{N2}} \right) \quad (\text{Eq.2})$$
where $M_j$ is the gas molar mass (kg/mol), R the universal gas constant (J mol⁻¹ K⁻¹) and T the temperature in Kelvin (K).

The concentration of the gas in the headspace $C_{jHS}$ can be calculated according to (Eq. 3)

$$RT \sum_{j=\{O_2,CO_2,N_2\}} \frac{C_{jHS}}{M_j} = P_T = 10^5 Pa \quad (\text{Eq. 3})$$

and similarly, for the partial pressure of the gas in the headspace $p_{jHS}$ (Eq. 4),

$$p_{jHS} = \frac{C_{jHS} RT}{M_j} \quad (\text{Eq. 4})$$

### 2.5.3 Gas permeation through the maturation foil

The gas permeation through the packaging film (maturation foil), assuming that the partial pressures in the atmosphere surrounding the cheese are constant, could be conveniently described by the Fick's first law (steady state regime). Therefore, the mass flow $\phi_f$ of gases j (N₂, O₂, CO₂) through the packaging film was calculated according to first Fick's law (Fick, 1855) as stated in Eq. 5.

$$\phi_{j_f} = M_j P e_j \frac{A_f}{l_f} (p_{j\infty} - p_{jHS}) \quad (\text{Eq. 5})$$

where $Pe_j$ is the permeability coefficient of the gas j through the film (mol m⁻¹ s⁻¹ Pa⁻¹), $l_f$ is the thickness of the film (m), $A_f$ is the film surface (m²) and $p_{j\infty}$ and $p_{jHS}$ are the partial pressure of the gas j in the surrounding atmosphere (denoted by symbol $\infty$) and the packaging headspace, denoted by HS.

### 2.5.4 Mass flow at the cheese rind/gaseous interface

The mass flow at the cheese/rind gaseous interface was calculated as follows (Eq. 6):

$$\phi_{jI} = kj(\frac{M_j A_I}{RT})(p_{jHS,I} - p_{jHS}) \quad (\text{Eq. 6})$$

Where $k(m/s)$ is the mass transfer coefficient at the interface between the cheese rind and the headspace and $A_I$ and $p_{jHS,I}$ are the surface at the interface and the partial pressure at the
immediate vicinity of the cheese surface. The mass transfer coefficient k is calculated according to Eq. 7

$$kj = \frac{Bi \times D_j}{L} \qquad (\text{Eq. 7})$$

Where Bi is the non-dimensional Biot number, assumed to be equal to $10^5$ (ratio between diffusivity of CO₂ in air and cheese) (Laridon, 2014), $D_j$ is the diffusivity of CO₂ in cheese (about $10^{-10} \text{ m}^2 \text{ s}^{-1}$) and L is the characteristic length of the considered material (length of the cheese equalled about 0.008m).

Assuming that the gaseous species in headspace are in thermodynamic equilibrium with the dissolved species at the cheese surface, the concentration of dissolved gas species j at the cheese surface $C_{jch,x=0}$ (kg/m³) relates to $p_{jHS,I}$ according to Henry's law:

$$p_{jHS,I} = \frac{c_{jch,x=0}}{M_j S_j} \qquad (\text{Eq. 8})$$

Where $S_j$ is the solubility coefficient of the considered gas j (mol m⁻³ Pa⁻¹).

### 2.5.5 Gas diffusion inside the cheese paste and CO₂ production

The CO₂ diffusion within the cheese paste was represented by making use of Fick's second law of diffusion for the transient state of diffusion and it was coupled to CO₂ production by adding production term v (kg m⁻³ s⁻¹) to Fick's second law, leading to Eq. 9.

$$\frac{\partial c_{CO2, ch}}{\partial t} = D_{CO2} \frac{\partial^2 c_{CO2, ch}}{\partial x^2} + v_{CO2} \qquad (\text{Eq. 9})$$

Where $\frac{\partial c_{CO2, ch}}{\partial t}$ is the partial derivative of the concentration of CO₂ in the cheese as regard to time, in the direction x (m). x represents the distance from the interface of the cheese rind to headspace and the considered position inside the cheese. x varies from 0 to 8 (cheese thickness in cm). $D_{CO2}$ is the effective diffusivity coefficient of CO₂ in cheese (m²/s).
The same equation without the production term was used to model O₂ diffusion within the cheese paste.

