M Jean-Marc

Rapporteur M Commenge

BOUGEARD Rapporteur Daniel

Kamelia Mme

Examinatrice M Christophe Boodhoo

Examinateur M Michel Gourdon

Directeur Cabassud

Nathalie De Thèse Mme

Directrice Raimondi

M De Thèse

Invité Fletcher David

This thesis is about the thermal characterization of compact heat-exchanger (HEX) reactors.

The objective is to measure heat transfer coefficients in these devices and understand the effect of intensification of heat transfer related to their geometry. The compact HEX reactor prototypes used are composed of three plates: two plates called process plate and utility plate where millimetric square straight or zigzag channels are engraved, separated by an aluminum plate. The heat transfer coefficient is measured experimentally and numerically. The influence of channel geometry, fluid properties and flow rates on heat transfer is discussed. An experimental method for measuring the local temperature is implemented, based on laserinduced fluorescence. Finally, the effect of scale-up of these devices for the implementation of competitive exothermic reactions is studied.

Résumé

Cette thèse porte sur la caractérisation thermique de réacteurs échangeurs de chaleur compacts.

Le but est de mesurer des coefficients de transfert thermique dans ces appareils et de comprendre l'effet d'intensification du transfert thermique lié à leur géométrie. Des prototypes de réacteurs échangeurs sont utilisés, composés de trois plaques : deux plaques dites « procédé » et « utilité » où sont gravés des canaux millimétriques droits ou en zigzag de section carrée, séparées par une plaque en aluminium. Le coefficient de transfert de chaleur est déterminé expérimentalement et numériquement. L'influence de la géométrie du canal, des propriétés des fluides et des débits sur le transfert de chaleur est discutée. Une méthode expérimentale de mesure de la température locale est mise en oeuvre, basée sur de la fluorescence induite par laser. Enfin, l'effet d'un changement d'échelle de ces appareils pour la mise en oeuvre de réactions compétitives exothermiques est étudié.

Introduction

The concept of process intensification emerged 40 years ago. Although the initial aim was to reduce the cost of units by reducing the size, there are other obvious benefits from size reduction, such as process safety, energy efficiency and environmental impact. Indeed the industrial pollution is nowadays singled out for its significant contribution to climate change, terrestrial and aquatic ecosystem degradation and public health deterioration. Therefore the industries have to modify their processes or develop new ones to insure their sustainability. The concept of the "factory of the future" has evidently emerged recently due to the significant technological progresses in terms of equipment manufacturing techniques, digitalization, robotics… The objectives of the factory of the future are to propose sustainable processes where the human work is facilitated and secured and that participate to the safe and clean territorial reindustrialization.

The intensification of the chemical processes requires the use of performant, flexible and compact technologies. For reaction engineering, numerous devices have been developed over the past decade such as compact heat-exchanger (HEX) reactors. They are miniaturized and multi-functionalized equipment with many advantages such as good heat transfer performances, better temperature control and reactive volume confinement. Despite the very successful demonstrations of these devices for the implementation of highly exothermic reactions, their use in industry is still marginal. The main obstacle to develop these technologies is the low capacity of production. Scale-up is then required to increase the capacity of the intensified reactors. For this step, the main challenge is the determination of scale-up laws to be able to estimate the size of the device that will provide the best compromise between production increase and performance degradation. The final objective will then be to guide the choice between different scale-up strategies based on device numbering-up and/or scaling-up.

This thesis is about the thermal characterization of compact HEX reactors with zigzag channels.

The objective is to measure the heat transfer coefficient in these devices by different techniques (experimental and numerical) and to evaluate the effect of intensification of heat transfer related to their geometry. The influence of the channel geometry in terms of hydraulic diameter and straight length between bends on the heat transfer coefficient has been particularly studied. The characterization methods have been adapted to compact devices to correlate the heat transfer coefficient in terms of fluid properties, channel geometry and operating conditions. viii In Chapter I, the concept of process intensification is firstly introduced. Some intensified reactors are presented, in particular the compact HEX reactors. Their global thermo-hydraulic performances are described and possible methods for the characterization proposed in the literature are discussed.

In Chapter II, the experimental method for thermal performance characterization of compact HEX reactors is described and discussed, based on temperature measurement by thermocouples at the channel inlet and outlet. A correlation of Nusselt number is proposed in terms of Reynolds number, Prandtl number and geometric parameters.

In Chapter III, a numerical method based on computational fluid dynamics (CFD) simulation is used to study the flow and heat transfer. It allows the acquisition of local information like velocity and temperature fields which can be used to analyze and understand the overall performances of compact HEX reactors.

In Chapter IV, an experimental technique based on laser-induced fluorescence thermometry with two-color/two-dyes is developed for local temperature measurement in compact HEX reactors. The accuracy of the method is discussed in comparison with the temperature calculated from a heat transfer model.

In the last chapter, a 1D plug flow model is used to evaluate different scale-up strategies of compact HEX reactors with zigzag channels. Conversion, selectivity and hot point are determined as a function of the channel hydraulic diameter for two reaction cases: temperature favors the formation of the main product and temperature favors the formation of the by-product.

Chapter I: Bibliography 1. Process intensification

The concept of process intensification emerged in the late 1970s, introduced by Ramshaw in the company ICI (UK) with the simple goal of reducing the plant capital cost by reducing the size or volume of the process plant components. In 1995, Ramshaw [START_REF] Ramshaw | Design Method for Plate Evaporators and Condensers[END_REF] made a definition of process intensification as a strategy for making dramatic reductions in the size of a chemical plant so as to reach a given production objective. This definition is quite narrow, for example the decrease of energy consumption and waste can also be considered as process intensification.

This concept was then broadened by Stankiewicz and Moulijn [START_REF] Stankiewicz | Process Intensification: Transforming Chemical Engineering[END_REF]: « any chemical engineering development that leads to a substantially smaller, cleaner, and more energy efficient technology is process intensification ».

At the beginning, process intensification is merely driven by the economic interest of reducing the equipment size in the plant to reduce the capital cost. However, the size reduction of equipment is often coupled to the reduction of the total amount of chemicals (toxic, flammable...) present in the production plant (from the raw materials to the final products) which reduces the inherent risk of the process. Then the reduction of the devices also brings flexibility of production and an easy control of the industrial processes. Of course, size reduction can also enhance mass transfer and/or heat transfer performances of the devices by increasing the relative importance of fluxes through surfaces (interfacial area, heat exchange area…) compared to the production of mass and/or heat in the volumes. In reaction engineering, this can make possible to reach higher yield and selectivity for transfer limited reactive processes without changing the chemical route of synthesis. Reduction in the energy consumption of separation operations can then be expected. Inspired by these advantages, lots of process intensification projects have been carried out in this century, such as the project of Rhodia [START_REF] Neveu | Potential benefits of process intensification for Rhodia[END_REF] and the 'project House' of Degussa [START_REF] Hahn | Market pull meets technology push[END_REF].

Many routes have been considered in recent years to intensify processes. Stankiewicz and Moulijn [START_REF] Stankiewicz | Process Intensification: Transforming Chemical Engineering[END_REF] divided them into two parts: by the use of intensified equipment (novel equipment for intensifying reaction, mixing and heat exchange) and/or intensified methods (integration of reaction with other unit operations, development of new hybrid separations and application of alternative energy sources in processes) as presented in [START_REF] Stankiewicz | Process Intensification: Transforming Chemical Engineering[END_REF] Following this classification, Ponce-Ortega et al. [START_REF] Ponce-Ortega | Chemical Engineering and Processing : Process Intensification Process intensification : New understanding and systematic approach[END_REF] presented a new process intensification classification as they found some "limitations" in the previous works that focused on unit-based intensification, not overall process-plant intensification: the strong interactions among unit operations within the process were ignored and the effect of one unit intensification on the rest of the process was not considered. Ponce-Ortega et al. firstly provided new definition of process intensification that lies in one or more of the following:

• Smaller equipment for given throughput, producing the same throughout using smaller equipment or less volume than commonly used today.

• Higher throughput for given equipment size or given process, producing higher throughput using the same equipment or process.

• Less holdup for equipment or less inventory for process of certain material for the same throughput, reducing holdup or inventory of certain material for the same production.

• Less usage of utility materials (heating or cooling utilities, solvents…) and feedstock for a given throughput and given equipment size.

• Higher performance for given unit size, improving unit performance (heat duty of heat exchanger, yield of reactor) for given unit size and throughput. [START_REF] Ponce-Ortega | Chemical Engineering and Processing : Process Intensification Process intensification : New understanding and systematic approach[END_REF] Gerven and Stankiewicz [START_REF] Van Gerven | Structure, Energy, Synergy, Time-The Fundamentals of Process Intensification[END_REF] presented a more comprehensive and fundamental view on process intensification (as shown in Figure I-3), which includes the generic principles and different approaches to realize these principles at all scales, as they found that most of the definitions only partly grasped the complete picture of process intensification. They defined four principles of process intensification are as follows:

Principle 1: Maximize the effectiveness of intra-and intermolecular events. This principle is primarily about changing reaction kinetics. Process intensification should look for the engineering methods to better control the factors that affect the reaction kinetics.

Principle 2: Give each molecule the same processing experience. This principle is not only about macroscopic residence time distribution, dead zones, or bypassing but also meso-and micromixing as well as temperature gradients. A plug-flow reactor with gradientless, volumetric heating is very close to the idea of this principle. Principle 3: Optimize the driving forces at every scale and maximize the specific surface area to which these forces apply. This principle is about the transport rates across interfaces.

Increased transfer areas (or surface-to-volume ratios) can be obtained by changing the length scale, e.g., from the millimeter to the micrometer scales. Principle 4: Maximize the synergistic effects from partial processes. Synergistic effects should be sought and utilized at all possible scales. Most commonly such utilization is the multifunctionality on the macroscale.

They distinguished four approaches in four domains: spatial, thermodynamic, functional, and temporal, which can realize the four principles presented above. They named these four approaches "structure", "energy", "synergy" and "time". They furtherly divided the process intensification into three scales: molecular scale (molecular collisions and diffusion), mesoscale (particles, bubbles, films, phases) and macroscale (processing units and plants).

Presentation of several intensified reactors

In chemical engineering, the reactor is the most important part in the process since it is the unit for carrying out chemical reactions and for the synthesis of various products. In this part, several continuous intensified reactors are presented which draw great concern in the process intensification. The technologies described are chosen in agreement with those identified in Figure I-1. They are rotating packed bed reactors, spinning disk reactors and microreactors, with different process intensification strategies. Static mixers are also included in this descriptive part as they are often used upstream of reactors for efficient reactant mixing. Finally heat-exchanger reactors will be presented more in details as they constitute the object of this thesis work.

Rotating packed bed reactors

Rotating packed bed reactors have drawn great attention as a method for process intensification in the last three decades. This technology (HiGee) was developed by Ramshaw and his coworkers [START_REF] Reay | Process Intensification: Engineering for Efficiency, Sustainability and Flexibility[END_REF] in the late 1970s at ICI, UK, which showed excellent mass transfer performances.

Since then, many researches on rotating packed bed reactors have been carried out and many applications have been installed in the industry for various productions, among which the nanoparticles are the most popular. Kang et al. [START_REF] Kang | Efficient preparation of monodisperse CaCO3 nanoparticles as overbased nanodetergents in a high-gravity rotating packed bed reactor[END_REF] presented a novel mass-transfer intensified approach for efficient preparation of monodisperse CaCO3 nanoparticles as overbased nanodetergents with rotating packed bed reactors. Fan et al. [START_REF] Fan | Continuous preparation of Fe3O4 nanoparticles through Impinging Stream-Rotating Packed Bed reactor and their electrochemistry detection toward heavy metal ions[END_REF] reported a continuous technique for the preparation of Fe3O4 nanoparticles using impinging stream-rotating packed bed reactor.

Lin and his coworkers reported the preparation of CuO nanoparticles and ZnO nanoparticles by the precipitation method in rotating packed bed reactors [START_REF] Lin | Production of CuO nanoparticles using a simple precipitation method in a rotating packed bed with blade packings[END_REF][START_REF] Lin | Preparation of ZnO nanoparticles using a rotating packed bed[END_REF].

Rotating packed bed reactors consist of an annular cylindrical rotor with a packed bed and a static housing. The rotor is driven by the motor, generating the centrifugal force which can tremendously improve micromixing as well as heat transfer. The schematic of a rotating packed bed reactor for liquid-liquid processes is shown in Figure I-4 [START_REF] Wenzel | Review and analysis of micromixing in rotating packed beds[END_REF]. The rotor is the most important part in rotating packed bed reactors which can be made of various packing types, like gas spheres, solid foam, and discs [START_REF] Visscher | Rotating reactors -A review[END_REF]. Rotating packed bed reactors are normally operated for gas-liquid processes (counter-currently) and liquid-liquid processes (co-currently).

Figure I-4: Schematic of a rotating packed bed reactor for liquid-liquid processes [START_REF] Wenzel | Review and analysis of micromixing in rotating packed beds[END_REF] Zhao et al. [START_REF] Zhao | High-gravity process intensification technology and application[END_REF] reviewed the advances on the high gravity (Higee) technology in rotating packed bed reactors, including the studies on liquid flow, gas-liquid mass transfer, residence time distribution and micromixing, and its applications in absorption, stripping, nanoparticles preparation, sulfonation, polymerization and emulsification. Wenzel and Górak [START_REF] Wenzel | Review and analysis of micromixing in rotating packed beds[END_REF] made a review on the fundamentals of liquid micromixing in rotating packed bed reactors and delivered an assessment of micromixing based on the Villermaux-Dushman protocol. Shi et al. [START_REF] Shi | CFD analysis of liquid phase flow in a rotating packed bed reactor[END_REF] performed numerical study on the liquid phase flow patterns in rotating packed bed reactors.

Neumann et al. [START_REF] Neumann | A guide on the industrial application of rotating packed beds[END_REF] made a review of industrial rotating packed bed reactor applications and introduced a guide (distinctive decision trees) to facilitate the identification of processes where it is beneficial to use rotating packed bed reactors.

Spinning disk reactor

Spinning disk reactor (SDR) is another type of reactor that uses Higee technology. Spinning disk reactor generally has a working disk with a diameter between 60 mm and 500 mm which can be heated or cooled (as shown in Figure I-5), it is connected to a motor which can generate centrifugal and shear forces and form a thin liquid film (20-300 μm). This can create a high surface-to-volume ratio and gives a heat transfer coefficient up to 25 kW.m -2 .K -1 [START_REF] Jacobsen | Micromixing Efficiency of a Spinning Disk Reactor[END_REF]. The rotational speed ranges from 100 to 6000 rpm (typically around 1000 rpm) [START_REF] Jachuck | Process Intensification for Responsive Processing[END_REF]. The reactants (process fluid) are introduced in the center of the disk, they travel across the surface and form a thin liquid film in the surface of the disk. The liquid leaves the edge of the disk and can be collected. The heat transfer fluid is introduced from the below of the disk and passes countercurrently with the process fluid, as presented in the following figure. [START_REF] Oxley | Evaluation of Spinning Disk Reactor Technology for the Manufacture of Pharmaceuticals[END_REF] Spinning disk reactor has the characteristics like intense mixing in the thin liquid film, short residence time, plug flow behavior and high solid/liquid, liquid/vapour heat and mass transfer [START_REF] Jachuck | Process Intensification for Responsive Processing[END_REF]. These properties allow the wide use of spinning disk reactor in chemical reactions.

Boodhoo and Jachuck [START_REF] Boodhoo | Process intensification: spinning disk reactor for styrene polymerisation[END_REF] studied the styrene polymerization (highly exothermic reaction) in a spinning disk reactor, their results showed SDR can significantly enhance the reaction rate, improve product quantity and inherent safety. Oxley et al. [START_REF] Oxley | Evaluation of Spinning Disk Reactor Technology for the Manufacture of Pharmaceuticals[END_REF] evaluated a spinning disk reactor for the manufacture of pharmaceuticals, they found SDR technology displayed many advantages over traditional batch processes in terms of reaction time, inventory, product purity and particle size distribution. Qiu et al. [START_REF] Qiu | Biodiesel synthesis in an intensified spinning disk reactor[END_REF] studied experimentally and numerically the synthesis of biodiesel in a two-disk spinning disk reactor and investigated the effects of disk speed, volumetric flowrate, temperature, disk design, etc. to the reactor performance. There are also many applications of SDR in the preparation of nanoparticles, such as AgI, TiO2, and BaSO4 [START_REF] Liu | Characterization of AgI nanoparticles synthesized in a spinning disk reactor[END_REF][START_REF] Mohammadi | Synthesis of TiO2 nanoparticles in a spinning disc reactor[END_REF][START_REF] Bagheri Farahani | Experimental investigation and process intensification of barium sulfate nanoparticles synthesis via a new double coaxial spinning disks reactor[END_REF].

Microreactors

Microreactors gained lots of interest in last two decades due to their excellent mixing and heat transfer performances. There are many researches on microreactors and their application for chemical synthesis carried out at lab-scale as well as at industrial-scale. Many industrial units are based on micro-structured devices in the chemical and pharmaceutical industry. The first company who installed a production plant based on microreactors is Merck, Germany, in 1998, for a organometallic reaction based on a carbonyl compound and a metallo-organic reagent [START_REF] Krummradt | Experiences with the use of microreactors in organic synthesis[END_REF].

The yield of this reaction at pilot-scale (as shown in Figure I-6) is 20% better than in their former batch process. A microreactor is a device with at least one characteristic dimension below 1 mm. It has a high ratio of surface area (for heat exchange) to reaction volume due to the size reduction, its magnitude is 10000 m 2 .m -3 [START_REF] Moreau | Methodologies pour la caracterisation hydrodynamique et l'extrapolation de réacteurs intensifiés millistructurés[END_REF]. Consequently, the mass transfer and heat transfer are significantly improved in microreactors and the operating conditions like temperature are well controlled in the resulting confined fluid media. Highly exothermic reactions can be carried out safely in microreactors. However, the main drawbacks of microreactors are the low reaction volume leading to low production capacity and fouling and clogging issues as microchannels are too small. Merck company designed wider channel with mixing elements for the reaction mentioned in the last paragraph, keeping good heat transfer performance and preventing the system from clogging [START_REF] Krummradt | Experiences with the use of microreactors in organic synthesis[END_REF]. It is applied at the pilot-scale production plant with parallel microreactors to ensure the desired production (as shown in Figure I-6).

Microreactors can be manufactured in different materials, including ceramics, polymers, stainless steel, glass and silicon [START_REF] Yao | Review of the applications of microreactors[END_REF]. There are many studies on the designing of microreactors with the purpose of better controlling the reaction parameters. Single phase reactions as well as multiphase reactions (gas-liquid, liquid-liquid, gas-liquid-solid) can be carried out in microreactors. Yao et al. [START_REF] Yao | Review of the applications of microreactors[END_REF] made a review of the recent microreactors structure and their applications on the synthesis of nanoparticles, organics, polymers and biosubstances in the last decades. Doku et al. [START_REF] Doku | On-microchip multiphase chemistry: a review of microreactor design principles and reagent contacting modes[END_REF] made a review of microreactor design principles and reagent contacting modes. Brian et al. [START_REF] Mason | Greener Approaches to Organic Synthesis Using Microreactor Technology[END_REF] made a review about the organic synthesis in microreactors. [START_REF] Kolb | Micro-structured reactors for gas phase reactions[END_REF] reviewed the application of micro-structured reactors for heterogeneously catalyzed gas phase reactions. Compared with conventional mixers, static mixers are generally cheaper because there is no moving part, it can reduce energy consumption as well as the cost of maintenance. Then the static mixers can homogenize the flow with a short residence time. They can also be built on different materials: plastics, metals, etc. Moreover, there are many geometric parameters in the inserts which can be adjusted to suit different applications.

Kolb and Hesselon

Static mixers

Static

As for the chemical synthesis, the advantage of the static mixers is first of all the good mixing of the fluid. In a tube, plug flow can be approached which can highly increase the reaction rate and the yield. Then the good radial mixing reduces the radial gradient of temperature, which also enhances heat transfer. There are many applications of static mixers for chemical synthesis in industry, most of them are polymerization, such as polystyrene, nylon, urethane and sulfur compound manufacture [START_REF] Schott | Motionless mixers[END_REF][START_REF] Khac Tien | Motionless mixers for the design of multitubular polymerization reactors[END_REF][START_REF] Baker | Motionless mixers stir up new uses[END_REF][START_REF] Myers | Avoid agitation by selecting static mixers[END_REF]. They are highly exothermic reactions which are difficult to control. If the heat is not removed quickly during the reaction, it will create a lot of byproducts due to the high temperature. There is also a big concern with safety. Moreover, polymerization can be carried out in fluid media of complex rheological behavior for which static mixers have performed better than other mixing technologies.

