Pape Sène

Florine Giraud

Mamadou Lamine Sow

Brice Tréméac
email: brice.tremeac@lecnam.net

Heat transfer coefficient correlations of water subatmospheric vaporization in a channel of a smooth plate heat exchanger, based on Vaschy-Buckingham theorem

Keywords: water vaporization, subatmospheric pressure, heat transfer coefficient, correlation, plate-type heat exchanger vapor density ratio[-]

Water as an environmentally friendly, safe and cheap natural refrigerant (R-718) requires vaporization conditions at sub-atmospheric pressures (~1 kPa). At these pressure ranges, studies on the specific vaporization of water are scarce and insufficient. For this reason, the engineering of heat exchangers using R-718 as working fluid remains empirical. The aim of this paper is thus to provide correlations to estimate water pool boiling and falling film evaporation heat transfer coefficient under subatmospheric pressure. Correlations developed are believed to be more widely used by providing dimensionless numbers consistent with phasechange phenomena occurring at sub and at-mospheric pressure. To develop these correlations, heat transfer phenomena during water vaporization at pressures ranging from 0.7 to 1.7 kPa in a scale-lab size (0.2 m wide x 0.5 m high) vertical smooth plate evaporator channel are studied.

Filling ratio vary from 1/2 to 1/10 of the total channel height, canal thicknesses are set to 2 mm, 4 mm or 6 mm. Thus, raw experimental data from 139 tests performed under various operating conditions of a thermosyphon loop mimicking a sorption system are exploited. Each test consists of more than 1200 values. A graphical and statistical comparison between the heat transfer coefficient (HTC) obtained from the experimental database and the HTC estimated by empirical correlations from the literature is made, calculating the percentage of predicted data within an error band of ± 30% (τ 30 ), the mean absolute error (MAE), the mean relative error (MRE) and the correlation coefficient (r). Two methodical correlations for predicting the HTC of pool boiling and liquid film evaporation, are developed. They predict 86% (with MAE = 19%, MRE = 3%, r = 0.79) and 83% (with MAE = 18%, MRE = 2%, r = 0.84) of the experimental data in τ 30 respectively. To extend their validity domains, the proposed correlations are evaluated using independent experimental data from the literature. Giving satisfactory results, it constitutes a first step towards the development of tools used for the sizing and the optimization of the design of two-phase heat exchangers.

NOMENCLATURE

Introduction

In a context where regulations on synthetic refrigerants are becoming more and more restrictive due to their negative impact on the environment, it is urgent to think about the development of compact, efficient and "clean" refrigeration technologies. Thus, through the improvement of components such as the evaporator or the compressor, systems using natural refrigerants like water (R-718), represent a promising alternative. R-718 is an efficient fluid (high vaporization enthalpy and heat capacity), available, non-toxic, non-flammable, inexpensive, inert for the environment (ODP: Ozone Depletion Potential and GWP: Global Warning Potential are zero) and easy to handle (minimizes safety precautions) [START_REF] Amir | Comparing water (R718) to other refrigerants[END_REF] [START_REF] Seiler | Capillary-assisted evaporation of water from finned tubes -Impacts of experimental setups and dynamics[END_REF]. Although the use of water as working fluid is widespread in sorption systems and is gaining ground in mechanical vapor compression systems equipped with turbo compressors [START_REF] Amir | Comparing water (R718) to other refrigerants[END_REF], systems using water as refrigerant still shows poor performances partly due to the poor performance of the heat exchanger [START_REF] Giraud | Vaporization of water at subatmospheric pressure:fundamentals of boiling phenomena and path towards the design of compact evaporators for sorption chillers[END_REF]. These poor performances could be mostly explained by the fact that using R-718 as refrigerant requires vaporization conditions at sub-atmospheric pressures (~1 kPa). Indeed, the evaporator is the essential component of thermal machines since the maximum permissible heat exchange rate depends on its pressure conditions (thermophysical properties of the refrigerant) and on the topography of its heat exchange surfaces. However, at subatmospheric pressure ranges, the specificity of the thermophysical properties of R-718 and in particular its low vapour density (ρ v = 0.009 kg. m -3 and ρ l = 999.68 kg. m -3 at 1.2 kPa versus ρ v = 0.598 kg. m -3 and ρ l = 958.34 kg. m -3 at 101,325 kPa), lead to atypical boiling behaviours such as the appearance of bubbles of centimetric size often referenced in the literature [START_REF] Giraud | Experimental study of water vaporization occurring inside a channel of a smooth plate-type heat exchanger at subatmospheric pressure[END_REF] [START_REF] Michaie | Towards a more generalized understanding of pool boiling at low pressure : Bubble dynamics for two fluids in states of thermodynamic similarity[END_REF]. For example, at 1.2 kPa, the typical working pressure of chillers using R-718, mushroom-shaped bubbles of 15 cm in diameter can be observed under pool boiling conditions, in contrast to the spherical or semi-spherical bubbles of a few millimeters that can be observed at atmospheric pressure [START_REF] Giraud | Vaporization of water at subatmospheric pressure:fundamentals of boiling phenomena and path towards the design of compact evaporators for sorption chillers[END_REF]. Due to this atypical behavior, correlations and design usually used to realize heat exchangers of thermal systems do not allow a proper sizing of R-718 evaporators.

