Dr Jocelyn Bonjour

Prof. Dr Giovanni Cortella

Dr Pedro Dinis

Mr Alexandre

Velocimetry system. Likewise, I am immensely grateful for the staffs of EPTA (France) for both their valuable insights on retail refrigeration industry and for allowing me to visit their factory.

I would like to express my deepest gratitude to my parents for their constant supports. I also would like to thank my girlfriend for her love

 Table 2.
Background

World population is predicted to increase from 7.3 billion in 2015 to 9.7 billion in 2050 (United Nations, 2017). This population growth would drive up food demand by approximately 59% to 98% [START_REF] Valin | The future of food demand: understanding differences in global economic models[END_REF]. Global food production therefore needs to increase to keep up with a rising demand (Hunter, 2017).

A cold chain is an essential part of production chains where several refrigeration facilities are used to keep food products, from production to consumption, at their good quality and low risk to consumer [START_REF] Badia-Melis | New trends in cold chain monitoring applications -A review[END_REF]. Temperature is a main determining factor of the food quality deterioration [START_REF] Mercier | The Canadian food cold chain: A legislative, scientific, and prospective overview[END_REF]. A failure in temperature control at any cold chain stage can markedly compromise the quality and safety of food products [START_REF] Ndraha | Timetemperature abuse in the food cold chain: Review of issues, challenges, and recommendations[END_REF]. High temperature accelerates the physiological changes of food, the growth of spoilage/pathogenic microorganisms, and eventually shortens the product shelf life [START_REF] Nunes | Environmental conditions encountered during typical consumer retail display affect fruit and vegetable quality and waste[END_REF][START_REF] Rediers | Evaluation of the cold chain of fresh-cut endive from farmer to plate[END_REF]. Therefore, effective temperature control along the cold chain is of paramount importance.

Governments and/or public authorities in various countries have established legislation and regulations regarding temperature control in the cold chain [START_REF] Carson | The cold chain in New Zealand -A review[END_REF][START_REF] Mercier | The Canadian food cold chain: A legislative, scientific, and prospective overview[END_REF]. In practice, product temperature is well controlled at the early stages e.g. production, storage, distribution and transport [START_REF] Baldera Zubeldia | Effectiveness of the cold chain control procedure in the retail sector in Southern Spain[END_REF]. However, with the exception of household practices, the temperature control at retail display appears to be problematic [START_REF] Evans | Retail display[END_REF][START_REF] Derens-Bertheau | Cold chain of chilled food in France[END_REF][START_REF] Derens | The cold chain of chilled products under supervision in France[END_REF]. In addition, food waste at supermarkets can be as high as 55% due to poor temperature conditions [START_REF] Nunes | Environmental conditions encountered during typical consumer retail display affect fruit and vegetable quality and waste[END_REF].

Open refrigerated display cabinets are widely used by retailers and account for almost half of the refrigerating equipment use in a typical supermarket [START_REF] Ashrae | Retail food store refrigeration and equipment ASHRAE Handbook: Refrigeration[END_REF]. The open display cabinets prove to be popular among retailers as they provide customers with almost unrestricted access to products [START_REF] Evans | Are Doors on Fridges the Best Environmental Solution for the Retail Sector? The Institute of Refrigeration[END_REF], whilst maintaining good product attractiveness [START_REF] Alzuwaid | The novel use of phase change materials in a refrigerated display cabinet: An experimental investigation[END_REF]. Nevertheless, their major limitations concern the thermal performance.

In open display cabinets, an air curtain creates a non-physical barrier between the cold air inside and the warm air outside. However, the air curtain cannot completely prevent the warm and humid ambient air infiltration [START_REF] Foster | The use of CFD to improve the performance of a chilled multi-deck retail display cabinet[END_REF], which leads to the increase of temperature and energy consumption [START_REF] Amin | Air curtains of open refrigerated display cases revisited: a new technique for infiltration rate measurements[END_REF][START_REF] Evans | Are Doors on Fridges the Best Environmental Solution for the Retail Sector? The Institute of Refrigeration[END_REF]Laguerre et al, 2012b). Several studies indicated that the ambient air infiltration has contributed approximately to 66-77% of total heat load of the open refrigerated display cabinets [START_REF] Faramarzi | Efficient display case refrigeration[END_REF][START_REF] Gaspar | Experimental analysis of the thermal entrainment factor of air curtains in vertical open display cabinets for different ambient air conditions[END_REF][START_REF] Tassou | Energy consumption and conservation in food retailing[END_REF]. It also causes internal temperature heterogeneity (Laguerre et al, 2012b).

Temperature differences of more than 5 °C was observed on cabinet shelves [START_REF] Willocx | A preliminary survey into the temperature conditions and residence time distribution of minimally processed MAP vegetables in Belgian retail display cabinets[END_REF], with the higher temperature commonly located at the front of the cabinets, and the lower temperature at the back [START_REF] Evans | Temperature and energy performance of refrigerated retail display and commercial catering cabinets under test conditions[END_REF].

Due to an increasing concern for energy efficiency and food safety, the issues of ambient air infiltration in open display cabinets have been subjects of attention.

Numerous experimental investigations and numerical simulations have been carried

out with the attempt to maximize the air curtain efficiency, thereby improving the overall performance of open display cabinets [START_REF] Chen | Parametric evaluation of refrigerated air curtains for thermal insulation[END_REF][START_REF] Chen | Experimental study of the performance of single-band air curtains for a multi-deck refrigerated display cabinet[END_REF][START_REF] Cortella | CFD simulation of refrigerated display cabinets[END_REF]D'Agaro et al., 2006;[START_REF] Foster | The use of CFD to improve the performance of a chilled multi-deck retail display cabinet[END_REF][START_REF] Gaspar | Detailed CFD modelling of open refrigerated display cabinets[END_REF][START_REF] Hadawey | Air curtain design optimization of refrigerated vertical display cabinet using CFD[END_REF][START_REF] Jouhara | Temperature and energy performance of open refrigerated display cabinets using heat pipe shelves[END_REF]Laguerre et al., 2012;[START_REF] Marinetti | Air velocity distribution analysis in the air duct of a display cabinet by PIV technique[END_REF][START_REF] Moureh | Numerical and experimental study of airflow patterns and global exchanges through an air curtain subjected to external lateral flow[END_REF][START_REF] Navaz | Jet entrainment rate in air curtain of open refrigerated display cases[END_REF][START_REF] Wu | A multi-scale approach for refrigerated display cabinet coupled with supermarket HVAC system -Part I: Methodology and verification[END_REF][START_REF] Yu | Modified two-fluid model for air curtains in open vertical display cabinets[END_REF]. [START_REF] Hammond | Application of short air curtain in retail display refrigerators[END_REF] proposed a novel short air curtain design, instead of a conventional single long air curtain, to improve temperature homogeneity and energy efficiency. In this study, despite multiple short air curtains were created to provide enclosure at the front of every shelf, they cannot eliminate the air infiltration.

Installation of doors on open-front display cabinets has become an alternative

design to address the air infiltration issue. It has been proven that the closed display cabinets show several favorable outcomes including potential energy savings [START_REF] Faramarzi | Efficient display case refrigeration[END_REF][START_REF] Fricke | Energy use of doored and open vertical refrigerated display cases[END_REF][START_REF] Kauffeld | Current and Future Carbon-saving Options for Retail Refrigeration[END_REF]Navigant Consulting Inc., 2009), reductions in internal temperature heterogeneity [START_REF] Evans | Performance of retail and commercial refrigeration systems[END_REF]Lindberg et al., 2010), and improved food quality [START_REF] Atilio De Frias | Improving spinach quality and reducing energy costs by retrofitting retail open refrigerated cases with doors[END_REF]. Additionally, doors can also provide better thermal comfort in retail stores due to less cold air spillage from the display cabinet (Lindberg et al., 2017). In view of these benefits, some major retailers in Europe have made closed display cabinets a default specification for their new and renovated stores [START_REF] Eia | Chilling Facts VII: Are Europe's supermarkets ready to quit HFCs? Environmental Investigation Agency[END_REF]. This trend is especially notable in France where the authorities and the supermarket stakeholders have signed an agreement to replace open display cabinets with closed ones. The country aims to increase the installation of closed display cabinets to 75% by 2020 [START_REF] Rpf | Fermeture des meubles réfrigérés: du positif pour les clients[END_REF].

Energy savings of closed display cabinets demonstrated in various studies (Figure 1.1) could be explained by a reduction in the entrainment of warm and moist ambient air into the cabinets. However, a wide variation of energy saving percentages among studies, ranging from 23-73% could be attributable to the difference of cabinet configurations and test conditions used in each study. This difference can be, for example, door integrity (retrofitted/factory-built), door type (hinged/sliding), door material (single/double/triple glazing windows with/without low emissivity coating), use of anti-sweat heaters, LED lightings, night covers, frequency and duration of door openings [START_REF] Evans | Are Doors on Fridges the Best Environmental Solution for the Retail Sector? The Institute of Refrigeration[END_REF]. The last one (frequency of door openings) is pointed out as one of the main factors contributing to an increased energy consumption due to the air entrainment (Li et al., 2007). The energy consumption of closed display cabinet during stable night condition was approximately 10% lower than that operating under periodical door openings [START_REF] Vallée | Energy saving potential at partial load for vertical glass door refrigerated display cabinets[END_REF]. Lindberg et al. (2010) showed that the heat extraction rate of the evaporator increased by 16% when the opening frequency increased from 10 to 30 openings per hour. Door openings appeared to be the largest contributor to the total heat load of closed display cabinet when the frequency reached 60 openings per hour [START_REF] Orlandi | CFD assisted design of closed display cabinets[END_REF].

Unlike energy consumption, the temperature performance of closed refrigerated display cabinets is less studied, only the internal temperature and its heterogeneity were reported. Lindberg et al. (2010) observed that an overall air temperature reduction of at least 2 °C was achieved in the cabinet retrofitted with doors. Atilio de [START_REF] Atilio De Frias | Improving spinach quality and reducing energy costs by retrofitting retail open refrigerated cases with doors[END_REF] reported that the temperature difference within the closed cabinet was 6 °C lower than that within the open one.

Research interests

The growing trend of closed refrigerated display cabinet use has contributed to the necessity to expand research in this field. However, to the author's knowledge, current literature only addresses the benefits of closed display cabinets over open ones. Studies on the mechanism of heat transfer and airflow within closed display cabinets are still limited. In fact, the airflow pattern influences the heat exchange between air and product, thus, product temperature. [START_REF] Faramarzi | Performance and energy impact of installing glass doors on an open vertical deli/dairy display case[END_REF] found that installing doors caused a reversal variation in product temperaturesproduct temperature at the front upper shelf became warmer than that at the bottom shelf.

To accommodate door openings, doors fitted to display cabinets often have gaps at the edges; however, these gaps allow ambient air infiltration [START_REF] Evans | Are Doors on Fridges the Best Environmental Solution for the Retail Sector? The Institute of Refrigeration[END_REF]. The advanced techniques of air velocity measurements, such as Particle Image Velocimetry (PIV), Laser Doppler Velocimetry (LDV), together with Computational Fluid Dynamics (CFD) simulations would provide an insight into the transport phenomena in closed refrigerated display cabinets.

Complex phenomena take place when door is open during which the disrupted air curtain causes the entrainment of warm and humid ambient air. A further investigation into the effects of frequency and duration of door openings on food temperature increase and energy consumption is therefore important.

A simplified heat transfer model based on a zonal approach should also be developed to obtain complementary knowledge of the mechanisms of heat transfer in a closed display cabinet. This model would be linked with the other wellestablished models for processing plant [START_REF] Lecoq | Simplified heat and mass transfer modeling in a food processing plant[END_REF], refrigerated vehicle [START_REF] Hoang | Heat transfer modelling in a ventilated cavity loaded with food product: Application to a refrigerated vehicle[END_REF], cold room [START_REF] Laguerre | Simplified heat transfer modeling in a cold room filled with food products[END_REF], open refrigerated display cabinets (Laguerre et al., 2012a) and household refrigerator [START_REF] Laguerre | Heat transfer by natural convection in domestic refrigerators[END_REF]. These models allow the prediction of product temperature in each equipment in short calculation time compared to CFD simulation. The closed display cabinets will soon be an additional component of the series of refrigeration equipment in the food cold chain. In conjunction with other equipment models, the closed display cabinet model would enable the prediction of time-temperature history of food products of the entire cold chain. Accordingly, the evaluation tool of consumer risk exposition can be constituted by integrating the proposed models with predictive microbial and/or quality models [START_REF] Duret | Identification of the significant factors in food quality using global sensitivity analysis and the accept-and-reject algorithm. Part II: Application to the cold chain of cooked ham[END_REF].

Aim and scope

Following the previous reasoning, the present thesis was conducted with the two main objectives.

Firstly, it aims to develop an experimental methodology to identify the thermal transport phenomena in a closed refrigerated display cabinet. Evolutions of spatial and temporal air/product temperatures and air velocity are the main parameters taken into investigations. The influence of different operating conditions on these parameters was then studied. These conditions include ambient air temperature, product-occupied volume, door-opening frequency and duration. Internal airflow pattern was characterized by both a hot-wire anemometer and PIV technique. The results obtained from these experiments were further used for the estimation of model input parameters and for the model validation purposes.

Secondly, it aims to develop numerical tools by both CFD simulation and simplified heat transfer model base on a zonal approach.

The CFD simulation allows the better understanding of airflow and heat transfer phenomena in the closed display cabinet, which cannot be observed by the experiments. However, the main objective is not to represent precisely all the flow and transfers details. Instead, the aim is to examine the capability of a 2D-CFD k-ε model to reproduce the main phenomena and to predict the product temperatures in the cabinet with acceptable accuracy. Although it is not in the scope of the thesis, after its validation with the experimental data, the model could be used to study the influence of unexplored operating conditions and design parameters.

The simplified model in both static and dynamic regimes permits, respectively, the predictions of both time-averaged air and product temperatures and the temperature fluctuations according to the on/off working cycles of the compressor regulation.

The model in the dynamic regime was solved with a spectral approach, thus the influence of different parameters on the damping of the temperature fluctuations can be identified with quick calculation.

Structure of the thesis

The structure of manuscript presented in Additional thermocouples (40 thermocouples) were inserted/attached to the core/surface of the test packages. It was observed that the trend of temperature distribution in the unloaded and loaded cabinet was the same (the highest temperature position at the front-top shelf and the lowest temperature position at the rear-bottom shelf). Finally, the benefit of door installation was studied by doing experiment using the unloaded cabinet with doors completely removed. Without doors, the mean air temperatures in the storage compartment were higher at all positions (minimum increase 0.4 °C and maximum increase 3.1 °C) compared to the case without doors. In addition to the temperature measurements, the air velocity was measured at the front of the closed-door and unloaded display cabinet for different heights from the discharge to return air grilles using a hot-wire anemometer. The shape of the air velocity in the curtain was presented (the air curtain was deviated towards the doors due to a Coanda effect with maximum air velocity of about 0.6 m•s -1 near the discharge grille). In the storage compartment, the air velocity was low, varying from 0.01 to 0.2 m•s -1 . The air velocity measurement in the rear duct was also carried out and the airflow distribution over the perforated back panel of different shelves was then quantified. More detail is presented in Article 2 (published). Because the precision of the velocity measurement using the hot-wire anemometer was limited, particularly for low velocity application (< 0.5 m•s -1 ), Particle Image Velocimetry (PIV) technique was used and presented in Chapter V.

Chapter IV Influence of operating conditions on the thermal performance. The thermal performance of the closed refrigerated display cabinet was presented under permanently closed-door condition in Chapter III (heat transfer analysis and airflow characterization using a hot-wire anemometer). This chapter continues to present the temperature performance under periodic door openings. An automatic door opening system developed and used in experiments enabled the investigation of effects of different door Opening frequencies Per Hour (OPH) on the air and product temperature variations. The studied frequencies were 0 (permanently closed), 10,20,40,60 OPH,and ∞ (permanently opendoors completely removed) and the studied opening duration was 15 s and 30 s. The frequency of 10 OPH with 15 s duration and 90° opening angle was programed with respect to the one prescribed in the standard test (EN ISO 23953-2, 2015), and this opening procedure was set to a baseline. Based on data available in the literature [START_REF] Fricke | Energy use of doored and open vertical refrigerated display cases[END_REF] and information provided by a display cabinet manufacturer, the opening frequency in supermarkets is averagely 6.3 OPH (12 s duration) and 12.1 OPH (13 s duration), respectively. Effects of ambient air temperature (14.6, 19.4, 24.3 and 29.2 °C) and product-occupied volume (half-loaded and full-loaded) on the air and product temperature distribution were experimentally investigated. It was observed that the product located at the front of mid-height of display cabinet was more influenced by the door opening frequency. However, the air and product temperatures in the cabinet with doors (whatever the frequency of door openings) remained lower, compared to the case without doors. More detail is presented in Article 3 (published).

Chapter V Airflow characterization using PIV and CFD techniques. Due to the precision limitation of the hot-wire anemometer for low velocity applications, the PIV technique was used to characterize the airflow pattern in the closed display cabinet. A Computational Fluid Dynamics (CFD) simulation was also developed in parallel due to the difficulties to implement the PIV technique experimentally at some areas in the cabinet. This chapter presents the air velocity measurement using a 2D-PIV system in a display cabinet with two configurations: closed and open doors (doors were open with 90° angle). In the closed configuration, experiments were conducted under two conditions: refrigeration system turned "off" or "on" so that its influence on the airflow pattern is highlighted. A zone of air recirculation was identified where ambient air infiltration was also observed through the door gaps, leading to an increase in the air curtain temperature). Because of the technical constraints, only the experiment with the refrigeration system turned "off" was Suggestions were also provided to generalize the model for other closed display cabinets and to take into account the air infiltration due to door openings in simple or more realistic manners. Like Chapter V (Airflow characterization using PIV and CFD techniques), some experimental data of Chapter III (Heat transfer analysis and airflow characterization using a hot-wire anemometer) were used for the model input and the model validation which showed a good agreement. More detail is presented in Article 5 (under review, second submission). Moreover, all input parameters determined in this chapter were further used in Chapter VII.

Chapter VII Simplified heat transfer modeling in unsteady state. This chapter presents the development of the simplified heat transfer model in transient state, which allows the prediction of the air and load temperature fluctuations according to the on/off working cycles of the compressor operation and the defrosting operation during which the compressor is "off". This dynamic model was solved with a spectral method by using Fourier transform and transfer functions. With this solving method, the model showed the influence of different parameters on the damping of the temperature fluctuations with quick calculation time. The estimation of the equivalent load thickness at which the temperature fluctuation was the same order as at the surface was also proposed. Some experimental data of Chapter III (Heat transfer analysis and airflow characterization using a hot-wire anemometer)

were used for the model input and the model validation which showed a good agreement. More detail is presented in Article 6 (under review, second submission).

Finally, the conclusions and perspectives are provided in Chapter VIII

Conclusions and perspectives.

A list of literature cited throughout the thesis is summarized in Reference.

List of publications

In peer-reviewed journals

Introduction

About a half of food products in retail food stores are presented in refrigerated display cabinets [START_REF] Bertrand | Consommation et lieux d'achat des produits alimentaires en 1991[END_REF]. Typically, for chilled products, vertical multideck display cabinets are often preferred because the limited space available in the stores can be utilized effectively. Food quality and consumer safety strongly depend on temperature control. Even though a great deal of legislation and regulations are extensively imposed, several surveys show that food products presented in retail cabinets encounter temperature abuse: this problem has been demonstrated in the UK [START_REF] Evans | Temperature and energy performance of refrigerated retail display and commercial catering cabinets under test conditions[END_REF], and in France [START_REF] Derens | The cold chain of chilled products under supervision in France[END_REF]. Around 360 million tons of food are lost annually worldwide because of the insufficient use of refrigeration (IIR, 2009).

Open-front refrigerated display cabinets are the most commonly used equipment, since they allow customers to access the products without any barriers. Despite good design of the air curtain, warm and humid air infiltration inevitably leads to temperature heterogeneity within the cabinets. This heterogeneity can cause food quality deterioration and high electrical energy consumption. In order to address these issues, numerous wide-ranging studies have been carried out with a view to improve the performance of display cabinets, particularly in terms of air curtain design, using both experimentation and numerical simulation. Nevertheless, infiltration can be limited using display cabinets with doors. Due to a possible reduction in product sales, customer interviews were carried out in supermarkets in England, and these interviews showed that half of the customers surveyed would prefer to purchase products from cabinets without doors [START_REF] The | The case for glass doors on refrigerated display cabinets is closed[END_REF].

On the contrary, [START_REF] Fricke | Energy use of doored and open vertical refrigerated display cases[END_REF] indicated that the utilization of closed display cabinets had no effects on product sales in the USA. As it is difficult to reach a clear conclusion regarding the influence of the presence of doors in terms of sales volume, and given the large amount of energy that can be saved, the use of closed refrigerated display cabinets is becoming more and more widespread, and closed cabinets will account for 75% of all display cabinets in retail stores by the end of 2020 in France [START_REF] Rpf | Fermeture des meubles réfrigérés: du positif pour les clients[END_REF]. This review article is divided into two main sections based on the type of refrigerated display cabinets: open or closed. Each section aims to present the state of the art of the investigations involving both field and laboratory studies on the air temperature evolution in the display cabinets and the food products inside the cabinets. These investigations were carried out by experimental and numerical approaches in order to study the heat and mass transfer, the airflow and the energy consumption of the cabinets. The objective of this review is to provide information on the current usage situation and to examine attempts to improve the performance of the display cabinets in terms of both temperature distribution and efficient energy consumption. Future trends in the field of refrigerated display cabinets are also highlighted.

Open display cabinets

Since energy saving has become a major concern worldwide during the past two decades, the performance of retail refrigeration systems must be optimized. Several experimental and numerical investigations have been carried out to determine the factors which have the greatest effects on system performance. These factors can be design parameters, operating conditions or ambient conditions. The knowledge acquired makes it possible to control the food temperature, to reduce temperature fluctuations and to achieve energy savings in refrigerated display cabinets.

Field investigation

A retail display cabinet constitutes the last stage in the food cold chain prior to purchase of food products by consumers. Effective temperature control is needed, as the quality of the products on the shelves can be adversely affected by inappropriate control and inadequate temperature levels [START_REF] Mariquele | Critical" Quality Points in fresh-cut vegetables supply chain[END_REF]. Field surveys have been carried out in various countries in order to investigate and identify the ongoing situation of cabinet operation. Most surveys produced the same results: chilled food products available in retail stores are subjected to temperature abuse as shown in Table 2.1.

A field study was carried out in France. The temperatures of three types of food products (dairy, prepared and packed meat products, with a total of 307 samples) throughout the entire food cold chain were monitored [START_REF] Derens | The cold chain of chilled products under supervision in France[END_REF]. The results showed that around 30% of the products were stored at temperatures that were at least 2 °C higher than the recommended values. Almost a decade later, a similar study within the framework of the European FRISBEE project was performed using a ham product [START_REF] Derens-Bertheau | Cold chain of chilled food in France[END_REF]. It was demonstrated that food preservation in display cabinets tends to take place under better temperature conditions than in the past due to a decrease in the mean product temperature and a lower percentage of products in which the temperature is higher than the specified temperature. However, the authors noted that the results should be interpreted carefully, given that only a small number of samples (83 samples) was considered in this study.

The position in which food products are placed inside the cabinet is also important.

Temperature differences of more than 5 °C were observed on the cabinet shelves by [START_REF] Willocx | A preliminary survey into the temperature conditions and residence time distribution of minimally processed MAP vegetables in Belgian retail display cabinets[END_REF]. When examining the effect of operating duration, they found that the temperature at one position increased by 4 °C towards the end of the day and by almost 7 °C towards the end of the week due to an increase in the store temperature. They also noticed that the top shelf had the highest temperature and that the opposite was true for the bottom shelf. This is quite different from [START_REF] Gill | The temperatures and ages of packs of beef displayed in multishelf retail cabinets[END_REF] who monitored the temperatures of beef products stored in open display cabinets in ten Canadian retail stores. The packages on the top and bottom shelves were similar and warmer than the packages on the middle shelves, while the packages at the front were warmer than the packages at the back. The results also showed that the maximum temperature of the selected product packages (5389 samples) from all stores was above 4 °C. Further investigation of this issue was performed in detail in a laboratory by [START_REF] Evans | Temperature and energy performance of refrigerated retail display and commercial catering cabinets under test conditions[END_REF]. [START_REF] Olsson | Chilled cabinet surveys[END_REF].

To gain an insight into the evolution of air and product temperatures in an open refrigerated display cabinet under unsteady conditions, Laguerre et al. (2012b) performed an experimental study. The air and product temperature fields in the cabinet and the temperature and velocity profiles of its air curtain were mapped using the Particle Image Velocimetry (PIV) technique. The results showed that the on/off compressor cycle promoted variations in the temperature of the air stream.

The air temperature varied readily, but the product temperature varied only slightly because of the thermal inertia of the products. The air temperature increased about 2.8 °C when the air flowed upwards along the back of the cabinet because of heat conduction through the wall. A temperature of 3.9 °C was observed at the supply air duct and the air temperature rose to 11.8 °C when the air reached the return air duct. As observed by [START_REF] Evans | Temperature and energy performance of refrigerated retail display and commercial catering cabinets under test conditions[END_REF], the product position in the cabinet is a determining factor of its temperature. For this reason, the products located at the front of the cabinet had higher temperatures than those at the back of the cabinet.

The product with the highest temperature was located at the front base of the cabinet and had a mean surface temperature of 9.5 °C, whereas at the back of this level, the lowest product temperature (0.9 °C) was observed. Based on the velocity profile mapped by PIV, the temperature fluctuations in the air curtain were affected by the introduction of ambient air via transient vortices which accelerate the infiltration rate. To determine the influence of operating conditions on the cabinet performance, a simplified heat transfer model was developed in a later study (Laguerre et al., 2012a), and these experimental results were used for the model validation.

The effect of ambient conditions

Howell et al. (1999a) monitored the relative humidity in eight supermarkets in the Tampa, Florida, area throughout the year (November 1997to October 1998). This study aimed to evaluate the existing levels of relative humidity in a retail store.

Overall, an annual average store relative humidity of 45% was observed: the lowest average relative humidity of 37% occurred in March and the highest average value of 57% was recorded in September. Howell et al. (1999b) demonstrated the influence of store relative humidity on display cabinet performance. These authors indicated that the performance based on energy requirements in the three components (cabinet refrigeration, defrost system and anti-sweat heaters) was strongly affected by the ambient relative humidity. Based on the results, the total energy demand of display cabinets was reduced by 10% when the store relative humidity was reduced by 5%, and consequently a 5% reduction in the total energy consumption of the entire store could be achieved. This can be explained by less frost formation on evaporator coils and less anti-sweat heat operation, and thus a lower latent heat load requirement. This observation is in agreement with that reported by [START_REF] Tassou | Frost formation and defrost control parameters for open multideck refrigerated food display cabinets[END_REF]. [START_REF] Axell | Field measurements in supermarkets[END_REF] conducted field measurement of temperature variations in refrigerated display cabinets in three supermarkets in Sweden. They found that the outdoor climatic conditions influenced the indoor conditions of the supermarkets, given that the total electrical power consumed by chilled display cabinets increased 55% during summer. This variation was mainly caused by variations in the ambient humidity. In the supermarkets, the mean air temperature in the cabinets during daytime operation was 4.1 °C in summer and 1.9 °C in winter.

The mean air temperature difference between the top and the bottom of the cabinets was 3.3 °C in winter and 6.7 °C in summer. A difference of more than 2 °C could be observed between daytime and night operation when a night cover was applied.

Regarding this observation, the authors pointed out that night covers should be placed on display cabinets during closing hours because 50% energy savings can be achieved. Moreover, the authors concluded that the vicinity of open display cabinets has provided poor thermal comfort to supermarket customers due to cold air spillage from the cabinets (a so-called cold leg effect). The recent study confirmed this finding by comparing the ambient temperature measurement with supermarket customers/staffs responses to the questionnaires (Lindberg et al.,

2017

).

An experimental study was carried out in a laboratory by [START_REF] Chen | Experimental study of the performance of single-band air curtains for a multi-deck refrigerated display cabinet[END_REF].

These authors found that room temperature and humidity caused an increase in the air temperature and heat gain in the cabinet refrigeration system. The temperature increase in the cabinet is mainly due to the increase in the room temperature, but this increase is also to some extent due to the humidity. However, higher humidity exerts effects on the latent heat gain, which in turn leads to higher energy demand.

The effect of Reynolds (Re) numbers (based on the supplied air velocity and the width of the air curtain) on the air temperature distribution and total heat gain was also shown. When the Re increased from 4100 to 5000, the temperature inside the cabinet decreased 1.0-1.5 °C approximately. When the Re increased from 5000 to 5700, the temperature changed only slightly. The authors proposed the thermal entrainment factor to characterize the air curtain performance; this factor is correlated to the Reynolds and Richardson (Ri) numbers. This study also showed the benefits of the application of night covers, which enable more homogeneous inside air temperatures to be achieved. It should be borne in mind that the Reynolds number characterizes flow regimes for forced convection (laminar or turbulent flow) and the Richardson number characterizes the importance of natural convection relative to forced convection.

In accordance with the results obtained by [START_REF] Gaspar | Experimental analysis of the thermal entrainment factor of air curtains in vertical open display cabinets for different ambient air conditions[END_REF], the ambient climate exerts a great impact on the total heat transfer rate and infiltration rate. The latent heat transfer rate component increased 42% when the humidity rose from 35% to 60%. The heat transfer rate due to air infiltration was the individual component with the greatest increase (from 67% to 77%). As humidity has an immediate impact on the latent heat transfer rate, the authors emphasized that the influence of humidity should be taken into account when calculating the thermal entrainment factor. [START_REF] Heidinger | Impact of environmental conditions on the performance of open multideck display case evaporators[END_REF] studied the influence of ambient conditions on the performance of the evaporators of refrigerated display cabinets. The climate classes 3 (25 °C, 60%RH) and 6 (27 °C, 70%RH) according to ISO 23953:2005 were compared in this study. A total heat load increase of 25% was observed when the climate changed from Class 3 to Class 6. This also caused thicker frost formation on the evaporator and led to less efficient heat exchange with air due to a reduction in the air velocity. Thus, the electrical energy demand of the display cabinets became higher.

By gaining an insight into the effect of ambient conditions, the design of display cabinets can be optimized, thus promoting energy savings and food safety.

The effect of design parameters and operating conditions

The infiltration between the refrigerated compartment of an open display cabinet and the ambient surroundings across its air curtain represents almost 67-77% of the total refrigeration load [START_REF] Faramarzi | Efficient display case refrigeration[END_REF][START_REF] Gaspar | Experimental analysis of the thermal entrainment factor of air curtains in vertical open display cabinets for different ambient air conditions[END_REF][START_REF] Tassou | Energy consumption and conservation in food retailing[END_REF].

Many researchers have endeavored to improve air curtain design in order to enhance the cabinet performance. [START_REF] Gray | Improvement of air distribution in refrigerated vertical open front remote supermarket display cases[END_REF] experimentally studied the effect of the proportion of holes in the perforated back plate on the performance of refrigerated display cabinets. The arrangement and the number of holes exerted a marked influence on the discharge airflow rate. These authors reported that a ratio of 70% airflow rate at the supply air duct and 30% at the perforated plate allows good cabinet performance. This condition allowed a higher Reynolds number (from 3740 to 4100) and a lower

Richardson number (from 0.13 to 0.11) compared with the original design. In this manner, the air temperature at the supply air duct decreased from -0.5 to -0.85 °C.

This Re number range, however, is lower than that reported by [START_REF] Chen | Experimental study of the performance of single-band air curtains for a multi-deck refrigerated display cabinet[END_REF] due to the different designs.

The influence of the width of return air duct of an open display cabinet performance was studied by [START_REF] Hammond | Investigation of return air grille width on the efficiency of recirculated air curtains[END_REF]. He studied the influence of the return air duct width in the range of 35 to 145 mm on the infiltration coefficient 𝐼 defined as 𝐼 = ( 𝑇 𝑟𝑎 -𝑇 𝑠𝑎 )/( 𝑇 𝑎𝑚𝑏 -𝑇 𝑠𝑎 ) where 𝑇 𝑟𝑎 , 𝑇 𝑠𝑎 and 𝑇 𝑎𝑚𝑏 are the return air, supply air and ambient temperatures, respectively. It was found that the optimal return air duct width was 70 mm (for a 300-mm high curtain with a jet velocity of 0.9 m•s -1 ). The author concluded that the effect of the width is relatively small on the cabinet performance; however, excessive width can cause poorer performance.

The temperature stability in the air curtains of open display cabinets is difficult to control because many parameters play simultaneous roles. [START_REF] Hammond | Application of short air curtain in retail display refrigerators[END_REF] proposed a new air curtain design to replace the conventional one, the so-called short air curtain. With this innovation, the front of the cabinet is enclosed by multiple short air curtains provided for each shelf rather than a single long air curtain. The experimental results showed that the cabinet with the short air curtain generated higher energy savings and greater temperature distribution uniformity.

The cooling load was reduced from 4.24 kW to 3.04 kW (reduction of 28.3%), and this represented a 35.9% reduction in electrical energy consumption. The range of maximum and minimum product temperatures was also narrower (a difference of 9.5 °C for the conventional design and 3.1 °C for the short air curtain). These results can assist display cabinet manufacturers or engineers when designing and manufacturing high-performance display cabinets. [START_REF] Yu | Experimental investigation on a vertical display cabinet with central air supply[END_REF] proposed an alternative design to maximize air curtain performance by supplying cold air from a central unit located outside the display cabinet. This system was then compared with a conventional display cabinet with an integrated cold production system. Based on the experimental results obtained by these authors, many benefits were obtained in the case of cold air supplied from the central unit, i.e. a better-controlled air curtain, longer intervals between defrost cycles (extended from once every 6 h to once every 9 h) due to less frost formation and lower maximum product temperature during defrosting periods. Moreover, these authors found that the optimal velocity of the air curtain for this cabinet type is 0.7-0.8 m•s -1 , corresponding to a Re number of 4350-5000. [START_REF] Kou | Open-refrigerated retail display case temperature profile and its impact on product quality and microbiota of stored baby spinach[END_REF] studied the influences of thermostat settings (at -0.5 °C and -2.2 °C with the same defrosting interval of 12 h and duration of 30 min) on the product temperature and quality of packaged baby spinach products at various locations.

With the -2.2 °C thermostat temperature, the products located on the top shelf at the front of the display cabinet had the highest temperature (a mean temperature of 6.5 °C) and the temperature decreased subsequently towards the back of the cabinet.

The lowest mean temperature was below zero (-0.6 °C), and the product located in this zone was thus subjected to freezing damage. To overcome this thermal problem, the warmer temperature of the thermostat was set at -0.5 °C. However, the products in the front rows then undergo temperature abuse. This represents the situation in supermarkets and the controlling of such temperature abuse to ensure that all products are preserved at suitable temperatures without freezing problems is becoming a major challenge. These results were confirmed by Laguerre et al.

(2012b): a warmer product temperature at the front and a cooler product temperature at the rear of the same shelf.

The temperatures of products preserved in display cabinets can rise several degrees during defrost operation and will take an hour to reach the correct temperature once again. Under such fluctuating temperature conditions, temperature-sensitive food products can deteriorate at rates that are different from those of products stored under normal conditions [START_REF] Wells | A quality-based inventory issue policy for perishable foods[END_REF]. The application of a heat pipe and phase-change materials (PCM) installed in the cabinet shelves was introduced to deal with this issue by Lu et al. (2010). Two cabinet prototypes, a shelf retrofitted with a heat pipe and combined heat pipe and PCM, were experimentally tested in their study. Based on their results, the authors claimed that the use of a heat pipe and PCM provides several benefits: a reduction in the core product temperatures (heat pipe and PCM) and the average temperature of food packages is reduced by 13.7 -32%.

Development of numerical models

A field investigation and laboratory experimentation are necessary in order to determine the operating conditions and problems occurring within a refrigerated display cabinet. However, such investigation is both costly and time-consuming.

For this reason, a numerical approach is complementary. Modelling of refrigeration systems can be applied to practical use in real situations. The model allows the operator to observe and predict the thermal phenomena under various conditions, and the knowledge acquired will allow the operator/engineer to find solutions designed to improve the performance. Thanks to recent advances in computer technology, modelling has become more reliable, and increasingly detailed and complex models can be developed. Computational Fluid Dynamics (CFD) is a tool that enables simulation of fluid flow problems by solving differential equation systems (mass, momentum, and energy equations). It is being successfully implemented for many applications worldwide, including refrigeration applications, as highlighted by [START_REF] Smale | A review of numerical models of airflow in refrigerated food applications[END_REF]. This review aims to further expand more recent developments in the area of refrigerated display cabinets. Table 2.2 summarizes the numerical studies conducted on open display cabinets using the CFD approach.