## 2.5.6 Boundary conditions

Equality of flows was considered at the interface between the cheese rind and the headspace:

$$D_j \frac{\partial c_{jch}}{\partial x} = \frac{\varphi_{jl}}{A_I} = kj(\frac{M_j}{RT})(p_{jHS,I} - p_{jHS}) \quad at \ x = 0 \quad (Eq. 10)$$

where j stands for either CO₂ or O₂.

Since the cheese is pressed on continuous impermeable material during ripening due to gravity, we assumed no diffusion at the cheese bottom rind (br), (x = br):

$$D_j \frac{\partial c_{jch}}{\partial x} = 0 \quad at \ x = br \quad (Eq. 11)$$

Assuming that the food sample was initially in equilibrium with a gas of fixed partial pressure p<sub>j0</sub> (Pa), the initial conditions take the form:

$$C_{jch}(x, 0) = M_j S_j p_{j0}(x, 0) \quad for \ t = 0 \quad (Eq. 12)$$

## 2.5.7 Prediction of model input parameters for different ripening conditions

The following empirical equation was used in the model for predicting the changes in CO₂ production rate, v<sub>CO2</sub> (mmol kg⁻¹ d⁻¹) as a function of salt to moisture ratio, SM and temperature, T (Acerbi et al., 2016a):

$$v_{CO2} = 8.527 + 0.0862 T^2 - 1.621 T + 5.367 SM - 0.415 SM T \quad (Eq. 13)$$

SM (g/100g) is calculated as the percentage of the mass ratio between sodium chloride and moisture.

The following empirical equation was used in the model for predicting the value of D<sub>CO2</sub> at different temperatures, T salt to moisture ratio, SM and PTATN (ratio of phosphotungstic acid soluble nitrogen on total nitrogen). The latter coefficient is a ripening indicator related to the
229 proteolytic phenomena (especially amount of free amino acids) which relates to structure changes
230 in the cheese during ripening (Acerbi et al., 2016d).

$$
\begin{array}{l}
231 \\
232 \\
233 \\
234 \\
235 \\
236 \\
237 \\
238 \\
239 \\
240 \\
241 \\
242 \\
243 \\
244 \\
245 \\
246 \\
247 \\
248 \\
249 \\
250 \\
251
\end{array}
$$

The value of S<sub>CO2</sub> (in mmol kg<sup>-1</sup> atm<sup>-1</sup>) was calculated by using a modified version of the
equation given by Acerbi et al. (2016b):

$$
S_{CO2} = 37.35879 - 0.83414 SM^2 + 4.4619 SM - 0.34768 T \quad (\text{Eq. 15})
$$

The modification consisted in the use of a quadratic term for the salt effect instead of a linear
term in the view of more recent results (figure 2).

Figure 2

238
239
240
241 The value of permeability (KP) of the gas j (O₂ CO₂, N₂) was estimated at different ripening temperatures by interpolating the results of permeability coefficients KP given by Acerbi et al. 242 (2016c) in units cm³ µm m⁻² d⁻¹ bar⁻¹. The KP values were converted into Pe values (mol m⁻¹ s⁻¹ 243 Pa⁻¹) to be used in the model (Eq. 5). 244

$$
245 \qquad KP_{CO2} = -108.03 T^2 + 4551.5 T - 3242.6 \qquad (\text{Eq. 16})
$$

$$
246 \qquad KP_{O2} = 386.53 T + 6313.6 \qquad (\text{Eq. 17})
$$

$$
247 \qquad KP_{N2} = 188.78 T + 3422 \qquad (\text{Eq. 18})
$$

248 **2.5.8 Numerical solving and identification procedure**

249 This system of equations governing the coupled diffusion and production of gas in the cheese and
250 the gas transfer and permeation in the cheese packaging headspace was solved using a dedicated
251 algorithm “ode15s” developed in Matlab® computing software (The Mathworks Inc., Natick,
Mass, USA) and adapted to stiff systems where each of unknown variables may exhibit radically different variation kinetics. This algorithm adjusted automatically the size of the time step used for numerical integration of the equations. To evaluate goodness of fit between experimental and predicted data, the coefficient of variation of the root mean square deviation (CVRMSD) was calculated as follows:

$$CVRMSD = \left( \frac{\sqrt{\frac{(ypred-yexp)^2}{(N)}}}{\mu_{exp}} \right) \times 100 \qquad (\text{Eq.19})$$

