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

Eye growth in cheese with intense CO 2 production during ripening mainly depends on CO 2 production and transfer properties. Despite CO 2 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 14 this study, experimentally assessed CO 2 production was coupled with transfer coefficients in a 15 mathematical model in order to predict CO 2 gradients formed inside the cheese during ripening. 16 The permeability coefficient of CO 2 through the multilayer barrier packaging which wraps the 17 cheese during ripening was also included in the model. The presented model was validated by 18 assessing the CO 2 concentration in the cheese and its partial pressure in the packaging headspace. 19 CO 2 production rate was found to be the most important input parameter affecting CO 2 gradients 20 formed in cheese during ripening whereas the other input parameters (solubility, diffusivity, 21 permeability) had little effect on the total CO 2 gradient.

into account mass transfer of gas (CO 2 produced by bacteria and responsible of eyes' growth), 45 production of the CO 2 and mechanical constraint imposed to cheese paste by this production. 46 This model was based on experimentally assessed rheological parameters (stress) and CO 2 47 production rate in simplified condition, but some input parameters such as CO 2 diffusivity and afterwards the higher influence of CO 2 production and diffusion parameters compared to the 51 rheological ones in semi-hard cheese (Laridon, 2014). 52 Faced to the importance of CO 2 diffusion and CO 2 production rate in the ripening of semi-hard 53 Swiss-type cheese, this paper aimed at deepening these two phenomena by coupling them in a 54 mathematical model in order to simulate and predict evolution with time of CO 2 gradients in the 55 cheese paste and in the packed cheese (including CO 2 permeation through the ripening foil). In used in this study is described in Table 1.

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Cheese blocks were about 15 to 8 cm square in shape (see Supplementary Material A) and were 76 sampled in their core, orthogonally to the interface exposed to headspace/surrounding atmosphere 77 (Figure 1), at least 3 cm away from side rinds, resulting in a cylinder of 8 cm height and about 2 78 cm of diameter. The sampling region was then cut in thin slices of minimum 0.5 cm of thickness 79 for assessment of chemical composition gradient or CO 2 gradient.  internal method based on high-performance liquid chromatography.
94 CO 2 determination in cheese was carried out with the protocol described in Acerbi et al. (2016b).

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The protocol included a first step where CO 2 was extracted from the cheese sample by immersing 96 it in acidic solution. The extracted CO 2 was then scavenged by a soda solution of known before each measurement. The cheese packs were not kept after the analyses because of the too 110 low amount of headspace volume. Therefore, new cheese packs produced from the same batch 111 were used for each analysis in order to follow the kinetics of headspace composition during 112 ripening. The gas chromatography unit was previously calibrated with gas bottles of known 113 compositions. At least two measurements were carried out per each sample.  The following assumptions were made in the present mono-directional modelling study:

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-Temperature in the packed cheese system and surrounding atmosphere is constant, 129 without gradients.

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

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-The solubilization of N 2 inside the cheese was considered negligible compared to CO 2 , 132 because of its lower solubility in water (about 50 times less soluble than CO 2 in water at

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-Gradients in chemical composition in the cheese from its upper rind to its core are 147 symmetric to the gradients from lower rind to the core.  Assuming that the gas mixture in the headspace obeys the ideal gas law, can be calculated as where . is the gas molar mass (kg/mol), R the universal gas constant (J mol -1 K -1 ) and T the 164 temperature in Kelvin (K).

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The concentration of the gas in the headspace can be calculated according to (Eq. 3) where > is the permeability coefficient of the gas j through the film (mol m -1 s -1 Pa -1 ), E = is the 177 thickness of the film (m), = is the film surface (m 2 ) and < B and < are the partial pressure of 178 the gas in the surrounding atmosphere (denoted by symbol ∞) and the packaging headspace, 179 denoted by . The mass flow at the cheese/rind gaseous interface was calculated as follows (Eq. 6): Where NO is the non-dimensional Biot number, assumed to be equal to 10 5 (ratio between 189 diffusivity of CO 2 in air and cheese) (Laridon, 2014), P is the diffusivity of CO 2 in cheese (about Where is the solubility coefficient of the considered gas (mol m -3 Pa -1 ).  where j stands for either CO 2 or O 2 .