In earlier numerical studies related to the refrigerated display cabinet application, CFD codes based on the finite element method [START_REF] Cortella | cfd-aided retail cabinets design[END_REF][START_REF] Cortella | CFD simulation of refrigerated display cabinets[END_REF] and finite difference method [START_REF] Ge | Simulation of the performance of single jet air curtains for vertical refrigerated display cabinets[END_REF] were developed individually in the authors' laboratories. Since computer technology is now more advanced and more reliable commercial CFD codes have been developed, many researchers tend to adopt them in their studies. The most common codes include FLUENT and CFX in which discretization is based on the finite volume method. CFD is proving to be a very useful tool at the design stage of refrigerated display cabinets as it enables knowledge of airflow and thermal phenomena to be acquired.

Among the typical thermal load contributions in an open display cabinet, the contribution of infiltration is responsible for the greatest thermal load [START_REF] Tassou | Energy consumption and conservation in food retailing[END_REF]. Reducing infiltration into the food preservation zone of the display cabinet is becoming the main priority in most studies. With validated CFD models, the efficiency of cabinet air curtains is quantified on the basis of the heat transfer rate across the air curtains [START_REF] Chen | Parametric evaluation of refrigerated air curtains for thermal insulation[END_REF][START_REF] Gaspar | Detailed CFD modelling of open refrigerated display cabinets[END_REF][START_REF] Ge | Simulation of the performance of single jet air curtains for vertical refrigerated display cabinets[END_REF][START_REF] Madireddi | Computation of three-dimensional flow field and heat transfer inside an open refrigerated display case with an air curtain[END_REF], or in terms of thermal entrainment (D'Agaro et al 2006; [START_REF] Moureh | Numerical and experimental study of airflow patterns and global exchanges through an air curtain subjected to external lateral flow[END_REF][START_REF] Navaz | Jet entrainment rate in air curtain of open refrigerated display cases[END_REF]. Considering this defined factor as a benchmark, researchers can then numerically modify cabinet design parameters and vary operating conditions so that optimal conditions can be obtained.

The findings of various studies provide a common principle even though different types of display cabinets were examined in the studies. The insulating ability of the air curtain can be maintained by selecting appropriate jet characteristics. To ensure that less ambient warm air is introduced, the jet must not generate a high turbulence intensity but must be able to reach the return air duct with minimum spillage to the external surroundings so that the open front is totally sealed, and customer comfort is ensured. The air curtain can be scientifically described by two dimensionless variables: the Reynolds number (Re) and the Richardson number (Ri). Reducing the Re of the curtain can achieve this characteristic. However, there is no specific optimum value as each particular cabinet establishes its own value. The Re in the range of 3200-3400 were recommended by [START_REF] Navaz | Jet entrainment rate in air curtain of open refrigerated display cases[END_REF], while [START_REF] Field | Entrainment of refrigerated air curtains down a wall[END_REF] reported that for conventional display cabinets, the Re range of 4200-8000 was typically found.

To represent the most accurate prediction of the real situation, a CFD model has to take into account many aspects. Some significant parameters to be taken into account for model development were underlined by several researchers. These parameters include ambient radiative heat transfer (Laguerre et al., 2012a;[START_REF] Madireddi | Computation of three-dimensional flow field and heat transfer inside an open refrigerated display case with an air curtain[END_REF] and geometric dimensions (2D or 3D) (D'Agaro et al., 2006). Moreover, some authors have introduced new modelling techniques to predict the performance of display cabinets: the integration of refrigeration devices in calculation domains [START_REF] Gaspar | Detailed CFD modelling of open refrigerated display cabinets[END_REF][START_REF] Ge | Simulation of multi-deck medium temperature display cabinets with the integration of CFD and cooling coil models[END_REF], calculation based on normal or modified two-fluid (TF or MTF) models in which the flow regime is assumed to consist of two fluids (turbulent and non-turbulent fluids) [START_REF] Cao | A novel strategy for predicting the performance of open vertical refrigerated display cabinets based on the MTF model and ASVM algorithm[END_REF][START_REF] Yu | Simulation of air curtains for vertical display cases with a two-fluid model[END_REF][START_REF] Yu | Modified two-fluid model for air curtains in open vertical display cabinets[END_REF]. Most of developed CFD models for display cabinet considered the flow regime at the exit of heat exchangers as an input model boundary. However, airflow maldistribution in the heat exchangers and cooling air ducts can cause an aggravated evolution of velocity profile at the discharge air grille of display cabinets. To obtain an insight of the transport phenomena in this region, numerical studies have been performed by [START_REF] Marinetti | Numerical and experimental analysis of the air flow distribution in the cooling duct of a display cabinet[END_REF]. After validation with the experimental results [START_REF] Marinetti | Air velocity distribution analysis in the air duct of a display cabinet by PIV technique[END_REF], the obtained results notified that there was a maldistribution of mass flow rate in the ducthigher velocity at the exit of cooling coils was observed on the right part. The study of Rossetti et al. (2015) showed that this air velocity has a great relationship with its temperature.

Due to the residual time inside the evaporator, higher air outlet temperatures were in the higher velocity areas whereas lower air outlet temperatures were in lower velocity area. With all obtained knowledges, a better model development can be promoted.

Several correlation models have been developed in order to assess the performance of refrigerated display cabinets in a rapid manner. Based on the results obtained using CFD models, simplified correlation models for the total heat transfer across the air curtain and the return air temperature were established and introduced by Ge and Tassou (2001). Using an approach similar to that of Yu et al. (2009a) (Laguerre et al., 2012b) were employed for the model validation. A comparison showed that the calculated values of both temperatures were slightly below the predicted values due to the simplification hypotheses.

Finally, the authors mentioned that the product time-temperature evolution throughout the entire food cold chain can be monitored by integrating this developed model with similar models that have already been developed for domestic refrigerators and refrigerated vehicles [START_REF] Laguerre | Using simplified models of cold chain equipment to assess the influence of operating conditions and equipment design on cold chain performance[END_REF]. As previously mentioned in Section 2.2.4, defrosting has a negative impact on the food shelf life of food products in display cabinets as well as energy consumption.

One notably solution using phase-change materials (PCM) has been proposed by many researchers. It was experimentally and numerically proven that the positive responses using PCM can be achieved including higher energy efficiency and more stabilization of air and product temperatures during defrosting [START_REF] Alzuwaid | The novel use of phase change materials in an open type refrigerated display cabinet: A theoretical investigation[END_REF][START_REF] Hoang | Potential use of phase change materials in a display cabinet: development of a dynamic modelling approach[END_REF]. Another technology to improve the performance is to apply a desiccant to refrigeration system to remove moisture in return air before entering to an evaporator [START_REF] Zhang | Theoretical study on a frost-free refrigerated display cabinet[END_REF]. Through theoretical analysis, the proposed system of a frost-free (open) refrigerated display cabinet can provide high coefficient of performance (COP) of 3.2. The authors noticed that the humidity and the temperature of return air have a significant effect on the COP. They reported that COP increased by 35% when humidity decreased from 80% to 50% and by 115% when return air temperature decreased from 14 to 8 °C.

Closed display cabinets

Ambient air infiltration is a major drawback in open-front display cabinets. In spite of efforts to provide better design, two problems are often observed in practice:

temperature control and energy efficiency. In addition, more tighten regulations are imposed i.e. the Food Code by the U.S. Food and Drug Administration (FDA). It requires all refrigerated food products need to be stored at the temperature at 5 °C or below (FDA, 2013). These technical issues bring more challenges to both manufactures and retailer to develop new design of display cabinet. An alternative is the installation of doors. It has been proven that energy savings of up to 70% can be achieved with this solution [START_REF] Fricke | Energy use of doored and open vertical refrigerated display cases[END_REF][START_REF] Rhiemeier | Comparative assessment of the climate relevance of supermarket refrigeration systems and equipment[END_REF][START_REF] Rolfsman | Changes of the refrigeration system in the dairy section of a Supermarket-Field measurements[END_REF] as well as the better temperature homogeneity (Atilio de [START_REF] Atilio De Frias | Improving spinach quality and reducing energy costs by retrofitting retail open refrigerated cases with doors[END_REF]Lindberg et al. 2010). Given these findings, the thermal performance of closed refrigerated display cabinets is attracting more and more attention.

Moreover, the use of doors by retrofitting an open fronted display cabinet can also provide environmental and economic benefits as highlighted in the review of [START_REF] Evans | Are Doors on Fridges the Best Environmental Solution for the Retail Sector? The Institute of Refrigeration[END_REF]. The thermal comfort of supermarket customers has been improved since around 2 K of mean temperature of air in front of the cabinet retrofitted with doors was increased (Lindberg et al., 2008). The prediction of annual financial saving of supermarkets by fitting doors to cabinets varies due to the store sizes (about 50,000-300,000 € for 2,500 to 18,000 m 2 ) and a payback period for door retrofitting was about 16 months. Nevertheless, the author [START_REF] Evans | Are Doors on Fridges the Best Environmental Solution for the Retail Sector? The Institute of Refrigeration[END_REF] noted that there were some limitations of the reported data, i.e. limitation of cabinet numbers, neglect of some significant factors (net demand of cooling capacity in stores, maintenance, labors, etc.) in cost estimation.

Field and laboratory investigations

Many comparisons of the performance of refrigerated display cabinets with and without doors have been performed by a number of researchers. The potential energy savings are usually taken into consideration in most studies, as shown in Table 2.3. Such savings are mainly achieved through a reduction in the entrainment of ambient warm and moist air into the preservation space, which leads to less frost formation on cooling coils and less compressor energy demand [START_REF] Faramarzi | Performance and energy impact of installing glass doors on an open vertical deli/dairy display case[END_REF]. However, findings published by [START_REF] Evans | Performance of retail and commercial refrigeration systems[END_REF] The advantages of the implementation of refrigerated display cabinets closed with doors compared with conventional open-type cabinets in terms of energy consumption, temperature variation as well as the food quality still constitute an issue which is generating discussion. Thus, further investigation is required in order to obtain additional data.

Development of numerical models

In order to understand the mechanism of transport phenomena occurring in a closed display cabinet, particularly when its doors are open, both experimental and numerical investigations are required. A review of the literature shows that little research involving extensive studies has been conducted, as shown in Table 2.4.

To the authors' knowledge, the only detailed numerical study applied to this type of cabinet is that performed by [START_REF] Orlandi | CFD assisted design of closed display cabinets[END_REF]. The incoming energy was numerically quantified during the door opening procedure (opening, holding and closing). In their study, two three-door cabinets were considered: one was equipped with sliding doors and the other one was equipped with hinged doors, and only the central door openings were analyzed. The comparative results showed that the internal temperatures were relatively similar for both door types during the holding step. However, during the opening and closing steps for the cabinet with sliding doors, the air temperature at the return air duct was markedly lower than in the case of the cabinet with hinged doors. For this reason, the display cabinet closed with sliding doors consumed 17% less energy than the other cabinet.

Moreover, the research team also highlighted the impact of the door opening frequency on energy consumption. According to ISO 23953 (10 openings, each with a duration of 15 seconds, per hour per door), the energy contribution due to door openings was responsible for 12% of the total heat extraction rate on evaporator coils, while the contribution of lights was greater (25%). Nevertheless, when a higher frequency was applied (60 openings per hour for each door), the contribution of openings became significant. It accounted for 44% of the total heat extraction rate. This very high frequency may not happen in real life situations. [START_REF] Fricke | Energy use of doored and open vertical refrigerated display cases[END_REF] found that the most frequent door opening duration was only 5 seconds and the daily mean door opening frequency was only 6.3 openings per hour, which means that one door is opened every 9.5 minutes. These data would indicate that the standard door opening cycle is more demanding than the most common situation in practice. Another numerical model of a closed display cabinet was used by D' Agaro et al. (2006). In this study, a model of the misting and demisting process in the glass doors of a closed frozen product display cabinet was developed. The main objective of this study was only to gain an insight into the physical mechanism of the thermo-fluid phenomena of fogging and defogging taking place during the door opening, as this is a very important issue for this type of cabinet. The high fogging level leads to poor transparency of the glass, thus influencing the visibility of the products from the customer's point of view.

Defogging with a heater is then required, and this process leads to higher energy demand and cooling load. However, no case study was shown to demonstrate the influence of these phenomena on the cabinet performance in terms of both energy efficiency and temperature distribution.

Conclusions

Many studies on retail refrigerated display cabinets have been carried out over the past two decades, and awareness of food product quality and energy efficiency is

Chapter III

Heat transfer analysis and airflow characterization using a hot-wire anemometer

Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 43 subjected to temperature abuse despite extensively imposed legislation and regulations. A field investigation conducted in France showed that around 30% of monitored products were stored at temperatures above their recommended values [START_REF] Derens | The cold chain of chilled products under supervision in France[END_REF]. Even if the situation tends to improve, and products have been maintained under better temperature conditions during the past decade [START_REF] Derens-Bertheau | Cold chain of chilled food in France[END_REF], insufficient temperature control remains problematic.

Nomenclature

A area [m -2 ] Dh hydraulic diameter = 4𝐴 𝑃 [m] Q power supply [W] P perimeter [m] Re Reynolds number [-] T temperature [°C or K] 𝑇 ̅ mean temperature [°C or K] 𝑇 * dimensionless temperature [-] ΔT mean temperature difference between front and back positions [°C or K] U overall heat transfer coefficient [W•m -2 •K -1 ] UA overall thermal transmittance [W•K -1 ] 𝑣 velocity [m•s -1 ] Greek letters 𝜌 density [kg•m -3 ] 𝜇 dynamic viscosity [kg•m -1 •s -1 ]
Vertical open refrigerated display cabinets are the most common refrigerated equipment adopted in many retail stores because they enable customers to have free access to food products. Regarding this open design, the food products stored inside the display cases are isolated from the external ambient environment only by one or more cold air curtains. Although the air curtain is well designed, warm and humid air infiltration is unavoidable. It is the greatest drawback of this type of display cabinet because it causes very high energy consumption: air infiltration contributes to approximately 66-77% of heat gain [START_REF] Faramarzi | Efficient display case refrigeration[END_REF][START_REF] Gaspar | Experimental analysis of the thermal entrainment factor of air curtains in vertical open display cabinets for different ambient air conditions[END_REF][START_REF] Tassou | Energy consumption and conservation in food retailing[END_REF] and it results in temperature heterogeneity inside the cabinets. As reported by [START_REF] Willocx | A preliminary survey into the temperature conditions and residence time distribution of minimally processed MAP vegetables in Belgian retail display cabinets[END_REF], temperature differences of more than 5 °C were observed on the cabinet shelves. [START_REF] Evans | Temperature and energy performance of refrigerated retail display and commercial catering cabinets under test conditions[END_REF] indicated that the maximum temperatures in cabinets were generally in the front areas which is the most exposed to ambient whereas the minimum temperatures were located in the back. These authors reported that for 135 investigated display cabinets, 97% of the packages of the maximum temperature were at the front of the cabinets (60% of them were on the bottom shelves), and 98% of the packages at the minimum temperature were at the back. These findings match those obtained in an experimental study conducted by Laguerre et al. (2012). The product with the highest temperature with a mean surface temperature of 9.5 °C was located at the front and at the bottom of the cabinet, while the lowest product temperature (0.9 °C) was observed at the back, on the same level.

Many studies have been conducted on open and closed display cabinets and were reviewed by [START_REF] Chaomuang | Experimental and numerical investigation of the performance of retail refrigerated display cabinets[END_REF]. Between them, experimentation (Amin et al., 2009;[START_REF] Chen | Experimental study of the performance of single-band air curtains for a multi-deck refrigerated display cabinet[END_REF][START_REF] Gaspar | Experimental analysis of the thermal entrainment factor of air curtains in vertical open display cabinets for different ambient air conditions[END_REF] and numerical simulation [START_REF] Cortella | cfd-aided retail cabinets design[END_REF]D'Agaro et al., 2006;[START_REF] Moureh | Numerical and experimental study of airflow patterns and global exchanges through an air curtain subjected to external lateral flow[END_REF] were carried out with a view to improve the performance of display cabinets. In these studies, the influence of significant factors on the display cabinet performance has been

Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 45

pointed out: ambient conditions, design parameters and operating conditions.

Nevertheless, the issue of air infiltration remains unsolved and is still attracting a lot of attention, and therefore a lot of research is being conducted on this issue.

The installation of doors on open display cabinets is becoming an alternative and has proven to provide many advantages including the achievement of energy savings [START_REF] Faramarzi | Performance and energy impact of installing glass doors on an open vertical deli/dairy display case[END_REF][START_REF] Fricke | Energy use of doored and open vertical refrigerated display cases[END_REF], reduced temperature fluctuations [START_REF] Evans | Performance of retail and commercial refrigeration systems[END_REF]Lindberg et al., 2010) and improved food quality (Atilio de [START_REF] Atilio De Frias | Improving spinach quality and reducing energy costs by retrofitting retail open refrigerated cases with doors[END_REF]. These results encourage many retailers to

replace their conventional open-front display cabinets with closed cabinets fitted with doors. Particularly in France, according to a voluntary convention signed between the government and the main retailer groups, it is expected that both new and refurbished closed display cabinets will account for up to 75% of all display cases in retail stores by 2020 [START_REF] Rpf | Fermeture des meubles réfrigérés: du positif pour les clients[END_REF]. This transition is generating a new research trend. The energy consumption of the refrigeration system of closed display cabinets has been widely studied, but few studies focused on the temperature performance (airflow and its corresponding temperature distribution),

which impacts food quality.

The present study was an attempt to bridge this research gap and was designed to investigate the air and product temperatures and airflow in a closed refrigerated display cabinet. With the acquired knowledge, real-life conditions and possible problems of food preservation in this equipment can be identified, thereby enabling design improvements and operating practices to be addressed. The air temperature and air velocity can be controlled in the test room, but it is technically not possible to control the air humidity. The measurement of the air characteristics (Ta and %RH) was undertaken with a hygrometer (Testo 174H, accuracy ±3%RH) and the air humidity ratio (kg of water vapor/kg of dry air) was then calculated using a psychrometric chart. It was found that the value of air humidity ratio in the test room was in the range of 0.003-0.004 kg of water vapor/kg of dry air for all room temperatures. This level of humidity is two times lower than that of EN 23953 testing condition (0.006-0.007 kg of water vapor/kg of dry air for a room temperature of 20 °C). As higher air humidity would accelerate frost formation on the evaporator, on one hand, this could affect the refrigeration load.

Material and methods

Description of the closed refrigerated display cabinet

On the other hand, a large amount of frost on the evaporator could decrease the air flow rate due to an increase of pressure drop and it could also cause higher temperature of the supply air due to less heat exchange efficiency. So, strictly speaking, the obtained results correspond to stores in relatively dry climate conditions. Nevertheless, for a given air flow rate and a given air temperature after the evaporator, the air and product temperatures should be the same for other air humidity values since there was no condensation observed in the cabinet during the experiments. The experimental results reported by [START_REF] Chen | Experimental study of the performance of single-band air curtains for a multi-deck refrigerated display cabinet[END_REF] affirmed that the variations in air humidity have only a moderate effect on the temperature rise inside the display cabinet (open type in their study). These authors observed that the refrigeration load increases when the humidity ratio increases.

Determination of the overall thermal transmission

The overall thermal transmission (𝑈𝐴) of the closed display cabinet (a low value indicates high insulating capacity) was evaluated by means of reverse heat loss rate measurement. The display cabinet, of which the refrigeration unit was switched off, was installed in a climate-controlled room in which the ambient temperature (𝑇 𝑒𝑥𝑡 ) was maintained at 5 °C. The internal temperature in the cabinet ( 𝑇 𝑖𝑛𝑡 ) was maintained at about 30 °C, by adjusting the power supplies of two electric heaters, placed on the left and right sides of the cabinet. In addition, small fans were installed at the end of each heater, and the cabinet ventilating fans were turned on to ensure a homogeneous air temperature inside the display cabinet. The temperature difference (𝑇 𝑖𝑛𝑡 -𝑇 𝑒𝑥𝑡 ) between the inside and the outside of the display cabinet (25 °C) is of the same order of magnitude that under real operating conditions in a supermarket. Temperatures at various positions inside (7 points) and outside (5 points) of the cabinet were measured with calibrated thermocouples (T-type, accuracy ±0.2 °C). Temperatures were recorded every minute until the cabinet temperatures reached a steady state (after 6 h), during which the power supplies of the heaters (𝑄 ℎ ), small fans (𝑄 𝑓ℎ ) and cabinet fans (𝑄 𝑓𝑐 ) were also recorded using digital wattmeters (Digiwatt, accuracy ±1% of the reading). The internal and external air temperatures and total power supplies were then averaged over a period of 5 h of steady state, and the overall thermal transmission of the display cabinet was calculated using the following equation:

𝑈𝐴 = 𝑄 ℎ +𝑄 𝑓ℎ +𝑄 𝑓𝑐 𝑇 𝑖𝑛𝑡 -𝑇 𝑒𝑥𝑡 (3.1)
According to the equipment accuracies, the relative accuracy of 𝑈𝐴 was estimated of which the value was about 3%. on, and the air temperatures were recorded every 10 seconds for at least 24 hours using two data loggers (Agilent 34970A). The mean values were then calculated at each measurement point during quasi-steady state (temperature fluctuations during defrost cycles were excluded from the calculation). Replications were also run in order to check repeatability.

Air temperature measurement

Another experiment was performed in the display cabinet loaded with test product packages (60% of total storage volume). The package was made of methylcellulose Zone 1, the airflow was mostly dominated by the air supply from the PBP, and the anemometer was oriented against the horizontal flow direction (x-axis). Both vertical and horizontal flow directions (x-and y-axes) were measured for Zones 2 and 3. To take into account the air velocity fluctuations with time, the air velocity at each position was averaged from 300 measurements (every second for 5 min).

The measurement began after 1 min since doors were closed to minimize the influence of door movement on flow perturbation. In fact, some preliminary tests had been run previously for different durations (3, 5, 7 and 10 min) and it was found that the measurement duration should be at least 5 min. to ensure the reliability of the mean air velocity.

Airflow rate in the vertical rear duct

To quantify the distribution of the airflow rate discharged through the PBP at various heights, measurement of the air velocity inside the vertical rear duct of the (y = 1.335 m) were considered, and at each height three holes were drilled in the back wall of the cabinet (at z = -0.26, 0, and +0.26 m). In each hole, the air velocity at three positions across the rear duct was measured using the hot-wire anemometer.

As mentioned in Section 3.2.4, the air velocity was recorded every second for 5 min and its mean value at each position was then calculated. As the cross-section of rear duct is known, it is possible to calculate the airflow rate at each height. The

Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 51

influence of the occupied volume of the load in the display cabinet on the airflow rate distribution was also studied: 0, 36% and 52% of total storage volume. Total plate area includes the area covered by shelf thickness (0.03 m 2 )

Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 53

Results and discussion

Overall thermal transmission of the display cabinet

There were apparently three main heat loss paths in the display cabinet: through glass walls (doors and side walls), through insulated walls (rear, top and bottom walls, conductive heat transfer coefficient of about 0.77 W•m -2 •K -1 ), and through the gaps (about 8 mm) along the door frame where air can flow out and be replaced by ambient air. The overall thermal transmission of the closed display cabinet was found to be 21.3 W•K -1 . This value corresponds to an overall average heat transfer coefficient of about 2.74 W•m -2 •K -1 , considering that the mean surface area of the display cabinet was 7.8 m 2 . To the authors' knowledge, the thermal transmission of closed display cabinets is not available in the literature. The value of the overall average heat transfer coefficient of the studied closed display cabinet was 5 -10 times higher than those of domestic refrigerators of which the value was about 0.28 [START_REF] Melo | Experimental evaluation of the heat transfer through the walls of household refrigerators[END_REF]. The poor insulating capacity of the display cabinet compared with that of domestic refrigerators can be mainly explained by greater heat transfer rate through the doors and side walls (made of double-glazing glass, with an conductive heat transfer coefficient of 1.13 W•m -2 •K -1 ) which are less insulated than those of domestic

W•m -2 •K -1 (Laguerre, 2010) and 0.53 -0.56 W•m -2 •K -1 (
Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 54 refrigerators (0.67 W•m -2 •K -1 ) [START_REF] Laguerre | Heat transfer by natural convection in domestic refrigerators[END_REF], and by convective losses through the gaps around the doors and through thermal bridges i.e. the connections between the internal and external components (evaporator, expansion valve and compressor) of the refrigeration system. Based on our estimation of heat losses through the walls and the doors (by knowing the thermal conductivity and thickness of wall and door materials as well as by estimating the convective resistances), gaps and thermal bridges were responsible for approximately 70% of the overall thermal transmission of the display cabinet. According to our estimation of room lighting with fluorescent tubes (~600 lux), transmission of visible radiation corresponds to less than 7% of the total transmission of conduction and convection through the glass doors and sides walls (about 25 W•m -2 ). an increase in the minimum temperature was observed (-7 °C at "a" to -6 °C at "c") while the maximum temperature was relatively similar in all positions (about 2 °C).

Temperature profiles

Empty closed display cabinet

The heat loss through the cabinet walls explained a slight increase in the mean

Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 56 temperature of 0.3 °C (-1.3 °C at "a" and -1.0 °C at "c"). The standard deviation, however, became slightly lower (2.5 °C at "a" and 2.1 °C at "c"). This can be explained by stabilization due to the thermal inertia of the wall of the rear duct. The same phenomena were observed in the upper horizontal duct (ceiling) from "c" to "d": an increase in the mean temperature (-1.0 °C at "c" and -0.6 °C at "d") due to heat losses and a lower standard deviation (2.1 °C at "c" and 1.6 °C at "d") due to exchanges with duct walls and the honeycomb of the DAG.

In the area in which food products are stored (positions e, f, g, h, i and j in Figure 3.7), the air temperature was below 2 °C, which is the recommended temperature for perishable foods (meat and fish). The lowest air temperature value was observed at the back of the bottom shelf (position "i"): mean temperature of -0.6 °C, standard deviation of 1.3 °C. This could cause freezing damage to foods with high moisture content like fruit and vegetables. A great deal of attention should be paid to the selection of the types of foods to store in such positions inside display cases.

The mean temperatures at the front of the shelves (positions f, h and j in Figure 3.7:

1.1 °C, 0.6 °C and -0.2 °C, respectively) were higher than at the DAG (-0.6 °C at "d"). This is because the air curtain exchanges heat with doors, through which heat is transferred from the external ambient (19 °C) and because of the infiltration of external air through the gaps. The mean temperatures at the back of the shelves (positions e, g and i in Figure 3.7: 0.5 °C, 0.3 °C and -0.6 °C, respectively) were lower than those at the front, especially on the bottom shelf. This is because of the air flowing through the PBP at an initial temperature of around -1.0 °C, particularly on the bottom shelf where the percentage of perforated area was the highest (8.3%).

Mixing of air coming from the back and from the air curtain probably occurred on each shelf as shown schematically in Figure 3.7.

In the area in which food products are stored, with the exception of the back of the bottom shelf (position "i"), the standard deviations were lower than 1 °C in spite of the large temperature fluctuations just after the evaporator (standard deviation of 2.5 °C, position "a"). The fluctuations in this zone are acceptable, and thus, a reduction in the amplitude of the regulation (-7 °C/ 2 °C) is not necessary for this display cabinet. In fact, such regulation would increase the number of on/off cycles, thus reducing the lifespan of the compressor. The ratio of high-low compressor

Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 57

pressure used for this display cabinet was 5 to 1 and the average evaporating temperature was about -7.5 °C.

The reduction in the standard deviation of the air temperature on the shelves in comparison with that at the DAG is due to heat exchange between air and the wall, door and shelves for which the thermal inertia damps the fluctuations. At the back of the bottom shelf, the standard deviation was the highest (1.3 °C) in the storage zone because a large amount of air came from the rear duct where the standard deviation was also high (2.5 °C) through the PBP which is perforated to a large extent (8.3%) at this level. In the storage zone (Table 3.2), the mean temperatures were slightly higher (about 0.5 °C) than at the middle plane (Figure 3.7) because there were additional heat losses near the lateral glass walls, but the trends are similar.

The temperature at a distance of 4 cm from the edge of the top shelf (position "k") was significantly higher than that at the DAG. This is because this position was located outside the air curtain as observed by air velocity measurements presented in the following section (Section 3.3). Figure 3.8a also shows the temperature at the RAG (1.0 ± 1.3 °C), which is important for compressor energy consumption: the higher the value (compared to the one at the DAG), the higher the refrigerating power. For this closed display cabinet 𝑇 ̅ 𝐷𝐴𝐺 -𝑇 ̅ 𝑅𝐴𝐺 is only 1.3 °C. To ensure that food products are stored at appropriate temperature in the display cabinet, a temperature is generally displayed on a monitoring screen [START_REF] Baldera Zubeldia | Effectiveness of the cold chain control procedure in the retail sector in Southern Spain[END_REF]. The display value corresponds to the instantaneous air temperature at a given position. In fact, air temperature depends on the positions in the cabinet. It should be preferable to display the temperature at the warmest and coldest positions: position "f" and "i" in our case, respectively.

Benefits of door installation

Doors of the studied display cabinet were removed to investigate the influence of the presence of doors on the temperature performance. and the standard deviations in the display cabinet with and without doors, respectively. The temperature profile in these two cases has the same trend: the highest temperature at the front of the top shelf (position "f") and the lowest temperature at the back of the bottom shelf (position "i"). Without doors, the mean air temperatures in the storage zone increase in all positions with a minimum increase of 0.4 °C (from -0.4 °C to 0.0 °C at position "i") and a maximum increase of 3.1 °C (from 1.0 °C to 4.1 °C at position "h") compared to the case with doors.

The comparison of the air temperature at 4 cm from the shelf edges (positions "k"

and "ℓ") shows a large increase: 8.2 °C (from 5.7 °C to 13.9 °C) at the top and 9.5 °C (from 2.4 °C to 11.9 °C) at the mid-height. was also observed at all positions in the cabinet without doors.

Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 61

For empty and same room conditions, the temperature performance of the closed display cabinet is better compared to the open one due to lower mean air temperature and less temperature heterogeneity in the storage zone. The mean temperature differences between the front and the back positions were less than 1 °C in the case with the doors whereas thereabout 2-3 °C in the case without. Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 63

Influence of ambient temperature

Closed display cabinet under loaded condition

The obtained results of the temperature profiles in the closed display cabinet under unloaded condition enable an insight into the thermal mechanism occurring in the equipment. To further investigate the temperature performance of this cabinet under real use condition, test product packages were placed in the cabinet with an occupied volume of about 60% of total storage volume. As reported in Figure 3.9a, the averages and standard deviations of product (core and surface) and air temperatures on the middle plane (z = -0.105 m) of the cabinet with doors were calculated during quasi-steady state. As expected, the temperature profiles of the closed display cabinet under loaded and unloaded conditions were consistent. The products placed at the front of the top shelf had the highest temperature (core 2.4

°C and surface 3.3 °C) whereas the products placed at the back of the bottom shelf had the lowest temperature (core -0.5 °C).

The incident of high temperature on the top shelf can be explained by interdependencies among various influencing factors. The products located at this position were mainly subject to heat exchanges with the air curtain and heat losses through the glass doors because less cooled air supplies from the back due to low percentage of perforation area (2.3%) on this shelf. In the vicinity of cabinet lighting, there was additional heat generation due to (visible) light absorption in the products. It can be noticed (Figure 3.9a) that product surface temperature was higher than that of the surrounding air (maximum difference of 2.6 °C) and was slightly higher than that of its core temperature (maximum difference of 0.9 °C).

At the lower shelves, more cooled air from the back instantly supplies to the storage space according to higher percentage of perforation areas. As a consequence, product temperature at the front gradually decreased from the top (core 2.4 °C and surface 3.3 °C) to the bottom (core 0.4 °C and surface 0.8 °C).

A temperature variation of products between the front and the back was also observed, of which the highest variation was on the top shelf (maximum difference of 2.1 °C). This results from a combination of convection between product and cold air coming from the PBP and conduction within and between the products.

Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 64 result, products at the middle shelf had highest temperature (core 4.9 °C and surface 6.5 °C). Due to higher percentage of perforation areas at the bottom shelf, the temperature of products on this shelf remained lowest (core 1.4 °C and surface 2.4 °C) compared to the other shelf. Indeed, highest product temperature is often found on the bottom shelf of most open display cabinets [START_REF] Evans | Temperature and energy performance of refrigerated retail display and commercial catering cabinets under test conditions[END_REF]. Allowing high perforation percentage at this shelf seems to be an effective remedy for this issue.

Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 65

Product temperature at the back of the cabinet without doors was slightly lower that of the cabinet with doors (maximum difference of -0.5 °C at the back of the bottom shelf). This is because higher refrigeration capacity is required to compensate additional heat loads due to warn air infiltration. As shown in Figure 3.9a and b, mean air temperature after the evaporator of the cabinet without doors (-1.3 °C) was lower that than of the one with doors (-0.7 °C). In opposite, the standard deviation was higher in the case without door (3.0 °C). Like the cabinet with doors, the product located at the back not only exchange heat with the supplied air from the PBP through convection but also exchange heat within and between the products themselves through conduction. As a result, the product temperature at the back was lower than that at the front of the cabinet.

Globally, higher temperature heterogeneity was observed in the storage space of the cabinet without doors, of which the temperature difference between the product at the front and at the back fell within the range of 2.7 -4.9 °C. In case of the cabinet with doors, this value was less than 2.1 °C. The findings confirm that the closed display cabinet has better temperature performance compared to the open one. Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 66 one position to another. This denotes that there is no plug flow from the PBP to the front but there is a more complex flow because of the interaction between the backair stream and the air curtain. There was probably air recirculation to a small extent between shelves as observed in a refrigerator [START_REF] Laguerre | Numerical simulation of air flow and heat transfer in domestic refrigerators[END_REF]. It is to be emphasized that the results presented in Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 67

Air velocity profile

Airflow rate in the vertical rear duct

The air velocity profiles across the width of the vertical rear duct on different planes (left, middle and right) and for three heights (H1, H4 and H6) are presented in to 0.58 m•s -1 (with a standard deviation of 0.12 m•s -1 ) downstream of the duct (H6).

This is because there is a perforated plate between the upstream and downstream of the rear duct which allows some air outflow to the food storage zone of the display cabinet. The standard deviation also decreased according to the height since the velocity variation among the three planes became smaller.

Because of the 90° elbow at the bottom of the duct and the high percentage of perforated area on the bottom shelf, the velocity profile across the duct width at the lowest level (H1) was not uniform: a lower velocity was observed near the perforated plate (x = -0.15 m), whereas a higher velocity was observed near the back wall of the cabinet (x = -0.45 m). However, it became more uniform at higher levels (H4 and H6) because of coarser perforated areas on the perforated plate and because of the well-developed flow at these levels. Indeed, the mean velocity of and52%). Based on the results, the presence of packages in the cabinet volume did not have a significant impact on the air velocity in the vertical rear duct. The velocity at all heights remained the same, whatever the loading percentage. This infers that the impact of slot-perforated density is more significant than the package-occupied volume as illustrated in Figure 3.12. The airflow distribution percentage decreased significantly with height because of a lower percentage of perforated area, while it changed indistinctly (<5% difference) whatever the percentages of productoccupied volume. It should be kept in mind that with respect to the design of the studied cabinet, a 10-mm gap from the PBP is spared so that products will not be completely placed against it to avoid product freezing. Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 69

Moreover, the flow distribution percentage from the DAG (air curtain) at the top was close to that from the PBP (about 50% in each case, i.e. a distribution ratio of 50%-50%). This ratio is lower than that recommended by [START_REF] Gray | Improvement of air distribution in refrigerated vertical open front remote supermarket display cases[END_REF] and [START_REF] Wu | The optimization and effect of back panel structure on the performance of refrigerated display cabinet[END_REF] for an open display cabinet. According to these authors, a 70%-30% distribution of flow between the air curtain and the rear duct perforations was appeared to yield a high performance. The open display cabinet requires a higher airflow rate from the DAG to generate a sufficient and stable air curtain, since it is the only thermal barrier that protects the food storage zone from the environment compared with a closed display cabinet. Based on the results presented in Section 3.3.2, this ratio is acceptable. A further study on the closed refrigerated display cabinet loaded with test products was experimentally investigated. The consistent temperature profiles were found under loaded and unloaded conditions. The front of the top shelf exposed to the highest temperature, whereas the back of the bottom shelf exposed to the lowest temperature. By comparing to the cabinet without doors, the result confirms that closed display cabinet can achieve better temperature performance.

The air velocity was measured by a hot wire anemometer; the results show that the air curtain is deviated toward the door by a Coanda effect. The flow between the shelves is far from plug flow because of the interaction between the air flowing through the PBP and the air curtain.

The occupied volume in the cabinet storage domain has no significant impact on the airflow rate from the PBP when a small gap is spared between the back products

Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 71

and the PBP. The air distribution between the DAG and the perforated plate is found to be split into two equal airflow rates.

