Combined deterministic and stochastic approaches for modelling the evolution of food products along the cold chain. Part II: A case studyApproches déterministes et stochastiques combinées de la modélisation de l’évolution des produits alimentaires tout au long de la chaîne du froid. Partie II : une étude de cas

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Abstract

The aim of the Part II is to apply the methodology combining deterministic and stochastic approaches to describe the evolution of food products along the cold chain. The proposed methodology, presented in Part I, takes into account different sources of variability (logistics, thermal conditions, consumer behaviour…) and uses deterministic models for equipment and product evolution. First, the data structure of a case study involving the three final links of the cold chain (display cabinet, shopping basket and domestic refrigerator) is presented. Then, the simulation results are analysed in order to highlight the potentialities of the modelling approach. These results are compared with survey data to validate the model. This part also underlines the sources of variability which exert the greatest influence on the microbial load development: the residence time of the product inside the refrigerator and the thermostat setting of the refrigerator.

Highlights

► The application of a methodology to predict food products evolution is presented. ► This methodology combines deterministic and stochastic approaches. ► Three cold chain links (display cabinet, basket, and refrigerator) are studied. ► A good agreement is obtained between numerical and survey data.

Introduction

In order to ensure the quality and safety of chilled food from production to consumption, management of the cold chain is of major importance. This task requires knowledge of food product evolution throughout the cold chain which can be obtained by field investigation or by numerical simulation. Even though investigations are necessary to identify the real conditions and problems of food preservation, they are costly and time-consuming. Numerical simulation is an alternative. It can constitute a tool that reproduces the cold chain and makes it possible to study the influence of different factors on the quality of a food product. In order to develop such a numerical tool, modelling of different equipment taking into consideration many sources of variability within the cold chain is necessary.

In the part I of this study (Flick et al., in press), a methodology using deterministic models for equipment and product evolution and considering different sources of variability of the cold chain (logistics, thermal conditions, consumer behaviour…) was proposed. The aim of the part II is to apply this methodology in a case study. The numerical results are compared with survey data (Cemagref and ANIA, 2004) in order to validate the model. The influence of the number of product items (I) on the results of temperature and microbial load in each link is studied in order to identify the minimum value of I for which the results become statistically stable. Analysis of the numerical results in terms of probability density and correlations between different variables is carried out to illustrate the potentialities of our approach. This part also highlights the sources of variability which have the most influence on the microbial load. Finally, it is shown that acceptable prediction of microbial growth can be obtained by knowing only the time-averaged product temperature and the residence time for each link.

Section snippets

Simulation methodology – part I (Flick et al., in press)

The simulation methodology that combines deterministic and stochastic approaches in order to predict the evolution of product items along the cold chain is composed of:

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    a stochastic description of the logistic chain: the variability of the logistic chain, of ambient thermal conditions, of operator and consumer behaviour, etc. It is characterised by independent random variables of given Probability Density Function (PDF).

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    a semi-deterministic description of the product and links evolution. The

Influence of number of products on the numerical results

The aim of this analysis is to find the minimal number of runs I for which the results become statistically stable. In our case, one run corresponds to one product item of index i, i varying from 1 to I. 7 values of I are considered: 10, 66 (number of product items in our survey), 100, 500, 1000, 10 000 and 100 000. For a low value of I (I = 10 for example), it is expected that two simulations (involving several random samplings) will not give the same statistical results. However, for a high

Time-averaged product temperature at each link T¯linkt

The cumulative distributions of time-averaged product temperature for each link (display cabinet T¯DCt, shopping basket T¯SBt and refrigerator T¯REFt) are shown in Fig. 5.

In our case study, the product of interest is pre-packaged meat which has a recommended preservation temperature between 0 and 4 °C. However, a temperature of up to 6 °C is permitted. For the display cabinet, both simulation and survey results show that the product temperature is higher than 6 °C for only 3% of products.

Overall microbial growth distribution

Fig. 7 presents the cumulative distribution of the overall Decimal Increase (DI, eq. (14)) in the microbial load of the cold chain for 100 000 products. It can be observed that for 50% of the products, the microbial load is multiplied by less than 3.31, which is reasonable. The microbial load increases over 200-fold in 5% of products, and this can be critical for consumers. An overall relative increase in the microbial load (NF/NI) of more than 104 is obtained for 1% of products, which may lead

Conclusion

This paper presents the application of a methodology combining deterministic and stochastic approaches in order to predict the evolution of food products along the cold chain and its variability. The evolution of a large number of food products within three successive links was studied: display cabinet, shopping basket and refrigerator. A comparison of the numerical results and the survey data (Cemagref and ANIA, 2004) shows good agreement in terms of mean value, standard deviation, and

Acknowledgements

The research leading to these results has received funding from the European Community’s Seventh Framework Programme (FP7/2007-2013) under grant agreement No. 245288.

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