Where $y_{pred}$ and $y_{exp}$ are respectively the experimental and the simulated residual value, N is the number of experimental measurements and $\mu_{exp}$ is the average of the experimental values. CVRMSD enables to compare the quality of the model with other models present in literature. A good quantitative model is considered to have $CVRSMD \le 10\%$.

## 2.6 Model validation

The experimental validation was made in three steps of increasing complexity which are summarized in Table 1.

1. Firstly, the mono-directional CO₂ diffusion within the cheese and transfer at the cheese rind interface was validated on an old ripened high salted cheese (namely “old cheese”) with no salt gradient and without CO₂ production (no CO₂ producing bacteria). The cheese was high salted via brining (about 5% NaCl/dm, salt content on dry matter) in order to avoid possible gas production from unwanted microorganisms in such old cheese. The cheese, apart for its upper rind, was fully covered with a gas impermeable membrane (see Supplementary Material A) and it was housed in a controlled temperature cabinet (set at 19°C) with a continuous flow of wet CO₂ (gas supposed to be equilibrated at 100% of relative humidity after bubbling in a flask containing pure water at the temperature of the experiment). After 3 days, the CO₂ dissolved in different positions (e.g. the sample is sliced parallel to the interface) from the upper rind (x=0) to the
275 bottom rind (x=8) was assessed with the CO₂ determination method described above. Two
276 repetitions were made in order to validate the diffusion phenomenon. For two other repetitions,
277 after 3 days of CO₂ sorption, the cheese samples were then kept 3 days in a continuous flow of
278 wet (100% RH) N₂ (desorption step). The gradient of dissolved CO₂ into the paste after these 3
279 days of desorption was then assayed as previously described for the sorption step (2 repetitions).

280 ② Secondly the coupling of CO₂ diffusion and production was experimentally assessed by using
281 a cheese with PAB and average salt concentration (about 2.5% NaCl/dm, salt on dry matter, in
282 the cheese, namely “young cheese”). This cheese also included a salt gradient due to its younger
283 age (15 days from renneting) when salt is still slowly diffusing from rind to core of the cheese
284 (brined cheese). The same experimental procedure previously described was used, apart for the
285 following: the cheese stayed 1 day at 19°C, before the experiment with the contact of wet CO₂
286 from upper rind (x=0) started and it lasted for 4.4 days.

287 The initial CO₂ concentration gradient in the cheese was measured just before the experiment
288 started (i.e. about 9.5 weeks from renneting for the first validation step and 15 days from
289 renneting for the second more complex validation step including CO₂ production). This initial
290 CO₂ gradient is indispensable to parametrize initial conditions in the numerical algorithm.

291 ③ Thirdly, in order to compare the output of gas concentration in the headspace given by the
292 model including permeation phenomenon, gas headspace composition was measured in the
293 headspace of packed cheeses (2 replications). The headspace gas analyses were repeated on
294 different cheeses during ripening (about at 10, 14, 16 ripening days at 19°C).

295
296 In each case of these aforementioned steps, the model was adapted accordingly to experimental
297 conditions: system of Eq. (6) to (15) for the first and second steps - without the production term
298 in Eq. (9) and related Eq. (13) – and system of Eq. (1) to Eq. (18) for the third, full case.
299

## 2.7. Statistics

Statistical tests were performed by using R software for statistical computing (R, 2014).

Comparisons between the chemical compositions were performed by pairwise comparisons using *t* test with pooled standard deviation. The Holm method for the multiplicity of error was used to adjust the p-value of multiple pairwise comparisons. Different letters were used for denoting significant difference between data sets (level of significance α = 0.05, unless stated).