In future research, the influence of the operating conditions (door opening frequency, occupied volume, etc.) on the product and air temperatures as well as on the air velocity will be studied. Laser Doppler Velocimetry (LDV) and/or Particle Image Velocimetry (PIV) will be used to map the velocity profiles in the display cabinet to overcome the limitations of hot-wire anemometer measurements due to complex flows. These experimental results will then be used to validate a simplified heat transfer model based on a zonal approach which requires less computer resources and calculation time compared with Computation Fluid Dynamics (CFD)

simulation. The developed model will be integrated into those already developed for other equipment such as a food production plant [START_REF] Lecoq | Simplified heat and mass transfer modeling in a food processing plant[END_REF], a refrigerated truck [START_REF] Hoang | Heat transfer modelling in a ventilated cavity loaded with food product: Application to a refrigerated vehicle[END_REF], a cold room [START_REF] Laguerre | Simplified heat transfer modeling in a cold room filled with food products[END_REF] and a domestic refrigerator [START_REF] Laguerre | Temperature prediction in domestic refrigerators: Deterministic and stochastic approaches[END_REF], and will make it possible to acquire knowledge of the product time-temperature history throughout the cold chain. , 2010). In open cabinets, a stable and homogeneous storage temperature is difficult to maintain since an air curtain alone is used to protect food products from the ambient surroundings, resulting in significant infiltration of warm and humid air [START_REF] Gaspar | Experimental analysis of the thermal entrainment factor of air curtains in vertical open display cabinets for different ambient air conditions[END_REF]. Air infiltration is frequently cited as one of the major factors causing temperature heterogeneity (Laguerre et al., 2012) and high energy consumption [START_REF] Tassou | Energy consumption and conservation in food retailing[END_REF].

Nomenclature

Temperature variations of up to 5 °C between locations within the cabinet were observed [START_REF] Willocx | A preliminary survey into the temperature conditions and residence time distribution of minimally processed MAP vegetables in Belgian retail display cabinets[END_REF]; the products in the front areas were usually exposed to a high temperature, whereas the products at the back were at lower temperatures [START_REF] Evans | Temperature and energy performance of refrigerated retail display and commercial catering cabinets under test conditions[END_REF]. A great deal of research has been carried out using both experimental investigations [START_REF] Chen | Experimental study of the performance of single-band air curtains for a multi-deck refrigerated display cabinet[END_REF][START_REF] Gray | Improvement of air distribution in refrigerated vertical open front remote supermarket display cases[END_REF] and numerical simulations [START_REF] Field | Entrainment of refrigerated air curtains down a wall[END_REF][START_REF] Moureh | Numerical and experimental study of airflow patterns and global exchanges through an air curtain subjected to external lateral flow[END_REF] in order to maximize the air curtain efficiency, thereby enhancing the overall performance of open display cabinets. In addition to efforts designed to improve the air curtain efficiency, a novel short air curtain design was proposed [START_REF] Hammond | Application of short air curtain in retail display refrigerators[END_REF]. In this study, multiple short air curtains used for each shelf provided enclosure of the front of the cabinet, rather than a single long air curtain. The experimental results showed that beyond potential energy savings, a more uniform temperature distribution can be achieved. A review of the current usage situation and attempts to improve the performance of refrigerated display cabinets can be found in [START_REF] Chaomuang | Experimental and numerical investigation of the performance of retail refrigerated display cabinets[END_REF]. Despite many improvements over the last few decades, higher display cabinet efficiency remains a necessary research topic for sustainable retail refrigeration (Al-Sahhaf, 2011).

Installation of doors on open-front display cabinets has become an alternative designed to address the air infiltration issue. It has been proven that the closed display cabinet achieves several favorable outcomes including energy savings of 20-70% [START_REF] Faramarzi | Efficient display case refrigeration[END_REF][START_REF] Fricke | Energy use of doored and open vertical refrigerated display cases[END_REF][START_REF] Kauffeld | Current and Future Carbon-saving Options for Retail Refrigeration[END_REF]Navigant Consulting Inc., 2009), reductions in internal temperature heterogeneity [START_REF] Chaomuang | Experimental analysis of heat transfer and airflow in a closed refrigerated display cabinet[END_REF][START_REF] Evans | Performance of retail and commercial refrigeration systems[END_REF]Lindberg et al., 2010), and improved food quality (Atilio de [START_REF] Atilio De Frias | Improving spinach quality and reducing energy costs by retrofitting retail open refrigerated cases with doors[END_REF]. Moreover, doors can also provide better thermal comfort in retail stores due to less cold air spillage from the display cabinet (Lindberg et al., 2017). Given these benefits, doors have been installed to an increasing extent in various countries. An agreement between the French authorities and the major supermarket stakeholders aims to replace open display cabinets by closed display cabinets, and to ensure that by 2020, 75% of all display cabinets in use will be closed models [START_REF] Rpf | Fermeture des meubles réfrigérés: du positif pour les clients[END_REF].

In contrast to the energy consumption aspect, the temperature performance of Their results showed that this factor becomes the predominant factor determining the refrigeration load when an opening frequency of 60 OPH is applied. To the best of our knowledge, the influence of the door-opening frequency on the temperature distribution (spatial and temporal) had rarely been studied.

The present study aims to investigate the influence of operating conditions, i.e. the frequency of door openings, ambient temperature and percentage of occupied volumes, on the thermal performance of a closed refrigerated display cabinet. This performance was evaluated using air and product data for design improvements and the practical handling of closed refrigerated display cabinets in supermarkets.

Material and methods

Closed refrigerated display cabinet

The closed refrigerated display cabinet was placed in a test room in which the ambient temperature was controlled at 19.4 °C (with a standard deviation of 0.5 °C)

and the air humidity varied between 0.007 and 0.009 kg water/kg dry air. As shown in Figure 4.1a, the test room was equipped with 2 fans to homogenize the room temperature. To avoid the influence of airflow in the room on the measured value inside the cabinet, the rear of the display cabinet was located against these fans. Based on data available in the literature [START_REF] Fricke | Energy use of doored and open vertical refrigerated display cases[END_REF] and information provided by a display cabinet manufacturer, the average frequency of door openings is, respectively 6.3 and 12.1 OPH. The frequency of 60 OPH was therefore set as an extreme condition as it is rarely encountered in reality. Note that the doors were completely removed from the display cabinet for the permanently open condition.

Ambient temperature

In addition to the room temperature of 19.4 °C considered as a baseline, the test room was controlled at different temperatures of 14.6 °C, 24.3 °C and 28.2 °C (with a standard deviation of less than 0.5 °C for all temperatures) to investigate its influence on the air and product temperatures in the display cabinet. The water content of the air in the test room varied from 0.008 to 0.013 kg water/kg dry air, corresponding to the variation in the relative air humidity of 50% and 75%, respectively. This small variation in water content in the air exerted little influence on the temperature performance, but it does affect the energy performance because of frost formation on the evaporator [START_REF] Chen | Experimental study of the performance of single-band air curtains for a multi-deck refrigerated display cabinet[END_REF]. The same experimental procedure (Section 4.2.3.2) was applied in all experiments.

Occupied volume

Two occupied volumes were used in the experiments: 30% (considered as half load)

and 60% (considered as full load). The difference in these occupied volumes concerns the number of layers of test packages in height only, while the arrangement remained the same (6 columns in width and 3 columns in depth, see (a) Baseline; (b) Permanently closed; (c) Permanently open; (d) To ascertain the thermal phenomena during the door openings, additional air temperatures at the A1, A3 and A5 positions were recorded with more frequent data acquisition (a time interval of 1 s). After 24 h of temperature stabilization, recording of the door-opening cycles with a frequency of 10 OPH was undertaken for 12 h. On the whole, the average air and product temperatures at most positions were not significantly different between lights on and off conditions when the doors were closed (Figure 4. 5a andc). The illumination effect is only significant at the front of the top shelf due to the proximity to the cabinet lighting.

Results and discussion

Evolution of air and product temperatures

During the door-opening period (Figure 4.5b), the average air temperature was slightly lower in the rear duct (about 0.1 °C) and higher at the front (up to 0.9 °C).

External warm air infiltration into the cabinet during the door openings explained an increase in the air temperature at the front. According to the air temperature increase, the compressor working cycles became more frequent, thereby lowering the air temperature in the rear duct. Compared with the closed-door periods, the number of compressor working cycles during the door openings was approximately 18% greater. Each door opening caused an increase in product temperatures at the front and a decrease in those at the back (Figure 4.5b). This resulted in a greater temperature difference between the front and the back products on the shelf.

Overall, the product temperature distribution in the closed display cabinet was consistent for every period. The highest temperature position was at the front of the top shelf and the lowest temperature position was at the back of the bottom shelf.

The top shelf was also the position at which the front and the back products had the maximum temperature difference. Because there were additional heat losses through the lateral glass walls, the product core temperature at the sides was higher than that at the center (Table 2). It should be emphasized that the measurement of product core temperatures at the side was taken at a distance of 100 mm from the lateral wall since 1-kg test packages were used in the study. There will be a tendency to observe higher temperatures at the sides in the EN23953 test because 500-g test packages (width × length × height: 100 × 50 × 100 mm) are usually used and the measurement is performed at a distance of 50 mm from the wall.

Effect of door-opening frequency

Different door-opening frequencies were applied to the closed display cabinet to investigate their effects on the time-averaged temperatures at different locations in the cabinet. The air and product temperatures at any given position inside the display cabinet are influenced by the operating conditions. Theoretically, these temperatures should be high when the ambient temperature and the door opening frequency are high. However, in practice, under such conditions (high ambient temperature and high door opening frequency), the compressor operates at a greater frequency (shorter "off" cycle) to compensate for thermal loads in the cabinet to a greater extent. Thus, the time-averaged air temperature after the heat exchanger (where the thermostat sensor is located: 𝑇 𝑡ℎ ) is lower, and this leads to lower air/product temperatures at the back and higher temperatures at the front. To overcome this complexity and to enable comparison of temperatures obtained from different operating conditions, the dimensionless temperature (𝜃 𝑖 ) was defined (Eq. it was found that the daily mean door-opening frequency was only 6.3 OPH with a mean opening duration of 12 s (the most frequent duration was 5 s). Based on these findings, food preserved in closed refrigerated display cabinets would be maintained under better temperature conditions, thereby prolonging the shelf life and reducing food loss. -0.4 0.7 1.6 -0.5 1.1 2.0 † Average (top and bottom packages on the stack) over 5 h of quasi-steady state excluding the defrost period; the standard deviation was 0.1 °C for all positions under both conditions.

In addition to the temperature performance, the closed refrigerated display cabinet also shows energy saving potential. Based on the total period of a complete compressor working cycle, the time fraction during the compressor "on" period increased from 23% for the closed-door condition to 25%, 32%, 37% and 44% for the door-opening frequencies of 10, 20, 40 and 60 OPH, respectively. The time fraction was 63% in the case of the cabinet without doors.

Effect of ambient temperature

Air and product temperature measurements in the closed refrigerated display cabinet were conducted for four different ambient temperatures (see Table 4.1) while a door-opening frequency of 10 OPH and an occupied volume of 60% were applied in all experiments. The same dimensionless temperature (𝜃 𝑝 ) defined in the previous section was used, and the results are reported in Figure 4.7. It was observed that 𝜃 𝑝 was almost independent of 𝑇 𝑒 -𝑇 𝑡ℎ for the products at the front of the cabinet (Figure 4.7a), while it decreased a little for most products at the back (Shelf 2-5) when 𝑇 𝑒 -𝑇 𝑡ℎ increased. This result is in agreement with that observed in our previous study [START_REF] Chaomuang | Experimental analysis of heat transfer and airflow in a closed refrigerated display cabinet[END_REF]. Despite door openings, forced

Chapter IV -Influence of operating conditions on the thermal performance 90 convection and conduction, which are linear phenomena, heat transfer modes remain predominant in the closed display cabinet. Free convection and radiation from the external walls, which are non-linear phenomena, can explain the slight influence of 𝑇 𝑒 -𝑇 𝑡ℎ on the dimension temperature. As reported in Table 4.4, when 𝑇 𝑒 -𝑇 𝑡ℎ rose from 15.0 °C to 30.4 °C, on the average 𝑇 𝑝 -𝑇 𝑡ℎ at the front and the back positions increased from 1.5 °C to 3.1 °C and from 0.5 °C to 0.9 °C, respectively. Under a door opening frequency of 10 OPH, the highest temperature position was at the front of the top shelf where 𝑇 𝑝𝑓 -𝑇 𝑡ℎ was approximately 14% of 𝑇 𝑒 -𝑇 𝑡ℎ . The lowest temperature occurred across the back of Shelf 3-5 where 𝑇 𝑝𝑏 -𝑇 𝑡ℎ was approximately 3% of 𝑇 𝑒 -𝑇 𝑡ℎ . These results highlight that the product temperature in a closed display cabinet is directly influenced by the external ambient temperature, and the front area is the position at which the products are the most exposed to a high temperature. The cabinet doors were periodically opened with a frequency of 10 OPH and with 60% occupied volume.

Effect of occupied volume

Two percentages of occupied volume in the closed display cabinet were studied: 30% and 60%. A door-opening frequency of 10 OPH and an ambient temperature of 19.4 °C were applied to both experiments. The same dimensionless temperature (𝜃 𝑝 ) defined in Section 3.2 was also used for analysis so that the variation in the air temperature after the heat exchanger (𝑇 𝑡ℎ ) due to different thermal loads was eliminated. As shown in where 𝑖 is the number of the shelf ranging from 1 (top) to 5 (bottom) and 𝜌 𝑎 is the air density (1.28 kg•m -3 at 0 °C). As shown in Figure 4.9, the air velocity above the products on Shelves 1-4 in the display cabinet with a 60% occupied volume could be about 5 times higher than that with a 30% occupied volume. It was about 1.7fold in the case of the bottom shelf. It should be borne in mind that the air velocity presented in Figure 4.9 was an average value supposing plug flow which is seldom observed in a real cabinet [START_REF] Chaomuang | Experimental analysis of heat transfer and airflow in a closed refrigerated display cabinet[END_REF].

Conclusions

Closed refrigerated display cabinets are increasingly used in retail stores because of their numerous advantages, particularly a considerable reduction in warm and humid air infiltration, compared with open cabinets. In this study, air and product temperature measurements in the closed display cabinet were carried out under various operating conditions to evaluate the thermal performance of the cabinet including door-opening frequency, ambient air temperature and product-occupied volume. The product position in the cabinet is a determining factor of its temperature. The product at the front always had a higher temperature than that at the back, whatever the operating conditions, and that on the top shelf most often had the highest temperature. Higher door-opening frequency and/or higher ambient temperature result in a substantial increase in the product temperature at the front.

The product temperature on the front middle shelf was the most affected when the door-opening frequency exceeded 60 OPH. On the other hand, each door opening causes a slight decrease in product temperature at the back because more refrigeration capacity was required to withstand higher thermal loads. Productoccupied volume also causes product temperature variations. A lower product temperature was observed at all positions when the cabinet volume was occupied to a greater extent. Overall, the closed refrigerated display cabinet fitted with doors provides better thermal performance, i.e. lower product temperature and more temperature homogeneity, as compared to the cabinet without doors despite a very high door opening frequency. Based on the time fraction of periods during which the compressor was operating, the closed refrigerated display cabinet also shows energy saving potential.

Chapter V Airflow characterization using PIV and CFD techniques

Nomenclature 𝑢 𝑝 settling velocity of particles = This factor was correlated to the Reynolds (Re) and Richardson (Ri) numbers which indicate the importance of momentum (depending on the discharge air velocity of the air curtain) and buoyancy effects, respectively. It was found that air entrainment was promoted when the momentum effect increased whereas it became less pronounced when the buoyancy effect was dominant. Because of the non-intrusive aspect and high spatial resolution accuracy, PIV measurement has become a technique used to study flow structures in various applications [START_REF] Adrian | Twenty years of particle image velocimetry[END_REF]. The turbulence intensity, the shape of the velocity profile at the DAG and 𝑅𝑒 were found to be the most significant parameters influencing the amount of air entrainment. Based on these studies, optimal jet characteristics should be attained in order to maintain the insulating capacity of the air curtain. In addition to efforts designed to optimize air curtain efficiency, [START_REF] Hammond | Application of short air curtain in retail display refrigerators[END_REF] proposed a new design featuring multiple short air curtains for each shelf rather than a single long air curtain and reported a reduction in electricity consumption of 35.9% and a smaller range of maximum and minimum product temperatures (a difference of 9.5 °C for the conventional design and 3.1 °C for the short air curtain design).

𝐴 area [m -2 ] 𝐶 𝑝 specific heat [J•kg -1 •K -1 ] 𝑑 𝑝 particle diameter [m] 𝑒 thickness [m] 𝑔 gravitational acceleration [m•s -2 ] 𝑘 turbulent kinetic
𝑔𝑑 𝑝 2 (𝜌 𝑝 -𝜌 𝑎 ) 18𝜇 𝑎 [m•s -1 ] 𝑈 heat transfer coefficient [W•m -2 •K -1 ] 𝑤 width [m] Greek symbols Γ diffusivity coefficient [m 2 •s -1 ] 𝜀 dissipation rate of 𝑘 [m 2 •s -3 ] 𝜆 thermal conductivity [W•m -1 •K -1 ] 𝜇 dynamic viscosity [Pa•s] 𝜌 density [kg•m -3 ] 𝜎 dimensionless diffusion parameter [-] 𝜐 kinematic viscosity [m 2 •s -1 ] 𝜙 generalized variable
Display cabinets retrofitted with doors constitute an energy saving alternative because they reduce the infiltration of external warm/humid air. Several studies have shown that a closed display cabinet generates 23-73% energy savings [START_REF] Evans | Are Doors on Fridges the Best Environmental Solution for the Retail Sector? The Institute of Refrigeration[END_REF][START_REF] Fricke | Energy use of doored and open vertical refrigerated display cases[END_REF]Lindberg et al., 2010;Navigant Consulting Inc., 2009) and the presence of doors was found to have no impact on the sales volume [START_REF] Fricke | Energy use of doored and open vertical refrigerated display cases[END_REF]. Given these advantages, display cabinets with doors have become default specification for new stores opened by various major retailers in Europe (EIA, 2017) and will account for about 75% of total display cases in French supermarkets by 2020 (RPF, 2016). [START_REF] Faramarzi | Performance and energy impact of installing glass doors on an open vertical deli/dairy display case[END_REF] reported that retrofitting a display cabinet with glass doors reduced the total cooling load by 68%

and the average product temperature by 0.2 °C. To the best of the authors' knowledge, the study of airflow in a closed display cabinet is rare even though door installation can affect the airflow pattern in display cabinets, which in turn influences the heat transfer and product temperatures [START_REF] Faramarzi | Performance and energy impact of installing glass doors on an open vertical deli/dairy display case[END_REF].

A review of the literature on experimental and numerical investigations of the performance of display cabinets was carried out by the authors [START_REF] Chaomuang | Experimental and numerical investigation of the performance of retail refrigerated display cabinets[END_REF]). An experimental study of heat transfer in a closed display cabinet was conducted and the effect of operating conditions on the product temperature was highlighted: room temperature, door-opening frequency and percentage of occupied volume (Chaomuang et al., 2019a(Chaomuang et al., , 2019b)). In order to build on our previous studies, the objectives of this study were:

※

To conduct air velocity measurement in a display cabinet using the PIV technique. It is to be emphasized that the originality of our study is the measurement of unstable and low air velocity (0 < 𝑉 𝑎 < 0.7 m•s -1 ) in closed and open cavities. Thus, several difficulties associated with implementation had to be overcome.

※

To analyze the airflow pattern of a closed display cabinet and to compare it with the airflow pattern in an open display cabinet. Thermal aspects based on the results obtained from the previous studies (Chaomuang et al., 2019a(Chaomuang et al., , 2019b) ) were discussed.

※

To develop a 2D-CFD model of a closed display cabinet in order to investigate the phenomena which could not be observed by the experiment.

The experimental and numerical methodologies developed in this study can be useful to scientists for understanding the phenomena and for the industry in order to improve the design under various operating conditions.

Materials and methods

Refrigerated display cabinet

A To characterize experimentally the airflow pattern inside a closed display cabinet, two conditions were studied: the refrigeration system was turned "off" and "on", while the cabinet fans operated continuously under both conditions. In such a manner, the influence of fans alone (the first condition) and combined fans/refrigeration (the second condition) on airflow can be highlighted. In addition, the experiment was conducted in the display cabinet with the doors open, and the refrigeration system was turned off.

PIV measurements

The air velocity measurement was performed using a two-dimensional PIV system (LaVision, FlowMaster 2D). The PIV system is composed of a double-pulsed

Principle of PIV measurement

The principle of PIV measurement involves the analysis of images of tracer particles present in a thin light sheet illuminated by a double-pulsed laser beam.

The light scattered by these particles is captured by a high-resolution digital camera positioned perpendicular to the light sheet and synchronized with the two laser impulsions. For air velocity evaluation, each of recorded images of the first and second illuminations is divided into small areas, called interrogation windows. The local particle displacement between these two successive illuminations is then evaluated by a cross-correlation statistical method at the level of each interrogation window. Given the time delay between the two illuminations, an instantaneous velocity vector field can be calculated. This operation was repeated iteratively for all interrogation areas to obtain a whole flow field across the recorded image. The calculation process of velocity vectors is summarized in [START_REF] Keane | Theory of cross-correlation analysis of PIV images[END_REF].

A comprehensive and more detailed explanation of the PIV measurement is widely available in the literature [START_REF] Prasad | Particle image velocimetry[END_REF][START_REF] Raffel | Particle image velocimetry: a practical guide[END_REF].

Measurement procedure

An entire measurement plane (z = 520 mm, Figure 5.1) was covered with 36 measured windows (112 mm × 112 mm) with a partial overlap between them.

Because of the opaque door frame, the measurement was performed at 30 mm from the bottom shelf. The camera and the laser were mounted on a displacement system allowing the position of the measured windows to be changed (precision of displacement ±1 mm). For each measured window, 1200 pairs of images were recorded at a frequency of 0.5 kHz. A reliable representation of mean velocity and other statistical information were verified. It is to be emphasized that a preliminary sensitivity study was conducted in order to determine the optimum number of pairs of images. A multi-pass algorithm with decreasing interrogation area was used to process instantaneous vector calculation. Two passes were used with interrogation dimensions of 64 × 64 pixels and 32 × 32 pixels, respectively. Both passes were defined with 50% overlap. Given the interrogation dimensions of the second pass, the spacing between two calculated velocity vectors was 16 pixels (about 3.5 mm) in both the horizontal and vertical directions. Any region near high reflection surface in the images (e.g. DAG, RAG, shelf edges and test packages) was always masked out and not included in the vector calculation. Referring to the guideline proposed by [START_REF] Keane | Theory of cross-correlation analysis of PIV images[END_REF], a time delay between two laser illuminations of 500 µs was the optimum for a good velocity estimation in our case since it allowed a mean particle displacement of less than 25% of the width of the interrogation window. After the mean velocity fields of all 36 measured windows were obtained, the entire measurement plane was established with the DaVis software by connecting all the windows together.

Measurement accuracy

There are various sources of errors in a PIV measurement [START_REF] Wilson | Uncertainty on PIV mean and fluctuating velocity due to bias and random errors[END_REF]. Typically, errors are classified into systematic (bias) and random errors [START_REF] Coleman | Experimentation, validation, and uncertainty analysis for engineers[END_REF]. Based on a recent implementation of a processing algorithm in DaVis 10.0.5, it is possible to quantify the measurement uncertainties of instantaneous velocity components by using a correlation statistics method [START_REF] Sciacchitano | PIV uncertainty propagation[END_REF]. This method accounts for most parameters that cause random errors from the vector computation process e.g. particle image disparity, particle background image noise and out-of-plane particle motion [START_REF] Sciacchitano | Collaborative framework for PIV uncertainty Reference 212 quantification: comparative assessment of methods[END_REF]. Nevertheless, the systematic error due to the so-called peak-locking effect is not taken into account since the magnitude of this error is relatively small, compared with the random error [START_REF] Wieneke | PIV uncertainty quantification from correlation statistics[END_REF]. To minimize this systematic error, a particle image diameter of at least two pixels is recommended [START_REF] Prasad | Particle image velocimetry[END_REF], and this diameter was approximately 3-4 pixels in our case. The settling velocity (𝑢 𝑝 ) of the water droplets used in the experiment was also negligible (~ 0.0015 m•s -1 ) compared with the air velocity in the display cabinet (0 < V < 0.7 m s -1 ). Thus, the particle velocity can be considered as representative of the airflow. Overall, the uncertainty of the measurement varied between 0.1 and 0.3% in most regions of interest and was slightly higher (0.6%) in the mixing or shear layers and at the level of the bottom shelf, particularly near the RAG because of higher turbulence levels. (5.1)

Experimental results

Closed display cabinet

𝑉 𝑟𝑚𝑠 = √ 1 𝑁 ∑ [(𝑢 𝑖 -𝑢) 2 + (𝑣 𝑖 -𝑣) 2 ] 𝑁 𝑖=1 (5.2)
where 𝑁 is the total number of measured windows (1200 in our experiment); 𝑢 𝑖 and 𝑣 𝑖 are the air velocities in 𝑥 and 𝑦 directions measured at the instant i, respectively [m•s -1 ] When the refrigeration system was turned "off" (Figure 5.2a), the inlet mean velocity magnitude of the jet normal to the DAG was approximately 0.57 m•s -1 , corresponding to a Reynolds number of about 2000 (based on the DAG width of 50 mm). This 𝑅𝑒 value was low compared with that of a typical open display cabinet (4200 and 8000, [START_REF] Field | Entrainment of refrigerated air curtains down a wall[END_REF]. From the DAG to the top shelf, the jet slightly accelerated because it was deviated by the test packages at the front.

The jet then spread out and the velocity gradually decelerated due to the diffusion processi.e. entrainment of neighboring air by the air jet. According to the discharge angle of the DAG (about 8° relative to the vertical axis), the air flowed toward the door. Under the top shelf level (y < 1120 mm), the velocity profile showed a Coanda effect (jet adhesion to the wall) which caused relatively high velocity close to the door surface. The air velocity increased progressively when it flowed downward because of additional air flowing from the back through the PBP at each shelf, thereby enhancing the momentum flux along the flow path. Because of a higher percentage of perforated area on the PBP over the bottom shelf, higher air velocity can be clearly observed at this position. When it approached the RAG, the airflow was accelerated again by the cabinet fans. A similar airflow pattern was observed in the display cabinet with the refrigeration system turned "on" (Figure 5.2d, e and f) compared with the case in which the refrigeration system was turned "off" (Figure 5.2a, b and c) despite slight discrepancies. The Coanda effect was less pronounced when refrigeration was applied because the air in the vicinity of the door surface had a relatively higher temperature, and thus an upward buoyancy force impeded downward flow for y < 1000 mm. As shown in Figure 5.2d, the velocity was relatively lower at the door when the refrigeration system of the display cabinet operated and reached its maximum near the shelf edges. Figure 5.2e shows the same regions with relatively high velocity fluctuations in the case of the refrigeration system turned "off" (top and bottom parts), but higher turbulence was generated by interaction between the air curtain and the shelf edges. The vector field depicted in Figure 5.2f shows similar flow patterns as in the case of the refrigeration system turned "off", with the exception of slightly more pronounced recirculation (upward flow) in the top part and the local maximum velocity situated at a greater distance from the door.

Comparison between closed and open display cabinets

The doors of the display cabinet were open with an opening angle of 90° and the PIV measurement was carried out with the same procedure as described in Section 5.2.2. The performing of measurements in the open display cabinet was particularly troublesome because it proved difficult to control the particle density. Particles Chapter V -Airflow characterization using PIV and CFD techniques 107 always flowed out to the ambient during the experiment, resulting in insufficient seeding to disperse the light. Because of this technical constraint, only the experiment with the refrigeration system turned "off" was carried out. Air inflow and outflow through the gaps along the door edges of the closed display cabinet were observed in our previous study (Chaomuang et al., 2019a). A PIV measurement was additionally conducted in the middle of the left door gap to quantify the volumetric air infiltration rate through the gaps (8 mm in width). By integration of the positive horizontal velocity component (𝑢) over the height of the door, the volumetric air infiltration rate through the gaps was estimated to be approximately 0.0029 m 3 •s -1 . It was assumed that the amount of outflow air was replaced by the ambient air infiltration and the rates of the air infiltration through the door gaps on the left and the right sides were the same. This result suggested that air infiltration under closed-door conditions accounted for about 8% of the supply air from the DAG, which is much less than the case of open display cabinets (~30-40%, Amin et al., 2009).

From a thermal point of view for the closed display cabinet, there is a small amount of convective exchange with the external ambient air through the gaps. The conduction and radiation exchanges through the doors are limited by the presence of double-glazing windowpanes coated with low-infrared emissivity materials. In addition, the low velocity fluctuations near the door reduce the convective heat transfer between the inner side of the door and the air curtain. The air entrained by the air jet at the top shelf level comes mainly from air recirculation near the door and it is, therefore, fairly cold. On the contrary, for the open display cabinet, the air entrained by the air jet is warm ambient air which mixes with the cold air in the outer mixing layer. While on the average air flows out at the intermediate height (100 mm < y < 1050 mm), ambient air is engulfed by the air curtain via the large unsteady eddies. Moreover, there is no radiation shield between the external walls and the products located at the front of the shelves. These findings explain the better thermal performance observed in closed display cabinets, compared with open ones (Chaomuang et al., 2019a).

CFD modeling

A two-dimensional CFD model was developed to investigate the airflow and its influence on the temperature distribution in the closed display cabinet. The main objective is not to simulate precisely all the flow and transfer details. Instead, the aim was to examine the capability of a 2D k-ε model to reproduce the main phenomena and to predict the product temperatures in the display cabinet with acceptable accuracy. An example of more precise models with a three-dimensional large-eddy simulation (LES) can be found in [START_REF] Moureh | Large-eddy simulation of an air curtain confining a cavity and subjected to an external lateral flow[END_REF]. The prediction of velocity profiles (with the refrigeration system turned "off" and "on") was compared with the results obtained from the PIV measurements presented in Section 5.3. The results of temperature measurements conducted in our previous study (Chaomuang et al., 2019a) were used to compare with the numerical prediction of temperature distributions in the present study (refrigeration system turned "on"). According to Chaomuang et al. (2019a), the air and product temperatures were measured by using calibrated thermocouples (±0.2ºC accuracy) in the middle section of a closed refrigerated display cabinet which operated under closed-door conditions and at a room temperature of 19 °C (humidity ratio of 0.003-0.004 kg of water vapor/kg of dry air). Timed-averaged temperatures were calculated over 5 h of quasi-steady state during which the defrosting period was excluded.

Geometry and mesh details

The dimensions used for the 2D model were those of the real closed display cabinet (Table 5.1) and only the internal storage space of the cabinet was considered in the computational domain (Figure 5.5). The domain consisted of fluid and solid regions, representing airflow and products in the cabinet, respectively. An incompressible ideal gas was used for the fluid region while the thermal properties of methylcellulose were specified for the solid region (density = 1071 kg•m -3 ; thermal conductivity = 0.510 W•m -1 •K -1 ; heat capacity = 3372 J•kg -1 •K -1 , [START_REF] Icier | The use of tylose as a food analog in ohmic heating studies[END_REF]. Both unstructured and structured meshes were used for the entire domain. Much finer meshes were particularly defined at the areas close to the DAG and the RAG. Several models with different mesh sizes (M1-3, Figure 5.6) were preliminarily tested for the grid independence study so that numerical accuracy was ensured. It was found that the mesh M2 with 134,968 nodes and 97,693 elements was satisfactory since the numerical results did not change when the mesh size was reduced. As a result, this mesh was used in all calculations.

Mathematical description

The CFD model in steady state was developed by using commercial ANSYS Code Fluent 17.2 software. The model is based on solving the governing equations which consist of the conservation of mass, momentum and energy equations [START_REF] Versteeg | An introduction to computational fluid dynamics: the finite volume method: Pearson education[END_REF]. Since the flow is turbulent, these governing equations are time-averaged (Reynolds-averaged Navier-Stokes, RANS) and the standard k-ε turbulence model was used together with these equations. Note that turbulence was firstly modeled with other k-ε turbulence models (i.e. Renormalization Group (RNG) and Realizable) and the results obtained were found to be almost identical.

The standard k-ε turbulence model was therefore used as it was the best compromise between accuracy and reasonable computational time where 𝜙 is a generalized transport variable, Γ 𝜙 and 𝑆 𝜙 are respectively the diffusion coefficient and the source term for the corresponding transport variable 𝜙 as presented in Table 5.2. Note that the energy equation as a function of temperature was activated only for the case of the refrigeration system turned "on". The complete description of these equations can be found in [START_REF] Tu | Computational fluid dynamics: a practical approach: Butterworth-Heinemann[END_REF].

Boundary conditions

Figure 5.5 shows the locations of boundary conditions imposed in the computational domain. Values used for boundary conditions were taken from the experimental data presented in this present study and in our previous work (Chaomuang et al., 2019a).

Inlet boundary conditions

Six velocity inlet boundary conditions were imposed in the computational domain:

one inlet for the flow at the DAG and the other five inlets for the flow from the PBP at the levels of different shelves (Figure 5.5). Uniform velocity profile and a direction normal to the inlet were assumed under all inlet boundary conditions. The velocity inlet at the DAG was the mean velocity magnitude obtained from the PIV measurement. The velocities at the PBP inlets were estimated based on the air distribution percentages (Chaomuang et al., 2019a) for each shelf. They represent the airflow rate averaging through the PBP at a given shelf divided by the total flow rate in the display cabinet. Given the velocity magnitude at the DAG, the corresponding velocities over these shelves were determined. In the case with the refrigeration system turned "on", thermal boundary conditions were activated, and the measured temperatures were applied. All the inlet boundary condition values are summarized in Table 5.3. lower turbulent intensity at the DAG was assumed due to the effect of the honeycomb.

(c) estimated from the honeycomb characteristics for the DAG and the hydraulic diameter of a slot for the PBP

Wall boundary conditions

Wall boundary conditions were used to confine the fluid and solid regions and nonslip boundary conditions were considered to be present at all walls. When the refrigeration system was turned "on", the thermal boundary conditions were defined at the walls considered in the heat transfer calculation. The walls in contact with fluid and solid regions were defined as coupled thermal interfaces (dotted lines in Figure 5.5). Although installation of doors can significantly reduce thermal radiation from the ambient by up to 94% [START_REF] Faramarzi | Performance and energy impact of installing glass doors on an open vertical deli/dairy display case[END_REF], a lot of heat generation due to visible radiation still occurs. Therefore, in addition to the coupled thermal interface, heat generation due to the absorption of visible radiation from the room and cabinet lights were applied directly at the front product surface level for all shelves (Figure 5.5). The heat dissipated from the room light (𝑄 ̇𝑟𝑙𝑖 , the same for all shelves) was estimated from the lighting power density of 1.0 W•m -2 , which is equivalent to a luminous intensity of 650 lux (an average value used in supermarkets, IESNA, 1991). The heat dissipated from the cabinet light (𝑄 ̇𝑐𝑙𝑖 ) was estimated from its power (10 W, manufacturer's data) with respect to the view factor for the different shelves.

For calculations concerning the door, the heat flux was calculated from the room temperature of 19 °C (experimental data) and the heat transfer coefficient of the door (𝑈 𝑑 ). The latter was determined by a thermal resistance method (Eq. (5.4)).

Based on the characteristics of the door (manufacturer's data), which is composed of two panes of tempered glass (4-mm thickness each) split by an argon gap (14mm thickness), we obtained

𝑈 𝑑 = 1 2𝑒 𝑔𝑙 𝜆 𝑔𝑙 + 𝑒 𝑎𝑔 𝜆 𝑎𝑔 = 1.13 W•m -2 •K -1
(5.4)

Since there was an inflow of warm air through the upper part of the door gaps (𝑚̇𝑔 𝑎𝑝 ≅ 0.0036 kg•s -1 , experimental data), an equivalent heat transfer coefficient (𝑈 𝑑,𝑒𝑞 ) was distinctly defined (Eq. (5.5)) for this part of the door (250 mm in height).

𝑈 𝑑,𝑒𝑞 = 𝑈 𝑑 + 𝑚̇𝑔 𝑎𝑝 𝐶 𝑝 𝐴 𝑑 = 12.7 W•m -2 •K -1 (5)
where 𝐴 𝑑 is the area of the upper part of the door (Figure 5.5).

The other walls were considered as adiabatic.

Numerical calculation

The model was solved iteratively with a SIMPLE algorithm for the coupling of

Numerical results

Air velocity profile

The air velocity profiles of the closed display cabinet with the refrigeration system turned "off" and "on" are shown in Overall, the predicted air and product temperature distributions showed relatively good agreement with the experimental data. For both experimental and numerical results, the air temperature was lower than 2.0 °C everywhere, except in the upper part beyond the air curtain where the temperature recorded at a measurement point was 6.1 °C (the simulation under-estimated by 4 °C). The air recirculation mixed with the warm air infiltration through the door gaps can explain the high air temperature at this position. Spatial temperature heterogeneity was observed between products located at the front and the back of the shelves. The higher product temperature at the front resulted from the heat exchanges with the air curtain which was warmer than the air coming from the PBP and (visible) radiation absorption due to the lights. Due to the proximity to the cabinet lights, the product at the front of the top shelf was the product with the highest temperature. The lowest product temperature was at the back of the bottom shelf because heat exchange was mainly governed by convection with the cold air from the PBP and conduction within the products. Since the cold air flowed from the PBP into the bottom shelf with a relatively higher mass flow rate (highest perforation percentage), the product at the back of the bottom shelf had the lowest temperature. In agreement with results published by [START_REF] Faramarzi | Performance and energy impact of installing glass doors on an open vertical deli/dairy display case[END_REF], these findings showed that the installation of doors could reverse the product temperature distribution in display cabinets since the highest product temperature was mostly located at the front of the bottom shelf of the cabinets without doors [START_REF] Evans | Temperature and energy performance of refrigerated retail display and commercial catering cabinets under test conditions[END_REF]. Nevertheless, closed refrigerated display cabinets still provide a better thermal performance (i.e. lower product temperature with small temperature variations between positions), compared with open ones (Chaomuang et al., 2019a).