To conclude, static mixers are widely used in industry for chemical synthesis, they have many advantages such as plug flow behavior, improvement of mixing and heat transfer, etc. Most of the applications of static mixers are for polymerization synthesis. Of course, inserts generally generate a large pressure drop which is the main disadvantage of static mixers. Today, there are still many studies on static mixers to build new designs for different specific needs.

Heat-exchanger reactors

Generalities

Recently numerous technologies of compact heat-exchanger (HEX) reactors have been developed to respond to the needs of safer, cleaner, more effective and less energy-consuming processes. They integrate two basic concepts of process intensification which are the miniaturization of the units and the multi-functionalization of the apparatuses. They exhibit many advantages such as good heat transfer performances, better temperature control and reactive volume confinement. Indeed, this type of reactor firstly has a large specific heat exchange area (A/V) due to the size reduction. Then this size reduction causes the reduction of the total amount of mass and heat inside the reactor which is favorable for safety concerns, especially for highly exothermic reactions. However, in order to have a sufficient residence time for the chemical reactions, the flowrate is relatively small in these devices. This results in laminar flow operating conditions which does not favor transport phenomena. Consequently, complex designs are required in order to enhance mass and heat transfer and promote plug flow behavior by interrupting the thermal boundary layer and in return enhance the heat transfer compared to technologies based on straight tubes. Compact HEX reactors can be found in various materials, such as glass, metals or silicon carbide. The objective in their conception is to design a performant heat exchanger suitable to carry out reactions, i.e. providing efficient reactant mixing, high interfacial area for multiphase reactions, and sufficient residence time to fit the reaction kinetics.

Many studies on HEX reactors can be found in the literature. Indeed, milli-structured HEX reactors have recently attracted a lot of attention in the chemical industry, especially to carry out fast and exothermic reactions. Their applications in processes are becoming wider and wider.

Anxionnaz et al. [START_REF] Anxionnaz | Heat exchanger / reactors ( HEX reactors ): Concepts , technologies[END_REF] made a review of different kinds of compact HEX reactors and presented their applications at pilot or industrial scale. Li et al. [START_REF] Li | Compact heat exchangers : A review and future applications for a new generation of high temperature solar receivers[END_REF] made a review on the performance of different compact HEX reactors and summarized their structures, heat transfer enhancement mechanisms, advantages and limitations. They compared these different heat transfer enhancement technologies and analyzed their thermo-hydraulic performances on the basis of available correlations. Niedbalski et al. [START_REF] Niedbalski | Study of a multi-phase hybrid heat exchanger-reactor (HEX reactor): Part II -Numerical prediction of thermal performance[END_REF][START_REF] Niedbalski | Study of a multi-phase hybrid heat exchanger-reactor (HEX reactor): Part I -Experimental characterization[END_REF] studied experimentally the hydrodynamic behavior and numerically the thermal performances of gas-evolving reacting flows in chevron plate heat exchangers. They developed a reduced-order semi-empirical model in order to predict the thermal performances. Chen et al. [START_REF] Chen | Thermal-hydraulic performance of printed circuit heat exchangers with zigzag flow channels[END_REF][START_REF] Chen | Annals of Nuclear Energy Experimental and numerical study of a printed circuit heat exchanger[END_REF] conducted experimental and numerical studies of thermal-hydraulic performances in a zigzag-channel as well as in a straight-channel printed circuit heat exchanger, they found good agreement between experimental and numerical results. Kwon et al. [START_REF] Kwon | Experimental investigation of heat transfer coefficient of mini-channel PCHE[END_REF] investigated experimentally the heat transfer coefficient of a mini-channel printed circuit heat exchanger with counter-flow configuration at different cryogenic heat transfer conditions, they furtherly proposed correlations for each heat transfer condition. Guo et al. [START_REF] Guo | Multi-channel heat exchanger-reactor using arborescent distributors : A characterization study of fluid distribution, heat exchange performance and exothermic reaction[END_REF] studied experimentally and numerically the flow distribution and heat exchange performance in a multi-functional HEX reactor comprising arborescent distributors and collector, they found the heat exchange capability (UA/V) was around 200 kW.m -3 .°C -1 , 20 times higher than traditional tank reactors. Théron et al. [START_REF] Théron | Chemical Engineering and Processing : Process Intensification Characterization of the performances of an innovative heat-exchanger / reactor[END_REF] characterized experimentally the thermal-hydraulic performance of a compact heat exchanger reactor. Their results showed the ratio between the reaction kinetics of sodium thiosulfate oxidation by hydrogen peroxide and the mixing time is over 100 and the heat exchange capability ranges from 5000 to 8000 kW.m -3 .K -1 . Zhang and Sethi [START_REF] Zhang | Compact heat exchange reactor for synthesis of mixed alcohols[END_REF] studied the synthesis of mixed alcohols in a stainless steel compact HEX reactor with 66 mL catalyst loading, demonstrated two to four times process intensification. Bakhtyari et al. [START_REF] Bakhtyari | Optimal conditions in converting methanol to dimethyl ether, methyl formate and hydrogen utilizing a double membrane heat exchanger reactor[END_REF] studied numerically a thermally integrated coupled membrane HEX reactor designed as a co-axial exchanger surrounded by a membrane. In a single device, they found optimal conditions to promote in the inner tube the exothermic dehydration of methanol to dimethyl ether (95.1% of conversion), in the outer tube the endothermic dehydrogenation of methanol to methyl formate and hydrogen (99.6% of conversion). The membrane allows the removal of the hydrogen from the second reactional media.

Most of the equipment manufacturers that have been interested in the design of compact HEX are companies that were formerly specialized in the construction of metal heat exchanger (Alfa Laval, Chart…) or that had an expertise in specific material handling for component manufacturing (glass and ceramics for Corning, silicon carbide for Boostec-Mersen…). In this part, some compact HEX reactors developed by these companies are presented. Most of them are designed as plate heat exchanger since this configuration allows flexibility in terms of residence time by modifying the number of plates. Unlike the devices presented previously, this HEX reactor is not commercialized. Its specificity mainly lied in the use of SiC for the conception of a chemical engineering equipment. This material is a hard ceramic, semiconductor, which presents very high chemical resistance and thermal conductivity (120 W.m -2 .K -1 ). It is mainly used in the fields of microelectronics, aeronautics, spatial and automotive. The features make it as a choice for HEX reactor design.

Alfa Laval Plate Reactor

Alfa

By the way, Boostec-Mersen now builds the Advanced-Flow TM Reactors in SiC commercialized by Corning.

One of the advantages of the zigzag pattern is that it is characterized by numerous geometric parameters (angle of the bends, curvature, straight section length between bends, cross-sectional shape and size of the channel…) which can be modified to adapt the design to different needs.

The HEX reactor presented in Figure I-11 and devices with similar design have been studied a lot at LGC in the last fifteen years. Their capabilities to handle some industrial applications have been assessed notably in the framework of collaborative projects involving the LGC and industrial partners [START_REF] Despènes | Intensification et passage en continu d'une synthèse pharmaceutique en réacteur-échangeur[END_REF][START_REF] Fustier | Développement d'un réacteur intensifié en Carbure de Silicium pour la transposition en continu de réactions d'hydrosilylation[END_REF][START_REF] Lachin | De la stabilité à la coagulation de latex acrylique : étude des mécanismes et mise en oeuvre en milliréacteur[END_REF]. Other studies have also been carried out to understand and model the influence of the operating conditions and geometric parameters of the zigzag channel on pressure drop, axial dispersion, mixing time, heat and interfacial mass transfer [START_REF] Moreau | Methodologies pour la caracterisation hydrodynamique et l'extrapolation de réacteurs intensifiés millistructurés[END_REF][START_REF] Anxionnaz | Etude de l'influence de la géométrie des canaux sur les performances d'un réacteur/échangeur[END_REF][START_REF] Roudet | Hydrodynamic and mass transfer in inertial gas-liquid flow regimes through straight and meandering millimetric square channels[END_REF]. The works described in the present thesis are in continuation of these studies and will mainly concentrate on heat transfer performance of HEX reactors with zigzag channels. In the following section, the particular hydrodynamics in millimetric zigzag channels and their performances are presented in more details.

Millimetric zigzag channel

Flow characteristics

To optimize yield and selectivity, it is better to carry out a reaction in an ideal plug flow. In this flow, all the elements of fluid have the same residence time, the dead zone can be avoided and a good mixing can be achieved. This flow can avoid hot spots as well for concern of safety, especially for highly exothermic reactions. Therefore, plug flow is sought in compact HEX reactors.

Laminar flow regime

In order to have a sufficient residence time for the chemical reactions, the flowrate in compact HEX reactors is relatively small, normally in laminar regime. This flow regime in straight channels leads to parabolic velocity profile (Poiseuille flow) that does not favor plug flow behavior (no velocity at the wall, maximal velocity at the channel center, no radial advection).

It is obtained for Reynolds number (Re) below 2100:

𝑅𝑒 = 𝜌𝑑 ℎ 𝑢 𝜇 (I-1)
where 𝜌, 𝑢, 𝜇 are the density, mean velocity, and dynamic viscosity of the flow respectively. 𝑑 ℎ is the hydraulic diameter of the channel, which is calculated by

𝑑 ℎ = 4𝐴 𝑃 (I-2)
where A and P are the area and the perimeter of the cross-section.

The laminar flow regime is not favorable for heat transfer and mixing as well, so complex design are necessary in milli-structured HEX reactors in order to be performant. Wavy geometries are often used in milli-structured HEX reactors, as mentioned before. This kind of geometry can generate Dean vortices (secondary flow) as well as chaotic flow even with small Reynolds number, which bring radial fluid motion and hence enhance the mass transfer and heat transfer.

Secondary flow

The secondary flow (Dean flow) is firstly theorized by Dean [START_REF] Dean | Note on the notion of fluid in a curved pipe[END_REF]. It corresponds to fluid motion in curved channels perpendicular to the main direction of the flow. It is due to centripetal forces generated in the curves. This motion can be described as radial vortex. It creates radial mixing that reduces the thickness of the velocity, concentration and temperature boundary layers at the walls and interfaces, enhancing in return mixing, mass transfer and heat transfer. This flow can be generated in wavy channels due to the presence of the bends. An example of a simulated secondary flow generated in a zigzag channel with square cross-section is presented in Figure I-12. This representation is obtained by projecting the velocity vectors perpendicularly to the main direction of the flow, i.e. over the cross-section of the channel. In this example, it can be clearly seen that there are 4 vortices. Their results suggested that the secondary flow created by channel curvature can increase the Nusselt number significantly (ratio of convective to conductive heat transfer at a boundary) and the friction factor moderately. Manglik et al. [START_REF] Manglik | Low Reynolds number forced convection in threedimensional wavy-plate-fin compact channels: Fin density effects[END_REF] studied numerically the steady forced convection in wavy plate-fin channels at low Reynolds number and found the generation of secondary flow which enhanced the heat transfer as well as the pressure drop penalty. Sui et al. [START_REF] Sui | Fluid flow and heat transfer in wavy microchannels[END_REF][START_REF] Sui | An experimental study of flow friction and heat transfer in wavy microchannels with rectangular cross section[END_REF] performed numerical and experimental studies for laminar liquid-water flow and heat transfer in three-dimensional wavy microchannels with rectangular cross section. Their results showed that the heat transfer performance in wavy microchannels was much better than that of straight microchannels with the same cross section due to the generation of secondary flow and the pressure drop penalty could be much smaller than the heat transfer enhancement. Fletcher and his coworkers [START_REF] Rosaguti | Laminar flow and heat transfer in a periodic serpentine channel with semi-circular cross-section[END_REF][START_REF] Gupta | Thermohydraulic performance of a periodic trapezoidal channel with a triangular cross-section[END_REF][START_REF] Rosaguti | Laminar flow and heat transfer in a periodic serpentine channel[END_REF][START_REF] Geyer | Thermohydraulics of squaresection microchannels following a serpentine path[END_REF][START_REF] Geyer | Laminar flow and heat transfer in a periodic trapezoidal channel with semi-circular cross-section[END_REF][START_REF] Rosaguti | Low-Reynolds number heat transfer enhancement in sinusoidal channels[END_REF][START_REF] Zheng | Chaotic advection in steady laminar heat transfer simulations: Periodic zigzag channels with square cross-sections[END_REF] have conducted a variety of numerical studies for steady laminar flow and heat transfer in wavy channels with different cross-sections, including trapezoidal, serpentine, and sinusoidal channels. They found the similar conclusion that Dean vortices can enhance heat transfer greatly with relatively small pressure drop penalty.

Chaotic flow

Zigzag channels can generate another effect for improving mixing and heat transfer which is called chaotic advection. It was firstly proposed by Aref in fluid mechanics in 1984 [START_REF] Aref | Stirring by chaotic advection[END_REF]. It describes unstable and aperiodic behavior in deterministic non-linear systems. This flow behavior can be achieved in two-dimensional unsteady flow or three-dimensional steady flow.

The chaotic flow can be characterized by the Lagrangian method. A lot of research on chaotic flow and its impact on mixing and heat transfer can be found in the literature. Liu et al. [START_REF] Olsen | Passive Mixing in a Three-Dimensional Serpentine Microchannel Passive Mixing in a Three-Dimensional Serpentine Microchannel[END_REF] presented a three-dimensional serpentine microchannel design as a means of implementing chaotic advection to enhance fluid mixing. Stroock et al. [START_REF] Stroock | Chaotic mixer for microchannels[END_REF] presented a passive method based on staggered herringbone mixer to generate chaos for mixing streams of steady pressure-driven flows in microchannels at low Reynolds number. Jiang et al. [START_REF] Jiang | Helical flows and chaotic mixing in curved micro channels[END_REF] investigated numerically and experimentally the mixing of helical flows in curved micro channels, their results showed that the hydrodynamic dispersion was substantially reduced compared to a straight channel, due to mass transfer enhancement originating from the chaotic flow. Kumar et Nigam [START_REF] Kumar | Laminar convective heat transfer in chaotic configuration[END_REF] studied numerically the heat transfer in a chaotic configuration of circular cross-section under laminar regime, their results showed a 25-36% enhancement of heat transfer due to chaotic mixing while pressure drop increase was 5-6%. Castelain et al. [START_REF] Castelain | Experimental study of chaotic advection regime in a twisted duct flow[END_REF][START_REF] Peerhossaini | Heat exchanger design based on chaotic advection[END_REF] have conducted experimental studies of chaotic advection in a twisted duct with square cross-section, they characterized the chaotic flow and found that it was sensitive to the initial conditions. A coiled heat exchanger based on chaotic advection was designed and it showed 15-18% higher efficiency than a helically coiled tube heat exchanger. Acharya et al. [START_REF] Acharya | Heat transfer enhancement in coiled tubes by chaotic mixing[END_REF] studied numerically and experimentally the heat transfer in an alternating axis coil. Their results showed 6-8% higher in-tube heat transfer coefficient due to chaotic mixing compared to a conventional constant axis coil, with a pressure loss increase of 1.5-2.5%.

Hydrodynamic performances

Residence time distribution and mixing

As mentioned before, plug flow is generally sought in continuous reactors. Péclet number Pe is often used to evaluate the gap to this ideal flow behavior in real continuous reactors. It is defined as follows:

𝑃𝑒 = 𝐿𝑢 ̅ 𝐷 𝑎𝑥 (I-3)
where L is the reactor length and Dax is the axial dispersion coefficient. This coefficient represents the overall spreading of a component in a fluid flow along the main direction of the flow (i.e. axial direction) which is due to molecular diffusion and radial velocity profile. It is commonly considered that plug flow behavior can be assessed for Pe > 50. A representation by an equivalent number of stirred tanks in series N can also be used (Pe ≈ 2N -1). Unlike to Poiseuille flow conditions with parabolic velocity profile that does not favor plug flow, zigzag millimetric channels generate radial fluid motion and plug flow is easily obtained for fluid that are not too viscous by adjusting the zigzag geometry [START_REF] Anxionnaz-Minvielle | Influence of the meandering channel geometry on the thermo-hydraulic performances of an intensified heat exchanger/reactor[END_REF]. Radial mixing as well is enhanced.

Mixing times of less than 1 s can be reached [START_REF] Anxionnaz-Minvielle | Influence of the meandering channel geometry on the thermo-hydraulic performances of an intensified heat exchanger/reactor[END_REF].

Axial dispersion coefficient can be estimated using Residence Time Distribution experiments or simulations. It consists of the injection of an inert tracer at the reactor entrance (solute or particles) in the main fluid flow, and the measurement of its concentration at the reactor outlet.

The comparison between the inlet and the outlet concentration profiles allows the axial dispersion coefficient calculation using deconvolution methods. A numerical method where this coefficient is calculated from the solute concentration and velocity fields at the outlet without applying deconvolution methods has also been used successfully for millimetric channels [START_REF] Moreau | A new numerical method for axial dispersion characterization in microreactors[END_REF].

Mixing times are generally obtained using colorimetric experiments or numerical particle tracking. Micro-mixing, that may influence very fast reactions, can also be characterized using chemical methods such as Villermaux-Dushman reaction scheme [START_REF] Kunowa | Characterization of Mixing Efficiency in Polymerization Reactors Using Competitive-Parallel Reactions[END_REF].

Pressure drop

Pressure drop is an important parameter in chemical process as it is related directly to the consuming power by the pump which is an economic concern. In HEX reactors, the wavy channels are often used to enhance heat transfer and mass transfer. However, the bends in wavy channels increases the pressure drop as well compared to straight channels. So when talking about the mass transfer and heat transfer enhancement, it is better to take the pressure drop into account for an overall evaluation factor [START_REF] Webb | Application of rough surfaces to heat exchanger design[END_REF].

When it comes to pressure drop, the Fanning friction factor cannot be skipped which is defined as:

𝑓 = 𝜏 𝜌 𝑢 2 2 (I-4)
which is the ratio between the local shear stress (𝜏) and the local flow kinetic energy density.

In a tube with circular cross-section of radius r, the shear stress can be calculated as

𝜏 = ∆𝑝 * 𝜋𝑟 2 2𝜋𝑟𝐿 = ∆𝑝𝑟 2𝐿 (I-5)
where 2𝜋𝑟𝐿 is the wall area, L is the length of the tube. With the two equations above, the pressure loss in a tube with circular cross-section can be calculated using the following equation.

It can be extended to other cross-sectional shape using the hydraulic diameter.

∆𝑝 = 𝑓 𝜌𝑢 2 𝐿 𝑟 = 2𝑓𝜌𝑢 2 𝐿 𝑑 ℎ (I-6)
In a tube with circular cross-section, the Fanning friction factor in fully developed laminar flow can be calculated by [START_REF] Shah | Laminar flow forced convection in ducts: A source book for compact heat exchanger analytical data[END_REF]:

𝑓 = 16 𝑅𝑒 (I-7)
while in a tube with square cross-section, the Fanning friction factor in fully developed laminar flow can be calculated by [START_REF] Shah | Laminar flow friction and forced convection heat transfer in ducts of arbitrary geometry[END_REF]:

𝑓 = 14.227 𝑅𝑒 (I-8)
Experimentally, the pressure drop is estimated using pressure sensors or differential pressure transmitters. Numerically, it is obtained from the resolution of Navier-Stokes equations, using the adequate models.

Thermal performances

Most of the chemical reactions depend on the temperature, highly or lightly. The temperature in the reactor can influence the reaction which furtherly affects the conversion and selectivity.

Besides, for the high exothermic reactions, it is very important to remove the heat timely for safety concern. For releasing the heat generated by the reactions (or offering heat for endothermic reactions), conventional reactors are equipped with serpentines (internal or external) or double-shells where circulates the cooling fluid or hot fluid for endothermic reactions. In large reactors the heat transfer efficiency is not very good and additional external heat exchangers may be used. The idea of heat-exchange reactors is to combine the reaction and efficient heat transfer in the same place (in the reactor). So normally in HEX reactors, the temperature is better controlled, the conversion and the selectivity are higher than conventional reactors.

The miniaturization of the equipment can enhance heat transfer mainly because the specific heat exchange area (heat exchange area divided by volume AH/V) increases by reducing the size of the equipment. The thermal characteristics of industrial HEX reactors presented in Part 2.5 are in the table below, and compared with other reactors. Heat exchanger reactors have large heat exchange capacity per volume, which allows to carry out highly exothermic chemical reactions in safe operating conditions. For characterizing the thermal performance of the heat exchanger reactors, there are some physical quantities needed to be presented. For a heat exchanger, the heat exchange flux (q) can be evaluated by:

𝑞 = 𝑚̇𝐶 𝑃 ∆𝑇 (I-9)
where 𝑚̇ is the mass flow rate and 𝐶 𝑃 is the specific heat of the fluid. ∆𝑇 is the difference between the inlet and the oulet temperatures of the fluid. One of the most important quantity, the overall heat exchange coefficient (U), can be calculated by:

𝑈 = 𝑞 𝐴 𝐻 ∆𝑇 𝑚𝑙 (I-10)
where 𝐴 𝐻 is the heat exchange area, ∆Tml is the logarithmic mean temperature difference. This overall coefficient corresponds to the inverse of a sum of thermal resistances that describes the heat transfer capacity by convection and advection of the different fluids and materials that contribute to the heat flux. The heat exchange coefficient in a fluid (h) is related to the fluid property, the velocity, etc. It is generally expressed in dimensionless form through the Nusselt number (Nu), which is defined as:

𝑁𝑢 = ℎ𝑑 ℎ 𝑘 (I-11)
where k is the thermal conductivity.