306

# 3. Results and discussion

Our model takes into account experimentally assessed parameters for describing CO₂ production and overall transfer in and out a cheese/packaging system during foil ripening. The focus was on the exhaustive description of the gas transport properties in the continuous cheese paste and at its interface with the surrounding atmosphere. We therefore used a model blind cheese (without eyes). We experimentally assessed the following parameters: transport of CO₂ in the cheese paste (effective diffusivity), transfer of CO₂ at the interface between cheese and headspace (solubility), oxygen, nitrogen and carbon dioxide transmission rate through the maturation foil (permeability) and CO₂ production rate from the bacteria. The cited parameters were previously assessed in function of the main ripening variables (different temperatures, ripening time, salt content amongst others) (Acerbi et al., 2016a, 2016b, 2016c, 2016d).

318

### 3.1 Compositional analyses

The chemical composition of the cheese without PAB (old cheese – data not shown) was not significantly different (α=0.05) as a function of the position in the cheese, whereas a relevant gradient in PTATN (ratio of phosphotungstic acid soluble nitrogen on total nitrogen) and SM

320

321

322
323 (salt in moisture ratio) was confirmed from analyses made on cheese with PAB analysed at 15
324 and 20 days from renneting (figure 3).

325 Although the transfer properties of nitrogen components (PTATN) and salt were not
326 experimentally assessed within this study and not included in the model, the initial heterogeneous
327 composition was considered and a linear interpolation was used for predicting the initial
328 composition in all positions of the cheese including PAB. The composition in the position 4 to 8
329 cm was considered symmetrical to the position 0 to 4 cm.

330

331

Figure 3

332

333 A relevant gradient in organic acid concentration was found in the cheese with PAB both at 15
334 and 20 days from renneting (figure 4). No propionate and acetate was found in the outer rind (0
335 cm) at 15 days from renneting. The observed delay in PAB fermentation in the outer part of the
336 cheese was probably due to the negative effect of high salt to moisture ratio on the bacteria cells
337 (Guinee, 2004) and it was already demonstrated by Huc et al. (2014) in similar cheeses. The
338 molar ratio between propionate and acetate in the core of the cheese equalled about 2. The ratio
339 of 2 is obtained if PAB convert lactate exclusively via the Fitz pathway (Fröhlich-Wyder and
340 Bachmann, 2004; Piveteau, 1999; Fedio et al., 1994). We can therefore confirm that, in the
341 studied cheese, PAB fermented lactate primarily via the Fitz pathway and that probably no other
342 metabolic activity (producing acetate) occurred.

343

344

Figure 4

345
346 The pH of the cheese was measured because pH changes may affect both the solubility of carbon dioxide dissolved in the water phase of the cheese (Chaix et al., 2014) and its diffusivity coefficient due to a different organisation of the casein matrix structure (Lawrence et al., 1987).
347
348
349 The pH in the studied cheese did not relevantly vary in the considered experimental time (figure 5a).
350

351 The enumeration of PAB cells confirmed the higher metabolic activity observed in the core of the cheese (figure 5b).
352
353

Figure 5

354

355

356 **3.2 Initial CO₂ concentration**

357 The initial CO₂ concentration in different positions (from upper to bottom rind) of the cheese was measured in 2 to 3 cheeses produced within the same production batch. Whereas the CO₂ distribution was low and rather homogeneous for the old ripened cheese with no intentionally added CO₂ producing bacteria (figure 6a, 3 repetitions), the initial CO₂ gradient was relevant in 358 the cheese with PAB (figure 6b, 2 repetitions). The low and homogeneous initial CO₂ 359 concentration observed in the older cheese was expected because this cheese did not contain 360 intentionally added CO₂ producing bacteria, while the CO₂ gradient observed in the cheese with 361 PAB at 15 days from renneting (14 days at 13°C and 1 days at 19°C) reflected the effect of salt 362 gradient on PAB. There is an inhibition of PAB growth (Figure 5b) by the higher salt content 363 close to the rinds with a consequently lower production of CO₂ and propionic acid close to the 364 rinds compare to the core of the product leading to this “bell like curve” for initial CO₂ gradient 365 in salted cheese with PAB. This evolution of salt content that is spatially-dependant, due to 366 diffusion of salt from the periphery towards the centre of cheeses, has been found in previous 367
368
369
studies (Huc et al., 2014; Guinee, 2004; Hollywood & Doelle, 1984) and before by Mocquot (1979), Geurts et al. (1974, 1980) and Pauchard et al. (1980). Therefore, the median value for initial CO₂ concentration was used (4 mmol/kg) for the model validated for the cheese without PAB and the real gradient was used as input vector for initial CO₂ concentration for the cheese with PAB.