Conclusions

Air velocity measurements using the PIV technique were conducted in a closed display cabinet. Two configurations were studied: closed-and open-door with a 90° opening angle. The measurements were carried out under two conditions (refrigeration system turned "off" and "on") for the closed configuration and under only one condition (refrigeration system turned "off") for the open one. According to the studied closed display cabinet, the 𝑅𝑒 (based on the DAG width) was approximately 2000. In the closed configuration, similar flow phenomena were observed in the case of the refrigeration system turned "off" or "on" despite small discrepancies. The air curtain deviated to the door through a Coanda effect (maximum velocity near the door surface). However, it was less pronounced in the case of the refrigeration system turned "on" for which the upward buoyancy force impeded the downward flow. The air curtain was quite stable in most regions, with the exception of the top and the bottom of the cabinet where relatively high velocity fluctuations occurred. At the top of the cabinet, air recirculation occurred and could have promoted external ambient air infiltration through the door gaps. In the open configuration, the air curtain was less stable. In the outer mixing layers of the air curtain, large unsteady eddies developed, causing the ambient air to be entrained in

Nomenclature 𝐶 𝑒𝑞 Equivalent thermal conductance [W•K -1 ] 𝐶 𝑝 Heat capacity [J•kg -1 •K -1 ] 𝑒 Thickness [m] ℎ Convective heat transfer coefficient [W•m -2 •K -1 ] 𝑚̇ Mass flow rate [kg•s -1 ] 𝑞 Heating power [W] 𝑄 Cooling load [W] 𝑅𝐻 Relative humidity [%] 𝑆 Exchange area [m 2 ] 𝑇 Temperature [°C or K] 𝑡 Time [s] 𝑈 Overall heat transfer coefficient [W•m -2 •K -1 ] Greek letters 𝛼, 𝛽 Exponential dimensionless heat transfer coefficient = exp (- 𝑈𝑆 𝑚̇𝐶 𝑝 ) 𝛿, 𝛾 Air distribution coefficient 𝜃 Dimensionless average temperature 𝜆 Thermal conductivity [W•m -1 •K -1 ] 𝜌 Density [kg•m -3 ] 𝜏 Period of

Introduction

Temperature is a well-known factor in determining food quality and safety across the logistic cold chain from production to final consumption. Failure in temperature control at any stage can trigger food quality deterioration, thus shortening food shelf life and consequently increasing consumers' health risk. Several field investigations have shown that temperature control in retail display cabinets is often subject to temperature abuse because food products are stored out of their recommended temperature limits [START_REF] Derens-Bertheau | Cold chain of chilled food in France[END_REF]Lundén et al., 2014). About half of the total retail display cabinet line-ups in a typical supermarket are open refrigerated display cabinets, of which the chilled-space storage is protected from the ambient surroundings by cold air curtain [START_REF] Ashrae | Retail food store refrigeration and equipment ASHRAE Handbook: Refrigeration[END_REF]. Warm and humid air infiltration is a primary cause of the heterogeneity of temperature distribution in the equipment [START_REF] Evans | Temperature and energy performance of refrigerated retail display and commercial catering cabinets under test conditions[END_REF][START_REF] Gaspar | Experimental analysis of the thermal entrainment factor of air curtains in vertical open display cabinets for different ambient air conditions[END_REF]Laguerre et al., 2012b). This draws a lot of attention with the view to minimize the air infiltration particularly

Chapter VI -Simplified heat transfer modeling in steady state 123 through the enhancement of air curtain efficiency as highlighted in a recent review by [START_REF] Chaomuang | Experimental and numerical investigation of the performance of retail refrigerated display cabinets[END_REF]. Despite the immense efforts in both academic and industrial research levels, the issue of air infiltration remains problematic.

Installation of doors on the open front of the display cabinets becomes a viable alternative. It has been proven to be an effective solution to mitigate the air infiltration problemachieving potential energy savings [START_REF] Fricke | Energy use of doored and open vertical refrigerated display cases[END_REF]Lindberg et al., 2010), improving temperature homogeneity [START_REF] Chaomuang | Experimental analysis of heat transfer and airflow in a closed refrigerated display cabinet[END_REF][START_REF] Evans | Performance of retail and commercial refrigeration systems[END_REF] as well as promoting food quality (Atilio de [START_REF] Atilio De Frias | Improving spinach quality and reducing energy costs by retrofitting retail open refrigerated cases with doors[END_REF]. In addition, the thermal comfort in the retail stores can be enhanced, which can be possibly translated into a positive sales impact (Lindberg et al., 2017).

These advantages encourage the implementation of closed refrigerated display cabinets in many retail stores. For instance, it will account for about 75% of all display cabinets in French supermarkets by the end of 2020 [START_REF] Rpf | Fermeture des meubles réfrigérés: du positif pour les clients[END_REF].

Due to a reduction of cooling load after door installation, some modifications of the refrigeration system can be made. Lindberg et al. ( 2010) exhibited that the performance of closed display cabinet was enhanced when refrigerant inlet temperature was increased. [START_REF] Faramarzi | Performance and energy impact of installing glass doors on an open vertical deli/dairy display case[END_REF] suggested to reduce the size of thermostatic expansion valves to avoid a hunting effect. In addition, the presence of doors can modify flow pattern and temperature distribution inside the closed display cabinets. In concordance with our previous study (Chaomuang et al., 2019a), the position of the warmest product temperature was different (temperature at the front of the upper shelf became higher than that of the bottom shelf in closed cabinets, in opposite to open ones). An understanding of mechanism of heat transfer and airflow in the equipment becomes necessary since it can demonstrate if these modifications will compromise the quality and safety of food products. An experimental investigation is therefore necessary to determine real conditions occurring within the display cabinet. However, such investigation is both costly and labor-intensive. For this reason, a numerical simulation with Computational Fluid Dynamics (CFD) model is complementary as one can see it is widely used in various refrigerated food applications [START_REF] Norton | Computational fluid dynamics (CFD)an effective and efficient design and analysis tool for the food industry: A review[END_REF][START_REF] Smale | A review of numerical models of airflow in refrigerated food applications[END_REF].

Two main advantages of the CFD numerical approach were emphasized by [START_REF] Laguerre | Simplified heat transfer modeling in a cold room filled with food products[END_REF]. Firstly, it enables the knowledge of local parameters (temperature, humidity and velocity) that allows the understanding of the [START_REF] Cortella | cfd-aided retail cabinets design[END_REF][START_REF] Gaspar | Detailed CFD modelling of open refrigerated display cabinets[END_REF][START_REF] Ge | Simulation of the performance of single jet air curtains for vertical refrigerated display cabinets[END_REF][START_REF] Yu | A correlation model of thermal entrainment factor for air curtain in a vertical open display cabinet[END_REF], whereas little research associated with closed display cabinets has been carried out. in the open display cabinet can be investigated. Moreover, demand response strategies during on-and off-peak periods for electricity usage can be devised with the model. Some authors in our team also proposed simplified heat and/or mass transfer models for other refrigerating equipment in the cold chain: production plant [START_REF] Lecoq | Simplified heat and mass transfer modeling in a food processing plant[END_REF], refrigerated vehicle [START_REF] Hoang | Heat transfer modelling in a ventilated cavity loaded with food product: Application to a refrigerated vehicle[END_REF], cold room [START_REF] Laguerre | Simplified heat transfer modeling in a cold room filled with food products[END_REF] and domestic refrigerator [START_REF] Laguerre | Heat Transfer and Air Flow in a domestic refrigerator[END_REF]. Nevertheless, to the best of our knowledge, such a model for a closed refrigerated display cabinet is not available.

The objective of the study is to develop a simplified heat transfer model of a closed refrigerated display cabinet based on a zonal approach. This development is divided The acquired information of the simulation enables the identification of the critical zones, i.e. too warm and too cold, which can lead to food quality deterioration and chilling injury, respectively. The influence of compressor operation on the load temperature fluctuation can also be evaluated with the dynamic model. In fact, the quality of the food products is affected not only by the average level of temperature but also by the temperature fluctuation [START_REF] Cai | Quality model of foodstuff in a refrigerated display cabinet[END_REF][START_REF] Tano | Comparative evaluation of the effect of storage temperature fluctuation on modified atmosphere packages of selected fruit and vegetables[END_REF].

Closed refrigerated display cabinets will be soon a new component of the series of refrigeration equipment in the food cold chain. To take into account this evolution, this developed simplified heat transfer model will be further integrated with the other models already developed by our teams for several refrigerating equipment.

By this mean, it allows the prediction of time-temperature history of food products from a production plant to a household refrigerator. Furthermore, an evaluation tool for consumer risk can be also established when these models are linked with predictive microbiological and/or quality models [START_REF] Duret | Identification of the significant factors in food quality using global sensitivity analysis and the accept-and-reject algorithm. Part II: Application to the cold chain of cooked ham[END_REF].

Experimental study

Temperature measurement in a closed refrigerated display cabinet was carried out;

the obtained result was then used for the model validation. The display cabinet was installed in a test room in which the ambient temperature was controlled at 14.6±0.4 °C, 19.5±0.6 °C, and 24.4±0.4 °C. The water content of the air in the test room varied from 0.008 to 0.012 kg water/kg dry air, corresponding to a variation in the relative air humidity between 60% and 75%. This small variation in water content in the air showed little influence on the temperature performance, but more impact on the energy performance because of the condensation and frost formation on the evaporator [START_REF] Chen | Experimental study of the performance of single-band air curtains for a multi-deck refrigerated display cabinet[END_REF]. As depicted in temperature in the rear duct successively increased from -0.8 °C after the heat exchanger to -0.2 °C at the DAG. On the other hand, the stabilization due to the thermal inertia of these walls caused a moderate decrease of the standard deviation (from 2.3 °C to 1.6 °C). The highest load temperature was observed at the front of the top shelf (average 2.3 °C). In fact, the load located at this position was not only subject to heat exchanges with the air curtain but also to heat generation due to

Chapter VI -Simplified heat transfer modeling in steady state 128 (visible) light absorption because of its proximity to cabinet lighting. As a result, the load temperature was higher than that of the neighboring air.

Despite the heat loss through the glass doors, the air and load temperatures decreased from the top (air 1.5 °C, load 2.4 °C) to the bottom shelves (air -0.2 °C, load 0.4 °C). This is because more cold air is flowing from the back at the lower shelves over which the percentage of perforation area is higher than that over the top shelf (Figure 6.1). For the sake of simplicity for presentation, the evolution of the temperatures is firstly assumed to follow a sinusoidal variation:

𝑇 𝑖 (𝑡) = 𝑇 𝑖 + ∆𝑇 𝑖 cos(𝜔𝑡 + 𝜑 𝑖 ) = 𝑇 𝑖 + ∆𝑇 𝑖 cos (𝜔(𝑡 -𝑡 𝑑𝑒𝑙,𝑖 )) (6.1)

where 𝑇 𝑖 (𝑡) is the temperature evolution at a given position 𝑖 and at time 𝑡 in the display cabinet; 𝑇 𝑖 is the time-averaged temperature at that position; ∆𝑇 𝑖 is the amplitude of temperature fluctuation defined as (𝑇 𝑖,𝑚𝑎𝑥 -𝑇 𝑖,𝑚𝑖𝑛 ) 2 ⁄ ; 𝜔 is the angular frequency defined as 2𝜋 𝜏 ⁄ where 𝜏 is the period of one complete compressor cycle and 𝜑 𝑖 is the phase shift related to the delay time 𝑡 𝑑𝑒𝑙,𝑖 (due to the effects of inertia) at which the temperature at the position 𝑖 reaches its maximum

Static model development

Static model is first developed to predict the time-averaged air and load temperatures under steady state condition, for which the following assumptions are established:

 All temperatures are constant with respect to time. In practice, the temperatures vary due to compressor working and defrosting cycles, night/day period etc.

 Air temperature just after the heat exchanger where the thermostat sensor is located (𝑇 𝑡ℎ ) and the external ambient air (𝑇 𝑒𝑥𝑡 ) are also constant and they are the input parameters.

 There are two loads of the same volume on each shelf (back and front) and the temperature of each load is considered homogeneous.

Dimensionless temperature (𝜃 𝑖 ) based on the time-averaged thermostat temperature (𝑇 𝑡ℎ ) and the external ambient air temperature (𝑇 𝑒𝑥𝑡 ) is defined (Eq. (6.2)) to generalize the model, and it will be used throughout the development.

𝜃 𝑖 = 𝑇 𝑖 -𝑇 𝑡ℎ 𝑇 𝑒𝑥𝑡 -𝑇 𝑡ℎ for { 𝑇 𝑡ℎ = 𝑇 𝑡ℎ 𝑇 𝑒𝑥𝑡 ≠ 𝑇 𝑡ℎ (6.2)
In fact, all heat transfer phenomena involved in the display cabinet (i.e. forced convection and conduction) are linear with respect to temperature difference.

Therefore, the properties of the dynamic linear system can be used: the timeaveraged output (𝑇 𝑖 ) for a fluctuating input (𝑇 𝑡ℎ ) is equal to the response for the time-averaged input (𝑇 𝑡ℎ ). Moreover, the defined dimensionless temperature 𝜃 𝑖 is also independent of 𝑇 𝑒𝑥𝑡 -𝑇 𝑡ℎ due to the linearity between the internal and external temperatures. As previously mentioned in Section 6.2, free convection and radiation from the external walls, which are non-linear phenomena, are limited because the display cabinet was closed with doors coated with low-emissivity materials.

Heat losses through the rear wall and the cabinet ceiling (Domain I)

In Domain I, the heat balance is established into seven zones from the upstream of the vertical rear duct to the DAG. Zones 1 to 6, i.e. EZ [1, 1…6], correspond to the heat exchange on the rear wall (exchange area, 𝑆 𝑤 ). It is assumed that a part of supply air is flowing through the PBP at the mid height of each shelf (with air distribution coefficient, 𝛾 ). In this way, the exchange areas of Zone 1 and 6 correspond to a half of the shelf height while the ones of Zone 2 to 5 correspond to two halves of the shelf height. Zone 7, i.e. EZ [1,7], represents the heat exchange on the cabinet ceiling (exchange area, 𝑆 𝑐 ).

Considering EZ [1, 1] (Figure 6.5a), the supply air (total mass flow rate, 𝑚̇𝑡), after being cooled down by the heat exchanger, is assumed to be at the temperature at the thermostat sensor (𝜃 𝑡ℎ ) which is located there. Flowing upward in the vertical rear duct, the supply air exchanges heat with the external air (𝜃 𝑒𝑥𝑡 ) through the rear wall by conduction and by convection (overall heat transfer coefficient, 𝑈 𝑜𝑣𝑤 ). The air temperature is then increased from 𝜃 𝑡ℎ to 𝜃 𝑎𝑟𝑑1 . As by integration along a part of the rear duct (Figure 6.5b), the heat balance equation of EZ [1, 1] is given by

𝑚̇𝑡𝐶 𝑝𝑎 𝑑𝜃 𝑎 = 𝑈 𝑜𝑣𝑤 (𝜃 𝑒𝑥𝑡 -𝜃 𝑎 )𝑑𝑆 𝑤 ⟹ 𝜃 𝑒𝑥𝑡 -𝜃 𝑎𝑟𝑑1 𝜃 𝑒𝑥𝑡 -𝜃 𝑡ℎ = exp (- 𝑈 𝑜𝑣𝑤 𝑆 𝑤1 𝑚̇𝑡𝐶 𝑝𝑎 ) 𝜃 𝑎𝑟𝑑1 = 𝛼 𝑟𝑑1 𝜃 𝑡ℎ + (1 -𝛼 𝑟𝑑1 )𝜃 𝑒𝑥𝑡 (6.3)
where 𝛼 𝑟𝑑1 = exp (- Because of the heat transfer from external air through the rear wall and the ceiling of the cabinet, the supply air temperature increased successively from 𝜃 𝑡ℎ to 𝜃 𝑎𝑟𝑑7 . 1, 1]). It is to be emphasized that cold air leakage through the gap is assumed to occur between Shelf 1 and 2. In reality, this leakage spreads over the height of the door. Nevertheless, this simplification is acceptable because the external warm air infiltration has more impact on the internal temperature than the leakage of cold air.

𝑈

Heat exchange between air and the perforated back panel (interface between Domain I and II)

Considering EZ [2, 1, 1] (Figure 6.6a), a part of the air (mass flow rate, 𝑚 ̇𝑎𝑠ℎ1 = 𝛾 1 𝑚̇𝑡 and temperature, 𝜃 𝑎𝑟𝑑5 ) flows horizontally from the back to the front at the mid height of Shelf 1. At steady state, the air temperatures before and after passing through the PBP are the same:

Chapter VI -Simplified heat transfer modeling in steady state 135 𝜃 𝑎𝑠ℎ11 = 𝜃 𝑎𝑟𝑑5 (6.4)

A similar equation applies for all shelves. In addition to this convective exchange with air, the back load also exchanges heat with the front load by conduction (heat transfer coefficient, 𝑈 𝑙𝑙1 ). At steady state, the heat balance equation for the back load is

𝑈 𝑙𝑙1 𝑆 𝑙𝑙1 (𝜃 𝑙𝑏1 -𝜃 𝑙𝑓1 ) + 𝑚̇𝑎 𝑠ℎ1 𝐶 𝑝𝑎 (𝜃 𝑎𝑠ℎ21 -𝜃 𝑎𝑠ℎ11 ) = 0 𝜃 𝑎𝑠ℎ21 = 𝜃 𝑎𝑠ℎ11 + 𝛽 𝑠ℎ11 (𝜃 𝑙𝑓1 -𝜃 𝑙𝑏1 ) (6.6)
where

𝛽 𝑠ℎ11 = 𝑈 𝑙𝑙1 𝑆 𝑙𝑙1 𝑚̇𝑎 𝑠ℎ1 𝐶 𝑝𝑎
An analogous analysis of heat balance at this zone was applied for all shelves, i.e.

EZ [2, 2, 1…5].

Heat exchange between the mixed air and the front load for Shelf 1

The air above the front load at Shelf 1 is a mix of three air streams as depicted in and temperature, 𝜃 𝑎𝑠ℎ51 ). After mixing, the air mass flow rate and temperature becomes 𝑚̇′ 𝑎𝑠ℎ1 and 𝜃 𝑎𝑠ℎ31 , respectively.

Mass balance:

𝑚̇𝑎 𝑠ℎ1 ′ = 𝑚̇𝑎 𝑟𝑑5 + 𝑚̇𝑎 𝑠ℎ1 + 𝛿 1 𝑚̇𝑎 𝑠ℎ1 ′ ⟹ 𝑚̇𝑎 𝑠ℎ1 ′ = 𝑚̇𝑎 𝑠ℎ1 +𝑚̇𝑎 𝑟𝑑5 1-𝛿 1 (6.7)
Energy balance:

𝑚̇𝑎 𝑠ℎ1 ′ 𝜃 𝑎𝑠ℎ31 = 𝑚̇𝑎 𝑟𝑑5 𝜃 𝑎𝑟𝑑7 + 𝑚̇𝑎 𝑠ℎ1 𝜃 𝑎𝑠ℎ21 + 𝛿 1 𝑚̇𝑎 𝑠ℎ1 ′ 𝜃 𝑎𝑠ℎ51 (6.8)
Then, this mixed air exchanges heat with the front load, EZ [2, 3, 1], and its temperature becomes 𝜃 𝑎𝑠ℎ41 (the integration along the exchange surface is the same as for Eq (6.3) and (6.5)).

𝜃 𝑎𝑠ℎ41 = 𝛼 𝑠ℎ21 𝜃 𝑎𝑠ℎ31 + (1 -𝛼 𝑠ℎ21 )𝜃 𝑙𝑓1 (6.9)

where 𝛼 𝑠ℎ21 = exp (-

𝑈 𝑙𝑓1 𝑆 𝑙1 𝑚̇𝑎 𝑠ℎ1 ′ 𝐶 𝑝𝑎 )
In addition to this convective exchange with the air, the front load also exchanges heat with the back load by conduction (heat transfer coefficient, 𝑈 𝑙𝑙1 ) and receives

Chapter VI -Simplified heat transfer modeling in steady state 137 heat from the conversion of supermarket and display cabinet lighting powers (𝑞 𝑙𝑖1 ).

At steady state: Mass balance:

𝑈 𝑙𝑙1 𝑆 𝑙𝑙1 (𝜃 𝑙𝑏1 -𝜃 𝑙𝑓1 ) + 𝑞 ′ 𝑙𝑖1 =
𝑚̇𝑎 𝑠ℎ2 ′ = 𝑚̇𝑎 𝑠ℎ2 + 𝛿 2 𝑚̇𝑎 𝑟𝑑4 (6.11)
Energy balance:

𝑚̇𝑎 𝑠ℎ2

′ 𝜃 𝑎𝑠ℎ32 = 𝑚̇𝑎 𝑠ℎ2 𝜃 𝑎𝑠ℎ22 + 𝛿 2 𝑚̇𝑎 𝑟𝑑4 𝜃 𝑎𝑠ℎ41 (6.12)

The mixed air then exchanges heat with the front load, EZ [2, 3, 2], and its temperature becomes 𝜃 𝑎𝑠ℎ42 .

𝜃 𝑎𝑠ℎ42 = 𝛼 𝑠ℎ22 𝜃 𝑎𝑠ℎ32 + (1 -𝛼 𝑠ℎ22 )𝜃 𝑙𝑓2 (6.13)

where 𝛼 𝑠ℎ22 = exp (-

𝑈 𝑙𝑓2 𝑆 𝑙2 𝑚̇𝑎 𝑠ℎ2 ′ 𝐶 𝑝𝑎 )
The load located at the front of Shelf 2 is subjected to the convection with the mixed air, the conduction with the adjacent load at the back (heat transfer coefficient, 𝑈 𝑙𝑙2 )

and the heat conversion of visible radiation from the supermarket lighting power (𝑞 𝑙𝑖2 ). At steady state: An analogous analysis of mass balance at MZ [2, 1, 2] and heat transfer at EZ [2, 3, 2] was applied to Shelf 3 to 5. glass doors (overall heat transfer coefficient, 𝑈 𝑜𝑣𝑑 ), and consequently its temperature increases to be 𝜃 𝑎𝑠ℎ52 .

𝑈 𝑙𝑙2 𝑆 𝑙𝑙2 (𝜃 𝑙𝑏2 -𝜃 𝑙𝑓2 ) + 𝑞 ′ 𝑙𝑖2 = 𝑚̇𝑎 𝑠ℎ2 ′ 𝐶 𝑝𝑎 (𝜃 𝑎𝑠ℎ42 -𝜃 𝑎𝑠ℎ32 ) 𝜃 𝑎𝑠ℎ42 = 𝜃 𝑎𝑠ℎ32 + 𝛽 𝑠ℎ22
𝜃 𝑎𝑠ℎ52 = 𝛼 𝑠ℎ32 𝜃 𝑎𝑠ℎ41 + (1 -𝛼 𝑠ℎ32 )𝜃 𝑒𝑥𝑡 (6.17)

where 𝛼 𝑠ℎ32 = exp (-

𝑈 𝑜𝑣𝑑 𝑆 𝑑2 (1-𝛿 2 )𝑚̇𝑎 𝑟𝑑4 𝐶 𝑝𝑎 )
In the mixing zone (i.e. MZ [2, 2, 2], Figure 6.6b), this air mixes with the air which exchanged heat with the load located at the front of the shelf (mass flow rate, 𝑚̇𝑎 𝑠ℎ2 ′ and temperature, 𝜃 𝑎𝑠ℎ42 ). The air temperature becomes 𝜃 𝑎𝑠ℎ62 .

Mass balance:

𝑚̇𝑎 𝑟𝑑3 = 𝑚̇𝑎 𝑠ℎ2 ′ + (1 -𝛿 2 )𝑚̇𝑎 𝑟𝑑4 (6.18)
Energy balance:

𝑚̇𝑎 𝑟𝑑3 𝜃 𝑎𝑠ℎ62 = 𝑚̇𝑎 𝑠ℎ2 ′ 𝜃 𝑎𝑠ℎ42 + (1 -𝛿 2 )𝑚̇𝑎 𝑟𝑑4 𝜃 𝑎𝑠ℎ52 (6.19)
An analogous analysis of heat and mass balances at these zones, i.e. EZ [2,4,2] and MZ [2, 2, 2], was applied to Shelf 3 to 5.

Equations in matrix form

In Domain I, the air temperature in the vertical rear duct is successively calculated and the equations can be generalized in the following form:

𝜃 𝑎𝑟𝑑,𝑖 = 𝛼 𝑟𝑑,𝑖 • 𝜃 𝑎𝑟𝑑,𝑖-1 + (1 -𝛼 𝑟𝑑,𝑖 ) • 𝜃 𝑒𝑥𝑡 for 𝑖 = 1, 2, … , 7 (6.20)
Note that the air temperature of 𝑖 = 0 refers to the temperature near the thermostat sensor (𝜃 𝑎𝑟𝑑0 = 𝜃 𝑡ℎ ).

Equation in a matrix form is used to summarize the heat balance equations in Domain II. Because of the difference of heat transfer phenomena, the equation of Shelf 1 (top shelf, 𝑗 = 1) is separately formulated:

𝐴 ̿ • 𝜃 ⃗ 1 = 𝐵 ⃗⃗ • 𝜃 𝑎𝑟𝑑7 + 𝐶 ⃗ • 𝜃 𝑎𝑠ℎ1,1 + 𝐷 ⃗ ⃗⃗ • 𝜃 𝑒𝑥𝑡 + 𝐸 ⃗⃗ • 𝑞 ′′ 𝑙𝑖1 (6.21)
where

𝐴 ̿ = [ 1 0 0 0 -(1 -𝛼 𝑠ℎ1,1 ) 0 1 0 0 0 𝛽 𝑠ℎ1,1 -𝛽 𝑠ℎ1,1 𝑚̇𝑎 𝑠ℎ,1 -𝑚̇𝑎 𝑠ℎ1 ′ 0 -𝛿 𝑚 𝑚̇𝑎 𝑠ℎ1 ′ 0 0 0 𝛼 𝑠ℎ2,1 -1 0 0 (1 -𝛼 𝑠ℎ2,1 ) 0 1 -1 0 𝛽 𝑠ℎ2,1 -𝛽 𝑠ℎ2,1 0 0 -𝛼 𝑠ℎ3,1 1 0 0 ] 𝜃 ⃗ 1 = [ 𝜃 𝑎𝑠ℎ2,1 𝜃 𝑎𝑠ℎ3,1 𝜃 𝑎𝑠ℎ4,1 𝜃 𝑎𝑠ℎ5,1 𝜃 𝑙𝑏,1 𝜃 𝑙𝑓,1 ] ; 𝐵 ⃗⃗ = [ 0 0 -𝑚̇𝑎 𝑟𝑑5 0 0 0 ] ; 𝐶 ⃗ = [ 𝛼 𝑠ℎ1,1 1 0 0 0 0 ] ; 𝐷 ⃗ ⃗⃗ = [ 0 0 0 0 0 (1 -𝛼 𝑠ℎ3,1 ) ] and 𝐸 ⃗⃗ = [ 0 0 0 0 -1 0 ]
The equations of Shelf 2 to 5 ( 𝑗 = 2 … 5 ) are generalized as follows:

𝜃 𝑎𝑠ℎ1,𝑗 = 𝜃 𝑎𝑟𝑑,6-𝑗 (6.22) 𝐹 ̿ • 𝜃 ⃗ 𝑗 = 𝐺 ⃗ • 𝜃 𝑎𝑠ℎ,𝑘,𝑗-1 + 𝐻 ⃗ ⃗⃗ • 𝜃 𝑎𝑠ℎ1,𝑗 + 𝐼 ⃗ • 𝑞 ′′ 𝑙𝑖,𝑗 { if 𝑗 = 2 → 𝑘 = 4 if 𝑗 = 3, 4, 5 → 𝑘 = 6 (6.23)
where

𝐹 ̿ = [ 1 0 0 -(1 -𝛼 𝑠ℎ1,𝑗 ) 0 1 0 0 𝛽 𝑠ℎ1,𝑗 -𝛽 𝑠ℎ1,𝑗 𝑚̇𝑎 𝑠ℎ,𝑗 -𝑚̇𝑎 𝑠ℎ,𝑗 ′ 0 0 0 0 𝛼 𝑠ℎ2,𝑗 -1 0 (1 -𝛼 𝑠ℎ2,𝑗) ) 0 1 -1 𝛽 𝑠ℎ2,𝑗 -𝛽 𝑠ℎ2,𝑗 ] 𝜃 ⃗ 𝑗 = [ 𝜃 𝑎𝑠ℎ2,𝑗 𝜃 𝑎𝑠ℎ3,𝑗 𝜃 𝑎𝑠ℎ4,𝑗 𝜃 𝑙𝑏,𝑗 𝜃 𝑙𝑓,𝑗 ] ; 𝐺 ⃗ = [ 0 0 -𝛿 𝑗 𝑚̇𝑎 𝑟𝑑,6-𝑗 0 0 ] ; 𝐻 ⃗ ⃗⃗ = [ 𝛼 𝑠ℎ1,𝑗 1 0 0 0 ] and 𝐼 ⃗ = [ 0 0 0 -1 0 ]
and 𝜃 𝑎𝑠ℎ5,𝑗 = 𝛼 𝑠ℎ3,𝑗 𝜃 𝑎𝑠ℎ,𝑘,𝑗-1 + (1 -𝛼 𝑠ℎ3,𝑗 )𝜃 𝑒𝑥𝑡 (6.24) 𝜃 𝑎𝑠ℎ6,𝑗 = (𝑚̇𝑎 𝑠ℎ,𝑗 ′ 𝜃 𝑎𝑠ℎ4,𝑗 + (1 -𝛿 𝑗 )𝑚̇𝑎 𝑟𝑑,6-𝑗 𝜃 𝑎𝑠ℎ5,𝑗 ) 𝑚̇𝑎 𝑟𝑑,5-𝑗 ⁄ (6.25)

Estimation of model input parameters

Input parameters used in the developed model were obtained from several sources: physical measurement, calculation based on literature and material properties (Table 6.1) and model fitting.

Air flow rate

The air flow rate at the upstream of the vertical rear duct (𝑚̇𝑡) and the proportion of the air flowing through the PBP at different shelves (represented by the air distribution coefficient 𝛾 𝑖 ) are input parameters of the model. The measurement of the air velocity in the rear duct was thus carried out by using a hot-wire anemometer (Testo 435-4) [START_REF] Chaomuang | Experimental analysis of heat transfer and airflow in a closed refrigerated display cabinet[END_REF], which was previously calibrated in the range of 0-2 m•s -1 (uncertainty 0.02 m•s -1 ). Three holes were drilled in the rear wall of the cabinet (at z = -260, 0, and +260 mm) at six height levels (at y = 15, 215, 465, 715, 965 and 1335 mm) to introduce the instrument. The flow perturbation due to the intrusive feature of the test probes can be considered negligible because the sensor is very small (7.5 mm in diameter) compared to the space in the rear duct (60 mm in width). There were nine measurement positions (three positions for each hole) across the cross-sectional area of the duct. At each position, the air velocity was recorded every second for 5 min. and its time-averaged value was then calculated. The mass flow rate and the proportion of the air flowing through the PBP at different heights were then estimated based on the average value obtained at the nine positions. The results are reported in Table 6.2. Given the proportion of the air flowing over the entire PBP, the mass flow rate at the DAG was attributed to about half of the total mass flow rate. In addition to the temperature measurements, the air velocity at the front of display cabinet (for different heights)

from the discharge to return air grille was also measured by a hot wire anemometer and the shape of the curtain was presented (Chaomuang et al., 2019a).

) 15 W•m -2 •K -1 Model fitting Front load (𝑈 𝑙𝑓 ) 24 W•m -2 •K -1 Model fitting Conductive Between the two loads (𝑈 𝑙𝑙 ) 3.4 W•m -2 •K -1 Calculation Overall
Rear wall and ceiling

(𝑈 𝑜𝑣𝑤 ) 0.77 W•m -2 •K -1 Calculation Door (𝑈 𝑜𝑣𝑑 ) 1.02 W•m -2 •K -1 Calculation Equivalent thermal conductance (𝐶 𝑒𝑞 ) 17.6 W•m -2 •K -

Conductive heat transfer coefficient between two adjacent loads

The conductive heat transfer coefficient between the back and front loads was calculated using a thermal resistance concept and it was supposed to have the same value for every shelf. In our study, the load was made of methylcellulose of which the thermal conductivity is about 0.51 W•m -1 •K -1 [START_REF] Icier | The use of tylose as a food analog in ohmic heating studies[END_REF] and the distance between adjacent loads (core to core) is 0.15 m.

𝑈 𝑙𝑙1 = 𝑈 𝑙𝑙2 = 𝑈 𝑙𝑙3 = 𝑈 𝑙𝑙4 = 𝑈 𝑙𝑙5 = 𝜆 𝑒 = 3.4 W•m -2 •K -1
The air space between two adjacent loads was considered negligible since the test packages were fully loaded in the display cabinet.

Overall wall heat transfer coefficient

The overall heat transfer coefficients due to the heat loss through the rear wall, ceiling and door of the display cabinet were also estimated based on the thermal resistance concept. Natural convection between the external air and the outer Laguerre et al., 2012a), conduction within the wall/door (𝑈 𝑤 or 𝑈 𝑑 ) and forced convection between the supply air and the inner Laguerre et al., 2012a) were taken into account.

wall/door (ℎ 𝑒𝑥𝑡 ~10 W•m -2 •K -1 ,
wall/door (ℎ 𝑖𝑛𝑡 ~20 W•m -2 •K -1 ,
The rear wall and the ceiling were composed of an insulation layer made of polyurethane foam (30-mm thickness) and an aluminum sheet (0.5-mm thickness), manufacturer data.

𝑈 𝑜𝑣𝑤 = 𝑈 𝑜𝑣𝑐 = 1 1 ℎ 𝑒𝑥𝑡 + 1 𝑈 𝑤 + 1 ℎ 𝑖𝑛𝑡 = 1 1 ℎ 𝑒𝑥𝑡 + 𝑒 𝒶ℓ 𝜆 𝒶ℓ + 𝑒 𝑝𝑢 𝜆 𝑝𝑢 + 1 ℎ 𝑖𝑛𝑡 = 0.77 W•m -2 •K -1
The door of the cabinet is a double-glazing window made of two panes of tempered glass (4-mm thickness each) split by an argon gap (14-mm thickness), manufacturer data.

𝑈 𝑜𝑣𝑑 = 1 1 ℎ 𝑒𝑥𝑡 + 1 𝑈 𝑑 + 1 ℎ 𝑖𝑛𝑡 = 1 1 ℎ 𝑒𝑥𝑡 + 2𝑒 𝑔𝑙 𝜆 𝑔𝑙 + 𝑒 𝑎𝑔 𝜆 𝑎𝑔 + 1 ℎ 𝑖𝑛𝑡 = 1.02 W•m -2 •K -1

Exchange area for the display cabinet in length of 1.25 m

The exchange area of the rear wall and door was split with respect to the shelf height: 0.25 m for Shelf 1 to 4 and 0.345 m for Shelf 5. The values are reported in Table 6.2. The exchange area between the loads located at the back and the front and the surrounding air was estimated as:

𝑆 𝑙 = 𝑆 𝑙𝑏 = 𝑆 𝑙𝑓 = (0.2 + 0.15) × 1.25 = 0.4375 m 2
The exchange area between the back and the front loads by conduction was estimated as: 𝑆 𝑙𝑙 = 0.2 × 1.25 = 0.25 m 2

Heat absorption due to the supermarket and display cabinet lightings

The illuminance level in a supermarket where refrigerated display cabinets are installed ranges from 300 to 1000 lux (IESNA, 1991) depending on types of designed activity areas. On average, the value of 650 lux (which is equivalent to lighting power density of about 1.0 W•m -2 ) was used and dispersed equally to the front loads at all shelves.

𝑞 𝑙𝑖_𝑒𝑥𝑡 = 𝑞 𝑙𝑖2…5 = (1.0 W • m -2 )(0.25 m 2 ) = 0.25 W Light-Emitting Diodes (LED) with input power of 13 W (manufacturer data) are used in the studied display cabinet. Due to its proximity to the cabinet lighting, the load at Shelf 1 would receive most of the heat dissipated from this lighting. The remainder lighting may be received by the loads on the other shelves and even escapes from the cabinet, which is difficult to quantify. As a first approach, 75% of the cabinet lighting power was only applied to the front load at Shelf 1.