For straight channels, there are many correlations available in publications to calculate the Nusselt number. Here two correlations are just presented. In laminar regime (Re<2100), the Nusselt number can be calculated with Schlunder correlation [START_REF] Gnielinski | Heat Exchanger Design Handbook[END_REF]:

𝑁𝑢 = (3.66 3 + 1.61 3 𝑅𝑒𝑃𝑟𝑑 ℎ 𝐿 ) 1/3 (I-12)
where L is the channel length, Pr is the Prandtl number defined as:

𝑃𝑟 = 𝐶 𝑝 𝜇 𝑘 (I-13)
The Schlunder correlation is valid when L/dhRePr is between 0.0001 and 10. In transient and turbulent regime, the Nusselt number can be calculated with Gnielinski correlations [START_REF] Gnielinski | New Equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow[END_REF]:

𝑁𝑢 = ℎ𝑑 ℎ 𝑘 = 0.012 (𝑅𝑒 0.87 -280) 𝑃𝑟 0.4 [1 + ( 𝑑 ℎ 𝐿 ) 2 3 ] ( 𝑃𝑟 𝑃𝑟 𝑤 ) 0.11 ( I-14)
This correlation is valid for Pr between 1.5 and 500 and Re between 2300 and 10 6 .

For innovative HEX reactors, correlations are not available in literature and the characterization of their thermal performances is required to allow the modelling and the scaling-up of these devices. This characterization is based on temperature measurements to calculate the heat fluxes and the temperature differences. Then the thermal characteristics like heat transfer coefficient (U and h), Nusselt number (Nu), etc. can be calculated. CFD can be used as far as the hydrodynamic and thermal models are adapted. This is a key point in millimetric channels since the transient flow regime (between laminar and fully-turbulent) is often observed in such geometry. This flow regime is complex and available models in common CFD codes can fail to represent it. Experimentally, there are many methods to measure the temperature. In terms of liquid flows, the methods often used are thermoelectric devices, platinum resistance thermometers and infrared thermography.

Thermocouples are the thermoelectric devices the most used in the research field or industry because of their low cost, simplicity, size and robustness. They can monitor temperatures between -270 °C and 3000 °C. The basis of thermocouples is the Seebeck effect: the junction between two metals generates a voltage that is a function of temperature. There are 8 standardized thermocouple types with different temperature ranges and sensitivities which are presented in Table I-2 [START_REF] Childs | Review of temperature measurement[END_REF]. Thermocouples provide high speed of response, however their measurements are not very accurate. They also have limited spatial resolution, e.g. some places may be difficult to reach in a fluid flow, and the probes may cause interactions with the flow (intrusive method). The resistance temperature devices (RTD) are other detectors commonly used to measure the temperature. The basis is that the resistance of a conductor is related to the temperature as the motion of free electrons and of atomic lattice vibration depends on the temperature. Materials like copper, gold, nickel, platinum can be used for the construction of RTD, among which the platinum is the most used because it has a big resistivity and a linear resistance-temperature relationship compared to other metals. The platinum resistance thermometers (PRT) can monitor temperatures between -260 and 962 °C. The most common PRT type is PT100 which has a resistance of 100 ohms at 0 °C and 138.4 ohms at 100 °C. For a PT100, a change of temperature of 1 °C will cause a 0.384 ohm change in resistance, so a small error in measurement of the resistance can cause a large error in the measurement of the temperature.

There are different circuit configurations but the four-wire configuration gives the best performance and can eliminate lead wire resistance uncertainties. One problem of the PRT is self-heating, the current through the sensor causes some heating which tend to report an artificially higher temperature. Attention should be paid to minimize the current and ensure good thermal contact with its environment to reduce the measurement errors. Also, the size of the probe is relatively large and it is difficult to measure the temperature of liquid flows in millimetric channels. Finally, this method is also intrusive and the probe can create significant disturbances in the fluid flow.

Infrared thermography (IRT) is a contact-free technique for temperature measurements, based on the emission of thermal radiation of an object which depends on the temperature. In the last decades, IRT has become a powerful means of thermo-dynamic analysis to measure convective To limit the impact of these disturbances the ratiometric technique has been developed. In this technique, two detection bands are used and the temperature can be deduced from the ratio of the fluorescence intensity of these two bands. In some studies, one dye with two bands (2 colors/1 dye) having different temperature sensitivities are utilized. Bruchhausen et al. [START_REF] Bruchhausen | Instantaneous measurement of two-dimensional temperature distributions by means of two-color planar laser induced fluorescence ( PLIF )[END_REF] studied a one-phase flow by using the two-colors/single-dye technique with rhodamine B.

Castanet et al. [START_REF] Castanet | Internal temperature distributions of interacting and vaporizing droplets[END_REF] measured the temperature field within droplets with the same technique using pyrromethene 597-C8. However, the most popular method uses two dyes with two detection bands which is called two-colors/two-dyes (2c/2d) technique. In this approach, a temperature-sensitive dye is associated with another little temperature-sensitive dye.

Sakakibara and Adrian [START_REF] Sakakibara | Whole field measurement of temperature in water using twocolor laser induced fluorescence[END_REF][START_REF] Sakakibara | Measurement of temperature field of a Rayleigh-Bénard convection using two-color laser-induced fluorescence[END_REF] conducted temperature measurements in a stable thermally stratified layer with the couple of dyes Rhodamine B/Rhodamine 110 and furtherly applied this technique to the measurement of temperature field in turbulent Rayleigh-Bénard convection flow. Chaze et al. [START_REF] Chaze | Spatially and temporally resolved measurements of the temperature inside droplets impinging on a hot solid surface[END_REF] utilized the couple of dyes fluorescein disodium/sulforhodamine 640 and obtained single-shot images of the temperature inside droplets. In some other studies, two dyes with opposite temperature sensitivities are utilized to obtain higher sensitivities to the temperature. Sutton et al. [START_REF] Sutton | A laser-induced fluorescence measurement for aqueous fluid flows with improved temperature sensitivity[END_REF] performed an improved temperature-sensitive LIF measurement in aqueous fluid flows using the couple of dyes fluorescein disodium/kiton red (also called sulforhodamine B). Behshad et al. [START_REF] Shafii | In situ LIF temperature measurements in aqueous ammonium chloride solution during uni-directional solidification[END_REF] reported an in situ whole-field measurements of the temperature field in aqueous ammonium chloride solution during uni-directional solidification using the same couple of dyes.

The LIF technique for fluid temperature measurement is used less often than thermocouples or resistance thermometers use because it requires expensive and large equipment (laser, cameras, etc.). However, it is interesting for local temperature measurements in small devices because it is non-intrusive and provides immediate response which can be adapted to unsteady flows. In this thesis, the laser-induced fluorescence thermometry with two dyes and two detection bands is developed for the temperature measurement in milli-structured heat-exchanger reactors which will be presented in Chapter IV. The two-color/two dye approach with opposite temperature sensitivities are utilized in order to have high sensitivity for the temperature measurement.

Conclusion

The concept of process intensification emerged 40 years ago and many technologies have been developed to intensify the process during the last two decades. These technologies can generally be divided into two parts: single unit operation intensification and intensification among different units. One method for unit operation intensification is the miniaturization of equipment which is very popular at this time, like the milli-structured and micro-structured devices. One method for intensification among different unit operations is the multifunctionality of the apparatus which has drawn great attention as well.

Several intensified reactors are then presented which are static mixers, spinning disk reactors, rotating packed bed reactors and microreactors. The intensification strategy is different among these different reactors. For static mixers, it is the immobile inserts that can redistribute the fluid in the radial direction. The spinning disk and rotating packed bed reactors use the centrifugal force to intensify the process. As for microreators, it is the miniaturization that enhances the mass transfer and heat transfer providing high surface-to-volume ratio.

Chapter II: Experimental method for the heat transfer study

In this chapter, the experimental method is utilized to study the heat transfer performance of a compact heat-exchanger reactor which is designed by the Laboratoire de Génie Chimique (LGC). It consists of a plate heat exchanger where the process fluid (the reaction may occur) circulates in a zigzag millimetric channel engraved in the plate. In the present work, several prototypes have been built to characterize the heat transfer coefficients in different millimetric channels. Two kinds of channel configuration are investigated which are zigzag channel and straight channel. For zigzag channels, the influence of fluid properties on heat transfer is also studied by working with water and a more viscous fluid (glycerol solution). The overall heat transfer coefficient of the heat exchanger reactor prototypes with zigzag channels and that with straight channels is characterized experimentally by measuring the inlet and outlet temperature of the two exchange fluids. The heat transfer coefficient in zigzag channels is then deduced and a correlation for Nusselt number estimated in zigzag channels is proposed.

Materials and methods

Experimental prototypes

In this work, six prototypes have been designed and constructed in LGC for the study of heat transfer in compact heat exchanger reactors. They consist of three plates: a plate made of aluminum sandwiched between two plates where channels are engraved for fluid circulation (Figure II-1). These two plates will be differentiated as "process plate" and "utility plate" according to the design of the channel engraved. They are pasted to the aluminum plate using double-sided adhesives cut to fit the channel designs. The adhesives have been realized by Atelier Graphique Saunière (France). The aluminum plate is of 1 mm thickness with a high thermal conductivity (k = 237 W.m -1 .K -1 ) and its thermal resistance is very low (e/k = 4.2⨯10 - 6 m 2 .K.W -1 ). Its surface has been anodized to obtain the black color which can allow the safe use of a laser for the temperature field measurement by LIF, this will be presented in Chapter IV. The process plate and utility plate are made of PolyMethylMethAcrylate (PMMA) with 10 mm thickness. This hard polymeric material has been chosen because it is cheap and can be easily engraved by milling. It is transparent which is convenient for flow observation. The transparence is also necessary for the temperature field measurement by LIF presented in chapter IV. Moreover, its thermal conductivity is very low which can largely avoid heat loss (k = 0.19 W m -1 K -1 , e/k = 0.05 m 2 .K.W -1 ). Finally PMMA presents a linear coefficient of thermal expansion with α = 7⨯10 -5 K -1 . Using Eq. II-1, less than 0.3% of dimensional change is estimated when working between 20°C and 60°C which is the temperatures range during the experiments. Therefore geometry changes due to temperature can be neglected. The straight section length between two bends is fixed at 7 mm, the angle (θ) between two straight sections is kept at 90°. This geometry pattern has been used to build compact heat exchangers-reactors designed by LGC, Boostec (France) and the Commissariat à l'Energie Atomique et aux Energies Alternatives (CEA, France). Such devices made of silicon carbide or stainless steel have been successfully used to carry out highly exothermic reactions [START_REF] Elgue | Direct fluorination of 1,3-dicarbonyl compound in a continuous flow reactor at industrial scale[END_REF][START_REF] Despènes | Impact of the material on the thermal behaviour of heat exchangers-reactors[END_REF][START_REF] Raimondi | Safety enhancement by transposition of the nitration of toluene from semi-batch reactor to continuous intensified heat exchanger reactor[END_REF]. In the utility plate, a straight channel with a cross-section of 12×6 mm² is engraved. Its geometry is unique for all the prototypes, whatever the process channel design is, and has been chosen so that:

-its width (12 mm) and length are slightly higher than the total width and length of all the zigzag channels in process plate. The process channel pattern that presents the largest width, 10.7 mm, is obtained for the zigzag channel of 4 mm width. The utility and process channels are face to face and centered in the PMMA plates. This way, the utility channel sufficiently covers the zigzag channel or straight channel in the process plate both in the flow direction and radial direction (Figure II-3).

-its depth (6 mm) allows a flowrate up to ten times higher than the process fluid flowrate for water. With this cross-section, the Reynolds number of utility fluid is higher than 10 000 in the experiment, which is in turbulent flow regime. Therefore, heat transfer is not mainly limited by the utility side and common Nusselt number correlations for straight channel can be used to estimate the heat transfer coefficient.

Experimental setup

The experimental setup is shown in Figure II-4. For each prototype, the process channel is connected to a gear pump (0 -100 L.h -1 ) and the utility channel is connected to a centrifugal pump (0 -1000 L.h -1 ), with two different mass flowmeters.

Water 𝑄 𝑝 = 1.5 -15 L.h -1 up = 0.1 -1 m.s -1
Rep = 280 -4000

𝑄 𝑝 = 2.8 -34 L.h -1 up = 0.09 -1 m.s -1
Rep = 430 -6200

𝑄 𝑝 = 5 -60 L.h -1 up = 0.09 -1 m.s -1
Rep = 640 -7200

𝑄 𝑢 = 300 -500 L.h -1 uu = 1.1 -1.

Method for heat transfer coefficient calculation

As the utility fluid flowrate is around ten times higher than the process fluid flowrate, the difference between inlet and outlet temperature of process fluid is ten times bigger than that of utility fluid. Therefore, the process fluid is taken into consideration for the calculation of heat fluxes (heat loss and heat exchanged with the utility fluid). From the temperature difference (∆Tp) of the process fluid, the heat flux exchanged between the two fluids (q) can be calculated using equation II-2. The thermal losses (ql) have been previously estimated by introducing just the process fluid (equation II-2 with q = 0), with no flow in the utility channel.

𝑞 = 𝑚 𝑝 ̇𝐶𝑃,𝑝 ∆𝑇 𝑝 -𝑞 𝑙 (II-2)
Where 𝑚 𝑝 ̇ is the mass flow rate and 𝐶 𝑃,𝑝 is the specific heat of the process fluid. The overall heat transfer coefficient (U) of the prototype can be calculated with the equation I-

(II-3)
For the sandwiched prototype, the overall conductance is: The heat capacity of glycerol at different temperature can be calculated by [START_REF] Righetti | Heat capacity of glycerol from 298 to 383 K[END_REF] :

𝐶 𝑃 = 988.0 + 4.71𝑇 (II-13)
The thermal conductivity of glycerol-water mixture between 20 °C and 60 °C varies little with the temperature with the value of around 0.337 W.m -1 .K -1 [START_REF] Bates | Thermal conductivity of liquids: Binary mixures of water and glycerol[END_REF].

For the calculation of all physicochemical properties, the mean temperature is considered which The heat loss in this case is: The heat transfer coefficient of utility fluid hu then is:

𝑞 𝑙 =
ℎ 𝑢 = 𝑁𝑢 𝑢 𝑘 𝑢 𝑑 ℎ,𝑢 = 85.2×0.622÷0.008 = 6624 W.m -2 .K -1
From equation II-4, the heat transfer coefficient of process fluid hp is:

ℎ 𝑝 = 1 ÷ [𝐴 𝐻,𝑝 ( 1 𝑈𝐴 𝐻 - 𝑒 𝐴𝑙 𝑘 𝐴𝑙 𝐴 𝐻 - 1 ℎ 𝑢 𝐴 𝐻,𝑢 )] = 1 ÷ ( 1 𝑈 - 𝑒 𝐴𝑙 𝑘 𝐴𝑙 - 𝐴 𝐻,𝑝 ℎ 𝑢 𝐴 𝐻,𝑢 ) (𝐴 𝐻,𝑝 = 𝐴 𝐻 , as considered) = 1 ÷ [ (1÷3796 -0.001÷237 -(0.004×0.747) ÷6624÷(0.012×0.588)] = 5121 W.m -2 .K -1
The Nusselt number is then:

𝑁𝑢 𝑝 = ℎ 𝑝 𝑑 ℎ,𝑝 𝑘 𝑝 = 5121×0.004÷6.3 = 32.3
The same calculation is done for six prototypes with different operating conditions.

Heat transfer performance of prototypes (zigzag and straight channels)

Overall heat transfer coefficient

The The asymptotic value can be obtained by the simplification of equation II-4 :

lim 𝑄 𝑝 →∞ 𝑈 = 1 𝑒 𝐴𝑙 𝑘 𝐴𝑙 + 𝐴 𝐻 ℎ 𝑢 𝐴 𝐻,𝑢 (II-14)
The same trends can be seen in the straight channels as shown in Figure II-5 (b). However, the increase to the asymptotic value is less obvious in the range of flowrate studied because the process side thermal resistance is higher in the straight channel than in the zigzag channel under the same flow condition. Indeed, by comparing both figures, it is observed that U in the prototypes with zigzag channels is around 2 times larger than that with straight channels for the same flowrates. It indicates that the presence of bends in the zigzag channel can enhance the heat transfer compared with straight channels, in particular at low flowrates, where the overall heat transfer coefficient can even be 4 times higher in zigzag channels than in straight channels.

Among the different channel cross-sections, U increases with the decrease of cross-section width both in zigzag channels and in straight channels. As for the influence of process flowrate to U, U increases with increasing utility flowrate, although it has little influence at low process flowrate (heat transfer limited by the process side).

Heat transfer coefficient in zigzag channels

To further illustrate the influence of the bends in zigzag channel to the heat transfer, the heat transfer coefficient of process fluid (water) hp in zigzag channel and straight channel is

Heat transfer coefficient in zigzag channels

The heat transfer coefficient in zigzag channel as a function of Reynolds number for two fluids is presented in The legend gives the hydraulic diameter of process channel (dh), the utility fluid flowrate (Qu) and the process fluid. Glycerol represents the glycerol solution of 74% in mass fraction.

Global heat transfer performances of HEX reactors with millimetric channels

The experimental results in the different prototypes are analyzed in terms of common performance criteria provided in Table I-1 to compare different technologies, they are the ratio of the heat exchange area to the volume and the heat exchange capacity per volume unit. A typical configuration of plate heat exchanger reactor is considered for the following comparison: the process plate (zigzag or straight) is surrounded by two utility plates and separated by two aluminum plates. In this case, the heat exchange area is two times of that discussed before (with two heat exchange walls in fact). The thermal characteristic result for such devices with two utility plates is presented in Table II-3. From this table, it can be clearly seen that the heat transfer performance of zigzag configuration is much better than that of straight configuration as process channel. As for the influence of channel cross-section length, the heat transfer coefficient range is similar among the three channels for both configurations. The intensification effect by size reduction is due to the increase of the ratio AH/V in fact. The thermal performances with zigzag configuration as process channel are of the same order of magnitude compared with industrial compact heat exchangers reactors, with overall heat exchange capacity per unit volume between 1 500 and 10 000 kW.m -3 .K -1 (as shown in Table I-1).

Dimensionless correlation for heat transfer coefficient in zigzag channels

Here the dimensionless quantity Nusselt number is presented and a correlation for its estimation is proposed since one important objective of this work is to establish correlations for the scale-

Conclusion

In this chapter, the heat transfer performance of compact heat exchanger reactor prototypes are experimentally characterized, in terms of overall heat transfer coefficient and heat transfer coefficient of the process side. The Reynolds number studied ranges from 16 to 7 500, covering laminar regime and transient regime. The Nusselt number is furtherly calculated and a correlation is proposed with the experimental data which should be very useful for the scale-up of HEX reactors.

For the HEX reactor prototypes, two kinds of process channel configuration are investigated:

straight channels and zigzag channels with 3 different square cross-sections (dh = 2mm, 3mm, 4 mm). The zigzag channels consist of periodic bends and straight sections. With water as process fluid, it is found that the heat transfer coefficient in zigzag channel is much larger than that in straight channel, with a ratio between 2 and 5 in the range of Reynolds number studied, demonstrating the heat transfer enhancement of zigzag channels compared with straight channels.

In order to study the influence of fluid property on the heat transfer in zigzag channels, another more viscous fluid is used in this work: a glycerol solution with a mass fraction of 74%. It presents a viscosity around 30 times higher than water at 20 °C. Its Reynolds number ranges from 16 to 415 in the experiment, which is totally in laminar regime. It is found that the evolution of heat transfer coefficient with glycerol solution as a function of Reynolds number is similar to that with water.

A correlation of Nusselt number in zigzag channels is then proposed with the consideration of fluid properties (Pr), channel geometries (dh/Ls), and fluid flowrates (Re). This correlation does not fit very well with the experimental data of glycerol solution compared with water, maybe due to experimental errors, since it takes more time to achieve the thermal equilibrium.