Figure 6

### 3.3. Investigating effect of salt content on CO₂ solubility

Prior model validation, additional points of solubility of CO₂ within the cheese paste were measured to complete the work of Acerbi et al. (2016b) and obtain more precise S<sub>CO2</sub> prediction (eq. 15). According to these new values (Figure 2), S<sub>CO2</sub> increased with light NaCl content, while it decreased at very high NaCl content, following a similar behaviour compared to salting in and salting out effect of protein. This phenomenon was ascribed to the possibly higher sorption of CO₂ in highly hydrated protein during salting in phase (Guinee, 2004), while the overall S<sub>CO2</sub> would decrease due to lower level of protein hydration at high salt concentration (salting out). Guinee (2004) suggested that salting out effect of protein in cheese (Mozzarella) may take place for salt to moisture ratio (S/M) higher than 6.3% w/w, which is close to the highest salt level investigated in Figure 2 (about 6.7% S/M).

### 3.4 Model validation

The simplest model form describing CO₂ diffusion inside the cheese paste (without PAB and CO₂ production) and the transfer at the cheese rind/gaseous interface (100% CO₂) was successfully validated because the difference between predicted and experimental data was below 10% (figure
393 7a). Same good validation with a 100% N₂ flux at the cheese rind surface was obtained
394 (CVRMSD < 8%) (see Supplementary Material B).

395 When increasing the complexity of the model, including CO₂ production rate and considering a
396 cheese with salt gradient, the observed error between experimental and predicted data was found
397 higher (30%) (figure 7b). In the latter more complex model, the predicted line was generally
398 underestimating the experimental gradient, leading to a high CVRMSD. The lack of fit of the
399 more complex model could be due to (1) the adoption of a mono-directional model which may
400 not be fully appropriate for diffusion in cheeses with gas production and/or (2) underestimations
401 in either the prediction of CO₂ production rate and/or in the initial CO₂ concentration gradient.
402 Underestimation of the CO₂ production rate may be due to the linear approximation used by
403 Acerbi et al. (2016a) to describe the effect of salt content. Indeed, lower salt contents were
404 investigated in the previous work, which provided the predictive equation for CO₂ production
405 rate, compared to the current study. Therefore, effect of high SM ratio is probably not well
406 predicted by Eq.13 which was never validated for such SM ratio. This underestimation may be
407 overcome by carrying out more experimental measurements of CO₂ production rate at higher salt
408 content. A possible underestimation of the initial CO₂ concentration gradient in the cheese may
409 be ascribed to natural deviations in metabolic activity of the PAB in different cheeses produced
410 from the same batch. Both hypotheses were confirmed by simulating a 1.7 folds higher CO₂
411 production rate (figure 8) and then, a 2 folds higher initial CO₂ concentration (figure 9). In both
412 cases, the predicted curve fit very well the experimental data.

413

414

Figure 7

415

416

Figure 8
417

418

419

420

421

422

423

424

425

426

427

428

429

430

431

432

433

434

435

436

437

438

Figure 9

Then, the full model considering permeation through ripening foil in addition to solubilisation/diffusion and production of CO₂ within the food was then used to predict the CO₂ partial pressure in headspace. A third set of experiments conducted on cheeses packed in ripening foils were conducted in the objective to validate it. Experimental difficulties arose because gas headspace analysis could not be assessed continuously in the pack (because of too high gas volume injected) therefore only a few experimental points have been collected (See supplementary material C). However, it has been noted that the model tended to underestimate the headspace partial pressure when CO₂ production occurs (in cheese with PAB). But, once again, considering a 1.7 fold higher CO₂ production rate, the prediction fitted better the experimental data. A better fit may also be obtained by reducing the value for CO₂ permeability of the packaging or increasing the initial CO₂ concentration gradient (results not shown). But this hypothesis does not sound well founded and therefore was not applied. The permeability values were especially assessed in the conditions encountered in the present work and were therefore considered relevant.