Static model validation

For model validation purposes, measurements of air and load temperatures at various positions in the closed display cabinet were carried out for three ambient temperatures (14.6 °C, 19.5 °C and 24.4 °C). As shown in Figure 6.8, the air temperature just after the heat exchanger decreased when the external ambient air temperature increased because the refrigeration system is required to provide more cooling capacity to compensate the additional heat gain. For this reason, different time-averaged air temperatures at the thermostat sensor (𝑇 𝑡ℎ ) corresponding to the ambient temperature were applied to the model. The results of calculated and measured air and load temperatures for three ambient temperatures were illustrated in Figure 6.9. Temperature of the air in the vertical rear duct progressively increased due to heat losses through the rear wall. As the rear duct wall adjacent to the shelf space is perforated, some cold air is permitted to cool down the load at the back of the cabinet. As a result, the temperature of the load located at the back tended to decrease from the top to the bottom shelf.

Similarly, the temperature of load located at the front also decreased from the top to the bottom shelves, but with higher temperature. This is because there were additional heat loads including heat losses through the glass door, the air infiltration through the door gap and the heat absorption corresponding to the cabinet lighting.

In addition, the temperature was affected by the different percentages of perforation area of the back panel on each shelf: lowest at the top and highest at the bottom.

These findings showed that the highest and lowest temperature positions in this closed display cabinet were respectively at the front of the top shelf and the back of the bottom shelf. Overall, both air and load temperatures in the closed display cabinet increased with an increase of external ambient temperature at all positions in the shelves-space storage (Table 6.4).

As depicted in Figure 6.10, comparative results provide a good agreement between the calculated and measured air and load temperatures even for the two external temperatures (14.6 °C and 24.4 °C) which were not used for the model fitting. The overall mean absolute error based on 60 paired-data of three ambient temperatures was about 0.2 °C (𝑅 2 = 0.96, Table 6.5). Mostly, the air temperature at the front of the shelves was slightly over-estimated. The measurement position should be mentioned as a possible source of uncertainty because a single point measurement may not be representative of the air at the front. Nevertheless, the model is likely to well represent the load temperature. The maximum difference of the calculated and measured load temperatures was less than 0.5 °C for every studied position.

Besides, the differences between the highest and the coldest load temperatures obtained from the experimental and the numerical data were less than 14% for all studied ambient temperatures.

. Effect of air infiltration through the door gaps

In this part, the model was used to investigate the effect of the air infiltration through the door gaps on the load temperature in the closed display cabinet. This effect on the energy consumption was also studied and the numerical results are presented hereafter.

Based on the results at the ambient temperature of 19.5 °C, it was found that load temperatures could be notably decreased (up to 0.7 °C) when the air infiltration through the door gap reduced by half as shown in Figure 6.11. This suggested that the temperature performance of closed refrigerated display cabinets could be improved by minimizing the air infiltration through door gaps of the cabinets.

6.5.2.

Effect of ambient air humidity

The influence of ambient humidity on the load temperature was considered as nonsignificant because of low water content in air (about 0.008 to 0.012 kg water/kg dry air) [START_REF] Chen | Experimental study of the performance of single-band air curtains for a multi-deck refrigerated display cabinet[END_REF]. However, the cooling load of the refrigeration system of the display cabinet is highly influenced by this humidity. The present model was further used to estimate the frost formation on the evaporator surface and the total cooling load of the refrigeration system by taking both sensible and latent heat loads into account. The closed display cabinet operated at the ambient temperature of 19.5 °C and 65%RH (𝜙 𝑒𝑥𝑡 = 0.009 kg•kg -1 ) was used to illustrate this model application. The air after the evaporator can be assumed to be almost saturated i.e. 𝜙 𝑡ℎ ≅ 𝜙 𝑠𝑎𝑡 𝑇 𝑡ℎ . In our case 𝑇 𝑡ℎ = -0.8 °C, thus 𝜙 𝑡ℎ = 0.003 kg•kg -1 , and the air infiltration rate through door gaps was 𝑚̇𝑔 𝑎𝑝 = 0.0051 kg•s -1 . For this example, the rate of frost formation on the evaporator was approximately 0.031 g•s -1 ≈ 100 g•h -1 .

The total cooling load (𝑄 𝑡 ) is the sum of the energy used to cool down the air from 𝑇 𝑅𝐴𝐺 to 𝑇 𝑡ℎ with the total mass flow rate and to convert the water vapor in the air into frost.

𝑄 𝑡 = 𝑄 𝑠 + 𝑄 ℓ = 𝑚̇𝑡𝐶 𝑝 (𝑇 𝑅𝐴𝐺 -𝑇 𝑡ℎ ) + 𝑚̇𝑔 𝑎𝑝 (𝜙 𝑒𝑥𝑡 -𝜙 𝑡ℎ )(∆𝐻 𝑣 + ∆𝐻 𝑓 ) (6.28)

Given the total mass flow rate 𝑚̇𝑡 = 0.1120 kg•s -1 , the total cooling load was about 249 W (𝑄 𝑠 = 169 W and 𝑄 ℓ = 80 W). When the external air infiltration through the door gaps decreased by half, the rate of frost formation on the evaporator was

Chapter VI -Simplified heat transfer modeling in steady state 154 reduced to 0.016 g•s -1 (≈ 58 g•h -1 ) and the total cooling load to 162 W (𝑄 𝑠 = 122 W and 𝑄 ℓ = 40 W). Based on this calculation, the total cooling load of the refrigeration system would be reduced by 35%.

Generalization of the model for other closed display cabinets

The simplified model proposed in this study was developed based on a closed refrigerated display cabinet of which the specific geometry is determined. The model is therefore applicable to closed display cabinets with similar configurations as described in Section 2 (i.e. the location of the heat exchanger and fans are underneath the bottom shelf, the lighting is at the top front of the cabinet, etc.).

Among all parameters, the number of shelves equipped in the display cabinet shows

Air infiltration during door openings

The developed model does not take into account the air infiltration due to door openings because the airflow during door openings is too complex to be predicted by the simplified model. CFD model is more appropriate to cope with this phenomenon. To keep the model simplicity, a simple way to include the air infiltration during door openings could be done by introducing a higher value of Retail display is often pointed out as the hotspot for temperature abuse in the cold chain [START_REF] Baldera Zubeldia | Effectiveness of the cold chain control procedure in the retail sector in Southern Spain[END_REF][START_REF] Derens-Bertheau | Cold chain of chilled food in France[END_REF][START_REF] Mercier | Time-Temperature Management Along the Food Cold Chain: A Review of Recent Developments[END_REF]. Air infiltration through the frontal opening of refrigerated display cabinets is the main cause of this issue. Dealing with it by closing the front of the display cabinets with doors become one of the simple and effective solutions because of better temperature performance [START_REF] Chaomuang | Experimental analysis of heat transfer and airflow in a closed refrigerated display cabinet[END_REF][START_REF] Evans | Temperature and energy performance of refrigerated retail display and commercial catering cabinets under test conditions[END_REF]Lindberg et al., 2010). As a result, the use of closed refrigerated display cabinets increases in retail stores [START_REF] Eia | Chilling Facts VI: Closing the door on HFCs[END_REF], and approximately 75% of total display cabinets in French supermarkets will be closed with doors by 2020 [START_REF] Rpf | Fermeture des meubles réfrigérés: du positif pour les clients[END_REF]. Most of dynamic models for refrigerating equipment are usually developed with the component sub-models of vapor compression systems (i.e. compressor, condenser, expansion device and evaporator) [START_REF] Zsembinszki | A novel numerical methodology for modelling simple vapour compression refrigeration system[END_REF]. A comprehensive review of dynamic modeling for the vapor compression systems is provided in the two-part article of [START_REF] Rasmussen | Dynamic modeling for vapor compression systems-Part I: Literature review[END_REF] in which both physics-based (e.g. lumped parameter, moving boundary, finite control volume,…) and data-based (e.g. linear, time-series,…) models for each system components are presented. Examples of the physics-based models can be found in numerous studies for different refrigerating equipment including cold-storage plants [START_REF] Eames | Description and validation of a computer based refrigeration system simulator[END_REF][START_REF] Hasse | Top-down model for dynamic simulation of cold-storage plants[END_REF]Lovatt et al., 1998), refrigerated trucks [START_REF] Rachek | Dynamic simulation of the refrigerated compartment of a truck coupled with its refrigeration system[END_REF] and domestic refrigerators [START_REF] Borges | Transient simulation of household refrigerators: A semi-empirical quasi-steady approach[END_REF]Borges et al., 2015;[START_REF] Chen | Dynamic simulation and optimal matching of a small-scale refrigeration system[END_REF][START_REF] Hermes | A first-principles simulation model for the start-up and cycling transients of household refrigerators[END_REF]. Nevertheless, these studies are dedicated to the dynamic response of the refrigerating equipment to enhance overall system energy efficiency [START_REF] Azzouz | Performance enhancement of a household refrigerator by addition of latent heat storage[END_REF], whilst bulk air/product temperatures in the equipment are often considered as heat reservoir. Despite very much higher computational time, the CFD approach is more appropriate choice for the analysis of spatiotemporal temperature fluctuations in high resolutions (Alfaro-Ayala et al.; [START_REF] Ambaw | Analysis of the spatiotemporal temperature fluctuations inside an apple cool store in response to energy use concerns[END_REF][START_REF] Gruyters | A numerical evaluation of adaptive on-off cooling strategies for energy savings during long-term storage of apples[END_REF][START_REF] Wang | A CFD Simulation of 3D Air Flow and Temperature Variation in Refrigeration Cabinet[END_REF]. A balance between simplicity and fidelity to select suitable modeling approach is therefore important.

W• K -1 ] 𝐶 𝑝 heat capacity [J•kg -1 •K -1 ] 𝑒 thickness [m] 𝐹𝑜 Fourier number = 𝜅𝜏 𝑒 2 𝐆 transfer function ℎ, 𝑈 heat transfer coefficient [W•m -2 •K -1 ] 𝐿 length [m] 𝑚 solid
In the present study, a simplified dynamic model was developed based on the zonal approach and solved with a spectral method by using Fourier transform and transfer

Model description

As described in Part I of the study, the air and load temperatures in the closed refrigerated display cabinet vary according to the compressor on/off cycle. As an input to the model, the temperature evolution of the air just after the heat exchanger at which the thermostat sensor is located (𝑇 𝑡ℎ ) is first assumed to be sinusoidal: The first term on the right side corresponds to the heat loss (overall heat transfer coefficient between inner wall and external ambient air, 𝑈 𝑜𝑣𝑤 ′

). The second term corresponds to the heat exchange with the supply air.

Because the thermal inertia of the air is very much lower than that of the wall, the heat balance equation for air along the inner side of the wall is given by

𝑚̇𝑡𝐶 𝑝𝑎 𝑑𝜃 𝑎 ′ = ℎ 𝑖𝑛𝑡 (𝜃 𝑤 ′ -𝜃 𝑎 ′ )𝑑𝑆 𝑤 ⟹ ln ( 𝜃 𝑤1 ′ -𝜃 𝑎𝑟𝑑1 ′ 𝜃 𝑤1 ′ -𝜃 𝑡ℎ ′ ) = - ℎ 𝑖𝑛𝑡 𝑆 𝑤1 𝑚̇𝑡𝐶 𝑝𝑎 𝜃 𝑎𝑟𝑑1 ′ = 𝛼 𝑑𝑦1 𝜃 𝑡ℎ ′ + (1 -𝛼 𝑑𝑦1 )𝜃 𝑤1 ′ (7.12)
where 𝛼 𝑑𝑦1 = exp (- Similar equations of the heat exchange at this zone were derived for all shelves.

Heat exchange between the air and the back load

In fact, temperature fluctuation in a solid structure steadily decreases along the distance from the surface due to its thermal inertia [START_REF] Carslaw | Conduction of heat in solids[END_REF]. To simplify the model, the temperature fluctuation is assumed to be the same from the load surface to a specific depth (equivalent thickness, 𝑒 𝑒𝑞 ), while the temperature fluctuation is neglected for the rest of the load (core); more details are presented in Analogous analysis of the heat balance was applied to all shelves.

Heat exchange with the door

The heat balance equation in this zone was developed in a similar manner to the heat exchange in the vertical rear duct (Section 7.2.2.1) which takes into account the attenuation due to the pulsation independent and dependent effects. For Shelf 2, i.e. EZ [2, 4, 2], the heat balance equation is given by

𝑚 𝑑2 𝐶 𝑝𝑑 𝑑𝜃 𝑑2 ′ 𝑑𝑡 = 𝑈 𝑜𝑣𝑑 ′ 𝑆 𝑑2 (𝜃 𝑒𝑥𝑡 ′ -𝜃 𝑑2 ′ ) -(1 -𝛿 2 )𝑚̇𝑎 𝑟𝑑4 𝐶 𝑝𝑎 (𝜃 𝑎𝑠ℎ41 ′ - 𝜃 𝑎𝑠ℎ52 ′ ) (7.34) 𝜃 𝑎𝑠ℎ52 ′ = 𝛼 𝑠𝑑𝑦22 𝜃 𝑎𝑠ℎ41 ′ + (1 -𝛼 𝑠𝑑𝑦22 )𝜃 𝑑2 ′ (7.35)
where 𝛼 𝑠𝑑𝑦22 = exp (-

ℎ 𝑖𝑛𝑡 𝑆 𝑑2 (1-𝛿 2 )𝑚̇𝑎 𝑟𝑑4 𝐶 𝑝𝑎 ) 𝐆 𝑑2 = 𝐆 𝑎𝑠ℎ41 • ( 1 (1+Ψ 𝑠𝑑𝑦2 )+𝑖𝜔𝜉 𝑠𝑑𝑦42 ) (7.36) 𝐆 𝑎𝑠ℎ52 = 𝐆 𝑎𝑠ℎ41 • ( (1-𝛼 𝑠𝑑𝑦22 ) (1+Ψ 𝑠𝑑𝑦2 )+𝑖𝜔𝜉 𝑑𝑦42 + 𝛼 𝑠𝑑𝑦22 ) (7.37)
where

Ψ 𝑠𝑑𝑦2 = 𝑈 𝑜𝑣𝑑 ′ 𝑆 𝑑2 (1-𝛿 2 )𝑚̇𝑎 𝑟𝑑4 𝐶 𝑝𝑎 (1-𝛼 𝑠𝑑𝑦22 )
and

𝜉 𝑑𝑦42 = 𝑚 𝑑2 𝐶 𝑝𝑑 (1-𝛿 2 )𝑚̇𝑎 𝑟𝑑4 𝐶 𝑝𝑎 (1-𝛼 𝑠𝑑𝑦22 )
Analogous analysis (Eq. (7.34) -(7.37)) was applied to Shelf 3 to 5.

Air mixing between the air flowing from the back and the air curtain

This incident emerges at all shelves except Shelf 1. Heat balance equation is therefore developed for these shelves. Considering MZ [2, 2, 2], the heat balance equation is given by

𝑚̇𝑎 𝑟𝑑3 𝐆 𝑎𝑠ℎ62 = 𝑚̇𝑎 𝑠ℎ2 ′ 𝐆 𝑎𝑠ℎ42 + (1 -𝛿 2 )𝑚̇𝑎 𝑟𝑑4 𝐆 𝑎𝑠ℎ52 (7.38)

Temperature variation of the air after the heat exchanger

In the study, the temperature of supply air after the heat exchanger is considered to be equal to the air temperature at the thermostat sensor (𝑇 𝑡ℎ ) which is located there.

Figure 7.2 shows the air temperature profiles obtained from the measurements with calibrated thermocouples during a period of the compressor operation (𝜏) for three ambient temperatures (14.6, 19.5 and 24.4 °C). It was observed that the duration of the compressor being "off" was affected by the ambient temperature: it decreased with an increase of the ambient temperature. This incident shows the implication of higher demand of cooling load from the refrigeration system (more frequency of the compressor operation). On the other hand, the "on" period of the compressor Referring to the earlier assumption, the temperature variation at any position was supposed to be in response to a sinusoidal temperature variation of the air after the heat exchanger ( 𝑇 𝑡ℎ ). However, in reality this temperature variation is not sinusoidal, but only periodic (Figure 7 In addition to the identification of influential parameters on the damping of the temperature fluctuations, the spectral method also permits quick calculation. the back and front loads, respectively. Based on the characteristic time of an "on/off" cycle of compressor regulation (340𝑠 < 𝜏 < 550𝑠, Fig. 2), the Fourier number for the load given by 𝐹𝑜 = 𝜅 𝑙 𝜏 𝑒 𝑙 2 varied between 1.2×10 -3 and 1.9×10 -3 (𝜅 𝑙 = 𝜆 𝑙 𝜌 𝑙 𝐶 𝑝𝑙 is the thermal diffusivity of the load). This very small 𝐹𝑜 indicates that heat is far to penetrate up to the core of the load during the "on/off" compressor working cycle. Since 𝐹𝑜 < 0.2 [START_REF] Bergman | Fundamentals of heat and mass transfer[END_REF], the load of thickness 𝑒 𝑙 can be considered as a semi-infinite Taking into account Eq. (7.44) and (7.45), the solution of the temperature fluctuation at the surface of the load can be calculated from [START_REF] Carslaw | Conduction of heat in solids[END_REF]: ) was approximately 0.1. However, both air and load temperatures at the front of the shelves were fluctuating around higher average temperature, compared to the ones at the back of the same shelves. This result implies that food quality and safety would be more influenced by the average temperature than the temperature fluctuation in the closed display cabinet in this case where massive products were considered (methylcellulose packages). In the case of product packages offering a high exchange surface and a low thermal inertia (e.g. vacuum-packed sliced ham) particularly when the product thickness is smaller than 𝑒 𝑒𝑞 (Eq. (7.48)), higher product temperature fluctuations are expected. More investigations should be carried out to study the influence of such temperature fluctuation on food quality and safety.

∆𝑇 𝑙 ∆𝑇 𝑎 = 1 [(1+

Temperature evolution during the defrost operation

Frost formation on the surface of evaporator coils is inherent in all refrigeration

systems. An increase of frost thickness causes a reduction of heat transfer and air flow rates, which ultimately deteriorates the overall display cabinet performance. Therefore, periodic defrost operation is always required to maintain the display cabinet performance. It is well known that the food quality and safety can be affected by temperature raise during the defrost operation which can sometimes exceed the desired limit [START_REF] Lawrence | Refrigerant flow instability as a means to predict the need for defrosting the evaporator in a retail display freezer cabinet[END_REF]. The validated simplified model was further used to simulate the air and load temperature evolutions during the defrosting process in the closed display cabinet. For this simulation, the Fourier transform was applied to the air temperature after the heat exchanger (𝑇 𝑡ℎ ) for a period of 6.5 h: 3 h before and after the defrost operation which lasts for 30 mins (Figure 7.8). Note that this input air temperature profile was selected based on the setting time for the defrost operation which takes place every 6 h (i.e. 4 cycles per day) for the studied display cabinet. The model well represented the area of large air temperature increase at the back of the cabinet (Figure 7.9a-7.9c), whereas it was likely to overestimate in the area of low air temperature increase at the front (Figure 7.9e-7.9f). These findings showed that the model prediction was more accurate in the zone with less interaction between the cold air from the back and the heat entering the cabinet at the front.

The overestimation was also observed for the load surface temperature increase.

This discrepancy can be explained by the approximation of the equivalent load thickness with Eq. ( 48) which assumed 𝐸√𝜔′ ≫ ℎ 𝑎 . The precision would be improved if the more complex equation (Eq. ( 47)) was applied. Nevertheless, the over-estimated temperature increase by the model can be considered as the worst scenario that may happen in real use condition. Note that the air and load temperature increase was calculated based on the maximum and minimum temperatures from the beginning to the end of the defrost operation as shown in This solving approach also permits the identification of influential parameters on the damping of the temperature fluctuations with quick calculation. Since 2-4 harmonics (k) are usually enough to describe the supply air temperature profile, this approach only needs to solve a limited number of algebraic complex equations, instead of ordinary differential equations. To take into account the temperature variation within the load, the load thickness in which the temperature fluctuates with the same order as at the surface was estimated in function of on/off compressor working frequency. Given the input parameters used in the static model (without any adjustment), a relatively good agreement between the experimental and numerical results for the air and load temperature fluctuations was observed: maximum difference of 0.5 °C and 0.1 °C for the air and load, respectively. The results show the air temperature fluctuation progressively decreases from the rear duct entrance to the DAG. Accordingly, the air temperature fluctuations at the lower shelf were greater than that of the upper one. Despite the exposure to the high air temperature fluctuation, the load temperature was not much fluctuating in the closed refrigerated display cabinet (∆𝑇 𝑙 ≤ 0.5 °C). This would be the same for all products so long as the product thickness is smaller than the equivalent thickness.

The model was further used to investigate the evolution of the air and load temperatures during defrost operation. It was found that the increase in temperature depends on the position in the display cabinet. The predicted temperature increase during the defrosting period was in a relatively good agreement with the experimental data, but with an overestimation at some locations. The ratio of the product to its surrounding air temperature variation (ΔTp/ΔTa) was approximately 0.1. A reduction in the amplitude of the compressor regulation for the studied cabinet may be unnecessary, thereby prolonging the lifespan of the compressor.

※

Defrosting process causes drastic air and product temperature increases. The ratio of product to supply air temperature increase was between 0.05 and 0.10, depending on the product position in the closed display cabinet.

※

Both air and product temperatures vary spatially in the closed display cabinet.

Based on the temperature profile at the center of the cabinet (symmetry plane), the front areas contribute to higher temperature, whereas the rear areas are at lower temperature. The front of the top shelf and the back of the bottom shelf are respectively the highest and lowest temperature positions in the cabinet.

The temperature distribution at the sides of the cabinet has the same trend as that at the center, but the temperature at the sides is slightly higher because of additional heat losses through the lateral glass walls. The product at the front always has a higher temperature than that at the back.

※

The air temperature evolution during the opening, holding and closing periods depends on its position in the cabinet. The nearer to the DAG, the quicker the air temperature stabilization after the door is opened. However, the air temperature is unstable at a certain distance from the DAG (i.e. from the middle shelf in this study). Due to this instability, the air temperature increases progressively during door openings and decreases after the door is closed.

※

An additional heat load due to higher door-opening frequency and/or higher ambient temperature resulted in a substantial increase in the product temperature at the front. The product temperature on the front middle shelf was the most affected when the opening frequency exceeded 40 openings per hour per door (OPH). This additional heat load also caused a slight decrease in the product temperature at the back because of more cooling capacity was needed (the average air temperature after the evaporator was lower). It is to be noticed that the highest temperature position was at the front-top shelf under closeddoor condition.

※

Under periodic door-opening (10 OPH) condition, lower product temperature at all positions when the storage compartment was more occupied could be explained by higher convective heat transfer coefficient (higher velocity in the air space over the products). was identified at the front of the top shelf level. This phenomenon was more pronounced when the refrigeration system was turned "on". The model was then used to explore the influence of the airflow pattern on the temperature distribution.

The trend of predicted product temperature profile was in a relatively good agreement with the experimental data.

Simplified heat transfer model based on the zonal approach

A simplified heat transfer model was developed based on a zonal approach to describe the evolution of product and air temperatures at different zones in the closed refrigerated display cabinet. The simplified model was developed in static and dynamic aspects, which permits, respectively, the predictions of both timeaveraged air and product temperatures and the temperature fluctuations according to the on/off working cycles of the compressor regulation.

According to the zonal approach, the display cabinet was divided into various zones, the heat transfers in each zone are described by algebraic and ordinary differential equations, corresponding to energy balances.

The input parameters were obtained by several manners: measurement (presented in the experimental study), calculation from correlations reported in literature and model fitting with the experimental time-averaged temperatures. Five input parameters were estimated by model fitting (for the ambient temperature of 19.5 °C) because of the difficulty to measure them: air circulation parameter at the top shelf, air distribution coefficients at the other shelves, air infiltration rate through the door gaps and convective heat transfer coefficients between the load surface and the surrounding air. The sensitivity study showed that the air infiltration rate through the door gaps is the most influencing parameter on the load temperatures.

A 20% increase/decrease of this fitted parameter caused the warmest load temperature at the front-top shelf to be increased/decreased up to 0.2 °C while it is less than 0.1 °C for other parameters.

I. Static model

Without any adjustment to the fitting input parameters, the predicted and measured time-averaged temperatures were in a good agreement for the other external ambient temperatures (14.6 °C and 24.4 °C). The overall mean absolute error based on 60 paired data (air/load temperatures) for three ambient temperatures was less than 0.2 °C. The maximum difference of the calculated and measured load temperatures never exceeds 0.5 °C for every studied position. Moreover, the differences between the highest and the coldest load temperatures obtained from the experimental and the numerical data differed less than 14% for all studied ambient temperatures.

In the storage compartment, the model represented well the load temperatures, whereas it over-estimated the air temperature in the air curtain. Possible explanation of the discrepancy between the model and the experiment is that the air curtain temperature was measured only at one position of a given height, which may not be representative. However, the load temperature ranges and trends in the closed display cabinet are well represented by the proposed model with a short calculation time (less than 1 s).

The model showed its ability to investigate the effect of the air infiltration through the door gaps on the performance of the closed display cabinet in both thermal and energy aspects. Based on the numerical simulation, a reduction of the air infiltration through the door gaps by half could lead to decreases in the load temperatures of up to 0.7 °C and in the total cooling load of the refrigeration system by 35%.

II. Dynamic model

A spectral method was used to solve the dynamic model in the study. The supply air temperature profile (air after the evaporator where the thermostat sensor situated) from the experimental data during a transient cycle of compressor regulation was formulated by using Discrete Fourier Transform (DFT). The dynamic response of air/load temperature was readily determined by the transfer functions specified at all positions of heat exchange zones. By solving with the spectral method, the model allows not only the prediction of air/load temperature fluctuations in the display cabinet but also the identification of different influencing parameters on the damping of the temperature fluctuations. Moreover, this solving method also permits a quick calculation since only 2-4 harmonics for the DFT are enough to describe the supply air temperature profile 𝑇 𝑡ℎ . This method needs only to solve a limited number of algebraic complex equations, instead of ordinary differential equations.

The estimation of the equivalent load thickness in which the temperature fluctuates with the same order as at the surface was derived in function of on/off compressor working frequency. This proposed method of estimation allows to take the temperature variation within the load into account in a simple manner compared to one-dimensional discretization of the load.

The numerical results of the air and load temperature fluctuations under periodic working conditions were in a good agreement with the experimental data: maximum difference of 0.5 °C for air and 0.1 °C for load. The high discrepancy at some positions in the storage compartment can be explained by the oversimplified assumptions concerning the flow distributions.

Based on the simulation during the compressor "on" and "off" regulation, the amplitude of the supply air temperature fluctuations progressively decreased when the air flowed in the rear duct and in the storage compartment. Despite the large surrounding air temperature fluctuations (∆Ta ~ 2-5 °C), the load temperature was not much fluctuating (∆Tl < 0.5 °C). This would be the same for all product types so long as the product thickness is larger than the equivalent thickness.

The air/load temperature fluctuations within the cabinet are very small compared to the supply air temperature fluctuations after the evaporator during the "on/off" cycles of compressor. Damping of temperature fluctuation is a result of heat loss with a constant ambient temperature (pulsation independent effect), and thermal inertia of the wall and product (pulsation dependent effect, increasing when the compressor on/off period decreases).

The model also showed its ability to investigate the temperature increase of the air and the loads during the defrost operation. Comparison between the predicted and measured temperature increases during the defrosting period showed good agreement, with a small overestimation at some locations. For the air, the model prediction was more accurate in the rear zone because of less interaction between the cold air from the back and the heat entering the cabinet at the front. For the load, the precision would be improved if a more complex equation was applied to estimate the equivalent load thickness. However, there are some other aspects not taken into considerations because of technical and time constraints. The suggestion for future work is listed below:

Effect of air humidity in a test room on product temperature

During the experimental study, the air temperature and air velocity can be controlled in the test room, but it is technically not possible to control the air humidity. The measurement of the air characteristics (dry-bulb air temperature and %relative humidity) was undertaken with a thermo-hygrometer (Testo 174H, accuracy ±3%RH). Then, the air humidity ratio (kg of water vapor/kg of dry air)

was calculated using a psychrometric chart. The experimental results reported by [START_REF] Chen | Experimental study of the performance of single-band air curtains for a multi-deck refrigerated display cabinet[END_REF] showed the moderate effect of the ambient air humidity variations on the temperature distributions inside the display cabinet (open type in their study), but the significant effect on the refrigeration load. The ambient air humidity effect would be less in the closed display cabinets; however, no justification is available in literature.

measurements of air velocity at the door during openings need to be carried out for model input.

Simplified heat transfer model based on the zonal approach

The air infiltration due to door openings was not considered in the simplified model because the airflow during door openings is relatively complex. To keep the model simplicity, the air infiltration during door openings could be included by introducing a higher value of the air infiltration rate through the door gaps.

Otherwise, some model modifications are required to be more realistic i.e.

subtraction of air recirculation zone and introduction of external air infiltration rate at a given mixing zone. Additional experiments should be carried out to quantify the air infiltration rate during the door openings for model input.

The transient effect of intermittent heat load cannot be simulated by the spectral method because periodicity is assumed for the dynamic model. To include the heat load during the door openings, a temporal discretization could be used to solve the model (e.g. Runge-Kutta).

Finally, the experimental and numerical methodologies developed in this thesis can be applied to other types of closed display cabinet i.e. horizontal air curtain, more or less shelves and doors with some additional experiments and equations.

Figure 1 . 2 Figure 3 . 1 Figure 3 . 2 Figure 3 . 3 Figure 3 . 4 Figure 3 . 5 Figure 3 . 6 Figure 3 . 7 Figure 3 . 8 Figure 3 . 9 Figure 3 . 10

 12313233343536373839310 Figure 1.1 Comparison of energy consumption between open and closed refrigerated display cabinets (source: Laguerre and Chaomuang, 2019). ............... 5 Figure 1.2 Structure of the thesis ............................................................................ 8 Figure 3.1 Location of the closed display cabinet in the climate-controlled room: (a) perspective view (b) side view ......................................................................... 46 Figure 3.2 Schematic view of the experimental setup for temperature measurement in the closed display cabinet: (a) unloaded and (b) loaded conditions .................. 49 Figure 3.3 Schematic view of the experimental setup for velocity measurement: (a) measurement positions on middle (z = 0 m) and right (z = +0.26 m) planes (b) position of a hot-wire anemometer and its support on a shelf ............................... 51 Figure 3.4 Velocity measurement in the vertical rear duct with three percentages of test package-occupied volumes in the display cabinet: (a) 0% (b) 36% and (c) 52% of total storage volume .................................................................................. 52 Figure 3.5 Perforation patterns on the back panel at different levels ................... 53 Figure 3.6 Evolution of air temperatures (on the middle plane, z = 0 m) at various positions in the closed display cabinet for a room temperature of 19 °C: (a) discharge air grille, (b) front of top shelf, and (c) back of the bottom shelf .......... 55 Figure 3.7 Evaluation of air temperature during quasi-steady state in various positions (on the middle plane) in the closed display cabinet for a room temperature of 19 °C .................................................................................................................. 59 Figure 3.8 Mean air temperature and standard deviation (°C) during quasi-steady state (average of the measurements on the three planes: z = -0.525, 0 and +0.525 m) of the display cabinet (a) with doors and (b) without doors for a room temperature of 19 °C .............................................................................................. 60 Figure 3.9 Air velocity profiles in the closed display cabinet measured by a hot wire anemometer on the middle (z = 0 m) and right (z = +0.26 m) planes: (a) vertical flow direction and (b) horizontal flow direction .................................................... 64 Figure 3.10 Air velocity profiles in the closed display cabinet measured by a hot wire anemometer on the middle (z = 0 m) and right (z = +0.26 m) planes: (a) vertical flow direction and (b) horizontal flow direction .................................................... 66 Figure 3.11 Air velocity profiles on three different planes (a) left, (b) middle and (c) right at three heights along the vertical rear duct of the unloaded display cabinet,

Figure 4 . 1 Figure 4 . 2 Figure 4 . 3 Figure 4 . 4 Figure 4 . 5 Figure 4 . 6 Figure 4 . 7 Figure 4 . 8 Figure 4 . 9 Figure 5 . 1 Figure 5 . 2 Figure 5 . 3 Figure 5 . 4 Figure 5 . 5 Figure 5 . 6 Figure 5 . 7 Figure 5 . 8 Figure 6 . 1 Figure 6 . 3 Figure 6 . 4 Figure 6 . 5 Figure 6 . 6 Figure 6 . 7 Figure 6 . 8 Figure 6 . 9 Figure 6 . 11 Figure 7 . 1 Figure 7 . 2 Figure 7 . 3 Figure 7 . 5 Figure 7 . 6 Figure 7 . 7 Figure 7 . 8 Figure 7 . 9 Figure 7 . 11

 414243444546474849515253545556575861636465666768696117172737576777879711 Figure 4.1 (a) Location of the closed refrigerated display cabinet in the climatecontrolled room (b) diagram showing the experimental setup for air and product temperature measurements inside the display cabinet ........................................... 79 Figure 4.2 Algorithm of automatic door openings ................................................ 80 Figure 4.3 Evolution of (a) air, and (b) product (core and surface) temperatures on the center plane of a closed refrigerated display cabinet over 36-hour operation for an ambient temperature of 19.4 °C. ....................................................................... 82 Figure 4.4 Air temperature evolution at 2 cm from the front product surface on the center plane of the closed display cabinet during 6 door openings (10 OPH frequency) with (a) 15 s and (b) 30 s opening durations (acquisition interval of 1 s) ............................................................................................................................... 83 Figure 4.5 Time-averaged (± standard deviation in °C) air and product temperatures in the closed refrigerated display cabinet for an ambient temperature of 19.4 °C during (a) closed doorlight off, (b) open door with a frequency of 10 OPHlight on, and (c) closed doorlight on conditions. Values are the average over 5 hours of quasi-steady state .......................................................................... 86 Figure 4.6 Effect of door-opening frequency on the dimensionless product core temperature (a) at the front, and (b) at the back of the five shelves ...................... 88 Figure 4.7 Variation of dimensionless product core temperature (𝜃𝑝) with the difference between the average of ambient air temperature 𝑇𝑒 and of air temperature after the heat exchanger 𝑇𝑡ℎ: (a) front products (b) back products (opening frequency of 10 OPH and occupied volume of 60%) ............................. 90 Figure 4.8 Effect of occupied volume on the product temperature in the closed display cabinet (opening frequency of 10 OPH and ambient temperature of 19.4 °C) ............................................................................................................................... 92 Figure 4.9 Variation in air velocity above the products on different shelves in the closed display cabinet (1.25 m in length) for two occupied volumes: (a) 30%, (b) 60%. ....................................................................................................................... 92

1

  Field investigations on chilled food products at retail displays in various countries ................................................................................................................. 19 Table 2.2 Numerical studies on heat transfer and airflow in open display cabinets ............................................................................................................................... 29Table 2.3 Comparison of the performance of open and closed display cabinets .. 37 Table 2.4 Numerical studies conducted on closed display cabinets ..................... 40 Table 3.1 Percentage of perforated area (PPA) on different shelves .................... 52 Table 3.2 Mean temperature and standard deviation (°C) during quasi-steady stateat various positions within the food storage zone on the left, middle and right planes for a room temperature of 19 °C ............................................................................ 57 Table 3.3 Effect of ambient temperatures on the internal air temperature of the closed display cabinet ............................................................................................ 62 Table 4.1 Operating conditions in a closed refrigerated display cabinet .............. 81 Table 4.2 Effect of lateral glass wall on the product core temperature in the closed display cabinet ....................................................................................................... 86 Table 4.3 Time-averaged product temperature † (in °C) at the back and front of the display cabinet with a door-opening frequency of 60 OPH and permanently open (without doors) condition for an ambient temperature of 19.4 °C and occupied volume of 60%. ...................................................................................................... 89 Table 4.4 Effect of external ambient temperature ( 𝑇𝑒 ) on the product core temperature (𝑇𝑝) in the closed display cabinet. ..................................................... 91 Table 5.1 Characteristics of the closed refrigerated display cabinet used in the study ............................................................................................................................. 101 Table 5.2 General form of the governing equations (a) ......................................... 112 Table 5.3 Inlet boundary conditions .................................................................... 113 Table 6.1 Thermal properties of materials .......................................................... 141 Table 6.2 Values of input parameters used in the heat transfer model of a closed refrigerated display cabinet .................................................................................. 142 Table 6.3 Values of the first top ten combinations of the input parameters with the mean absolute error (MAE) for the model fitting ................................................ 147 Table 6.4 Effect of ambient temperature on load temperature at various positions in a closed refrigerated display cabinet ................................................................ 151 xviii Table 6.5 Error of air and load temperatures calculated from the developed model for different ambient temperatures compared to the measured values ................ 151 Table 7.1 Weights and thermal mass of the load and the solid components of the display cabinet used in the model ........................................................................ 178

Figure 1 . 1

 11 Figure 1.1 Comparison of energy consumption between open and closed refrigerated display cabinets (source: Laguerre and Chaomuang, 2019).