Concerning the influence of channel geometry on Nusselt number, it is found that the ratio of channel hydraulic diameter divided by straight section length between two bends (

𝑑 ℎ 𝐿 𝑠
) plays an important role. In order to furtherly study this influence (

𝑑 ℎ 𝐿 𝑠
) on the heat transfer, a numerical method based on CFD simulation is then used, which is presented in the next chapter. It has the advantages like simple modification of the channel geometry, access to local information (velocity and temperature fields) in the channel. Numerical method is more efficient, since experimentally for changing this ratio (

𝑑 ℎ 𝐿 𝑠
), we need to construct many prototypes which takes time and is more expensive.

Chapter III: Numerical method for heat transfer study

In Chapter II, we characterized experimentally overall heat transfer coefficient and heat transfer coefficient of process fluid in 6 prototypes with zigzag channels as well as with straight channels (dh = 2, 3, 4 mm). A correlation of Nusselt number in zigzag channels was proposed for two process fluids (water, Pr~4, and glycerol solution, Pr~200) in a large range of Reynolds number (10 -7500). In terms of zigzag channels, we found a direct relation between the Nusselt number and the ratio of channel hydraulic diameter to the straight section length between two bends (

𝑑 ℎ 𝐿 𝑠
) for the channel geometries studied. In order to further study the influence of this ratio on the heat transfer, in this Chapter a numerical method based on the software ANSYS CFX is applied.

ANSYS CFX is a commercial software specialized in engineering simulation domain based on the finite element method. This software can simulate many physical phenomena, such as computational fluid dynamics (CFD), electronics, semiconductors and so on. In this Chapter, the CFD code is used to study the flow and heat transfer in zigzag channels with the crosssection of 2 mm, a straight channel with same cross-section is also studied as references.

Numerical method 1.1 Equations and solution method

The simulation is based on the resolution of equations for the conservation of mass, momentum and energy which deduces the velocity, temperature and the pressure of all the simulation domain. For an incompressible fluid and a steady state, these equations can be written as follows:

Continuity equation:

𝛻 • (𝜌𝒖) = 0 (III-1)
Momentum equation (Navier-Stokes):

𝛻 • (𝜌𝒖 ⊗ 𝒖) = -𝛻𝑝 + 𝛻 • (𝜇(𝛻𝒖 + 𝛻𝒖 𝑇 )) (III-2)
Energy equation:

𝛻 • (𝜌𝒖𝐻) = 𝛻 • (𝜆𝛻𝑇) (III-3)
u is the velocity vector, p the pressure, H the enthalpy and T the temperature. The fluid physicochemical properties, such as density , dynamic viscosity µ and thermal conductivity λ, are assumed to be constant (at 25 °C), which make the Prandtl number constant as well (Pr = 6.13). No phase change is considered.

At the inlet, the fluid temperature is constant (equal to 20 °C) and fully-developed flow is assumed. The Poiseuille velocity profile at the inlet is approximated using the expressions suggested by Shah and London [START_REF] Shah | Laminar flow forced convection in ducts: A source book for compact heat exchanger analytical data[END_REF]:

𝑢 𝑦,𝑧 = 𝑢 ̅ ( 2.2+1 2.2 ) 2 (1 -(𝑎𝑏𝑠 ( 2𝑦 𝑑 ℎ ))) 2.2 (1 -(𝑎𝑏𝑠 ( 2𝑧 𝑑 ℎ ))) 2.2 (III-4)
where y, z are the coordinates in the inlet cross-section. At the outlet, zero static gauge pressure is set. At the walls, the no-slip condition and constant heat flux (H2) boundary condition are applied. The value of heat flux is 1×10 5 W.m -², it comes from one simulation at constant temperature (80 °C) boundary condition at the wall, with inlet temperature of 20 °C and inlet mean velocity of 0.1 m.s -1 in the zigzag channel with straight section length of 7 mm.

The simulations were carried out by solving the system of equations with the finite volume CFD code ANSYS CFX 16. All the calculations were performed at steady state in the laminar flow regime. The convective terms were discretized using a second-order bounded differencing scheme. The system was deemed to having reached a converged state when all the locallyscaled residuals fell below 10 -6 .

Computational domain and meshing method

For all the geometries studied, the computational domain consists of a three-dimensional zigzag channel with square cross-section of 2 mm and a total length of roughly 0.1 m. Depending on the straight section length Ls, the number of bends varies from one geometry to another (in total 7 geometries). Table III-1 summarizes the number of bends Nbend and units Nunit for each case.

A unit consists of two straight sections and two bends (Figure III-1(b)). All the geometries start and end with a straight section. However, to avoid the potential effects near the outlet which may influence the results, the last straight section is not taken into account in post-processing calculations. Therefore the total length of the channel L considered in the rest of this chapter corresponds to a number of entire units. A swept hexahedral mesh is used for the computational domain to provide an efficient and accurate resolution. In the cross-section, the mesh was constructed to be finer near the wall than at the center, as shown in

Results and discussion

Parameters of interest: definitions

From the computed velocity, pressure and temperature fields, parameters of interest are calculated during post-processing. A coordinate reference that follows the main direction of the fluid is used. x is the axial coordinate, normal to the cross-section. At the channel inlet, x = 0; at the channel outlet, x = L. y and z are the coordinates along the channel width and depth, respectively. By convention, a bar above a parameter refers to an area or peripheral average value. The subscript m is used to refer to a mean value over the axial length. A represents the cross-sectional area of the channel and P the cross-section perimeter length. The heat transfer coefficient at axial location x is defined as:

ℎ(𝑥) = 𝑞 𝑤 𝑇 𝑤 (𝑥)-𝑇 𝑏 (𝑥) (III-5)
where 𝑞 𝑤 is the wall heat flux, 𝑇 ̅ 𝑤 is the peripheral mean wall temperature and 𝑇 ̅ 𝑏 is the bulk fluid temperature. The peripheral mean wall temperature is calculated as:

𝑇 𝑤 (𝑥) = 1 𝑃 ∫ 𝑇 𝑤 𝑑𝑙 𝑃 𝑥 (III-6)
where 𝑃 𝑥 refers to the cross-section perimeter at axial location x. The bulk fluid temperature is calculated using Eq. III-7. 𝐴 𝑥 refers to the cross-sectional area at axial location x. n is the normal vector of the cross-section towards the main direction of the flow.

𝑇 𝑏 (𝑥) = 1 𝐴𝑢 ̅ ∫ (𝒖 • 𝒏)𝑇 𝑑𝑎 𝐴 𝑥 (III-7) (a) (b) d h d h L s L c
The local Nusselt number 𝑁𝑢 is defined as:

𝑁𝑢(𝑥) = ℎ(𝑥)𝑑 ℎ 𝑘 (III-8)
The average Nusselt number 𝑁𝑢 𝑚 between inlet and axial location x is defined as:

𝑁𝑢 𝑚 (𝑥) = 1 𝑥 ∫ 𝑁𝑢 𝑑𝑋 𝑥 0 (III-9)
The local Fanning friction factor 𝑓 is defined as:

𝑓(𝑥) = - 1 2 𝑑 ℎ 𝜌𝑢 2 𝑑𝑝 𝑑𝑋 | 𝑋=𝑥 (III-10)
where 𝑢 ̅ is the mean fluid velocity, constant along the channel, and 𝑝̅ is the cross-sectional area average pressure, calculated by:

𝑝(𝑥) = 1 𝐴 ∫ 𝑝 𝑑𝑎 𝐴 𝑥 (III-11)
The average Fanning friction factor 𝑓 𝑚 between inlet and axial location x is defined as:

𝑓 𝑚 (𝑥) = 1 𝑥 ∫ 𝑓 𝑑𝑋 𝑥 0 (III-12)
The results will be discussed in terms of the Reynolds number Re defined as:

𝑅𝑒 = 𝜌𝑑 ℎ 𝑢 𝜇 (III-13)

Grid independence studies

Grid independence studies were carried out to make sure that the size of the grid did not influence the resulting solution for all the simulations carried out. as well, in fact the value is the same in two channels, so the difference is not made in this figure.

It can be observed that the difference of hot point and cold point in zigzag channel are smaller than in straight channel, suggesting the temperature field in zigzag channel is more homogeneous than in straight channel. In addition, this temperature difference decreases with increasing Reynolds number for both channels, which means that the temperature field is more homogeneous for high Reynolds number. To further compare the local temperature in zigzag channel and straight channel, the temperature fields of one cross-section at x = 0.075 m in these In the zigzag channel, the evolution of Nu and f in the entrance zone perfectly fits with those observed in the straight channel as the zigzag channel starts with a straight section. Then, oscillations appear that correspond to the periodicity of the geometry studied. Both Nu and f start to increase just before the bends and decrease in the straight sections. It can be observed that Nu and f in the zigzag channel are significantly higher globally than those in the straight channel, suggesting as expected that the bends enhance heat transfer but cause additional corresponds to the outer side of the bend studied.

Evolution of Nu and f versus Ls

Different zigzag channels with straight section lengths ranging from 3. Zigzag geometry 'e'

Re = 448 (a) (b) (c) (d) (e) (f) (g) (h) (i)
Relative velocity

Amplitude

Frequency/(Hz)

FFT (f-f m (L)) f 1 2f 1 3f 1 4f 1 f 1 3f 1 4f 1 5f 1 L s =

Amplitude

Frequency/(Hz)

FFT (f-f m (L)) L s = 3.5 mm Re = 224 f 1 2f 1 2f 1 3f 1 f 2 a b c L s = 3.5 mm Re = 224 f 1 2f 1 3f 1 f 2 a b c (a) (b) (c) (d)
simulation converged but the grid independence was not achieved. A transient solver has been tested to simulate the condition where Re = 560 and Ls = 7 mm as transient laminar flow was observed in analogous geometries [START_REF] Zheng | Transient laminar heat transfer simulations in periodic zigzag channels[END_REF]. In the present case, the transient simulations converge towards a steady solution but the converged solution still depends on the mesh grid size.

Reducing the mesh size is impractical. In the map, a boundary line between periodic and nonperiodic flow can be observed. As mentioned before, the non-periodic flow regime can be achieved by decreasing the straight section length between two bends or by increasing the fluid velocity.

Global results

Average Nusselt number and friction factor

The average Nusselt number Num and friction factor 𝑓 𝑚 in the different zigzag channels at different Reynolds numbers were also studied. Two cases of average Nusselt number at Re = 224 are presented in Figure III-12: one is in the steady periodic regime with a straight section length of 12 mm, whilst the other is in the non-periodic regime with a straight section length of 3.5 mm. The average Nusselt number in the straight channel is also presented. In both regimes, it can be seen that the Nusselt number quickly becomes stable while, in the straight channel, it decreases along the simulated length. The average Nusselt number stabilizes after several bends, Its variation for different Ls/dh seems to be random at the same Reynolds number, indicating the influence of the straight section length on the heat transfer is not obvious in the zigzag channel. This is contrary to the experimental result presented in Chapter II where it is found there is a direct relation between Nusselt number and the ratio (Ls/dh). However, it has to be noticed that the range of the Reynolds number numerically investigated (Re < 600) is much lower than that studied in the experiment (Re < 8 000). Even in the experiments, the influence

Heat transfer enhancement

As discussed above, the zigzag channel configuration enhances heat transfer compared with the straight channel with the same cross-section. This heat transfer enhancement is accompanied by higher pressure drop which has been noted. The heat transfer enhancement factor is defined as:

𝑒 𝑁𝑢 = 𝑁𝑢 𝑚 ∞ 𝑁𝑢 0 ∞ (III-16)
where 𝑁𝑢 0 ∞ is the Nusselt number in a straight channel. As mentioned before, for hydraulically and thermally developed laminar flow, 𝑁𝑢 0 ∞ = 3.091 for a square channel with H2 thermal boundary condition [START_REF] Shah | Laminar flow friction and forced convection heat transfer in ducts of arbitrary geometry[END_REF]. 𝑒 𝑁𝑢 for the studied zigzag configurations at different Reynolds numbers is plotted in Figure III-14. It is observed that the presence of the bends improves the heat transfer significantly, notably at high fluid velocity. 𝑒 𝑁𝑢 increases with increasing

Reynolds number for the different zigzag configurations. Its variation is random in terms of Ls/dh, indicating the impact of non-periodic flow on heat transfer enhancement is not obvious for the conditions simulated. Here the pressure drop penalty is not taken into account.

Thermo-hydraulic performance factor

Another factor PF (thermo-hydraulic performance) is then defined which considers the pressure drop penalty. This factor takes the straight tube as the reference and is very useful for comparing different heat transfer technologies. It is the ratio of the heat exchange capacity (global heat transfer coefficient h multiplying heat exchange area AH) in the considered technology to that in a straight tube divided by the ratio of the flow friction power in the technology 𝒫 to that in a straight tube 𝒫 0 :

𝑃𝐹 = (ℎ 𝑚 𝐴 𝐻 ℎ 0 𝐴 𝐻,0 ⁄ ) (𝒫 𝒫 0 ⁄ ) (III-17)
The flow friction power depends on the pressure drop p in the channel and the volumetric flowrate Q:

𝒫 = ∆𝑝 • 𝑄 (III-18)
Webb and Eckert have proposed different ways to compare the relative performance of heat exchangers with rough surface tubes compared with heat exchangers with smooth surface tubes [START_REF] Webb | Application of rough surfaces to heat exchanger design[END_REF]. They proposed a list of parameters of interest depending on user constraints (for instance the ratio of the surface areas while constant flow friction power and volumetric flowrate are imposed). By analogy, in the present study, each zigzag channel is compared with a straight channel of the same cross-section assuming that:  both configurations must provide the same heat exchange capacity in order to heat or cool a fluid from a fixed temperature to a desired one (equation III-19);

 the fluid flowrate is imposed (equation III-20).

ℎ 𝑚 ∞ 𝐴 𝐻 = ℎ 0 ∞ 𝐴 𝐻,0 (III-19) 𝑄 = 𝑄 0 (III-20)
Therefore, from equations III-19) and III-20, the performance factor can be expressed as follows:

𝑃𝐹 = ∆𝑝 0 ∆𝑝 (III-21)
In other words, the performance factor considered in the present study compares the pressure drop in a straight channel with that in a zigzag channel with the same cross-section and flow rate. In order to obtain the same heat exchange capacity, the length of both channels will be different. The pressure drop can be expressed as a function of the mean fluid velocity, the Fanning friction factor, the hydraulic diameter and the channel length:

∆𝑝 = 2𝜌𝑢 2 𝑓 ∞ 𝐿 𝑑 ℎ (III-22)
For same flowrate and hydraulic diameter, the performance factor simplifies to:

𝑃𝐹 = 𝑓 0 ∞ 𝐿 0 𝑓 ∞ 𝐿 (III-23)
where L0 is the straight channel length and 𝑓 0 ∞ the Fanning friction factor for fully-developed laminar flow in square straight channel [START_REF] Shah | Laminar flow friction and forced convection heat transfer in ducts of arbitrary geometry[END_REF]:

𝑓 0 ∞ = 14.227 𝑅𝑒 (III-24)
Considering the same cross-section in the zigzag and straight channels, the ratio of channel lengths is equal to the ratio of heat exchange areas. From equation III-19, the following expression is then obtained:

𝐿 0 𝐿 = 𝐴 H,0 𝐴 𝐻 = ℎ 𝑚 ∞ ℎ 0 ∞ = 𝑁𝑢 𝑚 ∞ 𝑁𝑢 0 ∞ (III-25)
From equations III-23, III-24, III-25, the performance factor can finally be expressed as follows:

and experiment (between 3.2 and 4.3) is different, it is presented as Nu/ Pr 0.33 for the comparison.

Conclusions

A numerical method based on the software ANSYS CFX has been developed to study the flow and heat transfer in millimetric zigzag channels with square cross-section of 2 mm. The simulations are carried out for the steady laminar flow regime with Reynolds numbers ranging from 112 to 560 and a Prandtl number of 6.13. The channel geometry in terms of straight section length between two bends from 2 mm to 12 mm has been investigated. The flow and heat transfer in a straight channel with the same cross-section is also studied as a reference. The The impact of other geometric parameters, such as the hydraulic diameter, the curvature radius of the bends and their angle should be furtherly studied.

In the simulation, problems of convergence or mesh independence have been met when Reynolds number increases (in laminar regime) or when the straight section length between two bends decreases. In addition, it is much more complicated to carry out the simulation in transient regime since there is not a satisfying thermal-hydrodynamic model available for this flow regime. In order to overcome these limitations of simulation, an experimental method based on laser-induced fluorescence thermometry is developed and described in the next chapter for the measurement of local temperature, which is not limited by the Reynolds number. Temperature measurement is very important for chemical process, as the temperature can directly influence the yield and selectivity. There is also a safety concern for highly exothermic reactions.

Chapter IV: Characterization of temperature field by laser-induced fluorescence

In the previous chapter, a numerical method based on the software ANSYS CFX has been used

to study the flow and heat transfer in the zigzag channels. The local results like temperature field, velocity field and pressure can be obtained by the resolution of the CFD code. However, as mentioned in that chapter, the simulation is limited by the Reynolds number, especially for the transient regime where classical models available fail to accurately represent this flow regime. So in this chapter, an experimental method based on laser-induced fluorescence thermometry is developed to measure local temperature in millimetric zigzag channels. The two-color/two dye approach with opposite temperature sensitivities are utilized in order to have high sensitivity for the temperature measurement. As mentioned in Chapter I, this method is particularly interesting for small devices because it is non-intrusive and does not disturb the flow. It provides immediate response which can also be adapted to unsteady flows.

The experiment is carried out in three heat-exchanger reactor prototypes with zigzag channels, which have been studied in Chapter II. The temperature measurement at different longitudinal positions and different height levels inside the zigzag channels are conducted. The temperature field is deduced from the fluorescence signal. The mean temperature is calculated and then compared to that calculated by a heat transfer model with the purpose of evaluating the reliability of this method.

Principle of laser-induced fluorescence thermometry with two dyes 1.1 Fluorescence signal equation

According to Chaze et al. [START_REF] Chaze | Spatially and temporally resolved measurements of the temperature inside droplets impinging on a hot solid surface[END_REF], the fluorescence signal dFλ emitted at a wavelength λ in an elementary volume dV of liquid can be expressed as:

𝑑𝐹 𝜆 = 𝜂 Ω 4𝜋 𝜀 0 𝜙 𝜆 𝐼 0 (1+ 𝐼 0 𝐼 𝑠𝑎𝑡 ) 𝐶𝑑𝑉 (IV-1)
where 𝜂 is the transmission efficiency of the fluorescence light of the detector, 𝛺 is the solid angle of the collection, 𝜀 0 is the molar absorptivity of the fluorescent molecules at the excitation wavelength of the laser beam, 𝜙 𝜆 is the fluorescence quantum yield, 𝐼 0 is the laser intensity, 𝐼 𝑠𝑎𝑡 is the saturation intensity of the fluorescent dye, C is the molar concentration of the dye. In this equation, 𝜀 0 , 𝜙 𝜆 , 𝐼 𝑠𝑎𝑡 are dependent on the temperature. 𝜙 𝜆 decreases with the temperature due to collisional quenching while 𝐼 𝑠𝑎𝑡 increases with the temperature [START_REF] Chaze | Spatially and temporally resolved measurements of the temperature inside droplets impinging on a hot solid surface[END_REF].

Temperature measurement based on two dyes

The analysis of the spectra of absorption and fluorescence emission of different dyes is necessary for choosing the couple of dyes. First of all, the dyes should display different sensitivities to temperature in terms of emission spectra. Fluorescein disodium (FL) is chosen as one of the dye in this thesis because of its high sensitivity (fluorescence intensity increases significantly with increasing temperature). Moreover, it retains the temperature sensitivity regardless of the laser irradiance while some fluorescent dyes lose some of their temperature sensitivity at high laser irradiance, like rhodamine B [START_REF] Chaze | Spatially and temporally resolved measurements of the temperature inside droplets impinging on a hot solid surface[END_REF]. Sulforhodamine B sodium (SRB) is another dye used in this work which has a negative temperature sensitivity compared to FL (as To avoid the use of a light source with a large spectral emission band that may also cause spectra conflicts and complicate data analysis, the two dyes should have a common spectral band of absorption. The optimal wavelength to excite both dyes should be in the range 470 -520 nm (in Figure IV-2). However, a laser that emits at 532 nm has been used because of its availability at the LGC. Therefore, as described afterwards, the concentration of FL in the experiment is much higher than that of SRB because of its relatively low intensity of absorption at such wavelength. Finally, the detection bands (2 colors) may be adequately chosen to improve the accuracy of the method. The two detection bands used in the experiment are (as showed in -Band 1 [543-549 nm]: FL has a predominant contribution in this band. SRB has some contribution as well, but as its concentration is very low compared to FL, its contribution can be neglected. Fluorescence is partially absorbed by SRB (𝜀=80000 M -1 .m -1 ).

shown
-Band 2 [>633 nm]: The emission of SRB is predominant. The emission of FL is not totally negligible and its contribution depends on the respective concentration of the dyes.

Neglecting the emission and absorption by SRB in band 1 and the emission of FL in band 2, the ratio R of the fluorescence signals in these two detection bands can be determined, based on equation IV-2.