The needs to improve the efficiency of said machines using R-718, makes it essential to size correctly, optimize and reduce the size of heat exchangers, especially the evaporator.

In the evaporator, the refrigerant exchange heat from a source, through a process of vaporization (boiling or evaporation). This exchanger can be made of tubes and shells, fins or plates. Among the latter, smooth plate heat exchangers can be good candidates. The use of vertical flat plate and falling film heat exchangers is especially attractive for sorption machines and potentially increases their compactness, reduces their weight and price compared to conventional vertical or horizontal tube configurations [START_REF] Giraud | Vaporization of water at subatmospheric pressure:fundamentals of boiling phenomena and path towards the design of compact evaporators for sorption chillers[END_REF] [START_REF] Triché | Étude numérique et expérimentale des transferts couplés de masse et de chaleur dans l'absorbeur d'une machine à absorption ammoniac-eau[END_REF]. The interest of these types of exchangers is also due to their unique advantages such as flexible thermal design (plates can be simply added or removed to meet different thermal services or process requirements), low manufacturing cost, ease of inspection and cleaning to maintain strict hygienic conditions, reduced load, better heat transfer performance due to fluid distribution over the plates, compactness, etc. [START_REF] Mota | Chapter 7 : Modeling and Design of Plate Heat Exchanger[END_REF] [9] [START_REF] Ayub | Literature survey and a universal evaporation correlation for plate type heat exchangers[END_REF]. Plate heat exchangers are also suitable for standardization of the manufacturing process [START_REF] Giraud | Experimental study of water vaporization occurring inside a channel of a smooth plate-type heat exchanger at subatmospheric pressure[END_REF]. Also, their deep channels, large ports and possible laser welding allow for vacuum vaporization [START_REF] Delaplace | Modélisation en génie des procédés par analyse dimensionnelle. Méthode et exemples résolus[END_REF]. This is consistent with the fact that the use R-718, with its high boiling temperature at atmospheric pressure (T = 100 °C, P = 101.325 kPa), requires to operate at sub-atmospheric pressures close to its triple point (T = 0.01 °C , P = 0.61 kPa) [START_REF] Giraud | Vaporization of water at subatmospheric pressure:fundamentals of boiling phenomena and path towards the design of compact evaporators for sorption chillers[END_REF] [11] in the part containing the evaporator. But, at these low pressures ranges, in addition to the observation of centimeter size mushroom bubbles, since the pressure at the free surface (P sat ) is small, the pressure induced by the liquid column (P static = ρ l gH l ) can be of the same order of magnitude as the system pressure [START_REF] Giraud | Experimental study of water vaporization occurring inside a channel of a smooth plate-type heat exchanger at subatmospheric pressure[END_REF] [START_REF] Giraud | Experimental evidence of a new regime for boiling of water at subatmospheric pressure[END_REF]. Several experimental studies ( [4] [13] [12] [14] [15]) confirmed the non-negligible influence of the liquid height (hydrostatic pressure) on the heat transfers during low-pressure pool boiling. The filling level (H l ) can therefore no longer be neglected in the analysis of heat transfer phenomena, as is commonly practiced for higher pressures. This is all the more important since the presence of this hydrostatic pressure causes the vapor bubbles to form in an inhomogeneous medium in pressure or subcooling [START_REF] Giraud | Vaporization of water at subatmospheric pressure:fundamentals of boiling phenomena and path towards the design of compact evaporators for sorption chillers[END_REF]. Thus, the thermophysical properties of the refrigerant present strong variations [START_REF] Michaie | Towards a more generalized understanding of pool boiling at low pressure : Bubble dynamics for two fluids in states of thermodynamic similarity[END_REF], leading to phenomena of a certain complexity regarding thermal couplings and two-phase flows.