  Figure 1.2 includes 8 chapters, which are described briefly below: Chapter I General introduction describes the background, research interests, aim/scope and structure of the thesis as well as list of publications. Chapter II Literature review provides the state of the art of the investigations involving experimental (carried out in fields and in laboratories) and numerical research studies on open and closed retail refrigerated display cabinets. The current usage situation of display cabinets and the problem of temperature control at retail display based on field investigations were presented. Studies aiming to improve the performance of the display cabinets were explored and the influencing factors were then analyzed in terms of temperature distribution and energy consumption. The growing trend of closed refrigerated display cabinet use in Europe was also highlighted. More detail is presented in Article 1 (published).

Figure 1 . 2

 12 Figure 1.2 Structure of the thesis

  carried out in the open configuration. The air curtain characteristics (mean velocity and velocity fluctuations) of the closed and open configurations were then compared and the implications on the internal temperature distribution was addressed. A cabinet with doors yields better thermal performance because the air curtain is more stable compared to the case without. Without doors, large unsteady eddies developed in the mixing layers, thereby promoting greater external air infiltration. Finally, a 2D-CFD model was developed to investigate the airflow and its influence on the temperature distribution in the closed display cabinet. The model showed the ability to reproduce the main flow phenomena observed in the experiment and allowed the prediction of product temperatures with acceptable accuracy. The experimental data of Chapter III (Heat transfer analysis and airflow characterization using a hot-wire anemometer) were used for the model input and validation. More detail is presented in Article 4 (under review, first submission). Chapter VI Simplified heat transfer modeling in steady state. Because of the calculation time-consuming using the CFD simulation, this chapter presents the development of a simplified heat transfer model in steady state based on the zonal approach. The model allows the prediction of time-averaged air and load temperatures at various positions in the closed display cabinet. Details of overall model conceptualization, static model development, determination of model input parameters as well as model validation by comparing with experimental data were described. In addition, the model was further used to estimate the frost formation on the evaporator surface due to the air infiltration through the door gaps.

( 3 .

 3 0 to 5.5 °C lower with a heat pipe alone), improved temperature distribution homogeneity (a small range of max.-min. temperatures during normal operation) and a reduction in the temperature rise occurring during the defrost process (0.3 °C rise with the combined structure) with no change in energy consumption. These findings are in corresponding with Wu et al. (2017) who reported that the temperature fluctuation of food packages is reduced by 53.3 -83.3% during defrosting of a vertical open display cabinet equipped with such a composite shelf

  rising continuously. Many researchers have made an effort to investigate and identify the key factors which influence the cabinet performance by means of both experimental (field-based and laboratory-based) and numerical (in-house code and commercial code) approaches. The knowledge acquired will provide an opportunity to optimize this equipment in terms of both temperature homogeneity and energy consumption. The thermal entrainment factor of air curtains seems to be the main performance indicator in the case of open display cabinets as it directly represents the amount of entrained ambient air. This issue still poses problems in many research and development contexts. The application of closed doors is becoming an alternative solution and several studies have demonstrated that closed doors can provide several advantages. Since there is no conclusion on the loss of sales of products due to the use of doors, many researchers are conducting investigations on the influence of doors. Nevertheless, most of these studies focused on the energy consumption perspective alone. The impact of temperature variations on food quality, particularly the effect of ambient operating conditions and door openings, requires further elucidation.

  cabinet is typically employed to display food products in retail stores. It is the last storage stage in the cold chain prior to purchasing by consumers. Product temperature control plays a vital role in maintaining food quality. Various studies have demonstrated that food products at this stage are Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 44

AFigure 3 . 1

 31 Figure 3.1 Location of the closed display cabinet in the climate-controlled room: (a) perspective view (b) side view

Figure 3 .

 3 Figure 3.2a depicts the experimental setup for air temperature measurement at various positions in the display cabinet using T-type thermocouples. Each thermocouple was calibrated in the temperature range of -5 °C to 30 °C with a high precision thermostatic oil bath (Fluke, model 7340) and the precision of ±0.2 °C was determined. The measurement was divided into three planes: at z = 0 m (middle), z = -0.525 m (left) and z = +0.525 m (right). On each plane, the air temperature was measured at three height levels (top, middle and bottom shelves) and three positions on each shelf (5, 30, and 44 cm from the PBP). The air temperatures at the DAG, RAG and the middle of the vertical rear duct at the same levels (3 cm behind the PBP) were also measured. The display cabinet was switched

(

  dimensions in width × height × length: 100 × 50 × 200 mm; thermophysical properties at 0 °C: density = 1100 kg•m -3 , thermal conductivity = 0.49 W•m -1 •K -1 , specific heat = 3372 J kg -1 •K -1 ). Thermocouples were positioned at the geometric center and the surface of some test packages as well as at 2 cm from the package surface as shown in Figure 3.2b.

Figure 3 . 2

 32 Figure 3.2 Schematic view of the experimental setup for temperature measurement in the closed display cabinet: (a) unloaded and (b) loaded conditions

  display cabinet was performed (Figure 3.4a). Six height levels, H1 (y = 0.015 m), H2 (y = 0.215 m), H3 (y = 0.465 m), H4 (y = 0.715 m), H5 (y = 0.965 m) and H6

Figure 3 .

 3 4b and c show the loading arrangement of test packages for each occupied volume, with 1-cm spaces between each stack of packages and between the back stacks and the PBP.

Figure 3 . 3 Figure 3 . 4

 3334 Figure 3.3 Schematic view of the experimental setup for velocity measurement: (a) measurement positions on middle (z = 0 m) and right (z = +0.26 m) planes (b) position of a hot-wire anemometer and its support on a shelf

Figure 3 . 5

 35 Figure 3.5 Perforation patterns on the back panel at different levels

Figure 3 . 6

 36 Figure 3.6 presents the air temperature evolution under closed-door conditions, at three different positions (DAG, front of the top shelf, and at the back of the bottom shelf) with the refrigeration system turned on, under ambient temperature conditions of 19 °C. The cooling duration of the system varied according to the positions in the display cabinet. The air temperature at the DAG took about 30 min to cool down from the onset of the quasi-steady state during which the min./max. temperature variation was between -4.0 °C and 1.3 °C (Figure 3.6a). Two temperature fluctuation cycles were observed: small and large fluctuations. The former (during the quasi-steady state) was due to the regulation of the on/off compressor working cycle which is automatically conducted by the thermostat sensor. The latter fluctuation was due to a time/temperature-based defrosting cycle, which occurred every 6 h with duration of about 25 min. An electric defrost system is used in this display cabinet. The compressor is switched off during defrosting and warm air heated by the electric strip heater is blown over the evaporator by the fans. This explains a rapid increase of temperature with a peak of 6.0 °C. Defrost termination is when either temperature threshold (6.0 °C) or maximum duration (30 min.) is reached. As depicted in Figure 3.6b and c, the air temperature takes about 58 min to cool down at the front of the top shelf (min./max. temperature variation of -0.3/+2.3 °C), while it requires a shorter cooling duration about 26 min at the

Figure 3 . 6

 36 Figure 3.6 Evolution of air temperatures (on the middle plane, z = 0 m) at various positions in the closed display cabinet for a room temperature of 19 °C: (a) discharge air grille, (b) front of top shelf, and (c) back of the bottom shelf

Figure 3 .

 3 Figure 3.8a presents the mean air temperature and standard deviation calculated based on the measurement on the three planes (z = -0.525, 0 and +0.525 m) for the storage zone of the closed display cabinet. It shows that, from the rear duct upstreamto the DAG, the air temperature after flowing across the cooling coils rose from -1.0 ± 2.2 °C to -0.3 ± 1.5 °C. This is very similar to the observation in the middle plane (Figure3.7). In the storage zone (Table3.2), the mean temperatures were

Figure 3 .

 3 8a and b show the mean air temperature calculated from the three planes (z = -0.525, 0 and +0.525 m)

  Figure 3.7 Evaluation of air temperature during quasi-steady state in various positions (on the middle plane) in the closed display cabinet for a room temperature of 19 °C

Figure 3 . 9

 39 Figure 3.9 Air velocity profiles in the closed display cabinet measured by a hot wire anemometer on the middle (z = 0 m) and right (z = +0.26 m) planes: (a) vertical flow direction and (b) horizontal flow direction

Figure 3 .

 3 Figure 3.10 presents time-averaged air velocity profiles in the closed display cabinet measured by a hot-wire anemometer. As shown in Figure 3.10a, the air velocity in the vertical flow direction (y-axis) varied from 0.01 m•s -1 to 0.59 m•s -1 in the middle plane (z = 0 m) and it was not significantly different from that of the right plane (z = +0.26 m). The highest velocity (≈ 0.59 m•s -1 ) was found near the discharge grille of the air curtain (x = 0.33 m, y = 0.065 m). At the y = 0.315 m position, the thickness of the air curtain increased because of the entrainment of neighboring air by the air jet. At the lower positions (y ≥ 0.565 m), maximal velocity was located close to the door due to the adhesion of airflow to the door surface (Coanda effect). This maximum velocity remained almost constant near the door (about 0.40-0.50 m•s -1 ). As can be seen from its corresponding temperature presented in Section 3.3.2, the air is almost stagnant in the area beyond the air curtain under the DAG (x > 0.40 m, y = 0.065 m), resulting in a high temperature at this position (Figure 3.8a).

Figure 3 .

 3 Figure 3.10b shows the profile of the horizontal component of air velocity between shelves. The low air velocity (0.0-0.10 m•s -1 ) on the top shelf compared with the others (0.10-0.20 m•s -1 ) can be explained by the lowest percentage of perforated area at this position (2.3%). As depicted in Figure 3.10b, the velocity changed from

Figure 3 .

 3 10b should be interpreted cautiously because of the limitations of the hot wire anemometer in the measurement of low air velocity and complex flow. More advanced measurement techniques, for example Laser Doppler Velocimetry (LDV) and Particle Image Velocimetry (PIV), are preferable when dealing with such complex flow, and generate more accurate data. These techniques allow the investigation of the air flow not only in x and y directions but also in z direction (a sign of complex flow).

Figure 3 .

 3 Figure 3.10 Air velocity profiles in the closed display cabinet measured by a hot wire anemometer on the middle (z = 0 m) and right (z = +0.26 m) planes: (a) vertical flow direction and (b) horizontal flow direction

Figure 3 .

 3 Figure 3.11a, b and c. The average value for the nine measurement positions at each height are also presented in these figures. The average air velocity decreased from 1.16 m•s -1 (with a standard deviation of 0.40 m•s -1 ) upstream of the duct (H1)

  about 0.6 m•s -1 (Reynolds number, Re = 𝜌𝑣 ̅ 𝑎 𝐷 ℎ 𝜇 ≈ 5100) observed at H4 and H6 leads to the conclusion that turbulent flow and almost uniform velocity profile can be expected inside the duct. Section 3.3.2 has shown that the temperature distribution in the display cabinet is asymmetric. Based on the velocity profile observed in the rear duct, this finding is confirmed. The air velocity on the left plane (z = -0.26 m) was markedly lower than those of the other two planes as depicted in Figure 3.11b and c. This can be explained by the effect of airflow maldistribution in the cooling duct located at the bottom of the display cabinet (from the fan discharge to downstream of the evaporator) because of the interaction between two air streams generated by fans which rotate in the same direction, leading to deviation of airflow on the same side of the channel and a slight asymmetric placement of the evaporator. In the present case, the flow streams mostly deviated to the right side of the cooling duct, and this explains the lower velocity on the left plane. This dissymmetry in flow was also Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 68 observed by Marinetti et al. (2012) and Marinetti et al. (2014) who performed experimental and numerical analysis of the flow distribution in a cooling duct model of a refrigerated display cabinet. They found a higher velocity in the right part of the duct.

Figure 3 .

 3 Figure 3.11d presents the mean air velocity calculated on the basis of nine measurement positions on the cross-section of the rear duct at each height for different percentages of product-occupied volume (0%, 36% and 52%). Based on

Figure 3 . 11

 311 Figure 3.11 Air velocity profiles on three different planes (a) left, (b) middle and (c) right at three heights along the vertical rear duct of the unloaded display cabinet, and (d) influence of percentages of productoccupied volume on the air velocity (average value of nine measurements at a given height)

Figure 3 .

 3 Figure 3.12 Percentage of airflow distribution through the perforated back plate at the level of each shelf for different percentages of product-occupied volume (0%, 36%, and 52%)

  closed refrigerated display cabinets is seldom investigated, particularly in terms of spatial and temporal temperature variations. A comparative investigation conducted by[START_REF] Evans | Performance of retail and commercial refrigeration systems[END_REF] revealed that the temperature variations within closed cabinets were lower than those within open cabinets.Lindberg et al. (2010) demonstrated that an overall air temperature reduction of at least 2 °C was achieved when the display cabinet was retrofitted with doors. Atilio de[START_REF] Atilio De Frias | Improving spinach quality and reducing energy costs by retrofitting retail open refrigerated cases with doors[END_REF] observed a substantial decrease in the spatial temperature differences of almost 6 °C and underscored the better visual quality and lower decay rate of minimallyprocessed vegetables achieved by the installation of doors on display cabinets.An experimental analysis of heat transfer and airflow in a closed refrigerated display cabinet under closed conditions was performed in our previous study[START_REF] Chaomuang | Experimental analysis of heat transfer and airflow in a closed refrigerated display cabinet[END_REF]. The knowledge acquired provides an insight into the thermal exchange mechanisms taking place in the equipment. It was found that convection and conduction were the predominant heat transfer modes, while radiation was rather weak due to the presence of low-emissivity coating materials on the glass doors. In addition, the main causes of air and product temperature variations in closed display cabinets are air infiltration through the door gaps and the proximity to the cabinet lighting. The authors highlighted that closed display cabinets can achieve better temperature performance in comparison with open cabinets.As much greater air infiltration occurs during door openings than through the door gaps when the cabinets are closed, the opening frequency plays an important role in the performance of closed display cabinets.Lindberg et al. (2010) found that when the opening frequency increased from 10 to 30 openings per hour (OPH), the heat extraction rate of the heat exchanger increased by 16%, while the difference between the maximum and minimum product temperatures (∆𝑇 = 𝑇 max -𝑇 min ) increased from 2.7 °C to 3.0 °C.[START_REF] Orlandi | CFD assisted design of closed display cabinets[END_REF] numerically highlighted the influence of this factor on the energy consumption of the closed display cabinet.

Figure 4 .

 4 Figure 4.1b depicts the closed refrigerated display cabinet (Offlip 2 Eco 125S, Costan) used in the study, which had external dimensions (width × length × height) of 650 mm × 1310 mm × 1980 mm and internal dimensions of 525 mm × 1250 mm × 1345 mm. The cabinet is an integral-type cabinet which is designed to meet the requirements of cabinet class M1 and climate class 3, i.e. product temperature varies between -1 °C and +5 °C for an ambient temperature of 25 °C and a relative humidity of 60% (EN ISO 23953-2, 2015). Two hinged doors (620 mm × 1330 mm) and the lateral walls are made of a double-glazed window with 22-mm thickness in which a 14-mm air gap filled with 90% argon gas is sandwiched between two panes of 4-mm tempered glass. The glass panes are also coated with a low-emissivity material to avoid the transmission of infrared radiation. There are 8mm gaps between the door and the lateral wall of the cabinet to assist door openings.

Figure 4 .

 4 Figure 4.1 (a) Location of the closed refrigerated display cabinet in the climate-controlled room (b) diagram showing the experimental setup for air and product temperature measurements inside the display cabinet

Figure 4 . 2

 42 Figure 4.2 Algorithm of automatic door openings

Figure 4 .

 4 Figure 4.1). The same experimental procedure (Section 4.2.3.2) was applied to all experiments.

Figure 4 .

 4 Figure 4.3 shows the evolution of the air and product (core and surface) temperatures in the center plane (520 mm from the left wall) of the closed refrigerated display cabinet over the 36-h operation comprising 12 h of closed-door and lights off conditions, 12 h of door openings with a frequency of 10 OPH and lights on, and 12 h of closed-door and lights on conditions. During the closed-door periods, the air temperature fluctuation was almost identical and followed the on/off regulation of the compressor (with a duration of about 10 min.) and the defrosting operation (every 6 h with a duration of about 25 min.) as shown in Figure 4.3a. During the door openings, a larger amplitude of fluctuation in the air temperature was observed at the front of the cabinet (A1, A3 and A5).

Figure 4 . 3

 43 Figure 4.3 Evolution of (a) air, and (b) product (core and surface) temperatures on the center plane of a closed refrigerated display cabinet over 36-hour operation for an ambient temperature of 19.4 °C.

Figure 4 . 4

 44 Figure 4.4 Air temperature evolution at 2 cm from the front product surface on the center plane of the closed display cabinet during 6 door openings (10 OPH frequency) with (a) 15 s and (b) 30 s opening durations (acquisition interval of 1 s)

Figure 4 . 5

 45 Figure 4.5 Time-averaged (± standard deviation in °C) air and product temperatures in the closed refrigerated display cabinet for an ambient temperature of 19.4 °C during (a) closed doorlight off, (b) open door with a frequency of 10 OPHlight on, and (c) closed doorlight on conditions. Values are the average over 5 hours of quasi-steady state

4. 1 )Figure 4 . 6

 146 Figure 4.6 shows the dimensionless product core temperatures (𝜃 𝑝 ) at the front and the back of the closed display cabinet for door-opening frequencies of 0, 10, 20, 40 and 60 OPH. The value ∞ represents the permanently open condition in which the doors were completely removed. As shown in Figure 4.6a, it was observed that 𝜃 𝑝 at the front of all shelves increased with an increase in the door-opening frequency.When the door-opening frequency was less than 40 OPH, 𝜃 𝑝 was the highest at the level of the top shelf (F1) and it was the lowest at the bottom (F5). Beyond this frequency, the door opening resulted in a relatively smaller increase in 𝜃 𝑝 at the top shelf (F1), as compared to its effect on the product at the level of the lower shelves (F2 and F3) for which 𝜃 𝑝 became the highest. This can be explained by the fact that the products at these positions (F2 and F3) are the most influenced by the introduction of external warm air when the door is open as observed in the open

Figure 4 . 6

 46 Figure 4.6 Effect of door-opening frequency on the dimensionless product core temperature (a) at the front, and (b) at the back of the five shelves

Figure 4 . 7

 47 Figure 4.7 Variation of dimensionless product core temperature (𝜃 𝑝 ) with the difference between the average of ambient air temperature 𝑇 𝑒 and of air temperature after the heat exchanger 𝑇 𝑡ℎ : (a) front products (b) back products (opening frequency of 10 OPH and occupied volume of 60%)

Figure 4 . 8

 48 higher percentage of occupied volume leads to a decrease in product temperature at all positions. When the cabinet volume was occupied to a greater extent, the space above the product top surface became smaller, resulting in higher air velocity flowing from the back. Cold exchange between the air and the products was therefore enhanced. By knowing the air mass flow rate (𝑚̇𝑎 ,𝑖 ,[START_REF] Chaomuang | Experimental analysis of heat transfer and airflow in a closed refrigerated display cabinet[END_REF] and the area above the product for each shelf (𝐴 𝑖 ), the mean air velocity flowing from the back (𝑢 𝑎,𝑖 ) was estimated by

Figure 4 . 8 Figure 4 . 9

 4849 Figure 4.8 Effect of occupied volume on the product temperature in the closed display cabinet (opening frequency of 10 OPH and ambient temperature of 19.4 °C)

  of the total heat load of open refrigerated display cabinets used in supermarkets were attributed to the entrainment of ambient warm and humid air into the system[START_REF] Gaspar | Experimental analysis of the thermal entrainment factor of air curtains in vertical open display cabinets for different ambient air conditions[END_REF][START_REF] Tassou | Energy consumption and conservation in food retailing[END_REF]. To address this issue, numerous studies have been carried out using both experimental and numerical approaches.[START_REF] Chen | Experimental study of the performance of single-band air curtains for a multi-deck refrigerated display cabinet[END_REF] conducted experiments to evaluate the effects of several important parameters on the performance of an open display cabinet and proposed the thermal entrainment factor to characterize the air curtain efficiency.

Field

  et al. (2002) implemented the PIV technique together with a hot wire anemometer to examine the entrainment characteristics of air curtains with an 𝑅𝑒 range of 1500-8500 and an 𝑅𝑖 range of 0.13-0.58. The entrainment of ambient air was found to be dominated by eddies that engulfed the ambient air entering the air curtain and the development of these eddies was related to 𝑅𝑒 . The authors suggested that a decrease in 𝑅𝑒 could reduce the amount of entrained air as long as the integrity of the air curtain was preserved. The effectiveness of the air curtain can also deteriorate as a result of external disturbances[START_REF] Heidinger | Experimental study of the influence of consumers' movement parallel to the frontal opening of a multideck display case on the evaporator's thermal performance[END_REF].[START_REF] Kaffel | Experimental investigation of a plane wall jet subjected to an external lateral flow[END_REF] performed experiments using both PIV and Laser Doppler velocimetry (LDV) techniques to demonstrate the effect of external lateral airflow on the effectiveness of the air curtain in a reduced-scale model of a display cabinet.Because of the difficulties involved in experimental implementation, several numerical studies on open display cabinets have been carried out[START_REF] Cortella | CFD simulation of refrigerated display cabinets[END_REF] D'Agaro et al., 2006; Laguerre et al., 2012a;[START_REF] Moureh | Numerical and experimental study of airflow patterns and global exchanges through an air curtain subjected to external lateral flow[END_REF].[START_REF] Ge | Simulation of the performance of single jet air curtains for vertical refrigerated display cabinets[END_REF] numerically studied the influence of design parameters and operating conditions on the air curtain performance: the width and length of the air jet, initial jet velocity (and turbulence), the position/dimensions of the return air grille (RAG). Foster et al. (2005) used a 3D-CFD model to study the airflow in an open refrigerated display cabinet. These authors found that the inclination of the discharge air grille (DAG) has a significant effect on the air curtain performance to protect food in storage zone and the optimum angle depends on the width and the air velocity of the curtain. Navaz et al. (2005) combined both experimental (PIV and LDV) and numerical (CFD Fluent) approaches to determine the effect of design parameters on the air entrainment across the air curtain of an open display cabinet.

Figure 5 . 1

 51 Figure 5.1 Experimental setup for PIV measurement in a closed refrigerated display cabinet: (a) perspective view; (b) front view; and (c) side view

  Nd:YLF laser (527 nm wavelength, 10 mJ pulse energy, 0.2-20 kHz repetition rate), a 12-bit CMOS camera (HighSpeedStar 4G, 1024 × 1024 pixels in resolution) equipped with a lens (Sigma, 105 mm, f/1:2.8) and a programmable timing unit for the synchronization of the device. To achieve light scattering, fine droplets of distilled water generated by a fogging system (Areco, model OD V7, with a median droplet diameter of 7 µm) were injected at the top and the back of the display cabinet. The overall experimental setup is illustrated in Figure 5.1. PIV measurement was conducted on a center plane (z = 520 mm, Figure 5.1b) along the air curtain of the display cabinet. Image acquisition and post-processing were conducted using a software package, DaVis 10.0.5. Chapter V -Airflow characterization using PIV and CFD techniques 102

Figure 5 .

 5 Figure 5.2a, b and c show respectively the mean air velocity profiles, root-meansquare (rms) velocity fluctuations and velocity vector fields at the middle plane (z = 520 mm) of the display cabinet running only with fans (refrigeration system

  High-velocity fluctuations were observed in two regions: the top and bottom parts of the cabinet (Figure5.2b). In the top parts near the DAG, high-velocity fluctuations can be explained by the development of eddies on both sides of the jet, i.e. in the mixing layers where high velocity gradient leads to large turbulent kinetic energy production. The fluctuations were further amplified because of the edge of the top shelf and the presence of the door which triggered air recirculation once air approached it. In the bottom part, fluctuations resulted from mixing between the back flow and the air curtain. Here, the flow was also turbulent in the air space above the test packages at the level of the bottom shelf and air recirculation also emerged near the front of the packages. Apart from these regions, small velocity fluctuations were observed. The vector field depicted in Figure5.2c shows the downward flow with recirculation at the front of the top shelf.

Figure 5 . 2

 52 Figure 5.2 Mean air velocity profiles (a); velocity fluctuations (b); and velocity vector fields (c); along the air curtain of a closed display cabinet running with fans only (refrigeration system "off"). Mean air velocity profiles (d); velocity fluctuations (e); and velocity vector fields (f); along the air curtain of a closed display cabinet running with fans and the refrigeration system "on".

  Figure 5.3 shows the air velocity profiles, the velocity fluctuations and the corresponding vector fields of the closed and open display cabinets. Note that the velocity profile, the velocity fluctuations and the vector field presented in Figure 5.3a, c and d are the same as those presented in Figure 5.2a, b and c, respectively. They are repeated here for the purposes of comparison with the open display cabinet.

Figure 5 . 3

 53 Figure 5.3 Velocity profiles, velocity fluctuations and vector fields on the middle plane of the closed (a, c and e) and open (b, d and f) display cabinets. The asterisk () represents a zone without signals (absence of particles in the external ambient air).

Figure 5 . 4

 54 Figure 5.4 Example of large unsteady eddy development at three consecutive instants (780 mm < y < 892 mm): (a) 0.10 s; (b) 0.18 s; and (c) 0.28 s.

Figure 5 . 5

 55 Figure 5.5 The computational domain of the 2D-CFD model for a closed display cabinet and the locations of boundary conditions.

Figure 5 . 6

 56 Figure 5.6 Velocity profiles without (a) and with (b) energy equation and temperature profile (c) in front of Shelf 3 for different meshes.

  . The model also used an enhanced wall treatment function in conjunction with the turbulence model to take into account viscous effects near the walls. For 2D, steady-state and incompressible fluid flow, the general form of the governing equations can

  pressure-velocity. The convection terms were estimated based on the second order upwind scheme. The default values of relaxation factors in Fluent software were applied for all variables. A numerical convergence criterion of 1 × 10 -6 was used for the residuals of components of velocity, turbulence kinetic energy, turbulent dissipation and energy equations and 2 × 10 -4 for the continuity equation. The simulation was driven using a computer server (Dell, Precision 3620, 64-bit operating system) with Intel Xeon E3-1240 Quad Core running at 3.50 GHz (8 Mb internal cache) and 64 Gb RAM. The computational time was four hours for the mesh used (M2, Figure 5.6).

Figure 5 .

 5 7a and b, respectively. The air curtain formed at the DAG was deviated to the door as it descended to the RAG. A large air recirculation region was observed in the upper part of the cabinet and some small recirculation regions were also detected around product packages. Near the DAG, the edge effect of the package on the top shelf caused a slight increase in the jet velocity. The diffusion process at the shear layer between the air curtain and the adjacent air caused a velocity decrease when flowing downward. Beyond the midheight between Shelves 1 and 2, the air velocity was maintained because the air curtain momentum was strengthened by the air flowing from the PBP. Unlike the open display cabinet, the thickness of the air curtain in the closed display cabinet was limited by the presence of the doors. Thus, flow integrity was preserved. In the case where the refrigeration system was turned "off" (Figure5.7a), the maximum velocity was close to the door due to the Coanda effect. However, the higher air temperature near the door deflected the maximum velocity slightly further from the door in the case where the refrigeration system was turned "on" (Figure5.7b).Since the mass flow rate gradually increased along the flow path, relatively high velocity was observed in the vicinity of the RAG. In comparison with the experimental results (Section 5.3), the developed model is capable of reproducing the main phenomena of the flow distributions in the closed display cabinet in both cases.

Figure 5 . 7

 57 Figure 5.7 Air velocity profiles in a closed display cabinet (a) refrigeration system turned "off"; and (b) refrigeration system turned "on".

Figure 5 . 8

 58 Figure 5.8 Air and product temperature distributions in a closed refrigerated display cabinet: (a) experiment (Chaomuang et al., 2019a); and (b) numerical prediction.

  [START_REF] Orlandi | CFD assisted design of closed display cabinets[END_REF] developed a three-dimensional CFD transient model to quantify incoming energy due to air infiltration during door openings. The model developed better understanding of the physical mechanisms of energy transport with various operating conditions (i.e. frequency of door openings) and equipment design (i.e. hinged or sliding doors). D'Agaro et al. (2006) developed a CFD model of the water condensation and evaporation processes on the glass doors of a lowtemperature (frozen food) closed display cabinet. This study aimed to understand the fogging and defogging mechanisms taking place during door openings since this incident is a very important issue for this cabinet type. Although the CFD model is a powerful tool, there are limitations because of the calculation time and computational resource requirements. Simplified heat transfer models based on a zonal approach become a complementary approach. Laguerre et al. (2012a) developed a model based on this approach in an open refrigerated display cabinet. This model was successful to predict load temperatures at different zones in the equipment with short calculation time. It also enabled the study of the influence of operating conditions. Further work by Ben-abdallah et al. (2018) proposed a similar heat transfer model in transient state under Modelica simulation environment through which the dynamic behavior of air and product temperatures

  into two parts: Part I proposes a steady state heat transfer model, which allows the prediction of time-averaged air and load temperatures in the closed display cabinet. Details of overall model conceptualization, static model development, determination of model input parameters as well as model validation by comparing with experimental data were discussed. Part II presents the model development in transient state, which allows the prediction of the temperature fluctuations according to the on/off working cycles of the compressor operation and to the defrosting operation.

  Figure 6.1, the closed display cabinet was loaded with test product packages made of methylcellulose (dimensions in width × height × length: 100 × 50 × 200 mm). Calibrated thermocouples (T-type, accuracy ±0.2 °C) were placed at various positions in the display cabinet for air temperature measurement (Fig.1, red stars): in the vertical rear duct from the duct upstream to the discharge air grille (DAG) and at 20 mm from the surface of the test packages. Some thermocouples were also instrumented at the core (Figure 6.1, blue circles) and the surface (Figure 6.1, green triangles) of the packages for load temperature measurement. Noting that each thermocouple was calibrated in a thermostatic oil bath (Fluke, model 7340, temperature stability of ±0.005 °C) with setting temperatures of -5, 0, 10, 20 and 30 °C and the precision of ±0.2 °C was determined.

Figure 6 . 1

 61 Figure 6.1 Characteristics of a closed display cabinet and schematic view of experimental setup for the temperature measurement

Figure 6 .

 6 Figure 6.2a also presents time-averaged air and load temperatures and their standard deviations in the closed display cabinet for the ambient temperature of 19.5 °C. Because of the heat losses through the cabinet walls, the time-averaged air

Figure 6 . 2

 62 Figure 6.2 (a) Experimental mean air and product core temperatures and the standard deviations during (quasi) steady state at various positions in a closed display cabinet for room temperatures of 19.5 °C (b) visualization of air flow at the front of the top shelf.

  value. The static model presented in this paper (Part I) allows the prediction of the time-averaged air and load temperatures. The dynamic model for the prediction of these temperature fluctuations will be presented in another part of the study (Part II). It is to be emphasized that the sinusoidal temperature variation in Eq. (6.1) is only to present the concept of the model development in a simple manner since the real temperature variation is periodic and not sinusoidal (Figure 6.3). For the model calculation, the temperature profile of the supply air (𝑇 𝑡ℎ , input parameter) is Chapter VI -Simplified heat transfer modeling in steady state 130 decomposed by using Discrete Fourier Transform. The temperature variations at other positions in the display cabinet are in response to this input temperature profile. This topic will be further discussed in more detail in Part II of the study.

Figure 6 . 3

 63 Figure 6.3 Experimental time-temperature profiles (a) air at the discharge air grille (DAG) (b) front load (core) at the top shelf and air at 2 cm from the load surface and (c) back load (core) at the bottom shelf and air at 1 cm above the load surface

Figure 6 . 4

 64 Figure 6.4 Schematic view of the simplified model composed of two domains: Domain I heat losses through the vertical rear duct and the cabinet ceiling and Domain II heat exchange in the shelves-space storage.

  𝑡ℎ = 0 and 𝜃 𝑒𝑥𝑡 = 1 A part of air flows into the shelf space while the other part (mass flow rate, 𝑚̇𝑎 𝑟𝑑1 ) continues flowing upward along the rear duct until it reaches the DAG. The heat balance of EZ [1, 2…7] was carried out in the same manner as that of EZ [1, 1].

Figure 6 . 5

 65 Figure 6.5 Heat balance in Domain I at the position after the heat exchanger in the vertical rear duct denoted as EZ [1, 1]: (a) schematic view (b) mathematical view

Figure 6 . 6

 66 Figure 6.6 Examples of heat and mass balances in Domain II: (a) Shelf 1 and (b) Shelf 2

Figure 6 .

 6 Figure 6.6a. This mixing zone, MZ [2, 1, 1] includes air from the DAG (mass flow rate, 𝑚̇𝑎 𝑟𝑑5 and temperature, 𝜃 𝑎𝑟𝑑7 ), air from the PBP (mass flow rate, 𝑚̇𝑎 𝑠ℎ1 and temperature, 𝜃 𝑎𝑠ℎ21 ) and a part of the recirculating air (mass flow rate, 𝛿 1 𝑚̇′ 𝑎𝑠ℎ1

  3.2.4. Heat exchange between the mixed air and the front load for Shelves 2-5Unlike Shelf 1, different analysis of heat balance is applied to the other shelves.The heat transfer mechanism on Shelf 2 was selected as a case sample of the model development and it is illustrated in Figure6.6b. Here, there are two air streams involved at the mixing zone, MZ [2, 1, 2], including the air from the PBP (mass flow rate, 𝑚̇𝑎 𝑠ℎ2 and temperature, 𝜃 𝑎𝑠ℎ22 ) and a part of the air curtain (mass flow rate, 𝛿 2 𝑚̇𝑎 𝑟𝑑4 and temperature, 𝜃 𝑎𝑠ℎ41 ).

  6.3.3.2.5. Heat losses through the door for Shelf 1The heat exchange through the door in front of Shelf 1,EZ [2, 4, 1], is different from the other shelves (Figure6.6a). The heat balance equation was specifically developed. Based on the experimental study, there was a formation of air recirculation at the front door of the display cabinet (see Figure6.2b). This air recirculation involves a part of the air curtain (mass flow rate, 𝛿 1 𝑚̇𝑎 𝑠ℎ1 and temperature, 𝜃 𝑎𝑠ℎ41 ) which interacts with the air flowing through the door gaps (mass flow rate, 𝑚̇𝑔 𝑎𝑝 ), and it is submitted to heat losses through the glass doors (overall heat transfer coefficient, 𝑈 𝑜𝑣𝑑 ). These exchanges result in an increase of air temperature to be 𝜃 𝑎𝑠ℎ51 . To simplify the model, an equivalent thermal conductance (𝐶 𝑒𝑞 ) was determined to take into account the effect of overall heat transfer coefficient through the glass door and the mass flow rate of the air infiltration through the gaps defined as 𝐶 𝑒𝑞 = 𝑈 𝑜𝑣𝑑 𝑆 𝑑1 + 𝑚̇𝑔 𝑎𝑝 𝐶 𝑝𝑎 (6.15)At steady state:𝜃 𝑎𝑠ℎ51 = 𝛼 𝑠ℎ31 𝜃 𝑎𝑠ℎ41+ (1 -𝛼 𝑠ℎ31 )𝜃 𝑒𝑥𝑡 (3.2.6. Heat losses through the door for Shelf 2-5 (development on Shelf 2 as a case sample) As mentioned earlier (Section 6.3.3.2), it was assumed that there is no air infiltration/leakage through the gaps located in front of the other shelves. The air curtain from the top shelf is split into two parts. The first part (mass flow rate, 𝛿 2 𝑚̇𝑎 𝑟𝑑4 and temperature, 𝜃 𝑎𝑠ℎ41 ) is mixed with the air flowing from the back of Shelf 2 and exchanges heat with the front load as described in Subsection iv. The second part (mass flow rate, (1 -𝛿 2 )𝑚̇𝑎 𝑟𝑑4 and temperature, 𝜃 𝑎𝑠ℎ41 ) continues flowing downwards and exchanges heat with the external ambient air through the Chapter VI -Simplified heat transfer modeling in steady state 139

  Figure 6.7, a 20-percent increase/decrease of the air flow ratio (𝛿 𝑔𝑎𝑝 ) caused the warmest load temperature to be increased/decreased up to 0.2 °C while it is less than 0.1 °C for other parameters.

Figure 6 . 7

 67 Figure 6.7 Sensibility study showing the variation in load temperatures with respect to the fitting parameters: air flow ratio (a), air distribution coefficients at Shelf 1 (b), air distribution coefficients at Shelf 2 to 5 (c), heat transfer coefficients over the back loads (d) heat transfer coefficients over the front loads (e).

Figure 6 . 8

 68 Figure 6.8 Temperature variation of the air after the heat exchanger (T_th) for three ambient temperatures.

  Figure 6.9 Comparison between numerical and experimental results of air and load temperatures (in °C) in a closed display cabinet for three ambient temperatures: (a) 14.6 °C, (b) 19.5 °C and (c) 24.4 °C. (Experimental values indicated for load temperature are the averages of the top and bottom layers)

Figure 6 .

 6 Figure 6.10 Calculated and measured time-averaged temperatures of (a) air and (b) load in a closed display cabinet for ambient temperatures of 14.6, 19.5 and 24.4 °C.