𝑅 = 𝑑𝐹 1 𝑑𝐹 2 ∝ 𝜀 0,𝐹𝐿 𝜀 0,𝑆𝑅B 𝐶 𝐹𝐿 𝐶 𝑆𝑅B 𝑅 𝑠𝑎𝑡 (IV-2)
where 𝑅 𝑠𝑎𝑡 = 1+𝐼 0 /𝐼 𝑠𝑎𝑡,𝑆𝑅𝐵 1+𝐼 0 /𝐼 𝑠𝑎𝑡,𝐹𝐿

. The quantities like 𝜂 and 𝜀 0 do not depend on dyes, they are only related to the laser. Experiments are carried out in the partially saturation regime of excitation (𝐼 0 ≈ 𝐼 𝑠𝑎𝑡 ), giving 𝑅 𝑠𝑎𝑡 ≈1 where 𝐼 𝑠𝑎𝑡,𝑆𝑅𝐵 is close to 𝐼 𝑠𝑎𝑡,𝐹𝐿 [START_REF] Chaze | The saturation of the fluorescence and its consequences for laser-induced fluorescence thermometry in liquid flows[END_REF]. Chaze et al. [START_REF] Chaze | Spatially and temporally resolved measurements of the temperature inside droplets impinging on a hot solid surface[END_REF] verified experimentally that R does not vary significantly with 𝐼 0 including in the partially saturated domain of excitation.

In this work, a reference temperature (T0) is needed for the temperature measurement and the fluorescence ratio R at one temperature will be normalized to that at the reference temperature.

Based on the normalized ratio, the temperature calibration at one plan of one zigzag channel can be directly applied to other plans and other channel geometries. This can avoid the repeating work of temperature calibration at each plan observed and each prototype studied. The normalized method can also get rid of potential disturbance effects on the fluorescence signal such as prototype surface defaults, optical reflection at different locations in the channels, etc.

In the experiment, the calibration is carried out at one plan in the zigzag channel of 4 mm width.

The normalized fluorescence ratio at one temperature T can be expressed as:

𝑅 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 = 𝑅 𝑅 0 = 𝜀 0,𝐹𝐿 (𝑇) 𝜀 0,𝑆𝑅𝐵 (𝑇) 𝜀 0,𝑆𝑅𝐵 (𝑇 0 ) 𝜀 0,𝐹𝐿 (𝑇 0 ) = f(𝑇) (IV-3)
where f is the calibration function, the temperature then can be deduced by:

𝑇 = 𝑓 -1 ( 𝑅 𝑅 0 ) (IV-4)
In this work, the reference temperature 𝑇 0 is chosen at 24.9 °C.

Materials

1 Chemicals

The chemicals used in this experiment are: fluorescein sodium technical (FL, VWR International SAS, CAS: 518-47-8); sulforhodamine B sodium salt (SRB, Alfa Aesar, CAS: 3520-42-1); carbonate buffer (pH=10, Acros Organics BVBA). The dye solutions are prepared with demineralized water.

In the experiment, 5µmol.L -1 of SRB solution and 0.2 mmol.L -1 of FL solution (400 mL demineralized water + 100 mL carbonate buffer) are prepared. The two solutions (500 mL for each) are mixed together during the experiment, their pH is measured at about 10 which makes sure that all the fluorescent molecules are in basic form.

Experimental prototypes and channel geometries

The prototypes are the three with zigzag channels which have been studied in Chapter II. All their information like materials, geometry parameters of the channels, etc. can be found in part 1.1 in Chapter II. The geometry parameters of zigzag channels in the process plates and of the straight channels (same for three prototypes) in utility plates are summarized in Table IV

Experimental setup

The experimental setup is the same as presented in part 1.2 of Chapter II. For each prototype, the process plate is connected to a magnetic drive pump (0 -100 L.h -1 ) and the utility plate is connected to a centrifugal pump (0 -1000 L.h -1 ). The process fluid (fluorescence solution) is heated by a thermostated bath through a heat exchanger before entering the prototype. 4

thermocouples are used to measure the inlet and outlet temperature of the two plates.

Temperature calibration experiment

The calibration is carried out near (about 0.01m) the entrance of process plate with zigzag channel of 4 mm width. The zone observed is in the middle of the channel (2 mm from the below edge). The flow rate of the fluorescent solution is 4 L.h -1 with no fluid introduced in the utility channel. The inlet and outlet temperatures of the process fluid are measured by thermocouples. During the experiment, the maximum difference between inlet and outlet temperatures is 0.5 °C, the inlet temperature therefore can be considered as the temperature of the zone observed. For the calibration experiment, the fluorescent solution enters the process channel at the temperature ranging from 17 °C to 60 °C (as shown in Table IV-2), where 24.9 °C is taken as the reference temperature since it is close to room temperature which is easier to keep for all the fluorescence experiment. For each temperature, 100 fluorescence images are taken and registered once thermal equilibrium is reached.

Temperature measurement in heat transfer experiment

In heat transfer experiment, the fluorescent solution is heated at about 60 °C before entering the process channel. Two flowrates of fluorescence solution are studied for the three prototypes, corresponding to Reynolds number of around 500 and 650. In the utility channel, water at ambient temperature is introduced with a flowrate of 300 L.h -1 . The two fluids circulate cocurrently. For each prototype, measurements at three longitudinal positions of zigzag channel are carried out. For each position, the fluorescence of one to three plans corresponding to different heights is detected, depending on the channel width. All the plans measured are presented in Table IV-3. For all the measurements, 500 images are taken and registered once heat transfer equilibrium is reached (overall heat transfer coefficient is constant with time).

Image registration between an experiment and that at the reference temperature

In the experiment, as the prototype moves slightly with time, the zone observed moves a little actually. Thus, another image registration is needed as the fluorescence signal ratio will be normalized with that at the reference temperature (24.9 °C). For a given plan, the zone observed for each experiment (calibration or heat transfer) are repositioned to that at 24.9 °C by another matrix which is obtained by calculating the affine transform functions between the image taken at the reference temperature and that taken at other temperature for the temperature calibration (and that taken in heat transfer for heat transfer experiment). With this matrix (different for each case), the fluorescence signal ratio can be normalized to that at the reference temperature.

Results

Temperature calibration

Calibration equation

The fluorescence ratio R is calculated by dividing its grey level of the two images taken by two cameras. This is done for all the pixels of the common zone, and for all the calibration experiments at different temperature. The ratio R obtained at other temperature is normalized to that obtained at the reference temperature (24.9 C°). A calibration equation for each pixel then can be obtained with a series of normalized ratio values and the corresponding temperature, using an exponential regression of the normalized ratio as a function of the temperature:

𝑅 𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 = 𝑅 𝑅 0 = f(𝑇) = 𝑎 • e 𝑏𝑇 (IV-5)
The coefficients a and b are calculated for all the pixels (i = 0-1376, j = 0-736). The calibration

Calibration verification

The temperature measurement at a high plan (3 mm) between the 5 th and 6 th bend (0.059 m from the entrance) in the zigzag channel of 4 mm width is carried out with only introducing the fluorescent solution in the process channel at three temperatures (without heat transfer): 24.9 °C, 36.4°C, 49.2 °C, which can be considered as the temperature at the zone observed as the difference of inlet and outlet temperatures is small (3 °C for 49. The temperature fields for all plans measured in three zigzag channels are obtained, the mean temperature (Tp,m) for all plans is calculated as well, which is presented in Table IV As expected, they are between the inlet and the outlet temperature. For a given channel, it can be observed that the mean temperature decreases when the location is away from the entrance. This is normal, because the process fluid is hot and cooled by a cold fluid, so the temperature far from the entrance is lower.

Among the three plans at one location, the low plan is closest to the cold fluid. However, it is not always the coldest, as can be seen for the data obtained in zigzag channels of 3 mm and 4 mm width. This can be explained by the presence of the Dean vortices which can generate tangential mixing. It can also be observed that the temperature varies more significantly from high plan to low plan for dh = 4 mm than for dh = 3 mm, indicating the temperature on the crosssection is more homogeneous when the channel width decreases. This is consistent with the results presented in Chapter II, where it has been observed an increase of the heat transfer coefficient in the zigzag channel when the hydraulic diameter decreases for same Reynolds number.

To conclude, the general trends observed tend to validate the consistency of the local thermometry method based on LIF technique with 2 dyes / 2 colors. However, because the inlet temperature of two fluids is not the same for each experiment, it is difficult to do more direct analysis for the results. In order to furtherly evaluate the reliability of this method, a heat transfer model is introduced which can estimate the local temperature of process fluid, it is then compared to the temperature obtained by LIF technique.

Comparison between temperature calculated with LIF and heat transfer model

The 1D heat transfer model can be expressed as follows (for process fluid): where 𝐿 𝑝 and 𝐿 𝑢 are the lengths of the process channel and the utility channel. where 𝑇 𝑝,𝑖𝑛 and 𝑇 𝑝,𝑜𝑢𝑡 are the inlet and outlet temperature measured during the heat loss experiment.

𝑢
s (utility channel)

x (process channel)

For the utility fluid, there is a similar equation to calculate the temperature: They need to be simulated with the inlet temperatures at these two locations.

𝑢

Conclusion

In this chapter, the local temperature measurement technique based on laser-induced fluorescence with two colors/two dyes is developed and applied to measure the fluid temperature in millimetric zigzag channels during heat transfer. Temperature fields at different height levels and at different locations in three zigzag channels are obtained. As expected, it is observed that the temperature decreases with the channel length. Concerning the temperature over the cross-section, tangential fluid motion seems enough significant to create mixing as the temperature from the top of the channel to the bottom does not evidently decrease. Moreover, cross-sectional temperature appears to be more homogeneous in the zigzag channel of 3 mm width than in the zigzag channel of 4 mm width.

A heat transfer model is then introduced as a way to estimate the temperature in zigzag channels and to evaluate the consistency of the LIF technique. It is found that the temperature difference between the experiments and the model increases when the channel width decreases. This is probably because the temperature calibration is just done in zigzag channel of 4 mm and it is better to do it for the other two zigzag channels. The evaporation of the dye solution in the experiment may also influence the measurements. Regular experiments for calibration check should be carried out to adjust the calibration equation.

In terms of problems or improvements, firstly, as the width of the zigzag channels is very small (2 to 4 mm), to have a good temperature measurement at different height levels inside the channel, it is better to use a laser with a smaller beam width keeping the similar intensity (around 0.5 mm in the present work). Another problem is that during the experiment, the prototype moves a little with time so the zone observed moves as well, that is why an additional image processing is needed to adjust the zone observed at one operating condition to that at the reference temperature. The prototype was placed on the bench where there are equipment like pumps, thermostatic bath which can cause vibration, they should be placed on a distinct support.

It should finally be concluded that the thermometry by LIF technique with 2 dyes/2 colors present a good potential for local temperature measurement in millimetric channels although the measurement is sensible to the spatial disturbances due to the small channel size. Unlike thermocouples, it does not disturb the flow, provides immediate response that allows the detection of fast dynamic effects and presents a very high spatial resolution. The trends observed related to the measurement in the present work are very consistent and some improvements have been proposed to increase the accuracy of this method.

Chapter V: Study of heat transfer performance in compact HEX reactors with implementation of chemical reactions

The heat transfer performance of heat-exchanger reactor with millimetric channels is studied experimentally and numerically in the last three chapters. The heat transfer coefficient is

Chemical reaction

In this chapter, a competitive consecutive reaction scheme is studied which is presented as:

𝐴 + 𝐵 → 𝑅 𝐵 + 𝑅 → 𝑆 (V-1)
where A and B are the reactants, R is the desired product and S is the undesired product (byproduct). The reaction rates 𝑟 1 and 𝑟 2 are calculated by:

𝑟 1 = 𝑘 1 𝐶 𝐴 𝐶 𝐵 (V-2) 𝑟 2 = 𝑘 2 𝐶 𝐵 𝐶 𝑅 (V-3)
where 𝑘 1 and 𝑘 2 are the reaction rate coefficients, which can be expressed as follows: For the simulation, the process fluid velocity is set at 0.7 m.s -1 , typical in millimetric channels.

For each utility plate, the width of the channel (wu) is the total width of one unit in zigzag channel (bend to bend), so the utility channel can completely cover the process channel.

Assuming a utility fluid velocity of 1 m.s -1 and a total flowrate 10 times larger than that of the process fluid, the depth of utility channel (du) then can be determined. The geometries of process channels and utility channels and the corresponding fluid flowrates are presented in the following table.

Assumptions

Before simulating the reaction, several assumptions have been made in this model. They are listed below:

1. The reactants are perfectly mixed at the inlet of HEX reactors and plug flow is assumed both in process channel and utility channels.

2. The influences of residence time distribution and mixing to the reaction are not taken into account.

3. Concentration and temperature gradients in the radial direction in zigzag channel and straight channel are assumed as zero.

4. The physicochemical properties of reactants and products are assumed to be the same, equal to those of water.

5. Fluid flowrates are supposed constant.

6. The heat loss in HEX reactors is assumed as zero.

Calculation of concentration and temperature

The temperatures of the process fluid and utility fluid are calculated with 1D thermal balances at steady state based on first order discretization on space: Since the process plate is surrounded by two utility channels, the total heat exchange area of the process channel is twice the surface of one side dA:

𝑑𝐴 = 𝑑𝐴 𝑝 = 𝑑 ℎ,𝑝 𝑑x (V-8)
From these equations, the temperatures in process channel and utility channels of HEX reactor can be deduced:

Results

The simulation of the model is carried out with Excel using the iterative solution method. With inlet reactant concentrations and process fluid and utility fluid temperatures, the concentration of the reactants and the products and the temperatures of the process and utility fluids at other locations can be deduced with all the equations presented above.

For the two reaction cases, two inlet concentrations are studied: CA0 = CB0 = 300 mol.m -3 with fast reaction rate and CA0 = CB0 = 100 mol.m -3 with relatively less fast reaction rate. The inlet temperature of the process fluid (Tp0) is fixed at 350 K for both cases.

Reaction case E1 > E2

For the reaction case E1 > E2, since temperature favors reaction 1 (desired reaction), it is better V-4) decreases when the channel cross-section size increases: the heat produced by the reactions is transferred less efficiently to the utility fluid. This can be clearly seen after the peak, once the reactions are completed: the process fluid is cooled slowly when the channel crosssection size increases although the same temperature is reached at the end. The difference between the temperature peak and the inlet temperature (ΔTm) in the three zigzag channels for both initial reactant concentrations (300 and 100 mol.m -3 ) is calculated and presented in Table V-5. The adiabatic temperature profile is also calculated by setting the overall heat transfer coefficient Ux = 0 in the simulation. The difference between outlet temperature and inlet temperature (ΔTa) is presented in this table. The conversion of reactant A and the selectivity of desired product R in three zigzag channels are also presented in Table V-5. The conversion of A and the selectivity of R are calculated by the following equations:

𝐶𝑜𝑛𝑣𝑒𝑟𝑠𝑖𝑜𝑛 = 𝐶 𝐴0 -𝐶 𝐴𝑒 𝐶 𝐴0 (V-16 ) 𝑆𝑒𝑙𝑒𝑐𝑡𝑖𝑣𝑖𝑡𝑦 = 𝐶 𝑅𝑒 𝐶 𝑅𝑒 +𝐶 𝑆𝑒 (V-17 )
where 𝐶 𝐴𝑒 , 𝐶 𝑅𝑒 and 𝐶 𝑆𝑒 are the concentrations of reactant A, desired product R and undesired product S at the end of the reactions. V-4) when channel cross-section width increases, so heat produced by the reaction is transferred less efficiently. This can be clearly seen after the peak, the process fluid is cooled less efficiently for the channel cross-section of 4 mm width compared to that of 3 mm and 2 mm, although they reach the same temperature at the end. The difference between the temperature peak and the inlet temperature (ΔTm) in three zigzag channels for both inlet reactant concentrations (300 and 100 mol.m -3 ) is calculated and presented in the Table V-6. The adiabatic temperature profile is also calculated with setting the mol.m -3 ) lower than in the adiabatic case. It shows the good heat transfer performance of HEX reactors.

To conclude, in this reaction case (E1 <E2), the reaction is finished after 10 m for inlet concentration of 300 mol.m -3 , but not for inlet concentration of 100 mol.m -3 where the reactant B still remains at the end. When the hydraulic diameter of zigzag channels varies from 2 mm to 4 mm, the selectivity of the desired product R decreases for both inlet concentrations, the conversion rate of reactant A decreases for the inlet concentration of 300 mol.m -3 and increases for the inlet concentration of 100 mol.m -3 due to uncomplete reaction. The difference between the maximum temperature and inlet temperature increases with increasing cross-section width for concentration of 300 mol.m -3 . That means the scale-up does not favor the conversion and the selectivity and a hotter point is created. So for this reaction case, it is better to use smaller channels in parallel than a large channel for the scale-up strategy.

Conclusion

In this chapter, a 1D model of plug flow is used to study the heat transfer performance of compact HEX reactors with introducing chemical reactions. The heat transfer coefficient is calculated with the Nusselt number correlation established in the previous chapters. The performances of HEX reactors in terms of conversion and selectivity for the implementation of an exothermic competitive consecutive reaction scheme are discussed. Two cases of reactions are investigated, one is the activation energy of desired reaction larger than undesired reaction (E1 > E2), the other case is the activation energy of desired reaction smaller than undesired reaction (E1 < E2). For the case E1 > E2, high temperature of utility fluid is introduced while for E1 < E2, low temperature of utility fluid is considered for the purpose of producing more desired product R. For each case, two inlet concentrations of reactants are simulated.

The compact HEX reactors simulated in this chapter are based on the geometry studied in the last three chapters, with the same zigzag channels in the process plate (dh = 2, 3, 4 mm). Two utility plates with straight channel are used with the consideration of industrial design. The influence of process channel hydraulic diameter on the hot point (temperature), the conversion rate and the selectivity is discussed. It is found that for the case E1 > E2, when the channel cross-section width varies from 2 mm to 4 mm, the conversion of the reactant A and the selectivity in the desired product R both increase which are favorable for the scale-up. The hot point increases as well which should be taken into account. For E1 < E2, the conversion of A and the selectivity in the desired product R decrease with increasing channel cross-section length which is not favorable for the scale-up, so is the hot point. In this reaction case, it is better to use smaller channels in parallel than larger channels for the scale-up strategy. In the simulation, problems of convergence or mesh independence have been met when Reynolds number increases (in laminar regime) or when the straight section length between two bends decreases. In addition, it is much more complicated to carry out the simulation in transient regime, since there is not a generic thermal-hydrodynamic model available for this particular flow regime. In order to overcome these limitations of simulation, an experimental method based on laser-induced fluorescence thermometry with two-color/two dyes is developed in Chapter IV for the measurement of local temperature. The temperature field at different levels and at different locations in the three zigzag channels is obtained. It is found that the temperature over the cross-section is more homogeneous in the zigzag channel of 3 mm than in the zigzag channel of 4 mm. Furtherly, a heat transfer model is introduced to evaluate the consistency of the LIF. It is found that the temperature difference gets bigger when the channel hydraulic diameter decreases. That is probably because the temperature calibration method has just been carried out in the zigzag channel of 4 mm. It is better to do it for the other two zigzag channels as well.

In last Chapter, a 1D model of plug flow is used to study the heat transfer performance of compact HEX reactors for the implementation of chemical reactions. Their performance in terms of conversion, selectivity for an exothermic competitive consecutive reaction scheme is discussed. Two cases of reactions are investigated: one is the activation energy of desired reaction larger than undesired reaction (E1 > E2), the other case is the activation energy of desired reaction smaller than undesired reaction (E1 < E2). For the case E1 > E2, high temperature of utility fluid is introduced while for E1 < E2, low temperature of utility fluid is considered for the purpose of producing more desired product R. The objective is to illustrate how such a model, based on the prior characterization of heat transfer performance of the devices, can be used for choosing optimal scale-up strategies. In the studied cases, the reactor scaling-up favors the production of the desired product in the first case while numbering-up is more appropriate for the second case.

This work allows the development of methods (experimental and numerical) for flow characterization and heat transfer coefficient measurements adapted to millimetric channels used in compact HEX reactors. These methods have been implemented to study the heat transfer performances of square zigzag channels. However it shows that for numerical method, the development of new numerical models or assessment of the reliability of available models is needed to simulate transient flow regime for numerous innovative design of compact HEX reactors which can define correlations for a large range of technologies. All these data will facilitate the process design (choice of the devices and optimization of the operating conditions)

and the process scale-up.

The proposed correlation for zigzag channels in Chapter II was established from the experimental data mainly with the variation of the channel width and flow rate of process fluid.

Other geometry parameters like the radius of curvature, the straight section length between two bends (although being numerically studied), the angle between two straight sections, should also be studied to enrich the data, so are process fluids with a wide range of physico-chemical

properties.