As a first approach towards the understanding of these complex phenomena and in order to design compact evaporators for sorption coolers and optimize their size, Giraud et al. [START_REF] Giraud | Experimental study of water vaporization occurring inside a channel of a smooth plate-type heat exchanger at subatmospheric pressure[END_REF] [16] [START_REF] Giraud | Experimental study of water vaporization occurring inside the channel of a smooth-plate type heat exchanger connected to an adsorber and comparison with trends observed in absorption configuration[END_REF] conducted experimental studies on the vaporization of water at subatmospheric pressure in a vertical smooth plate evaporator (VSPE) (scale-lab size: 0.2 m wide x 0.5 m high). Filling ratio (H l ) vary from 1/2 to 1/10 of the total channel height, the thicknesses (e) between two succeeding plates are set to 2 mm, 4 mm or 6 mm. The authors [START_REF] Giraud | Experimental study of water vaporization occurring inside a channel of a smooth plate-type heat exchanger at subatmospheric pressure[END_REF] mainly highlighted three different working zones: a boiling zone, a liquid film evaporation zone and a vapor zone. Their results are in perfect agreement with the observations of Chang et al. [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF]. The liquid film evaporation zone is due to the free surface moving due to boiling as well as the spread of the encapsulated liquid contain between the bubbles and the free surface. This liquid is spread over the top wall plate and forms the liquid film that evaporates immediately [START_REF] Giraud | Experimental study of water vaporization occurring inside a channel of a smooth plate-type heat exchanger at subatmospheric pressure[END_REF] [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF]. The liquid film evaporation zone is the seat of more than 70% of the heat transfer in the VSPE [START_REF] Giraud | Experimental study of water vaporization occurring inside the channel of a smooth-plate type heat exchanger connected to an adsorber and comparison with trends observed in absorption configuration[END_REF].

Studying the effect of the channel thicknesses on the VSPE performance, the authors also observed, for a confinement number Co = 1 D h • √ σ g(ρ l -ρ v ) = 0.69 (e = 2 mm), phenomena such as droplet entrainment by the steam flow [START_REF] Giraud | Influences of confinement on subatmospheric water vaporization phenomena in a vertical rectangular channel[END_REF] . They confirmed experimentally the impact of this observation on the heat transfer: higher performances are observed for a confinement number Co = 0.69 compared to a confinement number Co = 0.35 (e = 4 mm) and Co = 0.23 (e = 6 mm) when droplets entrainment is observed, i.e. for high rate of steam production obtained at high entrainment pressures and low filling rate. Then, as a first tool toward the design of compact subatmospheric VSPE, Giraud et al [START_REF] Giraud | Experimental study of water vaporization occurring inside the channel of a smooth-plate type heat exchanger connected to an adsorber and comparison with trends observed in absorption configuration[END_REF] developed dimensionless numbers depending on the measured cooling capacity, the thermophysical properties of the fluid and the operating conditions. They showed that, using these dimensionless number, similar evaporator behavior could be observed for different boiling phenomena kinetics since they show that, based on these numbers, the evaporator behavior is similar when the evaporator is included in a thermosyphon loop than when it is included in an ad-sorption system. Despite the efforts made, efforts still should be made to develop more universal correlations involving in particular a form factor and exchange coefficients. However, the knowledge on the vaporization of water at subatmospheric pressure remains scarce and insufficient and, given the great difference between the characteristics of subatmospheric vaporization and those observed at atmospheric pressure [START_REF] Giraud | Vaporization of water at subatmospheric pressure:fundamentals of boiling phenomena and path towards the design of compact evaporators for sorption chillers[END_REF] [START_REF] Michaie | Towards a more generalized understanding of pool boiling at low pressure : Bubble dynamics for two fluids in states of thermodynamic similarity[END_REF], the dimensionless numbers usually used at high pressure are not sufficient to properly predict phenomena observed at subatmospheric pressure.