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Since the cheese is pressed on continuous impermeable material during ripening due to gravity, Assuming that the food sample was initially in equilibrium with a gas of fixed partial pressure < U

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(Pa), the initial conditions take the form:  Table 1.

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Firstly, the mono-directional CO 2 diffusion within the cheese and transfer at the cheese rind 265 interface was validated on an old ripened high salted cheese (namely "old cheese") with no salt 266 gradient and without CO 2 production (no CO 2 producing bacteria). The cheese was high salted 267 via brining (about 5% NaCl/dm, salt content on dry matter) in order to avoid possible gas wet (100% RH) N 2 (desorption step). The gradient of dissolved CO 2 into the paste after these 3 278 days of desorption was then assayed as previously described for the sorption step (2 repetitions).

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

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The initial CO 2 concentration gradient in the cheese was measured just before the experiment 287 started (i.e. about 9.5 weeks from renneting for the first validation step and 15 days from 288 renneting for the second more complex validation step including CO 2 production). This initial 289 CO 2 gradient is indispensable to parametrize initial conditions in the numerical algorithm. In each case of these aforementioned steps, the model was adapted accordingly to experimental 296 conditions: system of Eq. (6) to (15) for the first and second steps -without the production term  The chemical composition of the cheese without PAB (old cheese -data not shown) was not

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The pH in the studied cheese did not relevantly vary in the considered experimental time ( figure   349 5a).

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The enumeration of PAB cells confirmed the higher metabolic activity observed in the core of the concentration observed in the older cheese was expected because this cheese did not contain

Model validation 389
The simplest model form describing CO 2 diffusion inside the cheese paste (without PAB and CO 2 390 production) and the transfer at the cheese rind/gaseous interface (100% CO 2 ) was successfully 391 validated because the difference between predicted and experimental data was below 10% (figure

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When increasing the complexity of the model, including CO 2 production rate and considering a 395 cheese with salt gradient, the observed error between experimental and predicted data was found 396 higher (30%) (figure 7b). In the latter more complex model, the predicted line was generally 397 underestimating the experimental gradient, leading to a high CVRMSD. The lack of fit of the 398 more complex model could be due to (1) the adoption of a mono-directional model which may 399 not be fully appropriate for diffusion in cheeses with gas production and/or (2) underestimations 400 in either the prediction of CO 2 production rate and/or in the initial CO 2 concentration gradient.

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Underestimation of the CO 2 production rate may be due to the linear approximation used by     shape of the curve. υ CO2 showed the highest effect on the CO 2 gradients, from -10 to +20 459 mmol/kg of difference for the lowest and highest υ CO2 respectively compared to the median value 460 in the core of the cheese. Concerning the effect of different permeability, the lower the gas 487 gradient formed in the cheese with, nevertheless, less precision when CO 2 production happens.

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This term was probably less accurately characterized for the conditions used in this paper and        Predicted CO 2 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 CO 2 gradient (19°C).
Red solid line and red error bars correspond to predicted and experimental CO 2 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 CO 2 solubility (a), diffusivity (b) permeability (c) and production rate (d) on the predicted CO 2 gradients in cheese ripened for 4 days at 19°C (age at beginning of ripening equalled 14 days from renneting).      Predicted CO 2 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 CO 2 gradient (19°C).
Red solid line and red error bars correspond to predicted and experimental CO 2 gradient after 4 days of contact.

Highlights
• We proposed the first validated model for the prediction of CO 2 gradient in cheese.
• CO 2 production is the most important parameter affecting CO 2 gradients in cheese.
• A variation of a factor 10 of CO 2 permeability of the packaging did not relevantly affect CO 2 gradients in cheese • CO 2 permeability of the packaging did not relevantly affect CO 2 gradients in cheese.