An original breakthrough in this thesis concerns the development of an experimental method based on laser-induced fluorescence thermometry with two-colors/two dyes. The result obtained in this work mainly showed the feasibility of the method. More studies need to be conducted for the future work to improve the reliability of this method for temperature field measurement inside the channel.

The work presented in this thesis concerns a monophase flow. However, in many chemical synthesis cases, the reaction takes place in a multiphase flow. Studies have already been conducted on the hydraulic behavior of these media as well as the determination of mass transfer coefficients. It would then be useful to extend the methodologies developed in this thesis for the characterization of heat exchange performance to more complex media, for example liquidliquid flow.

  Figure I-1.

Figure I- 1 :

 1 Figure I-1: Process intensification classification by Stankiewicz and Moulijn [2]

  Based on this definition, they proposed a new classification of process intensification which is divided into two main classes: unit intensification and plant intensification, as shown in Figure I-2. They furtherly proposed a general mathematical formulation for each intensification process with the consideration of the existing units and new units through a disjunctive formulation.

Figure I- 2 :

 2 Figure I-2: Process intensification classification by Ponce-Ortega et al[START_REF] Ponce-Ortega | Chemical Engineering and Processing : Process Intensification Process intensification : New understanding and systematic approach[END_REF]

Figure I- 3 :

 3 Figure I-3: Fundamental view on process intensification based on four principles andincluding four approaches within four domains[START_REF] Van Gerven | Structure, Energy, Synergy, Time-The Fundamentals of Process Intensification[END_REF]

Figure I- 5 :

 5 Figure I-5: Schematic of spinning disk reactor[START_REF] Oxley | Evaluation of Spinning Disk Reactor Technology for the Manufacture of Pharmaceuticals[END_REF]

Figure I- 6 :

 6 Figure I-6 : Pilot-scale production plant with microreactors (Merck, Germany) [25]

  mixers consist of identical and immobile inserts which can be installed in a tube, column or reactor. The function of the inserts is to redistribute the fluid on the radial direction with respect to the main flow direction to significantly enhance radial mixing. The effectiveness of this redistribution depends on the design and the number of inserts, the fluid properties and the volumetric flowrate. Many commercial designs are available in the industry, some of them are shown in Figure I-8 [31].

Figure I- 7 :

 7 Figure I-7: Inserts of different static mixers : (a) Kenics (Chemineer) ; (b) HEV (Chemineer) ; (c) Inlier series 45 (Lightnin) ; (d) Custody transfer mixer (Komax systems)[START_REF] Thakur | Static mixers in the process industriesða review[END_REF]

  Laval has developed a series of ART ® Plate Reactors which are used in the production of fine and specific chemicals, such as pharmaceuticals. The ART ® Plate Reactors are versatile reactors that can be easily reconfigured to suit different reaction types, such as liquid/liquid reactions, exothermic reactions, etc. They have the advantage of being easily openable for cleaning and inspection. The reactor shown in Figure I-8(a) is the ART ® Plate Reactor 37 [47]. It consists of a series of stainless steel plates where cranked channels are etched for the purpose of enhancing mixing and heat transfer. This geometry can bring plug flow and corresponding residence time distributions. The ART PR37 reactor can handle maximum flowrate of about 30 L.h -1 which corresponds to a residence time of 45 s per plate, with a temperature ranging from -40 ° C to 200 °C. The ART® Plate Reactor 49 reactor [48] (in Figure I-8(b)) is the extension of ART PR37 which accept flow rates up to 500 L.h -1 , with a residence time of 60 s per plate.

Figure I- 8 :

 8 Figure I-8: Alfa Laval Plate Reactors: (a) ART PR37; (b) ART PR49

Figure I- 9 :

 9 Figure I-9: Corning Advanced-Flow™ Reactors : G1-G3 [49]

Figure I- 10 :

 10 Figure I-10: Shimtech reactor of Chart industries [50]

Figure I- 11 :

 11 Figure I-11: Heat-exchanger reactor SiC/Al

Figure I- 12 :

 12 Figure I-12: Simulated secondary flow in zigzag channel with square cross-section

  Figure I-13 shows an example of the chaotic flow characterization in a zigzag channel with square cross-section obtained by Zheng et al. with CFD simulations [68]. They put the tracer particles in the cross-section at the entrance of a bend which is the straight line in Figure I-13 (a). After the numerical resolution, they recorded the tracer particles in the cross-section after 1 bend, 4 bends and 12 bends for three Reynolds numbers (150, 200, 400). For a Reynolds number of 150 (in Figure I-13(a)-(c)), they observed moderate stretching and folding of the tracer particle line. For a Reynolds number of 200 (in Figure I-13(d)-(f)), they observed stronger stretching and folding. For a Reynolds number of 400 (in Figure I-13(i)), they found the tracer particles had almost covered the entire cross-sectional area after 12 bends. They interpreted it as chaotic advection. The enhancement of mixing by chaotic flow is stronger than by the secondary flow.

Figure I- 13 :

 13 Figure I-13: Chaotic advection characterization in zigzag channel with square cross-section for Reynolds number of 150 ((a)-(c)), 200 ((d)-(f)), 400 (g)-(i)) at three cross-sections after 1bend, 4 bends and 12 bends. The initial tracer particles is the straight line in (a)[START_REF] Zheng | Chaotic advection in steady laminar heat transfer simulations: Periodic zigzag channels with square cross-sections[END_REF]

  heat fluxes as well as to investigate the surface flow field behavior over complicated body shapes. Leblay et al.[START_REF] Leblay | Infrared Thermography applied to measurement of Heat transfer coefficient of water in a pipe heated by Joule effect[END_REF] applied IRT to measure heat transfer coefficient of water in a round tube and in a multiport-flat tube. Freund et al.[START_REF] Freund | Investigation of local heat transfer coefficients in plate heat exchangers with temperature oscillation IR thermography and CFD[END_REF] developed a method for the measurement of local convective heat transfer coefficient from the outside of a heat-transferring wall based on IRT. IRT has advantages like high spatial resolutions of surface temperatures and immediate response. However, IRT is limited to outside surfaces, so it is difficult to apply it to fluid flow temperature measurement as inertial effects and temperature diffusion in the outside surface significantly reduce the accuracy of the method. At lab scale, laser-induced fluorescence thermometry can also be used. It is based on the temperature dependence of fluorescence emission of organic dyes and allows to measure the temperature in liquid flows. During the last two decades, different approaches to implement have been developed. The simplest technique uses a single dye and a single spectral band to measure the temperature. Nakajima et al. [88] and Lemoine et al. [89] used rhodamine B to measure the temperature in turbulent heat transfer. Sakakibara et al. [90] measured the temperature in thermally stratified pipe flow with rhodamine B. Coolen et al. [91] performed temperature measurements with rhodamine B in a natural convection flow. For this technique, the problem is that any disturbance of fluorescence signal can influence the measurement of temperature since the temperature is directly deduced from the fluorescence intensity. For example the temporal variation of laser intensity or a change of dye concentration affects significantly the measurement. The flow itself can influence the fluorescence signal as well, like in multiphase flows, the interfaces of droplets and bubbles can change light propagation.

  The heat-exchanger (HEX) reactors are presented in more details since they are the study object of this thesis. They combine two process intensification methods: the miniaturization and the multifunctionality. Several industrial HEX reactors are presented. The hydrodynamic performance of heat exchanger reactors are discussed, in terms of flow patterns, residence time, mixing and pressure loss. The thermal performance is presented as well. Several temperature measurement methods in liquid flows are introduced with the purpose of characterizing the thermal performance. The laser-induced fluorescence thermometry is presented in the last part which is utilized in this thesis to measure the local temperature in HEX reactors. This work concentrates on the heat transfer performance of millimetric zigzag channels which can be used for the design of compact HEX reactors. Experimental and numerical methods for the characterization are presented in the following Chapters.

l

  is the dimension, T is the temperature. One prototype with a zigzag channel engraved in the process plate is presented in Figure II-1.

Figure II- 1 :

 1 Figure II-1: Prototype composed of a PMMA process plate with a zigzag channel engraved (a) and a PMMA utility plate with a rectangular straight channel engraved (c) separated by an aluminum plate (middle in (b)). Top view of total length with of 65 cm long and 3 cm width (d).

Figure II- 2 :

 2 Figure II-2: Schematic of the zigzag channel: dh = 2 -4 mm, Ls = 7 mm, Rc = 1.5 -3 mm, θ = 90°

Figure II- 3 :

 3 Figure II-3: Illustration of the prototype top view for the zigzag process channel of 4 mm width.

  4 thermocouples are used to measure the inlet and outlet temperature of these two channels. The temperature measurement are Utility channel Process channel recorded by an acquisition system based on LabVIEW software (National Instruments, US)which can also control the gear pump easily.

Figure II- 4 :

 4 Figure II-4: Experimental set-up for the characterization of heat transfer coefficients

  9 m.s -1 Reu = 10 4 -2×10 4 Glycerol solution 𝑄 𝑝 = 1.2 -11 L.h -1 up = 0.08 -0.7 m.s -1 Rep = 20 -350 𝑄 𝑝 = 2.4 -28 L.h -1 up = 0.07 -0.9 m.s -1 Rep = 25 -420 𝑄 𝑝 = 2.4 -31 L.h -1 up = 0.04 -0.4 m.s -1 Rep = 15 -220

7 )

 7 𝐻,𝑢 (II-4) where hp and hu are the convection heat transfer coefficient of the process fluid and utility fluid, AH,p and AH,u are the heat transfer areas in the process plate and the utility plate. They are calculated by: 𝐴 𝐻,𝑝 = 𝑤 𝑝 𝐿 𝑝 (II-5) 𝐴 𝐻,𝑢 = 𝑤 𝑢 𝐿 𝑢 (II-6) where 𝑤 𝑝 , 𝑤 𝑢 and 𝐿 𝑝 , 𝐿 𝑢 are the channel width and channel length of process channel and utility channel. For the zigzag channel, channel length corresponds to the developed length of the geometry and it is longer than the channel length of utility. AH is considered equal to AH,p in this work. The heat transfer coefficient hu of the utility fluid (Re > 10 4 in the experiment) can be deduced from the correlation of Nusselt proposed by Gnielinski, reminded below: Pru is close to 4 in the experiment. Assuming the temperature of the wall is that of the process fluid (it is lower in fact), Prw is estimated to be around 6. So in the calculation, it is assumed that ( 𝑃𝑟 𝑢 𝑃𝑟 𝑤 ) 0.11 = 1. the heat transfer coefficient of the process fluid hp can then be deduced from equation II-4. The Nusselt number Nup can be furtherly obtained with Nu = hdh/k. The physicochemical properties (density 𝜌, specific heat capacity 𝐶 𝑝 , Prandtl number 𝑃𝑟) of water at different temperatures can be obtained by the following equations [102]: 𝜌[kg. m -3 ] = 330 + 5.92 × 𝑇 -1.63 × 10 -2 𝑇 2 + 1.33 × 10 -5 𝑇 3 (II-8) 𝐶 𝑝 [J. kg -1 . K -1 ] = 10 4 -5.14 × 10 2 𝑇 + 1.49 × 𝑇 2 -1.43 × 10 -3 𝑇 3 (II-9) 𝑃𝑟 = 900 -7.79𝑇 + 2.26 × 10 -2 𝑇 2 -2.2 × 10 -5 𝑇 3 (II-10) 𝑘[W. m -1 . K -1 ] = -12.2 + 1.18 × 10 -1 𝑇 -3.66 × 10 -4 𝑇 2 + 3.81 × 10 -7 𝑇 3 (II-11) The viscosity can be calculated with the three equations above (II-9, II-10, II-11): physicochemical properties of the water-glycerol solution as a function of temperature are obtained by different methods. The density and viscosity are measured at the LGC (presented in Appendix A). The density is measured by a density meter (Anton Paar, DMA 38), the viscosity is measured by Kinexus ultra+ rheometer (Malvern Instruments). The heat capacity of the mixture is calculated by the heat capacity of each component multiplying its mass fraction.

  is the average value between inlet and outlet temperatures of the fluid.The calculation of one experimental case in the prototype with the zigzag channel of 4 mm width in process plate is presented here. The flowrates of process fluid and utility fluid are 10.2 L.h -1 and 300 L.h -1 , the inlet and outlet temperatures of the process fluid and the utility fluid are measured at 59.1 °C (Tp,in), 45.2 °C (Tp,out), 36.6 °C (Tu,in) and 37.6 °C (Tu,out) respectively. The inlet temperature of the utility fluid is 36.6 °C, higher than ambient temperature, because the utility fluid circulates in closed loop and is heated all the time during the experiment, so does the process fluid. The heat exchange flux stabilized after nearly one hour and half. The zigzag channel length is 0.747 m, the utility channel length is 0.588 m. The thermal loss has been previously estimated by just introducing the process fluid and measuring the inlet and outlet temperatures(58.8 °C and 58 °C respectively). With all these data, the overall heat transfer coefficient U, the heat transfer coefficient ℎ 𝑢 and the Nusselt number 𝑁𝑢 𝑝 of the process fluid can be calculated.

  overall heat transfer coefficient U as a function of process fluid flowrate (water) at different utility fluid flowrates is presented in Figure II-5. In Figure II-5 (a), U increases with increasing flowrate in zigzag channels. It seems that the curves tend to asymptotic values, typical of the fact that the thermal resistance of the process side becomes negligible for high process flowrates.

Figure II- 5 :

 5 Figure II-5: Overall heat transfer coefficient as a function of process fluid flowrate (water) in (a) zigzag channels and (b) straight channels at three utility fluid flowrates. The legend gives the hydraulic diameter of process channel (dh) and the utility fluid flowrate (Qu).

  compared. hp in zigzag channels as a function of Reynolds numbers at different utility flowrates is presented in Figure II-6. As expected, hp is independent of the utility flowrate. It increases with increasing Reynolds number for the same channel. As for the influence of channel geometry, hp increases with the decrease of cross-section length at same Reynolds number.

Figure II- 6 :

 6 Figure II-6: Heat transfer coefficient of process fluid (water) as a function of Reynolds number in zigzag channels. The legend gives the hydraulic diameter of process channel (dh) and the utility fluid flowrate (Qu).

Figure II- 7 :

 7 Figure II-7: hp,zigzag/hp,straight as a function of Reynolds number at two utility flowrates. The legend gives the hydraulic diameter of process channel (dh) and the utility fluid flowrate (Qu).

3 .

 3 Influence of fluid properties on heat transfer in prototypes with zigzag channels 3.1 Overall heat transfer coefficient The overall heat transfer coefficient U as a function of process fluid flowrate and Reynolds number for two fluids (water and glycerol solution) in the prototypes with zigzag channels is presented in Figure II-8. In Figure II-8 (a), U with water as process fluid is much higher than that with glycerol solution at the same flowrate, generally 3 times. That is because the glycerol solution is much more viscous than water, resulting in a very small Reynolds number for the same flowrate (as shown in Figure II-8 (b)), which does not favor the heat transfer. In terms of channel size influence, U increases with the decreasing channel cross-section length for both fluids. In Figure II-8 (b), the evolution of U with glycerol solution as a function of Reynolds number almost follows that with water, except for the channel cross-section of 2 mm, U is a little larger than that with water at same Reynolds number.

Figure II- 8 :

 8 Figure II-8: Overall heat transfer coefficient for two process fluids in prototype with zigzag channels as a function of (a) flowrate, (b) Reynolds number. The legend gives the hydraulic diameter of process channel (dh), the utility fluid flowrate (Qu) and the process fluid. Glycerol represents the glycerol solution of 74% in mass fraction.

  Figure II-9. The similar trend can be seen compared to overall heat transfer coefficient (Figure II-8 (b)), that is the evolution of hp with glycerol solution generally follows that with water. For the same Reynolds number, when the channel cross-section decreases, hp increases for both fluids.

Figure II- 9 :

 9 Figure II-9: Heat transfer coefficient in zigzag channels as a function of Reynolds number.

  up of HEX reactors. With the heat transfer coefficient hp, the Nusselt number of process fluid Nup in zigzag channel is calculated and presented in Figure II-10. Nup increases with increasingReynolds numbers for both fluids in the three channels. When the cross-section length increases, Nup with water increases moderately at same Reynolds number. The increase between the crosssection of 3 mm and 4 mm is less significant than that between the cross-section of 3 mm and 2 mm for Nup with water. The influence of channel cross-section length on Nup with glycerol solution is not so evident.Therefore, the hydraulic diameter must be taken into account for Nusselt number correlation, in addition to the Reynolds number. Since the ratio (dh/Rc) is constant in the prototypes studied, so it is not taken into consideration in Nusselt number correlation. Although this ratio is usually used to characterize Dean number (De) in fluid hydrodynamics for the analysis of flow and heat transfer in curved channels, De is expressed as follows: ratio of hydraulic diameter to the straight section length between bends (dh/Ls) is considered in the Nusselt number correlation. It varies from 0.28 to 0.57 for the geometries studied. The influence of this ratio on the flow and heat transfer in zigzag channels will be discussed in detail in Chapter III with the numeric method based on CFD simulation.

Figure II- 10 : 16 )

 1016 Figure II-10: Nusselt number as a function of Reynolds number in zigzag channels. The legend gives the hydraulic diameter of process channel (dh), the utility fluid flowrate (Qu) and the process fluid. Glycerol represents the glycerol solution of 74% in mass fraction.

Figure II- 11 :

 11 Figure II-11: Comparison between experimental Nusselt number and that calculated by the correlation in zigzag channels, dashed lines represent 20% error.

  Figure III-1(a), since the fluid velocity and temperature gradients are much higher there. Along the axial direction, the mesh was constructed to be uniform, as shown in Figure III-1(b). A mesh is defined by three numbers of divisions:  along the channel depth and width, Nd, corresponding to a discretized length equal to dh;  along the straight section, Ns, corresponding to a discretized length equal to Ls;  along the curved section in the bend, Nc, corresponding to a discretized length in the center of the channel equal to Lc.

Figure III- 1 :

 1 Figure III-1: Illustration of the mesh for one repeating unit on (a) the cross-section and (b) the wall. dh = 2 mm, Ls = 2-12 mm, Lc = 2.4 mm.

Figure III- 2 : 2 . 3 results 2 . 3 . 1

 223231 Figure III-2: Mesh influence on the local Nusselt number in (a) geometry 'f' for Re = 560 and (b) geometry 'b' for Re = 224

  two channels (corresponding to the outlet of the 8 th bend for zigzag channel) for Re = 224 are presented in Figure III-4 (a) and (b). As expected, the temperature field over a cross-section in the zigzag channel is more homogeneous than in the straight channel. In the straight channel, characterized by a laminar Poiseuille flow (Figure III-4(c)), the heating of the successive concentric flow layers occurs by thermal diffusion. In the zigzag channel, the Dean vortices generated by the bends (Figure III-4(d)) create radial mixing which improves temperature homogenization over the channel cross-section.

Figure III- 3 :Figure III- 4 :

 34 Figure III-3: Hot point and cold point at different axial cross-sections in the zigzag channel 'e' and in the straight channel for Re: (a) 112, (b) 224, (c) 336

Figure III- 5 : 2 . 3 . 2 Figure III- 6 :

 52326 Figure III-5: (a) Relative maximum temperature difference and (b) relative standard temperature deviation over the cross-section at x = 0.075 m in the straight channel and the zigzag channel 'e' versus Reynolds number

  0

  Figure III-7: (a) Velocity fields and (b-i) relative tangential velocity vectors at different crosssections between two bends at Re = 224 in zigzag channel 'e'. The top side in (b-i) corresponds to the outer side of the bend studied.

Figure III- 8 :

 8 Figure III-8: (a) Velocity fields and (b-i) relative tangential velocity vectors at different crosssections between two bends at Re = 448 in zigzag channel 'e'. The top side in (b-i)

  5 mm to 12 mm are studied at Re = 224. The local Nusselt number Nu and friction factor 𝑓 from the channel entrance to the exit of the last unit of the zigzag channel are shown in Figure III-9(a-j). The local Nusselt number and friction factor in a straight channel with the same cross-section at the same Reynolds number are also presented as a reference. It is observed that the transition between periodic flow (Figure III-9(d-e, i-j)) and non-periodic flow (Figure III-9(a-c, f-h))occurs as the straight section length is decreased. By analogy with the previous part, where the non-periodic flow occurs by increasing the fluid velocity, it is confirmed that reducing the residence time between two bends favors the occurrence of a non-periodic flow regime.

Figure III- 9 : 2 . 3 . 4

 9234 Figure III-9: (a-e) Local Nusselt number and (f-j) Fanning friction factor profiles in the straight channel and the zigzag channel with Ls = 3.5-12 mm at Re = 224.