Figure 6 . 11

 611 Figure 6.11 Numerical result showing the variation of load temperatures with air infiltration rate through the door gaps

  the main limitation to the model applicability because the percentage of perforation area of the back panel over each shelf is different, resulting in different corresponding air distribution coefficients (𝛾 𝑖 ). For different patterns of the PBP and different shelf numbers, additional measurements or CFD simulations have to be conducted to obtain these coefficients. To apply the developed model to other closed display cabinets, all required input parameters are presented in Figure6.12.

Figure 6 .

 6 Figure 6.12 Input parameters required for application to another closed display cabinet

  𝑚̇𝑔 𝑎𝑝 . Otherwise, some model modifications are required to be more realistic i.e. subtraction of air recirculation zone and introduction of external air infiltration at a given mixing zone (e.g.MZ [2,1,1…5]).ConclusionsIn Part I of this study, a simplified heat transfer model in steady state was developed for a closed refrigerated display cabinet. Based on a zonal approach, the display cabinet was decomposed into various heat exchange zones allowing the prediction of air and load temperatures at different positions. Five parameters (𝛿 1 , 𝛿 2..5 , 𝑚̇𝑔 𝑎𝑝 , 𝑈 𝑙𝑓 and 𝑈 𝑙𝑏 ) were adjusted to fit the time-averaged temperatures for the external ambient temperature of 19.5 °C in the static model. For the other external ambient temperatures (14.6 °C and 24.4 °C), good agreement between the predicted and measured time-averaged temperatures were obtained without any adjustment. The overall mean absolute error based on 60 paired-data (air/load temperatures) for three ambient temperatures was less than 0.2 °C, and the maximum difference of the calculated and measured load temperatures never exceeds 0.5 °C for every studied position. In the shelves-space storage, timeaveraged load temperatures at 10 positions (front and back of the five shelves) were predicted and compared with the experimental results. It was observed that the load temperatures were well represented by the model whereas the air temperature in the air curtain was slightly less well represented. Possible explanation of the discrepancy between the model and the experiment is that the air temperature at a given height was measured only at one position, which may not be representative of the average air curtain temperature. Nevertheless, the load temperature ranges and trends in the closed display cabinet are well represented by the proposed model with a short calculation time (less than 1 s). The model also showed its applicability to investigate the effect of the air infiltration through the door gaps on the performance of the closed display cabinet in both thermal and energy aspects.Air and load temperatures vary according to the compressor working cycle of the refrigeration system. In Part II of the study, this dynamic aspect will be integrated in the developed model. All input parameters determined in Part I will be further used in the dynamic model. The model will be also used to investigate temperature increase during defrost operation.It is worth to remind that the developed model does not take into account the air infiltration due to door openings because the airflow during door openings is too complex to be predicted by the simplified model. CFD model is then more appropriate to cope with the phenomena. Nevertheless, suggestions were provided to include this phenomenon by both simple and more realistic manners. Besides, the main objective of the proposed model is to predict load temperatures in the equipment in short calculation time so that the model can be further integrated with others models already developed in the similar manner by the same research team (production plant, refrigerated truck, cold room, open display cabinet and domestic refrigerator). By this way, the integrated model will make it possible to acquire knowledge of the product time-temperature history across the entire food cold chain. By linking with quality and microbiological predictive models, the developed model can also be used as an evaluation tool of consumers' risk and can somewhat assist food operators to manage the logistic plan resulting in food loss reduction.

  This work on a closed refrigerated display cabinet is a continuity of the previous study (Part I) with the objective to present a simplified heat transfer model based on a zonal approach. Such a model is complementary to Computational Fluid Dynamics (CFD) model because less computational time and resources are required (Laguerre et al., 2014). The static model presented in Part I allows the prediction of time-averaged air and load temperatures at different positions in the closed display cabinet. Part II presents the development of the dynamic model which predicts the air and load temperature fluctuations according to the "on/off" regulation of compressor working cycles. Simplified models of open refrigerated display cabinets were already developed by our teams[START_REF] Ben-Abdallah | Modeling and experimental investigation for load temperature prediction at transient conditions of open refrigerated display cabinet using Modelica environment[END_REF] Laguerre et al., 2012). To our best knowledge, the development of a simplified model for a closed refrigerated display cabinet taking into account both the static and dynamic regimes is new.

  functions. The model did not include the refrigeration components, but formulated supply air temperature profile from the experimental data during a transient cycle of compressor regulation with Discrete Fourier Transform. The dynamic response of air/load temperature was readily determined by the transfer functions specified at different positions. By this way, the model not only allows the prediction of temperature fluctuations in the display cabinet but also shows the influence of different parameters on the damping of the temperature fluctuations with quick calculation. Moreover, this study proposes the estimation of the thickness at the surface of the load in which the temperature fluctuation can be observed. This thickness varies with the compressor working frequency while the dynamic models available in the literature often assume uniform load temperature, which is not realistic. Model development This section deals with the development of the model in the dynamic aspect. Based on the zonal approach, two domains of heat exchange zones in the display cabinet were defined and they are the same as the ones presented in Part I of the study. The first domain corresponds to the heat exchange in the vertical rear duct and the second one corresponds to the heat exchange in the storage space of the display cabinet. Figure 7.1 presents an example of one position of heat exchange zone in Domain I and heat exchange zones at Shelf 1 and 2 in Domain II. The descriptions of the symbols and indexes are also the same as those in Part I. Since the dynamic model was solved with the spectral method, transfer functions were defined at all positions of heat exchange zones.

Figure 7 . 1

 71 Figure 7.1 Schematic view of the simplified model: (a) heat exchange in Domain I at the position after the heat exchanger in the vertical rear duct denoted as EZ [1, 1], (b) heat exchanges in Domain II on Shelf 1 and (c) Shelf 2.

  𝑇 𝑡ℎ (𝑡) = 𝑇 𝑡ℎ + ∆𝑇 𝑡ℎ cos(𝜔𝑡) (7.1)Both air and load temperatures at any position in the cabinet are in response to the air temperature after the heat exchanger (𝑇 𝑡ℎ ) as well as the external ambient air temperature (𝑇 𝑒𝑥𝑡 ). Because all associated heat transfer phenomena in the display cabinet (i.e. forced convection and conduction) satisfy linear relationship with temperature difference, the properties of the dynamic linear systems can be employed for thermal analysis of the display cabinet. It is to be emphasized that free convection and thermal radiation from the external walls, which are non-linear phenomena, are limited because the doors were coated with low-emissivity materials (readers are referred to Part I of this study for more details). Therefore, the evolution of the temperature at a given position 𝑖 also follows a sinusoidal variation:𝑇 𝑖 (𝑡) = 𝑇 𝑖 + ∆𝑇 𝑖 cos(𝜔𝑡 + 𝜑 𝑖 ) = 𝑇 𝑖 + ∆𝑇 𝑖 cos (𝜔(𝑡 -𝑡 𝑑𝑒𝑙,𝑖 ))(7.2)where 𝑇 𝑖 is the time-averaged temperature at a position 𝑖 in the display cabinet and ∆𝑇 𝑖 is the amplitude of the temperature fluctuation at that position. From the superposition theorem, the temperature evolution at a given position is given by𝑇 𝑖 (𝑡) = 𝑇 𝑡ℎ + (𝑇 𝑒𝑥𝑡 -𝑇 𝑡ℎ ) • 𝜃 𝑖 + ∆𝑇 𝑡ℎ • 𝜃 𝑖 ′ (7.3)where 𝜃 𝑖 and 𝜃 𝑖 ′ are respectively the dimensionless static and dynamic temperatures. They are defined as follow:For the problem where 𝑇 𝑡ℎ = 𝑇 𝑡ℎ , 𝑇 𝑒𝑥𝑡 ≠ 𝑇 𝑡ℎ and 𝑞 𝑙𝑖 ≠ the problem where 𝑇 𝑡ℎ = 𝑇 𝑡ℎ + ∆𝑇 𝑡ℎ cos(𝜔𝑡), 𝑇 𝑒𝑥𝑡 = 𝑇 𝑡ℎ and 𝑞 𝑙𝑖 = can be obtained from the static aspect of the model of which the detail was presented in Part I of the study. The dynamic aspect is now integrated to obtain the solutions of Eq. (7.5) in the following form:𝜃 𝑖 ′ = Θ 𝑖 cos(𝜔𝑡 + 𝜑 𝑖 ) = Re(𝐆 𝑖 𝑒 𝑖𝜔𝑡 ) with Θ is the transfer function at a position 𝑖; Θ 𝑖 and 𝜑 𝑖 are the magnitude and the phase shift of 𝐆 𝑖 , respectively. be emphasized that the sinusoidal temperature variation in Eq. (7.1) is only to present the concept of the model development in a simple manner since the real temperature variation is periodic and not sinusoidal (Figure7.2). For the model calculation, the temperature profile of the supply air (𝑇 𝑡ℎ , input parameter) was decomposed by using Discrete Fourier Transform and the temperature variations at other positions in the display cabinet were in response to this input temperature profile. More detail is given in Section 7.2.3.7.2.2. Dynamic model developmentBased on Eq (7.5) and (7.6), the dimensionless air temperature fluctuation after the heat exchanger becomes: 𝜃 𝑡ℎ ′ = cos(𝜔𝑡) = Re(𝑒 𝑖𝜔𝑡 ) (7.9) Since homogeneous boundary condition (𝑇 𝑒𝑥𝑡 = 𝑇 𝑡ℎ ) is assumed for the dynamic problem: Attenuation of temperature fluctuation due to heat exchange with the rear wall (Domain I) In unsteady state condition, the temperature of the inner side of the rear wall of the display cabinet changes with time because it is exposed to fluctuating temperature of cold air. Considering EZ [1, 1] (Figure 7.1), the heat balance equation is given by

  operation is rather consistent (𝜏 ON ~ 90-100 s) due to the cooling capacity of the refrigeration system used in the studied display cabinet. To predict the temperature fluctuation in the closed display cabinet for different ambient temperatures, an air temperature profile with respect to the external temperature was applied to the developed model (Figure 7.2).

Figure 7 . 2

 72 Figure 7.2 Experimental temperature profiles of air after the heat exchanger for different ambient temperatures.

  ) are often enough to describe the supply air temperature profile 𝑇 𝑡ℎ . This method needs only to solve a limited number of algebraic complex equations, instead of ordinary differential equations. It is worth noting that the same model developed in this study could also be solved by temporal discretization (e.g. Runge-Kutta).Estimation of the equivalent depth in which temperature fluctuations occur in the productFor the dynamic model development in the present study, temperature gradients within the load cannot be neglected since the internal resistance to conduction within the load is much higher than external resistance to convection[START_REF] Bergman | Fundamentals of heat and mass transfer[END_REF] i.ethe thermal properties of the methylcellulose (thickness 𝑒 𝑙 = 200 mm; density 𝜌 𝑙 = 1071 kg•m -3 ; thermal conductivity 𝜆 𝑙 = 0.51 W•m -1 •K -1 and specific heat 𝐶 𝑝𝑙 = 3372 J•kg -1 •K -1 ,[START_REF] Icier | The use of tylose as a food analog in ohmic heating studies[END_REF]) and the estimated convective heat transfer coefficients 𝑈 𝑙𝑏 = 15 W•m -2 •K -

  solid hereafter for the analytical model development of temperature fluctuation in the solid. In real condition, periodic air temperature fluctuation provides temperature fluctuation in the load where the maximum amplitude is located at the load surface (𝑥 = 0, Figure 7.3a). Considering the load as a semi-infinite solid with constant thermal conductivity, the temperature fluctuation within the load decays exponentially with respect to the distance from the surface. Far from the surface, the amplitude of temperature fluctuation becomes null as schematically illustrated in Figure 7.3a. Since 1D discretization of the load would require many additional state variables, two layers of different thermal conductivities are assumed in our model for its simplicity (Figure 7.3b). A thin layer at the surface characterized by an equivalent thickness (𝑒 𝑒𝑞 ) is assumed to have a very high thermal conductivity (𝜆 = ∞) whereas the rest of the load (core) is assumed to have a very low thermal conductivity (𝜆 = 0, perfect insulation). The equivalent thickness is chosen so that the temperature fluctuation at the surface predicted by the simplified model is the same as for a semi-infinite solid.

Figure 7 . 3

 73 Figure 7.3 Illustration curves of temperature fluctuations in a solid (a) with constant conductivity (real condition) and (b) with inconstant conductivity split at the equivalent thickness (simplified condition).

  7.3.1. Development of temperature fluctuation model in a solidTwo analytical models were used to determine the equivalent thickness of which the solutions have to satisfy the same temperature fluctuations at the surface. For the first model, the heat transfer in the load with perfect conduction in the range of the equivalent thickness (0 < 𝑥 < 𝑒 𝑒𝑞 ) is considered (Model I). The second model involves the heat transfer in the load with constant conductivity with respect to the distance from the surface (Model II).The solution of the Model I depends on the heat transfer rate between the load and its surrounding air: 𝑚 𝑙 𝐶 𝑝𝑙 𝑑𝑇 𝑙 𝑑𝑡 = ℎ 𝑎 𝑆 𝑙 (𝑇 𝑎 -𝑇 𝑙 ) ⟺ 𝜌 𝑙 𝑒 𝑒𝑞 𝐶 the temperature fluctuation of the surrounding air as a sinusoidal variation (𝑇 𝑎 = 𝑒 𝑖𝜔𝑡 ), the solution of Model I can be calculated from , the solution is given by considering the conduction heat transfer rate in a semi-infinite solid with constant properties for which the heat balance surface (𝑥 = 0), the heat transfer by convection between the load and air is defined as:

Figure 7 . 4

 74 Figure 7.4 (a) Schematic of temperature measurement for the test package located at the back row (top layer) on the bottom shelf at different depths from the surface (x=0 mm) to the core (x=25 mm) and for the air at 10 mm above. (b) Decay of temperature fluctuations in the load from the surface to the core

Figure 7 . 5

 75 Figure 7.5 Experimental temperature evolutions of the air (10 mm above the load) and of the back load on the bottom shelf at the different depths from the surface (x=0 mm) to the core (x=25 mm): (a) overall view and (b) enlarged scale on the highlighted area

  1. Air and load temperature fluctuations in a closed refrigerated display cabinet After the integration of the dynamic regime, the simplified heat transfer model was employed to simulate the temperature evolutions of the air and the loads inside the closed refrigerated display cabinet. The predicted and measured results for the external ambient temperature of 19.5 °C are shown in Figure 7.6. The temperature fluctuation of the supply air decreased from the position just after the heat exchanger (∆𝑇 𝑡ℎ = 7.3 °C, Figure 7.2 Experimental temperature profiles of air after the heat exchanger for different ambient temperatures.) to the DAG (∆𝑇 𝐷𝐴𝐺 = 5.4 °C, Figure 7.6a) by 1.9 °C. While traveling to the DAG, a part of the air also flows horizontally into the shelve space through the PBP. The air temperature fluctuation at the back of the bottom shelf (Figure 7.6d) was more pronounced than that of the upper shelves (Figure 7.6b-c). This can be explained by the higher air flow rate through the PBP at the lower shelf and a higher air temperature fluctuation of this air (∆𝑇 𝑎 decrease along the rear duct). At the front of the shelves, the highest air temperature fluctuation is at the top shelf (Figure 7.6e). The high temperature fluctuation at this position resulted from the high temperature fluctuation of the air curtain from the DAG. Despite the exposure to high air temperature fluctuation, the load surface temperatures at all positions were not much fluctuating (∆𝑇 𝑙 ≤ 0.5 °C).It was found that the ratio of the load to its surrounding air temperature fluctuation (

Figure 7 . 6

 76 Figure 7.6 Temperature evolution of the air and the load surface at different positions inside the closed display cabinet for the ambient temperature of 19.5 °C. (a) air at the discharge air grille (DAG), (b-d) back and (e-g) front loads with the surrounding air on the top, middle and bottom shelves, respectively (the time-averaged load temperature at the front was extrapolated from the core temperatures between the back and the front loads).

Figure 7 . 7

 77 Figure 7.7 Comparison between the simulation and the measurement of temperature fluctuations of (a) air and (b) loads inside the closed refrigerated display cabinet. Data based on the external ambient temperature of 19.5 °C

Figure 7 . 8

 78 Figure 7.8 Input temperature profile of the air after the heat exchanger (𝑇 𝑡ℎ )

Figure 7 .

 7 Figure 7.9 and 7.10 respectively show the air and the load temperature evolutions during defrosting period at different positions (back and front of the top, middle and bottom shelves) in the closed refrigerated display cabinet for the external ambient temperature of 19.5 °C. Based on the experimental data, the air temperature increase (∆𝑇 = 𝑇 𝑚𝑎𝑥 -𝑇 𝑚𝑖𝑛 ) at these positions was between 2 and 6 °C (Figure 7.9) while the surface load temperature increased by only 0.5 -1.0 °C (Figure 7.10). The highest air and load temperature increases were observed at the front of the top shelf (Figure 7.9d and 7.10d).

Figure 7 . 9 Figure 7 . 10

 79710 Figure 7.9 Air temperature evolution at various position in a closed refrigerated display cabinet during the defrost operation (𝑇 𝑒𝑥𝑡 = 19.5℃)

Figure 7 . 8

 78 Figure 7.8 as an example.

Figure 7 . 11

 711 Figure 7.11 Comparison between the simulation and the measurement of temperature increase (𝑇 𝑚𝑎𝑥 -𝑇 𝑚𝑖𝑛 ) during defrosting period of (a) air and (b) load inside the closed refrigerated display cabinet for three external ambient temperatures

※※

  Although the proposed model was developed based on a specific closed display cabinet, it can be generalized to other display cabinets as addressed in Part I of the study. It needs to be emphasized that the air infiltration due to door openings is not considered in the developed model because of its complex airflow. This aspect could be integrated in a future work. The model will be further linked with others models already developed in similar manner by the same research team for different refrigerating equipment in the cold chain, i.e. production plant, refrigerated truck, cold room, open display cabinet and domestic refrigerator. By such a way, the prediction of the evolution of the product time-temperature across the entire food cold chain can be established. The developed models can then be used to evaluate the consumers' risk by integrating with quality and/or microbiological predictive models. cabinets are becoming increasingly used in supermarkets because of potential energy savings compared to open ones. This growing trend has contributed to an expanding research in the area of commercial refrigeration. However, most studies focused on improvements of energy efficiency after door installation while the influence of the presence of doors and the frequency/duration of door openings on product temperatures are much less studied.Like many refrigerating equipment, forced convection is usually a primary heat transfer mode in closed display cabinets; therefore, the temperature and its homogeneity are directly governed by the airflow patterns. A study on this matter becomes necessary to improve understanding of the main mechanism of these transport phenomena. Following this rationale, the work presented in this thesis was an attempt to bridge this research gaps by implementing both experimental and numerical approaches to characterize the airflow and heat transfer in a closed refrigerated display cabinet.8.1.1. Experimental studyA number of experiments were conducted in a closed refrigerated display cabinet under various operating conditions to study their influences on food temperature in the equipment. Thermocouples were distributed throughout the display cabinet to ascertain the internal air and the product temperature profiles. The implementation of an automatic door opening system made it possible to observe the air and product temperature evolutions according to different opening frequencies and durations.The use of a hot-wire anemometer and Particle Image Velocimetry (PIV) allowed to characterize the airflow pattern. It enables the knowledge of the velocity profile of the air curtain from the discharge air grille (DAG) to the return air grille (RAG), the airflow rate in the rear duct and the proportion of the air flowing from the perforated back panel (PBP) to the storage compartment at different shelves. The main results from the experimental study are summarized as follows:Chapter VIII -General conclusions and perspectives 1898.1.1.1. Airflow characteristic of a closed refrigerated display cabinet※ About 50%-50% airflow ratio between the DAG and the PBP was observed in the studied display cabinet. The variation in the occupied-volume percentages has insignificant effect on the airflow ratio.※ The presence of doors caused the air recirculation at the upper part of the closed display cabinet where the warm ambient air infiltration through the door gaps was also observed leading to an increase in the air curtain temperature. Based on the air velocity measurement in the middle of the door gaps using the PIV technique, the air infiltration under the permanently closed-door condition accounted for about 8% of the supply air from the DAG.※ At the lower part of the cabinet (from the top shelf level to the RAG), the presence of doors caused the air curtain to deviate toward the doors through a Coanda effect (maximum velocity observed near the door surface). This effect depends on the air temperature near the door surface; higher temperature enhances the upward flow because of higher buoyancy force.※ Overall, the air curtain was quite stable in most regions of the closed display cabinet, except for the top (neat the DAG) and the bottom (near the RAG) of the cabinet where relatively high velocity fluctuations occurred.8.1.1.2. Thermal characteristic of a closed refrigerated display cabinetI. Closed-door condition※The overall thermal transmission coefficient of the studied closed display cabinet was measured. Compared with domestic refrigerators, the insulating capacity of the closed display cabinet was relatively lower. The gaps at the door edges and the thermal bridges were found to be the most significant contributor (~ 70%) of the overall thermal transmission of the studied display cabinet.※Two temporal air temperature variations periodically emerge in the closed display cabinet: one according to the "on/off" cycles of compressor regulation and the other one according to the compressor "off" period during the defrost operation.The air temperature at the front-top shelf takes the longest time to reach the quasi-steady state (about an hour since the display cabinet started up). This indicates the minimum duration for the air temperature stabilization before product loadings. A quasi-steady state is confirmed when the time-averaged temperature over one "on/off" cycle of compressor regulation remains the same.Despite large air temperature fluctuations in the rear duct up to the DAG during the quasi-steady state (ΔTa ~ 8-9 °C), the air temperature fluctuations in the storage compartment were relatively low (ΔTa ~ 2-4 °C) because of the thermal inertia of solid cabinet components (e.g. rear wall, ceiling and PBP). This resulted in much lower fluctuations in the product temperatures (ΔTp < 0.6 °C).

※

  Forced convection and conduction are the predominant heat transfer modes in the closed display cabinet. This explains an almost linear relationship between the ambient temperature and the internal air temperature. Thermal radiation from the external walls is restricted by the glass doors shielded with lowemissivity coatings. However, visible radiation from the room and cabinet lightings still occurs and induces heat generation in the products, particularly at the front of the cabinet top shelf. When the cabinet doors are closed, free convection is insignificant in the cabinet because of small difference between the highest and coldest internal air temperatures (∆Ta < 5 °C).Chapter VIII -General conclusions and perspectives 191 II. Periodic door-opening condition ※ Spatial and temporal air and product temperature distributions under the periodic door-opening condition has the same trend as that under closed-door condition.

8. 1 ※※※

 1 .1.3. Comparison between the closed and open configurations I. Unloaded display cabinet (permanently closed and permanently open doors) The air temperature profile was similar during permanently closed and permanently open-door (doors of the studied cabinet were completely removed) conditions. As the case of loaded display cabinet, the front of the top shelf and the back of the bottom shelf are respectively the highest and lowest temperature positions. However, Chapter VIII -General conclusions and perspectives 192 Lower air temperature level at all positions in the closed configuration was observed: maximum difference of 3.1 °C at the middle shelf. ※ Lower spatial air temperature variations on the same shelf: the air temperature difference between the front and the back was less than 1 °C in the closed configuration, while it was about 2-3 °C in the open one. ※ Less heat load on the evaporator in the closed configuration: the air temperature difference between the RAG and the DAG was less than 1.3 °C, while it was about 5.9 °C in the open one. Therefore, the time fraction during compressor "on" period decreased from 49% (open configuration) to 24% (closed configuration), about 50% reduction. II. Loaded display cabinet (permanently closed, periodic door openings and permanently open doors) Slightly different air and product temperature profiles were observed during permanently closed and permanently open door conditions. For the permanently open-door condition, the air and product temperatures were highest at the front of the middle shelf, while the lowest temperature positions remained at the back of the bottom shelf. The same trend of the temperature profiles was observed when the periodic door openings were applied. The closed refrigerated display cabinet provides better temperature performance despite a very high door-opening frequency (i.e. 60 OPH), compared to the case of permanently open doors (or without doors). The following information was obtained for the closed display cabinet: Lower air and product temperature levels at all positions. The air and product temperatures respectively decreased up to 5.1 °C and 2.5 °C under permanently closed-door condition in comparison with permanently open-door condition. Under the opening frequency of 60 OPH, the product temperatures remained at least 1.0 °C lower than the ones under permanently open-door condition. ※ More uniform temperature distribution. The temperature difference between the highest and the lowest product (core) temperatures decreased from 4.0 °C (permanently open) to 2.0 °C (permanently closed) and 3.0 °C (opened with the frequency of 60 OPH). Chapter VIII -General conclusions and perspectives 193 Energy saving potential. The time fraction during the compressor "on" period decreased from 65% (permanently open) to 44% for the door-opening frequencies of 60 OPH and to 23% for permanently closed doors.8.1.2. Numerical modelingNumerical tools were developed in the thesis by both Computational Fluid Dynamics (CFD) simulation and a simplified model based on a zonal approach.8.1.2.1. CFD simulationA steady-state 2D-CFD model with the standard k-ε turbulence model was developed to investigate the airflow and its influence on the air and product temperature distributions in a closed display cabinet. Two conditions (refrigeration system turned "off" or "on") were considered. Despite the simplified hypothesis, the developed model has shown the ability to reproduce the main flow phenomena observed by the experiments. The zone of air recirculation (where the ambient air infiltration through the door gaps causes an increase in the air curtain temperature)

  Extensive experiments were carried out in the present study to characterize the airflow and temperature distributions in the closed display cabinet. The thermal cabinet performance under different operating conditions together with the comparison between the closed and open configurations was also examined.

  variation of relative humidity and temperature in supermarkets. Paper presented at the Proceedings of Twentieth International Congress of Refrigeration, Sydney, Australia.Howell, R., Rosario, L., Riiska, D., & Bondoc, M. (1999).Potential savings in display case energy with reduced supermarket relative humidity. Paper presented at the 20th International Congress of Refrigeration, Sydney, Australia. Hunter, M. (2017). We don't need to double world food production by 2050 -here's why. The Conversation. Titre de la thése : Caractérisation expérimentale et modélisation des écoulements d'air et transferts thermiques dans un meuble frigorifique fermé Résumé En France, dans le cadre des économies d'énergie, un accord a été signé entre les Ministères et la Fédération du commerce et de l'Industrie afin d'obtenir dans la grande distribution 75% de meubles fermés en 2020. La généralisation des meubles fermés pour produits réfrigérés entraîne la nécessité d'élargir les connaissances scientifiques concernant ce type de meuble. La plupart des études de la littérature ont porté sur l'amélioration de l'efficacité énergétique suite à l'installation des portes, tandis que les études sur les mécanismes de transfert de chaleur dans les meubles frigorifiques fermés sont rares. De plus, il convient d'étudier également la circulation d'air car elle influence les échanges de chaleur entre l'air et les produits, donc leur température. Cette thèse a pour objectif de mieux comprendre les phénomènes d'écoulement d'air et de transfert de chaleur dans les meubles frigorifiques fermés par la mise en oeuvre d'approches expérimentales et numériques. Des investigations expérimentales ont été menées sur un meuble frigorifique fermé avec un rideau d'air simple et des portes à double vitrage, placé dans une cellule d'essai à température contrôlée. Des thermocouples calibrés répartis dans le meuble ont permis d'observer les évolutions spatio-temporelles des températures de l'air et des produits dans différentes conditions de fonctionnement. Nous avons fait varier la température de l'air ambiant, le volume occupé par le produit et la fréquence d'ouverture des portes. Des expériences ont également été réalisées dans une configuration ouverte du meuble frigorifique (portes complètement retirées) afin de déterminer les avantages des portes sur les performances thermiques. Les résultats ont montré que lorsque les portes étaient fermées en permanence, les champs de température étaient similaires, quelle que soit la température ambiante et le pourcentage de volume occupé ; la température la plus élevée se situe à l'avant de l'étagère supérieure. Cette répartition change lorsque les portes sont périodiquement ou définitivement ouvertes ; la position où la température est la plus élevée migre vers l'avant de l'étagère du milieu. Même pour une fréquence très élevée d'ouverture des portes (60 ouvertures par heure), les températures de l'air et des produits dans le meuble fermé sont restées plus basses d'au moins 1,0 °C par rapport au meuble sans portes. Des mesures de vitesse d'air par anémométrie à fil chaud au niveau du rideau d'air ont permis d'observer son allure générale tandis que les mesures effectuées dans le conduit arrière ont permis de quantifier la répartition du flux d'air à travers le plaque arrière perforée au niveau des différentes étagères. L'utilisation d'une technique de vélocimétrie par image de particules (PIV) a permis de caractériser le rideau d'air avec une résolution et une précision spatiale supérieures. Les résultats ont notamment montré l'existence d'une zone de recirculation de l'air dans la partie supérieure du meuble où l'on observe également une infiltration d'air ambiant chaud par les fentes présentes autour des portes, ce qui induit une augmentation de la température du rideau d'air. Des simulations numériques bidimensionnelles d'écoulement (CFD Computational Fluid Dynamics) avec le modèle de turbulence k-ε ont permis de reproduire les principaux phénomènes d'écoulement observés par PIV et de voir leur influence sur la distribution de température dans le meuble. Enfin, un modèle simplifié des transferts de chaleur a été développé par une approche zonale en régime permanent et en régime transitoire ce qui permet de prévoir les températures moyennes de l'air et du produit et les fluctuations de température en fonction des cycles marche/arrêt du compresseur. Le modèle transitoire a été résolu avec une approche spectrale. L'influence de différents paramètres sur les niveaux de température et l'amortissement des fluctuations peut ainsi être identifiée par un calcul beaucoup plus rapide que par CFD. Mots clés : meuble frigorifique fermé, écoulements d'air, transferts thermiques, distribution de la température, ouverture des portes, Vélocimétrie par image de particules, modèle simplifié, fluctuation de température, Modèle CFD

Table 2.1 summarizes the

field studies conducted in several countries.

Table 2 .1

 2 Field investigations on chilled food products at retail displays in various countries

	Country	Sample	Temperature (°C) Min. Mean Max.	% of samples at specified temperature	Reference
	Canada	Beef	-2.0	2.6	10.6	18% > 4 °C	Gill et al. (2003)
		Fresh-cut lettuce		4.1			McKellar et al. (2014)
	Denmark	Chilled product			14.0	50% > 5 °C	Bøgh-Sorensen (1980)
	France	Yogurt	-2.1	4.2	9.8	18% > 6 °C	Derens et al. (2006)
		Processed meat	-1.1	3.3	10.2	36% > 4 °C
		Ham	0.2	2.8	7.0	12% > 4 °C	Derens-Bertheau et al. (2015)
		Bakery, pock and dairy			16.0	70% > 7 °C	Morelli et al. (2012)
	Finland	Fish	0.3	3.5	8.6	53% > 5 °C	Lundén et al. (2014)
		Minced meat	0.1	3.1	6.9	10% > 7 °C
	Germany	Chilled product			11.0	87.9% > 4 °C; 39.4% > 7 °C Murmann and Kuhlung (1987)
	Greece	Pasteurized milk	0	4.9	11.7	35% > 6 °C	Koutsoumanis et al. (2010)
		Smoked sliced turkey		4.0			Gogou et al. (2015)
	Ireland	Cooked sliced ham	1.5	5.7	12.7	14% > 5 °C	FSAI (2003)
	Slovenia	Meat	0	3.8	10.5		Likar and Jevšnik (2006)
		Dairy	0	5.6	16.0
	Spain	Meat	-1.8	4.3	9.9	43% > 4 °C	Baldera Zubeldia et al. (2016)
		Fish	2.9	6.2	10.0	75% > 4 °C
		Dairy	1.2	6.4	15.3	57% > 8 °C
	Sweden	Chilled product	-1.0	4.9	16.0	36% > 5 °C	Bøgh-Sørensen and Olsson (1990)
	UK	Cooked meat				28% > 5 °C	Sagoo et al. (2007)
		Chilled products				55% > 5 °C	Evans (2010)
	USA	Leafy green salad				40% > 7.2 °C	Brown et al. (2016)
		Meat, fish and dairy	-10.0	5.4	21.1	47% > 5 °C; 22% > 7 °C	Audits International (1999)
		Meat			20.0	38% > 4 °C	Rogers and Althen (1980)

most of the packages at the maximum temperature (97% of observed display cabinets) were at the front, and 60% of them were at the cabinet front base or on the bottom shelf. Meanwhile, most of the packages at the minimum temperature (98%) were located at the back of the cabinets. These findings match those obtained in a study conducted by Bøgh-Sørensen and

Table 2 .2

 2 Numerical studies on heat transfer and airflow in open display cabinets

	CFD codes	Turbulence models	Display cabinet schemes	Studied parameters	Main observations	References
	SOFIE	Standard k-ε	VORDC	 Load arrangement	 Velocity profile	Axell et al. (1999)
	(2D and 3D (1) )		Single curtain	 Hole characteristics of a
			5 shelves	perforated plate
			Loaded
	In-house	LES	VORDC	 Air velocities of both inner	 Temperature profile	Cortella et al.
	(2D)		Double curtain	and outer jets	 Velocity profile	(2001)
			6 shelves		 Flow pattern
			Loaded		 Air curtain efficiency
	In-house	n/a	VORDC	 Ambient enthalpy	 Total heat transfer	Ge and Tassou
	(2D)		Single curtain	 Temperature difference	 Return air temperature	(2001)
			5 shelves	between the jet and the air in a
			Empty	cabinet
				 Jet initial temperature
				 Jet initial velocity
				 Jet thickness
				 Air curtain height
	In-house	LES	VORDC	 Cabinet configurations:	 Temperature profile	Cortella (2002)
	(2D)		Double curtain	vertical and horizontal	 Velocity profile
			6 shelves	dimensions	 Flow pattern
			Loaded		 Load temperature (separate
			HORDC		code)
			Loaded

Table 2 .2

 2 Numerical studies on heat transfer and airflow in open display cabinets

	CFD codes	Turbulence models	Display cabinet schemes	Studied parameters	Main observations	References
	In-house,	Cebeci-Smith	VORDC	 Discharge air velocity	 Infiltration rate	Navaz et al. (2002)
	ROYA	(algebraic	Single curtain	 Discharge air temperature	 Velocity profile
	(2D)	model)
		Low-Re k-ɛ
	CFX	Standard k-ε	VORDC	 A number of design	 Temperature profile	Foster et al. (2005)
	(3D)		Single curtain	parameters:	 Flow pattern
			5 shelves	 Size and position of the	 Product temperature
			Loaded	cooling coils (evaporator)
				 The presence of a baffle plate
				within a back panel
	FLUENT	Standard k-ε	VORDC	 Radiation (based on discrete-	 Product temperature	Madireddi and
	(3D)		Single curtain	ordinates radiation model)	 Sensible heat load	Agarwal (2005)
			5 shelves	 Inlet mass flow rate
			Loaded
	FLUENT	n/a	VORDC	 Reynolds numbers (based on	 Infiltration rate	Navaz et al. (2005)
	(2D)		Single curtain	supply duct width)	 Temperature profile
			4 shelves	 Turbulence intensity	 Velocity profile
			Empty	 Presence of a perforated back
				plate
	CFX	Standard k-ε	VORDC	 3D effect (cabinet length)	 Temperature profile	D'Agaro et al.
	(2D and 3D)	RNG k-ε	Triple curtain	 Warm air curtain temperature	 Flow pattern	(2006)
			5 (inclined)	(the 3rd curtain exposed to	 Induction factor
			shelves	ambient)	 Return air temperature

Table 2 .2

 2 Numerical studies on heat transfer and airflow in open display cabinets

	CFD codes	Turbulence models	Display cabinet schemes	Studied parameters	Main observations	References
			Loaded	 Longitudinal ambient air	 Refrigerating capacity
				velocity
	In-house,	Standard k-ε,	VORDC	 Comparison of turbulence and	 Temperature profile	Yu et al. (2007,
	FORTRAN	(TF and MTF)	Double curtain	two-fluid models		2008)
	(2D)		No shelf	 Internal and external air
			Empty	curtain velocities
	FLUENT	Standard k-ε	VORDC	 Length-width ratio	 Stability of air curtain (based	Chen (2009)
	(2D)		Single curtain	 Discharge angle of air curtains	on temperature distribution
			6 shelves	 Height-depth of the cabinet	and sensible cooling load)
			Empty	 Dimensions and positions of
				internal shelves
	In-house,	Standard k-ε	VORDC	 Velocity and temperature of	 Thermal entrainment factor	Yu et al. (2009a)
	FORTRAN	(TF)	Single curtain	supply air (inner and outer in	 Return air temperature
	(2D)		6 shelves	case of dual curtains)
			Double curtain	 Air curtain width and height
			5 shelves	 Central distance between
			Empty	supply and return air ducts
				 Ambient temperature and
				humidity
				 Perforated back panel airflow
				ratio and its temperature

Table 2 .2

 2 Numerical studies on heat transfer and airflow in open display cabinets

	CFD codes	Turbulence models	Display cabinet schemes	Studied parameters	Main observations	References
	FLUENT	Standard k-ε	VORDC	 Ambient temperature	 Temperature profile	Ge et al. (2010)
	TRNSYS		Single curtain	 Ambient air humidity	 Flow pattern
	(2D)		5 shelves	 Cabinet air flow rate	 Refrigerant mass flow rate
			Loaded		 Cooling capacity
					 Product temperature
	FLUENT	RNG k-ε	VORDC	 Integration of air flow in the	 Temperature profile	Gaspar et al.
	(2D)		Single curtain	duct into simulation domain	 Velocity profile	(2012)
			5 shelves		 Relative humidity field
			Loaded		 Mass flow and heat transfer
					rate across air curtain
	In-house	n/a	VORDC	 Ambient temperature	 Air temperature	Laguerre et al.
	(2D)		Single curtain	 Radiation	 Product temperature	(2012)
			5 shelves	 Internal infiltration rate
			Loaded	 External infiltration rate
	n/a	Standard k-ε	VORDC	 Lighting	 Product temperature	Zhijuan et al.
	(2D)		Single curtain	 Air velocity at evaporator		(2013)
			6 shelves	outlet
			Loaded	 Ambient temperature and
				relative humidity
	CFX	Standard k-ε	Mock-up model of	 One study condition	 Velocity profile on different	Marinetti et al.
	(3D)		cooling air bottom		three heights	(2014)
			duct

Table 2 .2

 2 Numerical studies on heat transfer and airflow in open display cabinets

	CFD codes	Turbulence models	Display cabinet schemes	Studied parameters	Main observations	References
	FLUENT	n/a	VORDC	 Percentage of porosity on a	 Temperature profile	Wu et al. (2014)
	(2D)		Single curtain	perforated back plate	 Flow pattern
			6 shelves	 Location of perforations on a	 Air curtain velocity
			Loaded	plate
	CFX	Two-equation	Mock-up model of	 The presence of cooling coils	 Velocity profile in laminar and	Rossetti et al.
	(3D)	eddy-	cooling air bottom	in the bottom duct	turbulent flow regimes in the	(2015a)
		viscosity	duct		duct
					 Velocity profile on a single fin
					of cooling coils
	CFX	Two-equation	Mock-up model of	 One study condition	 Velocity profiles in the duct	Rossetti, et al.
	(3D)	eddy-	cooling air bottom		and on a single fin of cooling	(2015b)
		viscosity	duct		coils
					 Temperature profiles in the
					duct and on a single fin of
					cooling coils
	FLUENT	Standard k-ε	VORDC	 The application of a phase-	 Temperature profile	Alzuwaid et al.
	(2D)		Single curtain	change material (PCM)	 Defrost cycle	(2016)
			6 shelves		 On/off compressor cycle
			Loaded		 Compressor working time
					 Product temperature
	In-house	n/a	VORDC	 The application of PCM	 Air temperature	Hoang et al. (2016)
	(2D)		Single curtain		 Product temperature
			5 shelves

Table 2 .4

 2 Numerical studies conducted on closed display cabinets

	CFD code	Turbulence model	Display cabinet scheme	Studied parameters	Main observations	Reference
	Commercial	Realizable k-ε	CRDC	 Types of doors (hinged	 Temperature profile	Orlandi et al.
	code (3D)		3 doors	and sliding doors)	 Velocity profile	(2013)
	Open-source		6 shelves	 Operation time (day and	 Flow pattern
	code (3D)		Loaded	night)	 Return air temperature
				 The frequency of door	 Incoming thermal loads
				openings	through a door
					 Heat extraction rate
	CFX	n/a	CRDC	 Fogging (condensation)	 Temperature profile on a	D'Agaro et al.
	In-house		5 shelves	and defogging (with	glass surface	(2006)
	(2D and 3D)		Empty	electric heaters) processes	 Water layer height
				due to opening doors
	CRDC -a closed refrigerated display cabinet

Table 3 .