Empirical correlations of the literature developed, verified or adapted to low pressure conditions, are then limited mainly because they do not consider all the independent physical quantities that govern the confined vaporization at subatmospheric pressure such as the form factor (hydrostatic pressure), the inhomogeneous pressure condition of the medium (variation of the thermophysical properties of the refrigerant), the boundary conditions, etc. Moreover, the dimensional analysis adopted for the elaboration of the said correlations, in general, does not verify the independence of the physical quantities despite the necessity to ensure that no physical quantity, apart from the variable of interest, depends on the variation of the other quantities [START_REF] Delaplace | Modélisation en génie des procédés par analyse dimensionnelle. Méthode et exemples résolus[END_REF]. This independence is a key factor for dimensional analysis modeling since it allows an easily transferred results from one scale to another (similarity).

Referring to the observations of literature, the understanding of heat transfer phenomena during pool boiling and liquid film evaporation at sub-atmospheric pressure thus seems fundamental for optimizing design and performances of compact heat plates exchangers. For this purpose, heat exchanger designers inevitably have a great need for methodical HTC correlations based on rigorous dimensional analysis and offering higher accuracy than existing correlation. This is what the paper attend to do: offering correlations with dimensionless numbers requires for the proper design of subatmospheric water heat exchanger. It is believed, that although the obtention of the coefficient may remain empirical, the used of the chosen dimensionless numbers lead to universal correlations since it includes the independent physical properties influencing phase change phenomena at low and high pressure.

The approach starts with a presentation of the HTC correlations available in the open literature. Subsequently, the raw experimental database and the experimental set up used by Giraud et al. [START_REF] Giraud | Influences of confinement on subatmospheric water vaporization phenomena in a vertical rectangular channel[END_REF] to determine the heat transfer coefficients during pool boiling and liquid film evaporation of water at sub-atmospheric pressures is described. The constant and the exponents of the general or Dittus-Boelter type and other relevant correlations from the literature [START_REF] Bengt | Chapter Two -Advanced heat transfer topics in complex duct flows[END_REF] is fitted to the database obtained from the experimental results of Giraud et al. [START_REF] Giraud | Influences of confinement on subatmospheric water vaporization phenomena in a vertical rectangular channel[END_REF] for graphical and statistical comparison between the experimental HTC and the HTC estimated. Then, based on the scientific knowledge generated by previous studies, new methodical correlations for predicting HTC during pool boiling and evaporation of the liquid film falling from water in a VSPE is developed. These correlations are based on a rigorous dimensional analysis approach, following Vaschy-Buckingham theorem described by Delaplace et al. [START_REF] Delaplace | Modélisation en génie des procédés par analyse dimensionnelle. Méthode et exemples résolus[END_REF]. The news proposed correlations consider, among others, the form factor (canal thickness and filling ratio), the thermophysical properties of refrigerant (hydrostatic pressure and saturation pressure), the heat flux density and the parietal superheating. The obtained results are presented and discussed.

Then, in order to extend the range of validity and test the robustness of the developed correlations, these models are fitted and evaluated using independent experimental data from the literature.

HTC correlations of literature

In this section, different relevant correlations commonly used to predict HTC during water vaporization are presented and discussed. There are many correlations in the literature for estimating the HTC during water vaporization, but many of them are valid only for typical heat transfer surface topography and/or operating conditions (high pressure). The main parameters affecting nucleate boiling are heat flux, saturation pressure, thermophysical properties of the working fluid and characteristics of the boiling surface and material etc. [START_REF] Yu | Experimental research on water boiling heat transfer on horizontal copper rod surface at sub-Atmospheric pressure[END_REF]. Thus, Dittus-Boelter [START_REF] Bengt | Chapter Two -Advanced heat transfer topics in complex duct flows[END_REF], Labuntsov [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF], Kruzylin [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF], Stephan-Abdelsalam [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF], Chang (pool boiling region and intermittent region) [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF], Mc Nelly [START_REF] Baki | Boiling pure fluids at sub atmospheric pressures[END_REF] and Yu [START_REF] Yu | Experimental research on water boiling heat transfer on horizontal copper rod surface at sub-Atmospheric pressure[END_REF] correlations consist of parameters that influence water vaporization.