### 3.5 Exploratory analysis of model simulations

Simulations were carried by considering the min and max values of 4 parameters in their range of variation (table 2): CO₂ solubility (figure 10a), diffusivity (figure 10b), permeability (figure 10c) and production coefficients (figure 10d). During the simulation, only one of the mentioned input parameters was varied from the lowest to the highest value described in Table 2. The boundary
conditions used during these simulations included that the packed cheese was in contact with the atmosphere (20.9% O₂, 0.03% CO₂, 78.1% N₂) from one rind (upper rind) and in contact with a non-permeable support from the below rind (shelf). The range of values for solubility and diffusivity was chosen in agreement with the min and max observed in the experimental campaign described by Acerbi et al. (2016b, 2016d). The values for permeability were decided to vary of factor 10 of the predicted value of Equation 16, 17, 18. The value of production rate was decided to vary of 4 units compared to the predicted value of equation 13, because it was considered a realistic variation for the different salt contents observed in different positions of the studied cheese at 19°C.

<caption>**Table 2**</caption>

The other input parameters were kept fixed (medium value) as respect of their position. This simplified sensitivity analysis had the goal of highlighting which input parameter had the strongest effect on the output (CO₂ gradients formed in cheese). Logically, changing SCO₂ led to light differences in CO₂ concentration only close to the gaseous interface (figure 10a) because this parameter only intervenes in the boundary condition at cheese/headspace interface: the lower the solubility, the lower the CO₂ content at the interface. Changing DCO₂ slightly affected the shape of the CO₂ gradient close to the cheese rinds, probably because these positions were characterized by lower vCO₂ due to the higher salt content (figure 10b) and it affected the overall shape of the curve. vCO₂ showed the highest effect on the CO₂ gradients, from -10 to +20 mmol/kg of difference for the lowest and highest vCO₂ respectively compared to the median value in the core of the cheese. Concerning the effect of different permeability, the lower the gas
462 permeability of the packaging, the higher was the CO₂ concentration at the cheese rind, but the
463 overall change in CO₂ gradient due to different permeability was negligible.

464

465

466

467 This paper presents an unprecedented modelling approach that successfully describes the
468 mechanisms of CO₂ diffusion, solubilisation and production by Swiss-type cheese with intensive
469 PAB-based CO₂ production and also CO₂ permeation through the ripening foil. All mechanisms
470 were dynamically coupled and experimentally validated permitting to achieve the initial objective
471 of this paper which was to predict evolution with time of CO₂ gradients into the packed cheese.
472 The developed model is the most complete one compare to precedent similar approach of the
473 literature. For instance, the one of Jakobsen and Risbo, (2009), developed for prediction, among
474 others, of the changes in solubilised carbon dioxide in semi-hard cheese packed in modified
475 atmosphere packaging, neglected all mechanisms of CO₂ diffusion and production in their
476 approach.

477 This model could be further used to predict CO₂ gradients into blind cheese or, once coupled with
478 a mechanical model of bubble growth such as the one proposed by Laridon et al., (2014), could
479 be used to predict eye growth in Swiss type cheese and then pilot the ripening step.

480

481 **4. Conclusion**

482 We presented the first experimentally validated model which couples the phenomena of CO₂
483 production, solubilisation/diffusion and permeation in a packed cheese system for predicting the
484 CO₂ gradients formed in the cheese during ripening. A three steps validation procedure enabled
485 to build a robust model for a quantitative description of CO₂ gradient formed in a cheese with or
486 without CO₂ production by PAB and including the phenomena of CO₂ diffusion in the cheese

487 paste and transfer at the gaseous interface. The model was able to describe the shape of the CO₂

488 gradient formed in the cheese with, nevertheless, less precision when CO₂ production happens.