Figure III- 10 :

 10 Figure III-10: Fast Fourier transform analysis for Nusselt number and friction factor in zigzag channels with straight section lengths of (a,c) 7 mm and (b,d) 3.5 mm at Re = 224.

Figure III- 11 :

 11 Figure III-11: Map of periodic flow and non-periodic flow in the zigzag channels studied for Re = 112-560.

  boundary layer has been fully-developed for both flow regimes within a zigzag channel length of few centimeters.

Figure III- 12 :

 12 Figure III-12: Average Nusselt number in straight channel and zigzag channel at Re = 224: (a) Ls = 12 mm; (b) Ls = 3.5 mm.

  of the ratio Ls/dh on the Nusselt number is small at low Reynolds number (in Figure II-10). Concerning the friction factor, it decreases with increasing Ls/dh or Reynolds number, as shown in Figure III-13(b), since the number of singularities (i.e. bends) increases with decreasing straight section length between two bends for a fixed channel developed length.

Figure III- 13 :

 13 Figure III-13: Fully-developed average Nusselt number (a) and friction factor (b) in the zigzag channels.

Figure III- 14 :

 14 Figure III-14: Heat transfer enhancement in the zigzag channels studied.

  The power factor of Pr is 0.33 as suggested by Sieder and Tate correlation in laminar flow or Colburn correlation in turbulent flow[START_REF] Incropera | Fundamentals of heat and mass transfer[END_REF]. Nu/ Pr 0.33 is plotted as a function of Re as shown in Figure It is observed that the simulation results fit well with the experimental data which validates the simulation results.

Figure III- 16 :

 16 Figure III-16: Comparison of Nu/ Pr 0.33 from simulation and experiment for the zigzag channel 'e'. The experimental error is estimated from the accuracy of the sensors (±0.2 kg.h -1 for the process fluid flowrate, ±3 kg.h -1 for the utility fluid flowrate, ±0.3 °C for the temperatures).

Figure III- 17 :

 17 Figure III-17: Comparison of characterized Nusselt number (by simulation and experiment) and calculated by the proposed correlation in zigzag channels, dashed lines are with 20% error.

  numerical method is validated by comparing the results with the experimental data for Nusselt number obtained in Chapter II.The results show that local Nusselt numbers and Fanning friction factors vary along the zigzag channels in form of oscillating curves. The amplitude of the oscillations is constant with periodic flow but varies in the case of non-periodic flow. Based on Fourier analysis, a map has been provided which allows to distinguish the conditions in terms of Reynolds number and straight section length that gives a periodic flow or a non-periodic flow. Non-periodic flow is obtained by increasing the fluid velocity or by decreasing the straight section length. For Re = 560 and Ls/dh = 3.5, the flow instability created by the bends was too great for steady laminar flow to develop. The heat transfer enhancement and thermo-hydraulic performance factor are quantitatively discussed, showing the zigzag channel enhances the heat transfer strongly and increases pressure drop less significantly compared with the straight channel. It appeared that non-periodic flow is not particularly interesting in terms of thermo-hydraulic performances. All these data should be very useful for the design of high-performance heat exchangers-reactors.

  in Figure IV-1, KR in the legend is SRB). The use of two dyes with opposite temperature sensitivity can improve the accuracy of the measurement as presented in Chapter I. Moreover the absorption and emission spectra of SRB are red-shifted which can reduce the spectra conflicts in the two-dyes approach, problem mentioned by Coppeta and Roger [114].

Figure IV- 1 :

 1 Figure IV-1: Variation in the fluorescence signal as a function of the temperature for differentdyes where KR is SRB in the legend[START_REF] Chaze | The saturation of the fluorescence and its consequences for laser-induced fluorescence thermometry in liquid flows[END_REF]

Figure IV- 2

 2 Figure IV-2):

A

  pulsed Nd:YAG laser at 532 nm (Litron Lasers, Nano L 120-20 PIV, pulse energy E= 60 mJ, pulse duration dt = 6 ns) is used in this work for the excitation of the fluorescent molecules. The laser beam is about 500 µm of width. The pulsation frequency of the laser is 10 Hz. The fluorescence is detected by two CCD cameras (PC, 1376x1040 pixels, 10 Hz, 12 bits) which are placed above the prototype and synchronized with the laser. The two cameras are equipped with two different interference filters to separately obtain the fluorescence signals emitted in the two bands of interest as shown in Figure IV-3. The optical detection system is based on a binocular microscope (Leica, M651) with an objective lens which can detect the fluorescence signal. There are two tubes over the objective lens to support the two cameras. The fluorescence signals detected by the cameras are registered by a computer with the software Camware from PCO company (German). The cameras and the computer are from the regional research federation FERMAT (Midi-Pyrénées).

Figure IV- 3 :

 3 Figure IV-3: Optical detection system

Figure IV- 4 :

 4 Figure IV-4: Optical system for the visualization of fluorescence emission in the zigzag channel

  Two images observed simultaneously by two cameras in the zigzag channel of 4 mm width are shown in Figure IV-5. It can be seen that the zone observed by the two cameras is not the same. It is due to the slight angular offset of the two optical axes of the binocular microscope which slightly deviate the light trajectory. Therefore a mathematical method is first implemented to superimpose the two images (by translation and rotation) to obtain a common observation area (code developed and launched in Matlab). A matrix is obtained by calculating the affine transform function between these two images which uses the normalized cross-correlation to adjust each pair of control points of the two images. The effects of the image vignetting and the background noise to the grey scale of the image are taken into account, since the zone observed by two cameras is different. With this matrix, the fluorescence signal ratio (in terms of grey level ratio) between the two cameras at each pixel can be calculated.

Figure IV- 5 :

 5 Figure IV-5: Fluorescence images simultaneously observed by two cameras in the zigzag channel of 4 mm width

  curve at the pixel (1000, 500) is presented in Figure IV-6. The coefficients of calibration calculated at this pixel are: a=0.4499, b=0.0323. It can be seen that the calibration equation fits well with the experimental data.

Figure IV- 6 :

 6 Figure IV-6 : Calibration curve at the pixel (1000, 500)

Figure IV- 7 :

 7 Figure IV-7: Distribution of coefficients a and b in calibration equation for all pixels

Figure IV- 8 :

 8 Figure IV-8:Temperature field of a high plan (0.059 m) in the zigzag channel of 4 mm width with inlet temperature of (a) 36.4°C and (b) 49.2 °C. Flow circulates towards the left.

Figure IV- 9 :

 9 Figure IV-9: Temperature field of a low plan at the location of 0.034 m in the zigzag channel of 4 mm width with Re = 503. Flow circulates towards the left.

  -4. The inlet and outlet temperatures of process fluid (Tp,in, Tp,out), inlet temperature of utility fluid (Tu,in) for the measurement at each location are also presented in this table. They are the mean value of the three plans for a given location except for zigzag channel of 2 mm width where only one plan is measured at one location.

  𝑝 𝜌 𝑝 𝑐 𝑃,𝑝 𝑑𝑇 𝑝,𝑥 𝑑𝑥 = 𝑈𝑑𝐴(𝑇 𝑢,𝑠 -𝑇 𝑝,𝑥 )-𝑈 𝑙 𝑑𝐴(𝑇 𝑝,𝑥 -𝑇 𝑎𝑖𝑟 ) 𝑑𝑉 𝑝 (IV-7)where 𝑢 𝑝 , 𝜌 𝑝 and 𝑐 𝑃,𝑝 are the velocity, density and specific heat capacity of the process fluid respectively. 𝑇 𝑝,𝑥 is the mean temperature at the location x of the process fluid and 𝑇 𝑢,𝑠 is the mean temperature at location s of utility fluid (Figure IV-10). As the channel lengths of two plates are different (process plate and utility plate), in order to calculate the temperature, the reference location for two channels should be the same. Therefore, a transformation is needed which is presented as:

Figure IV- 10 :

 10 Figure IV-10: Illustration of the axial coordinates in the process and utility channels

  ,𝑥 -𝑇 𝑢,𝑠 ) (IV-11) where 𝑢 𝑢 , 𝜌 𝑢 and 𝑐 𝑃,𝑢 are the velocity, density and specific heat capacity of the utility fluid respectively. 𝑑𝑉 𝑢 is the elementary volume of utility fluid corresponding to the channel length element ds. The thermal loss in the utility side is neglected since its temperature is close to room temperature and the temperature difference between inlet and outlet is less than 1°C during the experiment. Finally, the overall heat exchange coefficient is calculated from the inlet out outlet temperatures measured by the thermocouples: ,𝑖𝑛 -𝑇 𝑢,𝑖𝑛 )-(𝑇 𝑝,𝑜𝑢𝑡 -𝑇 𝑢,𝑜𝑢𝑡 ) ln ( 𝑇 𝑝,𝑖𝑛 -𝑇 𝑢,𝑖𝑛 𝑇 𝑝,𝑜𝑢𝑡 -𝑇 𝑝,𝑜𝑢𝑡 ) (IV-13) The temperatures of process fluid and utility fluid can be deduced from equations IV-7 and IV-11, which is expressed as: 𝑇 𝑝,𝑥+d𝑥 = 𝑇 𝑝,𝑥 + ( ,𝑥 -𝑇 𝑢,𝑠 ) d𝑠 𝑢 𝑢 𝜌 𝑢 𝑐 𝑝,𝑢 (IV-15) Based on these equations above, the temperatures of process fluid and utility fluid at different locations can be calculated by an iterative differential method using the two equations alternately with inlet temperature of two fluids (where x = s = 0). The temperature profiles of process fluid and utility fluid can be obtained finally. The local mean temperature calculated with the LIF technique is then compared to that deduced by the heat transfer model. The temperature profiles of process fluid and utility fluid calculated with heat transfer model in the prototype with zigzag channel of 3 mm width at Re = 494 is presented, as shown in Figure IV-11. For the measurement at location x = 0.103 m, the inlet temperatures of process fluid and utility fluid are 56.4 °C and 23.1 °C. It should be noted that the two inlet temperatures are different for the measurement at different location, so the calculation is done for each location with corresponding inlet temperatures of two fluids. In the heat transfer model, the mean value of U at three locations for one Reynolds number is taken for the calculation where for each location, U can be calculated with inlet temperature and outlet temperatures of two fluids using equation IV-12. The heat loss for process fluid is measured at 16.3 W for Re = 494 by just introducing the process fluid, the inlet and outlet temperatures are measured at 57.2 °C and 52.9 °C. With heat transfer model, the outlet temperature of process fluid and the utility fluid are estimated at 28.5 °C and 23.4 °C where in the experiment they were measured at 28.6 °C and 23.7 °C, showing the good reliability of the model for the temperature estimation. The three locations of temperature measurements by LIF technique are marked in this figure as well, corresponding to 0.029 m, 0.050 m and 0.103 m, where the mean temperature estimated by the heat transfer model are 54.2 °C, 52.6 °C and 49.1 °C respectively. However, as mentioned before, the inlet temperatures of process fluid and utility fluid are different for different locations, the temperatures estimated at 0.029 m, 0.050 m (54.2 °C, 52.6 °C) are not considered.

Figure IV- 11 :

 11 Figure IV-11: Temperature profiles of the process fluid and the utility fluid calculated with heat transfer model in the prototype with the zigzag channel of 3 mm width at Re = 494 (the black lines are the locations of temperature measurement by LIF technique)

  scheme. The objective is to study the influence of heat transfer performance on the reaction (conversion and selectivity) in zigzag channels to illustrate the impact of scale-up for HEX reactors. A 1D model of plug flow is introduced and the influence of channel geometry in terms of hydraulic diameter on the temperature distribution, conversion rate and selectivity is discussed which is very useful for industrial applications of compact heat-exchanger reactors.

  (E1 < E2), where high temperature favors reaction 2. They are presented in the following table.

  to keep a high temperature. So in this case, an inlet temperature of utility fluid (Tu0) at 350 K is considered. The concentrations of the desired product R, of the undesired product S and of the reactants A and B at different locations in the three zigzag channels are calculated with two initial concentrations of reactants (300 and 100 mol.m -3 ). The concentration profiles in the zigzag channel of 2 mm width with inlet concentrations of 300 mol.m -3 are presented as an example (in FigureV-1). The concentration profiles in other channels and with inlet concentration of 100 mol.m -3 are similar, so here they are not presented. It can be seen that the reaction is very fast and that they are completed at the channel length of 0.2 m. The reactant B is totally consumed. The concentration of reactant A is equal to undesired product S at the end.

Figure V- 1 :

 1 Figure V-1: Concentration profiles in the zigzag channel of 2 mm width, withCA0 = CB0 = 300 mol.m -3

Figure V- 2 :

 2 Figure V-2: Temperature profile of process fluid and of utility fluid in HEX reactor with zigzag channel of 2 mm and CA0 = CB0 = 300 mol.m -3

Figure V- 3 :

 3 Figure V-3: Temperature profile of process fluid in three zigzag channels with CA0 = CB0 = 300 mol.m -3

  utility fluid at 305 K is then set. The concentrations of the desired product R, the undesired product S and the reactants A and B at different locations in three zigzag channels are calculated with two inlet reactant concentrations (300 and 100 mol.m -3 ) and presented in FigureV-4. As the reaction rates are slower in this case, the reactions do not finish for inlet concentration of 100 mol.m -3 for the channel length studied (as shown in Figure V-4 (d), (e), (f)). Therefore, it is better to present the concentration profiles for both inlet concentrations for the discussion. It can be seen that the reactions finish after 10 m for inlet concentration of 300 mol.m -3 (in Figure V-4 (a), (b), (c)). For the inlet concentration of 100 mol.m -3 , the reactant B is not consumed completely after 10 m. When B is totally consumed, the concentration of reactant A should be equal to that of undesired product S. The temperatures of process fluid and of utility fluid in HEX reactor with zigzag channel of 2 mm width with inlet concentrations of 300 mol.m -3 is presented in Figure V-5. The temperature of the process fluid increases a little at the beginning, then it is cooled rapidly by the utility fluid and reaches the same temperature as utility fluid. That is because the reaction rate is fast when the reactants just enter the channel, this produces a hot point (temperature peak) which vanishes rapidly when the reaction rate becomes slower, as can be seen in Figure V-4. The temperature profiles of the process fluid in the three zigzag channels for the inlet concentration of 300 mol.m -3 are shown in Figure V-6. In Figure V-6, the temperature peak increases with increasing channel hydraulic diameter. That is because the heat transfer performance decreases (shown in Table

Figure V- 4 : 3 Figure V- 5 : 3 Figure V- 6 :

 43536 Figure V-4: Concentration profiles in zigzag channels of (a, d) 2 mm, (b,e) 3 mm and (c,f) 4mm with inlet reactant concentrations of (a, b, c) 300 mol.m -3 and (d, e f) 100 mol.m -3

For the resolution of

  the model, the correlation of Nusselt number in zigzag channels established in previous Chapters is applied in the simulation. Here it is shown as an example how the correlation can be used. The reliability of the simulation result shows the accuracy of the correlations. Plus the simulation results with reactions in this Chapter, all the data will be very useful for the scale-up of compact heat-exchanger reactors and facilitate their industrial applications. compared to straight channel configuration. With the purpose of studying the influence of fluid property on the heat transfer in zigzag channels, another more viscous fluid is used in this work: a glycerol solution with a mass fraction of 74%. It presents a viscosity around 30 times higher than water at 20 °C. Its Reynolds number is in laminar regime. It is found that the evolution of heat transfer coefficient with glycerol solution as a function of Reynolds number is similar to that with water. A correlation of Nusselt number in zigzag channels is proposed from the experimental data, in terms of fluid properties (Pr), channel geometries (dh/Ls), and fluid velocity (Re). This correlation does not fit very well with the experimental data of glycerol solution compared with water, maybe due to experimental errors. Concerning the influence of channel geometry on Nusselt number, the ratio of the channel hydraulic diameter divided by the straight section length between two bends (dh /Ls) plays an important role.In order to furtherly study the influence of this parameter (dh/Ls) on the heat transfer, a numerical method based on CFD simulation is then used in Chapter III. The simulations are carried out in zigzag channels with square cross-section of 2 mm with Reynolds numbers ranging from 112 to 560 (laminar flow regime). The straight section length between two bends ranges from 2 mm to 12 mm are investigated. The heat transfer performance in a straight channel with the same cross-section is also studied as a reference. The numerical method is validated by comparison with experimental results in terms of Nusselt number obtained in Chapter II. The results show that local Nusselt numbers and Fanning friction factors vary along the zigzag channels in the form of oscillating curves. The amplitude of the oscillations is constant with periodic flow but varies in the case of non-periodic flow. Based on Fourier analysis, a map has been provided which allows the distinction of the conditions that give a periodic flow or a non-periodic flow. Non-periodic flow can be obtained by increasing the fluid velocity or by decreasing the straight section length between two bends. The heat transfer enhancement and thermo-hydraulic performance factor are quantitatively discussed, showing the zigzag channel enhances the heat transfer strongly and increases pressure drop less significantly compared with the straight channel. It appeared that non-periodic flow is not particularly interesting in terms of thermo-hydraulic performances.

E1 activation energy of reaction 1 E2 activation energy of reaction 2 fk1 reaction rate coefficient of reaction 1 k10 frequency factor of reaction 1 k2 reaction rate coefficient of reaction 2 k20 frequency factor of reaction 2 LU

 121122 Cpspecific heat of the fluid (J.kg -1 .K -1 ) da element of cross-sectional area (m²) 𝑑 ℎ hydraulic diameter (m) Dax axial dispersion coefficient (m 2 .s -1 ) dl element of perimeter length (m) dX element of development length (m) e thickness of aluminum plate (m) eh heat transfer enhancement factor local Fanning friction factor fi frequency i (Hz) F fluorescence signal (A.U.) h heat transfer coefficient (W.m -2 .K -1 ) H enthalpy (J.kg -1 ) ∆𝐻 𝑟1 heat of reaction for reaction 1 (J.mol -1 ) ∆𝐻 𝑟2 heat of reaction for reaction 2 (J.mol -1 ) I0 laser intensity (W.m -2 ) Isat saturation intensity of the fluorescent dye (W.m -2 ) k thermal conductivity (W.m -1 .K -1 ) channel length (m) Lc curved section length (m) Ls straight section length (m) 𝑚̇ mass flow rate (kg.s -1 ) 𝒏 normal vector at a cross-section Nbend number of bends Nc number of divisions along the curved section Nd number of divisions for the channel depth and width Ne number of computational volume elements Ns number of divisions along the straight section Nunit number of periodic units Nu global heat transfer coefficient (W.m -2 .K -1 )

Table I -

 I 1: Comparison of thermal characteristics for different types of reactors[START_REF] Moreau | Methodologies pour la caracterisation hydrodynamique et l'extrapolation de réacteurs intensifiés millistructurés[END_REF]

Table I

 I

		-2: Standardized thermocouples
	Thermocouple	Temperature	Output
	type	range (°C)	(µV/°C)	Cost	Stability
	T	-262 to 850	15 at -200 °C	Low	Low
			60 at 350 °C
	J	-196 to 700	26 at -190 °C	Low	Low
			63 at 800°C
	E	-268 to 800	68 at 100 °C	Low	Low to mid
			81 at 500 °C
			77 at 900 °C
	K	-250 to 1100	40 from 250 to	Low	Low
			1000 °C
			35 at 1000 °C
	N	0 to 1250	37 at 1000 °C	Low	Mid to high
	B	100 to 1750	5 at 1000 °C	High	High
	S	0 to 1500	6 from 0 to 100 °C	High	High
	R	0 to 1600	10 at 1000 °C	High	High

Table II -

 II 1: Physical properties of water and 74% glycerol solution at 20 °C The flowrate in process channel and utility channel for three prototypes is presented in Table II-2. The corresponding velocity and Reynolds number are also presented.