The general or Dittus-Boelter type correlation (Eq. 1) [START_REF] Bengt | Chapter Two -Advanced heat transfer topics in complex duct flows[END_REF] which relates the Nusselt number (Nu) to the Reynolds (Re) and Prandtl (Pr) numbers, is frequently used to estimate the HTC in heat exchangers. Several authors use this ready-made correlation and adjust the coefficients (C) and exponents (a, b) to be close to their experimental results.

h δ λ l = C • ( ρ l U δ μ l ) a • ( μ l c p λ l ) b (1)
As a first approach toward a generalization of correlations, efforts are made by [13] [11] [21] and [START_REF] Yu | Experimental research on water boiling heat transfer on horizontal copper rod surface at sub-Atmospheric pressure[END_REF] in developing empirical models of HTC or testing existing correlations in pool boiling at low pressure conditions. However, from these studies, only those in [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF] could be apply to heat exchange in the context of a plate evaporator.

Zajaczkowski et al. [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF] conducted experimental studies on the influence of subatmospheric pressure and heat flux density on water nucleate boiling. Experiments were conducted above a copper plate in an insulated glass cylinder. Pressure ranges between 1 to 10 kPa, and heat fluxes between 10 and 45 kW/m 2 . As a main result, they showed that accurate approximations of the experimental heat transfer coefficient are obtained using the Labuntsov correlation (Eq. 2)

(Mean Absolute Deviation (MAD 0.12-0.89), the Kruzylin correlation (Eq.3) (MAD 0.25-0.35) and the Stephan-Abdelsalam correlation (Eq. 4) (MAD 0.16). Thus, Labuntsov correlation (Eq.

2) and Kruzylin correlation (Eq. 3) directly determines the heat transfer coefficient as a function of some thermophysical properties of the fluid, the saturation temperature and the heat flux.

Stephan-Abdelsalam correlates (Eq. 4) the Nusselt number as a function of dimensionless numbers that take into account the heat flux and several thermophysical properties of the fluid.

h = 0.075 • q̇0 .67 • (1 + 10 • ( ρ v ρ l -ρ v ) 0.67 ) • ( k l 2 ν • σ • T sat ) 0.33 (2)
h = 0.0777 • ( ρ v ∆h lv ρ l -ρ v ) 0.33 • ( ρ l σ ) 0.33
• ( k l 0.75 . q̇0 .7 μ l 0.45 • c p,l 0.12 • T sat 0.37 )

h = 0.23 • k l d b • ( q̇ d b k l T sat ) 0.674 • ( ρ v ρ l ) 0.297 • ( ∆h lv d b 2 ( k l ρ l c p,l ) 2 ) 0.371 • ( ( k l ρ l c p,l ) 2 ρ l σ d b ) 0.35 • ( ρ l -ρ v ρ l ) -1.73 (3)
Stephan-Abdelsalam correlation (Eq. 4) is one of the most used correlations in the literature. However, in this equation, bubble diameter (d b ) is sometimes very difficult to predict.

The authors recommend using the Fritz correlation (Eq. 5) for prediction of the bubble diameter as the following [START_REF] Alavi Fazel | Pool boiling heat transfer to pure liquids[END_REF].

d b = 0.0146 • θ • ( 2 σ g(ρ l -ρ v ) ) 0.5 (5)
In which the contact angle θ is specified equal to 45° for water and 35° for other liquids studied by the author [START_REF] Alavi Fazel | Pool boiling heat transfer to pure liquids[END_REF]. In their calculations, Zajaczkowski et al. [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF] set a contact angle θ equal to 35°.

Chang et al. [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF] studied the thermal performance and behavior of an evaporator consisting of a pair of parallel stainless steel plates of 138 x 20.8 mm² each with 10 mm channels. The evaporator is inserted in a two-phase thermosiphon loop. The pressure varied between 2.1 kPa and 66.2 kPa. In their experiments, the authors observed three distinct flow regions with different heat transfer properties. They developed a set of three HTC correlations for each region : pool boiling region (Eq. 6), intermittent region (Eq. 7) and vapour region Eq. ( 8). Correlations for the pool boiling region (Eq. 6) and the intermittent region (Eq. 7) will be used. They allow the HTC of pool boiling and liquid film evaporation to be calculated from the heat flux and reduced pressure.