489 This term was probably less accurately characterized for the conditions used in this paper and

490 induces less performant prediction. A simplified sensitivity analysis highlighted CO₂ production

491 as the most important input parameter affecting the CO₂ gradients formed in cheese during

492 ripening. Results presented in this paper represent a solid basis for the description of the most

493 important phenomena affecting the quality of cheese with intense CO₂ production during

494 ripening.
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592
Figure 1: Scheme of the simplified food / packaging system with the four phenomena considered (mono-directional transfer through the headspace/cheese interface, br =cheese bottom-rind)

Figure 2. CO₂ solubility in semi-hard Swiss-type cheese as a function of salt content at 13°C.

The point at intermediate NaCl content (1.5% w/w) is added to the ones reported by Acerbi et al. (2016b) and it was measured with same protocol on the same cheese. Different letters a, b, c, ... denote significant difference for α = 0.05.

Figure 3. (a) PTATN - ratio of phosphotungstic acid soluble nitrogen on total nitrogen and (b) SM - salt in moisture ratio- in different positions of the semi-hard Swiss-type cheese with PAB, from rind (0 cm) to core (4 cm) at 15 and 20 days from renneting, representing starting time and ending time of the validation experiment.

Figure 4. Propionate (a) and acetate (b) concentrations in different positions of the semi-hard Swiss-type cheese with PAB at 15 and 20 days from renneting (young cheese).

Figure 5. pH and PAB count in different positions of the semi-hard Swiss-type cheese with PAB at 15 and 20 days from renneting.

Figure 6. Median values for initial CO₂ concentration measured in 9.5 weeks after renneting for old semi-hard Swiss-type cheese without PAB (a) and in 2 weeks after renneting for young semi-hard Swiss-type cheese with PAB (b). Horizontal and vertical error bars represent cheese position used and min and max values for assessed CO₂ respectively.

Figure 7. Experimental (red dots) and predicted (solid line) CO₂ concentration in the semi-hard cheese without PAB (a) and with PAB (b) CO₂ production after about 4 ripening days at 19°C. Root mean squared error (CVRMSD) is indicated on each figure. Error bars in experimental data represent standard deviations.

Figure 8. Predicted CO₂ gradients in cheese (solid lines) after 1, 2, 3, 4 and 10 days of contact with 100% CO₂, calculated considering 1.7 folds higher CO₂ production rate (19°C). Red
solid line and red error bars correspond to predicted and experimental CO₂ gradient after 4
days of contact.

Figure 9. Predicted CO₂ gradients in cheese (solid or dotted lines) after 1, 2, 3, 4 and 10 days of contact with 100% CO₂, calculated considering 2 folds higher initial CO₂ gradient (19°C). Red solid line and red error bars correspond to predicted and experimental CO₂ gradient after 4 days of contact.

Figure 10. Effect of the intensity (low, medium and high, as stated in Table 2, in black, blue and red respectively) of the input parameter CO₂ solubility (a), diffusivity (b) permeability (c) and production rate (d) on the predicted CO₂ gradients in cheese ripened for 4 days at 19°C (age at beginning of ripening equalled 14 days from renneting).
<table><caption>Table 1. Target chemical composition of “old” and “young” cheeses and steps of model validation</caption><thead><tr><th>Type of cheese</th><th>Time after renneting</th><th>Target moisture % w/w</th><th>Target fat absolute % d.m.</th><th>Target salt content % NaCl/d.m.</th><th>Target pH</th><th>PAB</th><th>Validation step</th></tr></thead><tbody><tr><td>“Old cheese”</td><td>15 days</td><td>42</td><td>40</td><td>5 %</td><td>5.45</td><td>No</td><td>① CO<sub>2</sub> diffusion only</td></tr><tr><td>“Young cheese”</td><td>9.5 weeks</td><td>42</td><td>8</td><td>2.5 %</td><td>5.45</td><td>Yes*</td><td>② CO<sub>2</sub> diffusion + production by PAB<br>③ CO<sub>2</sub> diffusion + production by PAB + CO<sub>2</sub> permeation through ripening foil</td></tr></tbody></table>