 3

	Shelves	Plate area (m 2 )	Percentage of perforation (%)
	Shelf 1 (top)	0.28	2.3
	Shelf 2-4	0.28	3.8
	Shelf 5 (bottom)	0.44	8.3
	Total plate	1.68 (a)	4.4
	(a)

1 Percentage of perforated area (PPA) on different shelves

Table 3 .2

 3 Mean temperature and standard deviation (°C) during quasisteady state at various positions within the food storage zone on the left, middle and right planes for a room temperature of 19 °C

	Planes	e	f	g	Positions h	i	j	All
	Left	0.9±0.6	1.8±0.5	0.4±0.5	0.9±0.6 -0.6±1.1 0.1±1.2	0.6±1.1
	Middle 0.5±0.7	1.1±0.8	0.3±0.6	0.6±0.6 -0.6±1.3 -0.2±1.0 0.3±1.0
	Right	0.8±0.5	2.0±0.4	0.6±0.6	1.6±0.7 -0.1±0.9 1.1±1.1	1.0±1.0
	All	0.7±0.6	1.6±0.7	0.4±0.6	1.0±0.7 -0.4±1.1 0.3±1.2	0.6±1.1

Chapter III -Heat transfer analysis and airflow characterization using a hot-wire anemometer 58

Table 3 .

 3 3 shows the influence of the ambient temperature(15 °C, 19 °C and 24 °C) on the air temperature in the food storage zone (mean value and standard deviation over 5 h of quasi steady state periods). The mean temperature at the thermostat position (𝑇 𝑡ℎ ) was slightly different for these three different ambient temperatures.

	An increase in ambient temperature causes also an increase in the frequency of
	compressor working cycles (shorter "off" period) in order to maintain the desired
	thermostat temperature. The number of compressor "on" cycles over a 5-hour
	period rose from 36 (for 15 °C) to 52 (for 24 °C). This implies an increase in the
	energy consumption of the display cabinets installed in stores without air-

To compare the results obtained under several ambient temperatures, temperature difference (𝑇 -𝑇 𝑡ℎ ) was shown in Table

3

.3. Averagely for all the positions 𝑇 -𝑇 𝑡ℎ increases from 1.1 to 2.0 °C when 𝑇 ̅ 𝑎𝑚𝑏 -𝑇 𝑡ℎ rises from 15.2 to 25.4 °C. In order to determine if the influence of the ambient temperature is linear, a dimensionless temperature was also calculated defined as

𝑇 𝑖 * = 𝑇 𝑖 -𝑇 𝑡ℎ 𝑇 ̅ 𝑎𝑚𝑏 -𝑇 𝑡ℎ

where i is a given position. It appears that 𝑇 𝑖 * is almost independent of 𝑇 ̅ 𝑎𝑚𝑏 -𝑇 𝑡ℎ . This is due to the linearity (between 𝑇 𝑎𝑚𝑏 and 𝑇 𝑖 ) since the forced convection and conduction are predominant heat transfer modes. In the closed configuration, free convection is negligible because of the rather high velocities at the DAG and PBP.

Radiation is also limited because the glass doors shield the radiation from the external walls in opposite to the open configuration. The highest value of 𝑇 * is at the position f (the front of the top shelf) where 𝑇 𝑓 -𝑇 𝑡ℎ is about 12% of 𝑇 ̅ 𝑎𝑚𝑏 -𝑇 𝑡ℎ . The lowest value of 𝑇 * is at the position i (the back of the bottom shelf) where the temperature is slightly higher than to the thermostat one.

conditioning systems in summer. It should be borne in mind that these experiments were carried out under closed conditions, with less ambient air infiltration into the cabinet than for an open cabinet.

Table 3 .3

 3 Effect of ambient temperatures on the internal air temperature of the closed display cabinet

	Ambient	Temperature	Number of
	temperature	difference	compressor
			"on"

cycles/5h Temperature difference and time-averaged dimensionless temperature at a given position

											All
	Tamb ± SD [°C]	Tamb-Tth [°C]			e	f	g	h	i	j	positions
	15 ± 0.5	15.2	36	T-Tth [°C] T * [-]	1.1 0.07	1.7 0.11	1.0 0.06	1.4 0.09	0.3 0.02	0.8 0.06	1.1 0.07
	19 ± 0.4	20.3	44	T-Tth [°C] T * [-]	1.5 0.08	2.4 0.12	1.2 0.06	1.8 0.09	0.4 0.02	1.1 0.06	1.4 0.07
	24 ± 0.3	25.4	52	T-Tth [°C] T * [-]	2.3 0.09	3.7 0.14	1.6 0.06	2.4 0.09	0.6 0.02	1.5 0.06	2.0 0.08
		All ambient temperatures T * [-]	0.08	0.12	0.06	0.09	0.02	0.06	0.07

Table 4 .1 Operating conditions in a closed refrigerated display cabinet Experiment Door-opening frequency Ambient temperature (STD)

 4

	Percentage of
	occupied volume

  Number of openings per hour per door

Table 4 .2

 4 Effect of lateral glass wall on the product core temperature in the closed display cabinet

	Plane	Time-averaged product core temperature † [°C] F1 F3 F5
	Left	3.0	2.3	1.5
	Center	2.0	1.4	0.8
	Right	3.6	3.5	3.3

Table 4 .

 4 3 Time-averaged product temperature † (in °C) at the back and front of the display cabinet with a door-opening frequency of 60 OPH and permanently open (without doors) condition for an ambient temperature of 19.4 °C and occupied volume of 60%.

			60 OPH			Permanently open
	Shelf no.	Back	Front	Back		Front
		Core	Core	Surface	Core	Core	Surface
	1 (top)	0.0	2.4	3.6	0.2	2.7	4.7
	2	-0.2	2.7	-	0.0	3.4	-
	3	-0.4	2.6	5.1	-0.3	3.6	8.7
	4	-0.4	1.7	-	-0.4	2.2	-
	5 (bottom)

Table 4 .4

 4 Effect of external ambient temperature (𝑇 𝑒 ) on the product core temperature (𝑇 𝑝 ) in the closed display cabinet.

	Temperature difference	𝑻 𝒆 -𝑻 𝒕𝒉	[°C]	15.0	20.0	25.2	30.4	Average
	Front product (pf)
	Shelf 1			2.0	2.6	3.6	4.4	3.1
	Shelf 2			1.7	2.3	3.1	3.8	2.7
	Shelf 3	𝑇 𝑝𝑓 -𝑇 𝑡ℎ [°C]	1.6	2.1	2.7	3.4	2.5
	Shelf 4			1.3	1.7	2.1	2.4	1.9
	Shelf 5			1.0	1.3	1.4	1.7	1.3
			Average	1.5	2.0	2.6	3.1	2.3
	Back product (pb)
	Shelf 1			0.6	0.9	1.2	1.4	1.0
	Shelf 2			0.6	0.8	0.9	1.1	0.8
	Shelf 3	𝑇 𝑝𝑏 -𝑇 𝑡ℎ [°C]	0.5	0.7	0.6	0.8	0.6
	Shelf 4			0.5	0.6	0.6	0.6	0.6
	Shelf 5			0.5	0.6	0.5	0.6	0.6
			Average	0.5	0.7	0.8	0.9	0.7

Table 5 .1

 5 Characteristics of the closed refrigerated display cabinet used in the study

	Dimensions of the storage space	1250 mm × 1345 mm × 525 mm
	(width × height × depth)
	Dimensions of shelves
	Shelves 1-4	1250 mm × 25 mm ×350 mm
	Shelf 5	1250 mm × 25 mm × 475 mm
	Dimensions of a test product package	200 mm × 50 mm × 100 mm
	Dimensions of a double-glazed door	620 mm × 1330 mm × 22 mm
	Discharge air grille width	50 mm
	Return air grille width	30 mm
	Dimensions of the perforated back panel	1250 mm × 1345 mm × 3 mm
	(width × height × thickness)
	Perforation percentage of the back panel	Shelf 1: 2.8%
	(based on total panel area)	Shelves 2-4: 3.8%
		Shelf 5: 8.3%
	Number of propeller fans	2

Table 5 .2

 5 General form of the governing equations(a)

	Equation	Mass	Momentum x-direction y-direction	Energy	k	ε
	𝜙	1	𝑢	𝑣		𝑇		𝑘	𝜀
	Γ 𝜙	0	𝜐 + 𝜐 𝑇	𝜐 + 𝜐 𝑇	𝜐 𝑃𝑟	+	𝜐 𝑇 𝑃𝑟 𝑇	𝜐 𝑇 𝜎 𝑘	𝜐 𝑇 𝜎 𝜀
	𝑆 𝜙	0	𝑆 𝑢	𝑆 𝑣		𝑆 𝑇	𝑆 𝑘	𝑆 𝜀
	(a)

Table 5 .3

 5 Inlet boundary conditions

						PBP
	Parameter	Unit DAG	Shelf	Shelf	Shelf	Shelf	Shelf
				1	2	3	4	5
	Air distribution percentage (a)	%	50	2	2	3	5	38
	Velocity	m•s -1	0.57	0.005	0.005	0.008	0.013	0.096
	Temperature (a)	°C	-0.1	-0.5	-0.5	-0.5	-0.6	-0.7
	Turbulent intensity (b)	%	3	5	5	5	5	5
	Length scale (c)	m	0.002 0.005	0.005	0.005	0.005	0.005
	(a) taken from the experimental data of Chaomuang et al. (2019a)
	(b)

  Secondly, it permits the prediction of influence of operating conditions and equipment design, which is rather cost-inefficient or impractical to perform by experiments. Extensive studies applied such model to open refrigerated display cabinets

Chapter VI -Simplified heat transfer modeling in steady state 124 phenomena.

Table 6 .1

 6 Thermal properties of materials

	Materials	Density [kg•m -3 ]	Heat capacity [J•kg -1 •K -1 ]	Thermal [W•m -1 •K -1 ] conductivity	Reference
	Air (at 0 °C)	1.287	1006	0.024	(Bergman et
	Aluminium	2700	910	237	al., 2011)
	Polyurethane foam	70	1045	0.026
	Glass pane	2500	750	1.4
	Methylcellulose	1071	3372	0.510	(Icier and
					Ilicali, 2005)
	Argon gas (at 20 °C)			0.017	(Annaratone,
					2010)

Table 6 .2

 6 Values of input parameters used in the heat transfer model of a closed refrigerated display cabinet

	Parameters	Value	Unit	Reference
	Mass flow rate
	Rear duct upstream (𝑚̇𝑡)	0.1120	kg•s -1	Measurement
	Gaps (𝑚̇𝑔 𝑎𝑝 )	0.0051	kg•s -1	Model fitting
	Air distribution coefficients †
	Air supply from PBP (𝛾 1..5 )	(0.02, 0.02, 0.04,	-	Measurement
		0.05, 0.36)
	Air supply at the air curtain	(0.90, 0.70, 0.70,	-	Model fitting
	(𝛿 1..5 )	0.70, 0.70)
	Heat transfer coefficients
	Convection
	Back load (𝑈 𝑙𝑏

Table 6 .3

 6 Values of the first top ten combinations of the input parameters with the mean absolute error (MAE) for the model fitting

	Parameter	Unit	1 †	2	3	4	5	6	7	8	9	10
	𝛿 1	-	0.9	0.9	0.8	0.8	0.9	0.9	0.8	0.9	0.3	0.9
	𝛿 2-5	-	0.7	0.7	0.7	0.7	0.7	0.7	0.7	0.7	0.7	0.7
	𝑈 𝑙𝑏	W•m 2 •K -1	15	15	15	15	15	16	15	15	15	15
	𝑈 𝑙𝑓	W•m 2 •K -1	24	25	23	24	26	24	25	27	22	23
	𝛿 𝑔𝑎𝑝	-	0.09	0.09	0.09	0.09	0.09	0.09	0.09	0.09	0.1	0.09
	MAE	°C	0.1472	0.1476	0.1478	0.1479	0.1481	0.1483	0.1484	0.1484	0.1485	0.1486
	† The combination of parameters used in the developed model in this study

Table 6 .4

 6 Effect of ambient temperature on load temperature at various positions in a closed refrigerated display cabinet

	𝑻 𝒆𝒙𝒕	𝑻 𝒕𝒉	𝑻 𝒍_𝒆𝒙𝒑 -𝑻 𝒕𝒉 [°C]
	[°C]	[°C]

Table 6 .5

 6 Error of air and load temperatures calculated from the developed model for different ambient temperatures compared to the measured values

	𝑻 𝒆𝒙𝒕
	[°C]

Mean absolute error [°C] Coefficient of determination † , 𝑹 𝟐 [-]

		Both	Load	Air	Both	Load	Air
	14.6	0.14	0.13	0.15	0.95	0.96	0.95
	19.5	0.15	0.14	0.16	0.96	0.97	0.97
	24.4	0.24	0.24	0.23	0.96	0.96	0.96
	Overall	0.17	0.18	0.17	0.96	0.96	0.95
	† The coefficient of determination is defined as 𝑅 2 =	[∑ (𝑇 𝑛𝑢𝑚,𝑖 -𝑇 𝑛𝑢𝑚 )(𝑇 𝑒𝑥𝑝,𝑖 -𝑇 𝑒𝑥𝑝 ) 𝑁 𝑖=1 ] ∑ (𝑇 𝑛𝑢𝑚,𝑖 -𝑇 𝑛𝑢𝑚 ) 𝑁 𝑖=1 2 ∑ (𝑇 𝑒𝑥𝑝,𝑖 -𝑇 𝑒𝑥𝑝 ) 2 𝑁 𝑖=1

  . The attenuated fraction (1 -𝛼 𝑑𝑦1 ) increases with the air/wall heat exchange intensity and decreases with the air flow rate.The pulsation independent effect (Ψ 𝑑𝑦1 ) is related to the increase of heat loss with a constant ambient temperature. The pulsation dependent effect, corresponding to the characteristic time (ξ 𝑑𝑦1 ), is proportional to the thermal inertia of the wall. This effect increases if the compressor on/off period decreases (i.e. 𝜔 increases).

		𝐆 𝑎𝑟𝑑1 = 𝐆 𝑎𝑠ℎ11 = 𝐆 𝑎𝑟𝑑5 • ( (1-𝛼 𝑑𝑦1 ) (1+Ψ 𝑑𝑦1 )+𝑖𝜔𝜉 𝑑𝑦1 1-𝛼 𝑠𝑑𝑦11 1+𝑖𝜔𝜉 𝑠𝑑𝑦11	+ 𝛼 𝑑𝑦1 + 𝛼 𝑠𝑑𝑦11 )	(7.15) (7.19)
	Similar equations were derived for all heat exchange zones in Domain I. 𝑚 𝑝𝑙1 𝐶 𝑝𝑝𝑙 where 𝜉 𝑠𝑑𝑦11 = 𝑚̇𝑎 𝑠ℎ1 𝐶 𝑝𝑎 (1-𝛼 𝑠𝑑𝑦11 )
	The implication of Eq. (7.15) shows that a part of the fluctuation of 𝜃 𝑡ℎ ′ (fraction
	𝛼 𝑑𝑦1 ) is not attenuated whereas the remaining part (fraction 1 -𝛼 𝑑𝑦1 ) is damped
	with a pulsation independent effect (Ψ 𝑑𝑦1 ) and a pulsation dependent effect
	ℎ 𝑖𝑛𝑡 𝑆 𝑤1 𝑚̇𝑡𝐶 𝑝𝑎 From Eq. (7.11) and (7.12), the heat transfer equation at EZ [1, 1] becomes: ) 𝑚 𝑤1 𝐶 𝑝𝑤 𝑑𝜃 𝑤1 (ξ 𝑑𝑦1 ) 7.2.2.2. Attenuation of temperature fluctuation in the shelves-space storage ′ 𝑑𝑡 = 𝑈 𝑜𝑣𝑤 ′ 𝑆 𝑤1 (𝜃 𝑒𝑥𝑡 -𝜃 𝑤1 ′ ) + 𝑚̇𝑡𝐶 𝑝𝑎 (1 -𝛼 𝑑𝑦1 )(𝜃 𝑡ℎ ′ -𝜃 𝑤1 ′ ) (7.13) (Domain II)
	Since 7.2.2.2.1. Heat exchange between the transversal air flow and the PBP (interface
	𝜃 𝑡ℎ ′ = Re(𝑒 𝑖𝜔𝑡 ) between Domain I and II) A part of the air flows into the shelf space through the perforated back panel (PBP)
	𝜃 𝑤1 ′ = Re(𝐆 𝑤1 𝑒 𝑖𝜔𝑡 ) of which the thermal inertia leads to a pulsation dependent attenuation of the air
	𝜃 𝑎𝑟𝑑1 ′ temperature fluctuation. The heat balance equation is developed in a similar manner = Re(𝐆 𝑎𝑟𝑑1 𝑒 𝑖𝜔𝑡 ) to the heat exchange in the vertical rear duct, except that there is no exchange with
	Eq. (7.13) is reduced to the external ambient air; therefore, there is no pulsation independent attenuation.
	𝑚 𝑤1 𝐶 𝑝𝑤 Considering EZ [2, 1, 1], the heat balance equations are given by 𝑑 𝑑𝑡 (𝐆 𝑤1 𝑒 𝑖𝜔𝑡 )
		= -𝑈 𝑜𝑣𝑤 ′ 𝑚 𝑝𝑙1 𝐶 𝑝𝑝𝑙 𝑆 𝑤1 (𝐆 𝑤1 𝑒 𝑖𝜔𝑡 ) + 𝑚̇𝑡𝐶 𝑝𝑎 (1 -𝛼 𝑑𝑦1 )(𝑒 𝑖𝜔𝑡 -𝐆 𝑤1 𝑒 𝑖𝜔𝑡 ) 𝑑𝜃 𝑝𝑙1 ′ 𝑑𝑡 = 𝑚̇𝑎 𝑠ℎ1 𝐶 𝑝𝑎 (𝜃 𝑎𝑟𝑑5 ′ -𝜃 𝑎𝑠ℎ11 ′ ) (7.16)
	thus,	𝜃 𝑎𝑠ℎ11 ′	= 𝛼 𝑠𝑑𝑦11 𝜃 𝑎𝑟𝑑5 ′	+ (1 -𝛼 𝑠𝑑𝑦11 )𝜃 𝑝𝑙1 ′	(7.17)
	where 𝛼 𝑠𝑑𝑦11 = exp (-	𝐆 𝑤1 = ℎ 𝑖𝑛𝑡 𝑆 𝑝𝑙1 𝑚̇𝑎 𝑠ℎ1 𝐶 𝑝𝑎 )	1 (1+Ψ 𝑑𝑦1 )+𝑖𝜔𝜉 𝑑𝑦1	(7.14)
	where Ψ 𝑑𝑦1 = Eq. (7.16) and (7.17) yield 𝑈 𝑜𝑣𝑤 ′ 𝑆 𝑤1 𝑚̇𝑡𝐶 𝑝𝑎 (1-𝛼 𝑑𝑦1 )	and 𝜉 𝑑𝑦1 =	𝑚 𝑤1 𝐶 𝑝𝑤 𝑚̇𝑡𝐶 𝑝𝑎 (1-𝛼 𝑑𝑦1 )
				𝐆 𝑝𝑙1 = 𝐆 𝑎𝑟𝑑5 • ( 1+𝑖𝜔𝜉 𝑠𝑑𝑦11 1	)	(7.18)
						167

Replacing Eq. (7.14) into Eq. (7.12) yields Chapter VII -Simplified heat transfer modeling in unsteady state

7.20) and (7.21) yield

  For Shelf 2-5, there are only two air streams mixing above the front load including the air from the back (mass flow rate, 𝑚̇𝑎 𝑠ℎ2 ) and a part of the air curtain (mass

	𝑚 𝑎𝑠ℎ1 ′	𝐆 𝑎𝑠ℎ31 = 𝑚̇𝑎 𝑟𝑑5 𝐆 𝑎𝑟𝑑7 + 𝑚̇𝑎 𝑠ℎ1 𝐆 𝑎𝑠ℎ21 + 𝛿 1 𝑚 𝑎𝑠ℎ1 ′	𝐆 𝑎𝑠ℎ51 (7.24)
	flow rate, 𝛿 2 𝑚̇𝑎 𝑟𝑑4 ). Considering MZ [2, 1, 2], the heat balance equation is given
	by
		𝑚 𝑎𝑠ℎ2 ′	𝐆 𝑎𝑠ℎ32 = 𝑚̇𝑎 𝑠ℎ2 𝐆 𝑎𝑠ℎ22 + 𝛿 2 𝑚̇𝑎 𝑟𝑑4 𝐆 𝑎𝑠ℎ41	(7.25)
	Analogous analysis of heat balance at MZ [2, 1, 2] was applied to Shelf 3 to 5.
	7.2.2.2.4. Heat exchange between the mixed air and the front load
	The heating due to light is constant and it does not contribute to temperature
	fluctuations. The heat balance at EZ [2, 3, 1] is reduced to
	Section 7.3.
	Considering EZ [2, 2, 1], the heat balance equations are given by 𝑚 𝑙𝑓1 ′ 𝐶 𝑝𝑙 𝑑𝜃 𝑙𝑓1 ′ 𝑑𝑡 = 𝑚̇𝑎 𝑠ℎ1 ′ 𝐶 𝑝𝑎 (𝜃 𝑎𝑠ℎ31 ′ -𝜃 𝑎𝑠ℎ41 ′ )	(7.26)
		𝑚 𝑙𝑏1 ′ 𝐶 𝑝𝑙 𝜃 𝑎𝑠ℎ41 ′	𝑑𝜃 𝑙𝑏1 ′ 𝑑𝑡 = 𝛼 𝑠ℎ21 𝜃 𝑎𝑠ℎ31 = 𝑚̇𝑎 𝑠ℎ1 𝐶 𝑝𝑎 (𝜃 𝑎𝑠ℎ11 ′ ′ + (1 -𝛼 𝑠ℎ21 )𝜃 𝑙𝑓1 -𝜃 𝑎𝑠ℎ21 ′ ′	)	(7.20) (7.27)
	where 𝑚 𝑙𝑏1 ′ where 𝛼 𝑠ℎ21 = exp (-= 𝜌 𝑙 𝑆 𝑙 𝑒 𝑒𝑞	𝑈 𝑙𝑓1 𝑆 𝑙1 𝑚̇𝑎 𝑠ℎ1 ′ 𝐶 𝑝𝑎	)
		𝜃 𝑎𝑠ℎ21 ′ 𝐆 𝑙𝑓1 = 𝐆 𝑎𝑟𝑑31 • ( = 𝛼 𝑠ℎ11 𝜃 𝑎𝑠ℎ11 ′ + (1 -𝛼 𝑠ℎ11 )𝜃 𝑙𝑏1 ′ 1 1+𝑖𝜔𝜉 𝑠𝑑𝑦31 )	(7.21) (7.28)
	where 𝛼 𝑠ℎ11 = exp (-𝐆 𝑎𝑠ℎ41 = 𝐆 𝑎𝑠ℎ31 • ( 𝑈 𝑙𝑏1 𝑆 𝑙1 𝑚̇𝑎 𝑠ℎ1 𝐶 𝑝𝑎 )	1-𝛼 𝑠ℎ21 1+𝑖𝜔𝜉 𝑠𝑑𝑦31	+ 𝛼 𝑠ℎ21 )	(7.29)
	Eq. (𝐆 𝑙𝑏1 = 𝐆 𝑎𝑠ℎ11 • ( 𝑚 𝑙𝑓1 ′ 𝐶 𝑝𝑙 where 𝜉 𝑠𝑑𝑦31 = 𝑚̇𝑎 𝑠ℎ1 ′ 𝐶 𝑝𝑎 (1-𝛼 𝑠ℎ21 )	1 1+𝑖𝜔𝜉 𝑠𝑑𝑦21	)	(7.22)
		𝐆 𝑎𝑠ℎ21 = 𝐆 𝑎𝑠ℎ11 • (	1-𝛼 𝑠ℎ11 1+𝑖𝜔𝜉 𝑠𝑑𝑦21	+ 𝛼 𝑠ℎ11 )	(7.23)
	where 𝜉 𝑠𝑑𝑦21 =	𝑚 𝑙𝑏1 ′ 𝐶 𝑝𝑙 𝑚̇𝑎 𝑠ℎ1 𝐶 𝑝𝑎 (1-𝛼 𝑠ℎ11 )
	7.2.2.2.3. Air mixing above the front load
	For Shelf 1, there are three air streams mixing above the front load including air
	from the DAG (mass flow rate, 𝑚̇𝑎 𝑟𝑑5 ), air from the back (mass flow rate, 𝑚̇𝑎 𝑠ℎ1 )
	and a part of the recirculating air (mass flow rate, 𝛿 1 𝑚 𝑎𝑠ℎ1 ′	). Considering MZ [2,
	1, 1], the heat balance equation is given by
	Chapter VII -Simplified heat transfer modeling in unsteady state	169

  |∆𝑇 𝑡ℎ,𝑘 | and 𝜑 𝑡ℎ,𝑘 are the magnitude and the phase shift related to the k-th complex Fourier's transform coefficient. 𝑁 is the number of measured data during a period of the compressor operation (𝜏). The coefficient 𝑘 = 1 represents the fundamental 𝑇 𝑖 is the average temperature at a given position 𝑖 retrieved from the static aspect of the model (presented in Part I) and Θ 𝑖,𝑘 and 𝜑 𝑖,𝑘 is the magnitude and the phase shift of that position obtained from the solution of the dynamic aspect

	pulsation while the other coefficients 𝑘 = 2, 3, … ,	𝑁 2	represent the harmonic
	pulsations.
	Based on Eq. (7.39), the temperature variations at other positions in the closed
	display cabinet can be estimated by
	𝑇 𝑖 (𝑡) = 𝑇 𝑖 + ∑	𝑁/2 𝑘=1	Θ 𝑖,𝑘 • |∆𝑇 𝑡ℎ,𝑘 |	cos(𝑘𝜔𝑡 + 𝜑 𝑡ℎ,𝑘 + 𝜑 𝑖,𝑘 )	(7.40)
	where
			.2). Discrete Fourier Transform was
	therefore applied to decompose the temperature profile of the supply air in the
	following form:
	𝑇 𝑡ℎ (𝑡) = 𝑇 𝑡ℎ + ∑ |∆𝑇 𝑡ℎ,𝑘 | 𝑁/2 𝑘=1	cos(𝑘𝜔𝑡 + 𝜑 𝑡ℎ,𝑘 )	(7.39)

explained in Section 7.2.2.

  This equation shows that the equivalent thickness (𝑒 𝑒𝑞 ) depends not only on the thermal diffusivity (𝜅 𝑙 ) and the angular frequency (𝜔) but also the convective heat transfer coefficient between the load surface and the air (ℎ 𝑎 ). For the sake of simplicity, it can be assumed that

			2			1/2	(7.46)
		√ 𝜆 𝑙 𝜌 𝑙 𝐶 𝑝𝑙 𝜔/2 ℎ 𝑎	)	+	𝜆 𝑙 𝜌 𝑙 𝐶 𝑝𝑙 𝜔 2ℎ 𝑎 2	]
	Let the temperature fluctuations obtained from the solutions of the two models (Eq.
	(7.43) and (7.46)) have the same amplitude, the equivalent thickness can be
	estimated by
	𝑒 𝑒𝑞 = [	ℎ 𝑎 𝜌 𝑙 𝐶 𝑝𝑙 𝜔 √ 𝜌 𝑙 𝐶 𝑝𝑙 𝜔 𝜆 𝑙 /2	+	𝜆 𝑙 𝜌 𝑙 𝐶 𝑝𝑙 𝜔	1/2 ]	(7.47)
		√𝜆 𝑙 𝜌 𝑙 𝐶 𝑝𝑙 𝜔 ℎ 𝑎	≫ 1 or Ε√𝜔 ≫ ℎ 𝑎 where Ε =
	√ 𝜆 𝑙 𝜌 𝑙 𝐶 𝑝𝑙 is the thermal effusivity or heat penetration coefficient. Based on a period
	of compressor working cycle (340𝑠 < 𝜏 < 550𝑠) and the thermal properties of the

methylcellulose, the order of magnitude of Ε√𝜔 is about 145-185 W•m -2 •K -1 which is high compared to the estimated convective heat transfer coefficients (15 W•m -

Table 7 .1

 7 Weights and thermal mass of the load and the solid components of the display cabinet used in the model

	Components	Weight kg	Thermal mass J•K -1	References
	Load (𝑚 𝑙 )
	𝑚 𝑙𝑓 ′ = 𝑚 𝑙𝑏 ′	1.44	4855.7	Calculation
	Rear walls (𝑚 𝑤 )
	𝑚 𝑤1	0.29	263.9	Calculation
	𝑚 𝑤2 = 𝑚 𝑤3 = 𝑚 𝑤4 = 𝑚 𝑤5	0.42	382.2	Calculation
	𝑚 𝑤6	0.21	191.1	Calculation
	Ceiling
	𝑚 𝑐	0.68	618.8	Calculation
	Doors (𝑚 𝑑 )
	𝑚 𝑑1 = 𝑚 𝑑2 = 𝑚 𝑤3 = 𝑚 𝑑4	3.13	2347.5	Calculation
	𝑚 𝑑5	4.31	3232.5	Calculation
	Perforated back panel (𝑚 𝑝𝑙 )
	𝑚 𝑝𝑙1	2.47	2247.7	Calculation
	𝑚 𝑝𝑙2 = 𝑚 𝑝𝑙3 = 𝑚 𝑝𝑙4	2.43	2211.3	Calculation
	𝑚 𝑝𝑙5	3.20	2912.0	Calculation

†Average (top package on the stack) over 5 h of quasi-steady state excluding the defrost period.

and 𝑈 𝑙𝑓 =

W•m -2 •K -1 (Part I of the study), 𝐵𝑖 = 5.9 and 𝐵𝑖 = 9.4 are determined for

•K -1 and 24 W•m -2 •K -1 for the back and front loads, respectively). Therefore, the

Firstly, I gratefully acknowledge the financial support provided by the King Mongkut's Institute of Technology Ladkrabang (KMITL, Thailand), the National Research Institute of Science and Technology for Environment and Agriculture (Irstea, France) and the Embassy of France in Thailand through the Franco-Thai scholarship program.

A steady-state 2D-CFD model with the standard k-ε turbulence model was also developed in this study to investigate the airflow and its influence on the thermal distributions in a closed display cabinet. Qualitatively, good agreement was found between numerical predictions and experimental results. Despite its simplifications, the model has shown the ability to reproduce the main flow phenomena observed by the experiments. The predicted product temperature profile trend also agreed with the experimental data. In future studies, the developed model will be used to examine the influences of unexplored design parameters and the operating conditions on the air and product temperature distributions, e.g. airflow rate, air temperature at the DAG and product occupied volume.

Even though a specific display cabinet was used, the results provide general information on the phenomena taking place in the display cabinet. The presence of doors causes air recirculation which could induce the infiltration of external warm air through the gaps along the door frames. Sealing these gaps would certainly mitigate this issue. Nonetheless, the design of closed display cabinets should probably be revisited. For example, the air curtain might not be necessary due to the replacement of the physical barriers. [START_REF] East | Potential for energy cost savings by utilising alternative temperature control strategies for controlled atmosphere stored apples[END_REF][START_REF] Jol | Issues in time and temperature abuse of refrigerated foods[END_REF][START_REF] Platenius | Effect of temperature on the rate of deterioration of fresh vegetables[END_REF][START_REF] Tano | Comparative evaluation of the effect of storage temperature fluctuation on modified atmosphere packages of selected fruit and vegetables[END_REF]. Nevertheless, periodical temperature fluctuation is unavoidable because of automatic "on/off" cycles of compressor operation of refrigeration system. Knowledge of spatial and temporal equivalent thickness can be simply calculated from the following equation:

and the ratio of the temperature fluctuation between load surface and air is given by

Based on the compressor period and the material thermal properties mentioned earlier, the temperature fluctuation ratio calculated from Eq. (7.49) was approximately 0.09 for the back load.

Case study

To verify the accuracy of the estimation of the equivalent thickness by using Eq.

Effect of air humidity in a closed display cabinet on product quality

For unwrapped products such as fruits and vegetables displayed in the cabinet, air humidity is a determining factor of their quality (freshness, color, etc.). High ambient humidity allows better food preservation; however, it leads to more frost formation on the evaporator, thus, less energy efficiency. Optimization of these two criteria should be undertaken by modification of design and operating conditions.

Effect of air humidity in a test room on energy consumption

The scope of the present study was to address the thermal performance of closed refrigerated display cabinets. The frequency of "on/off" compressor regulation was only used as an indicator of energy performance. Measurements of electrical energy consumption along with temperature measurements under different operating conditions would provide additional useful information.

Air infiltration rate during door openings

Air infiltration during door openings is an important factor determining the overall performance of closed refrigerated display cabinets. Implementation of a tracer gas technique to quantify the air infiltration rate for different opening frequencies would provide additional useful information, which can be further used in numerical simulations and design improvements.

Modeling

A 2D-CFD and a simplified heat transfer models based on a zonal approach were developed in the closed-door condition i.e. the air infiltration due to door openings was not considered. Because of this limitation, suggestions are therefore proposed for future work. Finally, a simplified heat transfer model was developed based on a zonal approach in both static and dynamic regimes which permits, respectively, the predictions of time-averaged air and product temperatures and temperature fluctuations according to the on/off cycle of the compressor regulation. The dynamic model was solved with a spectral approach, thus the influence of different parameters on the damping of the temperature fluctuations can be identified.

CFD simulation