h pbr = (1.951 • 10
h ir = (2 • 10 6 • q̇-0.325 ) • ( P sat P crit ) (0.449-4.72•10 -6 • q) (6)
h vr = (2.06 • 10 6 • q̇-0.412 ) • ( P sat P crit ) (0.441-7.08•10 -6 • q) (7)
Baki [START_REF] Baki | Boiling pure fluids at sub atmospheric pressures[END_REF] performed a review on experimental boiling data at subatmospheric pressures for three different fluids, among them water, to compare them to four known correlations. His results showed poor accuracy between experimental and predicted data for three of the correlations. Only Mc Nelly correlation (Eq. 9) that determines the HTC from the heat flux, some thermophysical parameters of the fluid and the working pressure, gives satisfactory results

(mean error of 40% and correlation coefficient (r) of 0.91).

.

h = 0.225 • ( q c 𝑝 ∆h lv ) 0.69 • ( P λ σ ) 0.31 • ( ρ l ρ v -1) 0.32 (9)
Yu et al. [START_REF] Yu | Experimental research on water boiling heat transfer on horizontal copper rod surface at sub-Atmospheric pressure[END_REF] experimentally estimated water boiling heat transfer coefficient obtained on a horizontal copper bar surface for a pressure range of 1.8-3.3 kPa and a heat flux range of 4000 -10000 W • m -2 . Based on their experimental data, the authors obtained, by regression, the following correlation (Eq.10). The correlation developed is a pressure and heat flux corrected version of the convective boiling correlation [START_REF] Schnabel | Water as refrigerant -Experimental evaluation of boiling characteristics at low temperatures and pressures[END_REF].

h = 0.226 • P r 1.497 • q r 1.023 • ( g β ∆T L c 3 μ l 2 ) -0.891 • ( μ l c p l λ l ) 0.578 (10)

Test bench

Giraud et al. [START_REF] Giraud | Influences of confinement on subatmospheric water vaporization phenomena in a vertical rectangular channel[END_REF] performed an experimental study on a prototype refrigeration system developed at Lafset to simulate the operating conditions that could occur in machines using water as refrigerant (Fig. 1). The test bench is a closed-loop thermosiphon, assisted by a pump.

It consists mainly of two heat exchangers: an oversized condenser and a smooth plate evaporator. To form the classic circuit of a refrigerating machine, a liquid supply line and a steam exhaust line connect these two main components. Type K thermocouples (named T + a number), pressure sensors (named P + a number) and flowmeters (F1 and F2) are installed in the refrigerant circuit and in the two auxiliary circuits. The saturation temperatures inside the evaporator and the condenser are regulated by means of two heating/cooling devices (B1 and B2). A third heating device (B3) is used to adjust the fluid temperature at the inlet of the expansion valve to match the temperature of the refrigerant at the outlet of the condenser in a chiller. The condenser is a cylindrical steel vessel (304L stainless steel) with an inside diameter of 300 mm and a height of 720 mm insulated with the outside. An oversized copper tube coil heat exchanger is placed inside the container to condense the steam along the outer surface of the tube and store the refrigerant in liquid state. The condenser is used here to set driving pressures and simulate an absorber or compressor that would exist in a real chiller.

The entire test bench is made up with ISO-K component to allow high gas tightness. To limit the introduction of incondensable gas, before filling the condenser with water, the entire experimental setup is vacuumed during at least 12 hours and the condenser is then filled with an amount of water that was previously boiled. The condenser and then the entire experimental setup are regularly expunged of a part of the vapor phase, until the pressure measured is equal to the water saturation pressure at the measured temperature during at least one week.

Figure 1 : Drawing of the pump-assisted thermosiphon loop used [START_REF] Giraud | Influences of confinement on subatmospheric water vaporization phenomena in a vertical rectangular channel[END_REF].

Evaporator

The present study focuses on the vaporization phenomena occurring in the evaporator (Fig. 2a) of the test bench (Fig. 1). This heat exchanger consists of three smooth plates 500 mm high and 200 mm wide. Two of these plates are made of steel (304L stainless steel), the third plate is made of polymethyl methacrylate (PMMA). The plates are placed in parallel so that they constitute vertical channels in which a primary fluid or refrigerant and a secondary fluid (hot source) circulate. These fluids, circulate in co-current from bottom to top. The central plate, separating the two channels, ensures heat exchange between the two fluids. The channel formed between the two steel plates in which the secondary fluid flows is fixed at 1 mm. The thickness of the refrigerant circulation channel between the outer PMMA plate and the central plate can be set at 2, 4 or 6 mm according to the needs of the experiment, using an O-ring and the spacer.