	Process fluids	ρ (kg.m -3 )	Cp (J.kg -1 .K -1 )	µ (mPa.s)	k (W.m -1 K -1 )	Pr
	Water	1000	4180	1	0.6	7
	Glycerol solution	1192	2842	32.6	0.337	286
			Process channel	Utility channel
		dh = 2 mm	dh = 3 mm	dh = 4 mm	dh = 8 mm

Table II-2: Flowrate in process channel and utility channel for three prototypes with corresponding velocity and Reynolds number

  𝑞 = 𝑚 𝑝 ̇𝐶𝑃,𝑝 ∆𝑇 𝑝 -𝑞 𝑙 = 10.2÷ 3600 ×4184×(59.1-45.2) -9 = 156 W The mean logarithmic temperature difference ∆𝑇 𝑚𝑙 is:

	∆𝑇 𝑚𝑙 = [(59.1-36.6) -(45.2-37.6)] ÷ ln[(59.1-36.6) ÷ (45.2-37.6)] = 13.7 °C
	The overall heat transfer coefficient U of the prototype is:
	𝑈 =	𝑞 𝐴 𝐻 ∆𝑇 𝑚𝑙	= 156 ÷ (0.004×0.747) ÷13.7 = 3796 W.m -2 .K -1
	The Nusselt number of utility fluid is calculated by the Gnielinski correlation:
	𝑁𝑢 𝑢 = 0.012 (𝑅𝑒 𝑢	0.87 -280) 𝑃𝑟 𝑢	0.4 [1 + (	𝑑 ℎ 𝑢 𝐿 𝑢	)	2 3 ] (	𝑃𝑟 𝑢 𝑃𝑟 𝑤	) 0.11
		= 0.012×(13600 0.87 -280)×4.5 0.4 × [1 + (0.008 ÷ 0.588) 2/3 ] ×1 = 85.2

𝑚 𝑝 ̇𝐶𝑃,𝑝 ∆𝑇 𝑝 = 10.2÷ 3600 ×4184×(58.8-58) = 9 W The heat exchange flux between the two fluids is:

Table II -

 II 3: Thermal characteristics of HEX reactors with millimetric channels

			Zigzag channels			Straight channels
	dh (mm)	2	3	4	2	3	4
	Specific area
	AH/V	1000	667	500	1000	667	500
	(m².m -3 )
	Heat exchange
	coefficient hP (*)	2800 -14000	3100 -15500	3100 -14300	800 -4900	900 -6800	800 -
	(W.m -2 .K -1 )
	Heat exchange
	capacity per	2800 -14000	2060 -10300	1 550 -7150	800 -4900	600 -4530	400 -
	volume hPAH/V
	(kW.m -3 .K -1 )
	Overall heat
	exchange	2 500 -9000	2600 -8800	2 700 -7100	800 -4200	900 -5200	780 -
	coefficient U (**)
	(W.m -2 .K -1 )
	Overall heat
	exchange	2500 -9000	1700 -5900	1300 -3600	800 -4200	600 -3470	390 -
	capacity per
	volume UAH/V
	(kW.m -3 .K -1 )

(*) process fluid is water, velocity range is 0.1 -1 m.s -1 (**) flowrate is 300 L.h -1 in each plate with two utility plates, 2 aluminum plates with thickness of 1 mm

Table III -

 III 1: Number of bends and units in the zigzag channels with different straight section

	lengths (dh = 2 mm)
	Geometry 'a'	'b'	'c'	'd'	'e'	'f'	'g'
	Ls (mm)	2	3.5	4.5	5.5	7	9	12
	Nbend	22	16	14	12	10	8	6
	Nunit	11	8	7	6	5	4	3
	L/(m)	0.096	0.094	0.096	0.094	0.094	0.091	0.086

  can be seen in Figure III-2. This will be discussed further in section 2.3 of this chapter. Table III-2: Characteristics of the meshes used for geometry 'f' at Re = 560 and geometry 'b'

			for Re = 224
		Geometry 'f' (Ls = 12 mm)	Geometry 'b' (Ls = 3.5 mm)
			Re = 560		Re = 224
	Mesh	(1)	(2)	(3)	(1)	(2)
	Nd	40	60	80	40	60
	Ns	120	180	240	120	180
	Nc	40	60	80	40	60
	Ne (million)	2.2	7.6	18	4.3	14.5
	Num	20.19	19.84	19.77	20.19	19.84
	The variation of the local Nusselt number for different meshes, whose characteristics are given
	in Table III-2, is shown in Figure III-2. For geometry 'f' at Re = 560, from mesh (2) to mesh
	(3), the local Nusselt number hardly varies and the global average Nusselt number (Num) varies
	by only 0.35%. Therefore mesh (2) is adequate for this case study with 7.6 million volume
	elements (Ne) in the computational domain. For geometry 'b' at Re = 224, Num varies by 0.54%

The local Nusselt number 𝑁𝑢 and local Fanning friction factor 𝑓 were assessed for different mesh densities. Grid independence studies for geometry 'f' at Re = 560 and geometry 'b' at Re = 224 are presented. while Ne is significantly increased. These two cases have been considered to illustrate the grid independence study because they present two different levels of complexity in terms of local thermal behaviors as

Table IV -

 IV 2: Inlet temperatures of fluorescent solution for the calibration experiment

	T (°C)	17.5	24.9	30.3	36.7	42.7	49.5	52.5	55.9	58.8

Table IV

 IV As an example, for the experiment at the location x = 0.103 m in the zigzag channel of 3 mm width with Re = 494, the inlet temperatures of the process fluid and the utility fluid are 56.4 °C and 23.1 °C, the outlet temperatures are 28.6 °C and 23.7 °C. The mean temperatures of the process fluid at the three plans (high, medium and low) measured by LIF are 46.8 °C, 46.5 °C and 46.3 °C.

	(b) Zigzag channel 4 mm, Re = 695 (f) Zigzag channel 2 mm, Re = 631
	Location Location		Tp,m (°C) Tp,m (°C)		Tp,in Tp,in	Tp,out Tp,out	Tu,in Tu,in
	(m) (m)	High plan High plan	Medium plan Medium plan	Low plan Low plan	(°C) (°C)	(°C) (°C)	(°C) (°C)
	0.045 0.033	53.4 -		53.6 49.4	52.6 -	57.5 55.7	35.6 26.5	27.0 22.0
	0.092 0.052	54.5 -		51.4 47.8	49.5 -	57.5 55.7	33.3 26.2	23.6 21.7
	0.175 0.108	47.4 -		47.2 43.8	45.0 -	57.4 54.8	32.1 22.7	22.0 19.0
	(c) Zigzag channel 3 mm, Re = 494
	Location		Tp,m (°C)		Tp,in	Tp,out	Tu,in
	(m)	High plan	Medium plan	Low plan	(°C)	(°C)	(°C)
	0.029	51.0		50.5	50.9	56.2	26.1	19.7
	0.050	50.1		49.2	49.8	56.4	28.4	22.8
	0.103	46.8		46.5	46.3	56.4	28.6	23.1
	(d) Zigzag channel 3 mm, Re = 685
	Location		Tp,m (°C)		Tp,in	Tp,out	Tu,in
	(m)	High plan	Medium plan	Low plan	(°C)	(°C)	(°C)
	0.030	52.4		52.5	52.1	57.1	33.0	25.5
	0.072	49.7		49.7	50.2	57.0	31.7	25.1
	0.134	46.4		46.0	46.2	57.1	30.8	23.6
	(e) Zigzag channel 2 mm, Re = 461
	Location		Tp,m (°C)		Tp,in	Tp,out	Tu,in
	-4: Mean temperature of all plans measured in three zigzag channels by the LIF technique (m) High plan Medium plan Low plan (°C) (°C) (°C)
	(a) Zigzag channel 4 mm, Re = 503 0.024 -	47.9	-	54.7	21.9	18.2
	Location 0.043	-	Tp,m (°C) 46.3	-	Tp,in 54.6	Tp,out 22.0	Tu,in 18.3
	(m) 0.108	High plan -	Medium plan 41.5	Low plan -	(°C) 54.7	(°C) 22.6	(°C) 18.9
	0.034	52.2		53.1	51.8	56.9	28.3	18.9
	0.069	55.2		50.1	51.1	56.9	28.5	19.5
	0.138	49.4		46.6	47.0	56.9	29.6	21.1

Table IV -

 IV 5: Temperatures calculated with LIF technique (Tp,m) and heat transfer model (Tmod). Zigzag channel 3 mm with Re = 494 and Um = 2860 W.K -1 .m -²

	(c) Location		Tp,m (°C)		Tmod
	(m)	High plan	Medium plan	Low plan	(°C)
	0.029	51.0	50.5	50.9	53.8
	0.050	50.1	49.2	49.8	52.6
	0.103	46.8	46.5	46.3	49.1
	(d) Zigzag channel 3 mm with Re = 685 and Um = 3350 W.K -1 .m -²
	Location		Tp,m (°C)		Tmod
	(m)	High plan	Medium plan	Low plan	(°C)
	0.030	52.4	52.5	52.1	55.2
	0.071	49.7	49.7	50.2	52.6
	0.134	46.4	46.0	46.2	48.9
	(e) Zigzag channel 2 mm with Re = 461 and Um = 3700 W.K -1 .m -²
	Location		Tp,m (°C)		Tmod
	Um is the mean value of heat transfer coefficient at three locations. (m) High plan Medium plan Low plan	(°C)
	0.024	-	47.9	-	52.0
	(a) Zigzag channel 4 mm with Re = 503 and Um = 2320 W.K -1 .m -² 0.043 -46.3 -	50.1
	Location 0.108	-	Tp,m (°C) 41.5	-	Tmod 44.5
	(m)	High plan	Medium plan	Low plan	(°C)
	0.034 (f) Zigzag channel 2 mm with Re = 631 and Um = 4260 W.K -1 .m -² 52.2 53.1 51.8	54.5
	0.069 Location	55.2	50.1 Tp,m (°C)	51.1	52.2 Tmod
	0.138 (m)	49.4 High plan	46.6 Medium plan	47.0 Low plan	48.5 (°C)
	0.033	-	49.4	-	52.7
	(b) Zigzag channel 4 mm with Re = 695 and Um = 2730 W.K -1 .m -² 0.052 -47.8 -	51.2
	Location 0.108	-	Tp,m (°C) 43.8	-	Tmod 45.8
	(m)	High plan	Medium plan	Low plan	(°C)
	0.045	53.4	53.5	52.5	55.1
	0.092	54.5	51.4	49.5	52.4
	0.173	47.4	47.2	45.0	48.2

Table V -

 V 2: Specific dataset for the two reaction cases studied

		Symbol	Unit	E1 > E2	E1 < E2
	Frequency factor for reaction 1	k10	m 3 .mol -1 .s -1	5×10 7	3.5×10 7
	Frequency factor for reaction 2	k20	m 3 .mol -1 .s -1	1.5×10 2	1.58×10 20
	Activation energy for reaction 2	E2	J.mol -1	24000	150000
	2. Simulation model
	2.1

Geometry of HEX reactors

In this chapter, a steady state 1D plug flow model is used to simulate the competitive consecutive reaction scheme in heat-exchanger reactors with zigzag channels. The heatexchanger reactors simulated are based on the ones studied in the Chapters II and IV. the geometry of zigzag channels in the process plate is exactly the same as before with a square cross-section of 2 mm, 3 mm and 4 mm width (dh,p). With the consideration of industrial design, two utility plates with a straight channel engraved are used and put in the two sides of the process plate forming a sandwich. They are separated by two aluminum plates (1 mm thickness).

Table V

 V

			-3: Geometries and flowrates for the simulation of HEX reactors
	dh,p (m) up (m.s -1 ) Qp (L.h -1 ) Qu (L.h -1 ) uu (m.s -1 )	wu (m)	du (m)	dh,u (m)
	0.002	0.7	10.1	101	1	0.008	0.0018	0.0029
	0.003	0.7	22.7	227	1	0.010	0.0032	0.0048
	0.004	0.7	40.3	403	1	0.011	0.0051	0.0070

  reaction term in the calculation of process fluid. The overall heat transfer coefficient Ux can be calculated by The heat transfer coefficient of the process fluid hp,x is calculated by the correlation (equation II-16) which has been established in Chapter II. The heat transfer coefficient of the utility fluid hu,x is calculated with the Gnielinski correlation (equation II-7). The density 𝜌, specific heat capacity 𝐶 𝑝 , Prandtl number 𝑃 𝑟 and viscosity µ of the process fluid and utility fluid at different temperatures can be calculated by the equations II-8 to II-7. The range of Ux calculated for each HEX reactor is presented in TableV-4. The UxA/V in terms of process channels is also presented.Since the utility temperature in the simulation is different for two reaction cases (see next part), hence Ux for the same HEX reactor is different in two cases. It can be seen that Ux is generally the same for three HEX reactors, however, when process channels varies from 2mm to 4 mm,

	𝐶 𝑆,𝑥+∆𝑥 = 𝐶 𝑆,𝑥 + 𝑟 2	∆𝑥 𝑢 𝑝			(V-15 )
	1 𝑈 𝑥 𝑑𝐴	=	1 ℎ 𝑝,𝑥 𝑑𝐴 𝑝	+	𝑒 𝐴𝑙 𝑘 𝐴𝑙 𝑑𝐴	+	1 ℎ 𝑢,𝑠 𝑑𝐴 𝑢	(V-11)
	UxA/V decreases greatly.
									E1 > E2	E1 < E2
									Ux	UxA/V	Ux	UxA/V
									(W.m -2 .K -1 )	(kW.m -3 .K -1 )	(W.m -2 .K -1 )	(kW.m -3 .K -1 )
	HEX reactor 2mm			7400-8000	3700-4000	6000-6500	3000-3250
	HEX reactor 3mm			7000-7700	2333-2567	6000-6500	2000-2167
	HEX reactor 4mm			6500-7000	1625-1750	5500-6100	1375-1525
	The concentration of reactants and products in process plate is calculated by the following
	equations:
	𝑇 𝑝,𝑥+𝑑𝑥 = 𝑇 𝑝,𝑥 + ( 𝐶 𝐴,𝑥+∆𝑥 = 𝐶 𝐴,𝑥 -𝑟 1 ∆𝑡 = 𝐶 𝐴,𝑥 -𝑟 1 2𝑈 𝑥 𝑑𝐴 𝑑𝑉 𝑝 (𝑇 𝑢,𝑠 -𝑇 𝑝,𝑥 ) -(∆𝐻 𝑟1 𝑟 1 + ∆𝐻 𝑟2 𝑟 2 )) ∆𝑥 𝑢 𝑝	𝑑𝑥 𝑢 𝑝 𝜌 𝑝 𝑐 𝑃,𝑝	(V-9) (V-12)
	𝑇 𝑢,𝑠+𝑑𝑠 = 𝑇 𝑢,𝑠 + 𝐶 𝐵,𝑥+∆𝑥 = 𝐶 𝐵,𝑥 -(𝑟 1 + 𝑟 2 ) 𝑈 𝑥 𝑑𝐴 𝑑𝑉 𝑢 (𝑇 𝑝,𝑥 -𝑇 𝑢,𝑠 ) ∆𝑥 𝑢 𝑝	𝑑𝑠 𝑢 𝑢 𝜌 𝑢 𝑐 𝑃,𝑢	(V-10) (V-13 )
	𝐶 𝑅,𝑥+∆𝑥 = 𝐶 𝑅,𝑥 + (𝑟 1 -𝑟 2 )	∆𝑥 𝑢 𝑝	(V-14 )

These two equations are similar to that used in Chapter IV (equations IV-14 and IV-15) for estimating the local temperature with comparison to that measured by fluorescence, with just introducing the Table V-4: Ux and UxA/V calculated in three HEX reactors for two reaction cases. "HEX reactor 2mm" represents HEX reactor with zigzag channel of 2mm width

.K -1 ) Rep hp (W.m - 2 .K -1 ) Nup

.K -1 ) Nup

.K -1 )

Re = 224 (Figure III -7). The only change is the direction, suggesting the flow behavior in the next bend returns to that of the previous bend. For Re = 448, the flow behavior in the coming bend does not return to that of the previous bend, as the instability created in this bend is still significant (Figure ). This flow regime is more irregular and unstable, different from that at a Reynolds number of 224 where the fluid spends more time in the straight section. PF for the channel configurations studied at different Reynolds numbers is presented in Figure For all the cases studied, PF is greater than one, suggesting that the bends significantly enhance the thermo-hydraulic performances of the channel. That means the zigzag channel brings less significant pressure drop compared with the heat transfer enhancement. PF increases with increasing Reynolds number or Ls/dh. The increase with Reynolds number is particularly important at low Re. From Re = 336, the increase becomes moderate with increasing Reynolds number. Concerning the impact of the straight section, decreasing its length to reach nonperiodic flow regime is not interesting in terms of thermo-hydraulic performances in the cases studied. Finally, it has to be noted that by increasing Ls/dh, the zigzag design will get closer to the straight channel. Therefore, the different curves should converge towards PF = 1 for very high Ls/dh, depicting a bell-shaped curve with an optimal range of Ls/dh values depending on Re. The observed plateaus for Re = 448 and Re = 560 suggest that this optimum is between Ls/dh = 4.5 and 6 for these conditions.

Comparison between simulation and experimental results

Here the simulation results of the geometry 'e' is compared with the experimental results in Chapter II in terms of Nusselt number. Nu in zigzag channel at two small Reynolds numbers (285, 576) in the experiment is introduced. As the Prandtl number Pr in the simulation (6.13) The fluorescence signal of all the plans measured at reference temperature (24.9 °C) is also detected and 100 images are registered, with just passing the fluorescent solution in process plate and inlet temperature of 24.9 °C (without heat transfer).

Image processing

Image registration between two cameras

where 𝑘 10 , 𝑘 20 , 𝐸 1 , 𝐸 2 are the frequency factors and activation energy of reaction 1 and 2. 𝑅 ̅ is the universal gas constant (𝑅 ̅ = 8.314 J.mol -1 .K -1 ).

Chemical reactions are influenced by many variables and in single liquid phase systems, composition, temperature, mixing and residence time distribution will determine the composition of the final products. In this chapter, the influence of inlet concentrations and utility temperature on the reaction rate and product distribution is discussed. The dataset of two reactions is presented in the following table. The physicochemical properties of process fluid where the reactions occur and of utility fluid presented here are at 20 °C. These data correspond to case studies proposed by Shelat et al. [START_REF] Shelat | Conceptual analysis to evaluate suitability of compact heat exchanger reactors for rapid and complex chemical reactions[END_REF] who simulated different reaction schemes in HEX reactors with multiple injections of reactants.

Heat capacity of the utility fluid

Cp,u J.kg -1 .K -1 4184

Thermal conductivity of the utility fluid ku W.m -1 .K -1 0.6

Two reaction cases are studied in this chapter, one is the activation energy of the desired reaction larger than that of the undesired reaction (E1 > E2), where high temperature favors reaction 1. In the other case, the activation energy of the desired reaction is smaller than that of From this table, it can be seen that the conversion of reactant A and the selectivity of desired product R both increase with increasing channel hydraulic diameter for the two concentrations, although for the present study conditions the increases are relatively small. This is because the heat is released less efficiently when the channel cross-section increases, resulting in higher temperatures in process fluid. Since temperature favors reaction 1 in this case, the conversion of A and selectivity of R both increase with increasing channel cross-section length.

The difference between the maximum temperature and the inlet temperature ΔTm increases as well with increasing channel cross-section length for both concentrations. Compared to the temperature increase ΔTa in adiabatic condition, the hot point of the three channels is 2-3 K (for CA0 = CB0 = 300 mol.m -3 ) and 11-12 K (for CA0 = CB0 = 100 mol.m -3 ) lower than adiabatic condition. As expected, the faster the reaction is, the more difficult it is to release the heat.

To conclude, in this reaction case (E1 > E2), the reaction is very fast and reactant B is totally consumed rapidly for the three zigzag channels. When the hydraulic diameter of zigzag channels varies from 2 mm to 4 mm, the conversion of reactant A and the selectivity of desired product R both increase, the difference between the maximum temperature and the inlet temperature increases as well. It means that scale-up favors conversion and selectivity for this reaction case. However it should be noticed that the hot point increases as well. This should be taken into account in the evaluation of scale-up strategies for the safety concern or product decomposition at high temperature.

Reaction case E1 < E2

For the reaction case E1 < E2, temperature favors reaction 2 (undesired reaction). Thus, it is better to keep a low temperature in order to form more desired product. An inlet temperature of overall heat transfer coefficient Ux = 0 in the simulation. The difference of outlet temperature and the temperature (ΔTa) is calculated and presented in the same table. The conversion of reactant A and the selectivity of desired product R in three zigzag channels are also presented in Table V-6.

Table V- From this table, the conversion of the reactant A and the selectivity of the desired product R both decreases with increasing channel cross-section width for inlet concentration of 300 mol.m -3 . This is because when the channel hydraulic diameter increases, the heat transfer performance decreases as discussed before, which leads to higher temperature in the zigzag channel of 4 mm compared to that of 2 mm. Since temperature favors reaction 2 in this case, the selectivity of R hence decreases with increasing channel cross-section width. Thus the production of the undesired product S and the consumption of reactant B by reaction 2 increases as well, which leads to the decrease of the conversion of reactant A (consumed by reaction 1).

For the inlet concentration of 100 mol.m -3 , the selectivity of R decreases with increasing crosssection width as well. As for the conversion of reactant A, it increases with increasing channel cross-section width due to uncomplete reaction. It should decrease when the reactant B is totally consumed.

The difference between the maximum temperature and inlet temperature ΔTm increases with increasing cross-section width for concentration of 300 mol.m -3 . For the concentration of 100 mol.m -3 , the temperature decreases all the time in three channels because heat transfer is faster than heat production. Compared to the temperature increase ΔTa in adiabatic condition, the hot point of three channels is 8-12 K (for CA0 = CB0 = 300 mol.m -3 ) and 15 K (for CA0 = CB0 = 100

General conclusion

The