<footnote>*10⁶ CFU ml⁻¹ milk</footnote>
<table><caption>Table 2. Range of values used in the simulations</caption><thead><tr><th>Parameter/level</th><th>low</th><th>medium</th><th>high</th></tr></thead><tbody><tr><td>Solubility (mmol kg<sup>-1</sup> atm<sup>-1</sup>)</td><td>25</td><td>36</td><td>40</td></tr><tr><td>Diffusivity (10<sup>-10</sup> m<sup>2</sup>s<sup>-1</sup>)</td><td>1</td><td>4</td><td>8</td></tr><tr><td>Permeability (cm<sup>3</sup> µm m<sup>-2</sup> d<sup>-1</sup> bar<sup>-1</sup>)</td><td>4000</td><td>44000</td><td>440 000</td></tr><tr><td>Production rate (mmol kg<sup>-1</sup> d<sup>-1</sup>)</td><td>2</td><td>4</td><td>8</td></tr></tbody></table>
<figure><img src="image_3.png" /><figcaption>Figure 1: Scheme of the simplified food / packaging system with the four phenomena considered (mono-directional transfer through the headspace/cheese interface, *br* = cheese bottom-rind)</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 10. Effect of the intensity (low, medium and high, as stated in Table 2, in black, blue and red respectively) of the input parameter CO₂ solubility (a), diffusivity (b) permeability (c) and production rate (d) on the predicted CO₂ gradients in cheese ripened for 4 days at 19°C (age at beginning of ripening equalled 14 days from renneting).</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 2. CO₂ solubility in semi-hard Swiss-type cheese as a function of salt content at 13°C.</figcaption></figure>

The point at intermediate NaCl content (1.5% w/w) is added to the ones reported by Acerbi et al. (2016b) and it was measured with same protocol on the same cheese. Different letters a, b, c, ...denote significant difference for α = 0.05.
<figure><img src="image_6.png" /><figcaption>Figure 3. (a) PTATN - ratio of phosphotungstic acid soluble nitrogen on total nitrogen and (b) SM - salt in moisture ratio- in different positions of the semi-hard Swiss-type cheese with PAB, from rind (0 cm) to core (4 cm) at 15 and 20 days from renneting, representing starting time and ending time of the validation experiment.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 4. Propionate (a) and acetate (b) concentrations in different positions of the semi-hard Swiss-type cheese with PAB at 15 and 20 days from renneting (young cheese).</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 5. pH and PAB count in different positions of the semi-hard Swiss-type cheese with PAB at 15 and 20 days from renneting.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 6. Median values for initial CO₂ concentration measured in 9.5 weeks after renneting for old semi-hard Swiss-type cheese without PAB (a) and in 2 weeks after renneting for young semi-hard Swiss-type cheese with PAB (b). Horizontal and vertical error bars represent cheese position used and min and max values for assessed CO₂ respectively.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 7. Experimental (red dots) and predicted (solid line) CO₂ concentration in the semi-hard cheese without PAB (a) and with PAB (b) CO₂ production after about 4 ripening days at 19°C. Root mean squared error (CVRMSD) is indicated on each figure. Error bars in experimental data represent standard deviations.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Figure 8. Predicted CO₂ gradients in cheese (solid lines) after 1, 2, 3, 4 and 10 days of contact with 100% CO₂, calculated considering 1.7 folds higher CO₂ production rate (19°C). Red solid line and red error bars correspond to predicted and experimental CO₂ gradient after 4 days of contact</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Figure 9. Predicted CO₂ gradients in cheese (solid or dotted lines) after 1, 2, 3, 4 and 10 days of contact with 100% CO₂, calculated considering 2 folds higher initial CO₂ gradient (19°C). Red solid line and red error bars correspond to predicted and experimental CO₂ gradient after 4 days of contact.</figcaption></figure>
**Highlights**

* We proposed the first validated model for the prediction of CO₂ gradient in cheese.

* CO₂ production is the most important parameter affecting CO₂ gradients in cheese.

* A variation of a factor 10 of CO₂ permeability of the packaging did not relevantly affect CO₂ gradients in cheese

* CO₂ permeability of the packaging did not relevantly affect CO₂ gradients in cheese.