The refrigerant is fed through three capillary tubes with an inner diameter of 2 mm each, located at the bottom of the plate. This creates a water column, the height of which is deliberately set according to the needs of the experiment by means of monitoring.

The central plate is 6 mm thick and is the seat of 44 K type thermocouples (22 pairs facing each other -Fig. 2b). A detailed configuration of a pair of thermocouples is shown in Fig. 3. The thermocouples in each pair are recessed at intervals l 1 = 4 mm to measure the temperature distribution in the evaporator center plate (Fig. 2a). The depths of the slots are identical and equal to l 2 = 1 mm (distance between the wall and the thermocouple T 1,i and T 2,i ). In this figure ,T Further information about the experimental procedure is presented by [START_REF] Giraud | Influences of confinement on subatmospheric water vaporization phenomena in a vertical rectangular channel[END_REF].

Experimental Uncertainties

A summary of the uncertainties is reported in Table 1. All instruments are connected to an acquisition system. The maximum error of the acquisition system is 0.02%. The uncertainties of the data are estimated by assuming that the instrumental error was uniformly distributed and could be estimated by Eq.11 as in [START_REF] Zhang | Experimental investigations on single-phase convection and steam-water two-phase flow boiling in a vertical rod bundle[END_REF] cited by [START_REF] Giraud | Influences of confinement on subatmospheric water vaporization phenomena in a vertical rectangular channel[END_REF].

u(a) = φ instr -φ acq √3 (11)

Experimental program and data reduction

In order to estimate the impact of the thermophysical properties of the fluid, the heat flow, the parietal superheating, the filling height of the channel and the geometrical parameters on the HTC during the confined vaporization of water at subatmospheric pressure in a smooth plate heat exchanger, the raw experimental data obtained on the test bench (Fig. 1) by Giraud et al. [START_REF] Giraud | Influences of confinement on subatmospheric water vaporization phenomena in a vertical rectangular channel[END_REF] is exploited. These data are determined as a function of the thickness e of the refrigerant vaporization channel. The database consists of 43 points for e = 2 mm, 48 points for e = 4 mm and 48 points for e = 6 mm. A total of 139 tests with 1200 values collected for each of them during at least 20 minutes once steady-state is obtained are exploited. Data obtained are then used to develop dimensionless number since they allow to vary internal measurements representing the causes (dimensionless number) and measuring their consequences on the heat transfer phenomena occurring in the smooth plate evaporator (Fig. 2).

The experimental conditions covered are summarized in table 2. -temperatures measured by the thermocouples on the secondary fluid side T2 and on the refrigerant side T1 etc.

Based on this data, a local heat flux is estimated assuming one directional heat flux and quasi-steady state conditions. This local heat flux is the heat flux estimated at the area where a pair of thermocouples is located. It is determined by the derivative of the one-dimensional heat conduction (Eq. 12).

q̇i = λ p T 2,i -T 1,i l 1 (12)
The average heat flow from the secondary fluid to the refrigerant for each zone (boiling zone, liquid film evaporation zone and vapor zone) is then determined by the Eq. 13.

q̇z = S l p H z ∑ (λ p T 2,i -T 1,i l 1 ) n i=1 (13)
Where S = l p * 0.05537m 2 is the area of the mesh around a thermocouple (elementary area), l p = 0.2 m is the width of the plate, n is the maximum number of thermocouples in the considered zone, λ p = 16.3 W. m -1 . K -1 is the thermal conductivity of stainless steel, assumed to be constant in the temperature range studied and H z is the height of the considered zone. This considered zone is based on the previous observation of Giraud et al. [START_REF] Giraud | Experimental study of water vaporization occurring inside a channel of a smooth plate-type heat exchanger at subatmospheric pressure[END_REF], those identify four different working zones: a zone at the entrance of the channel where the thermal equilibrium is reached between the refrigerant and the secondary fluid and in which the incoming bubbles collapse, a boiling zone, a liquid film evaporation zone and a vapour zone (Fig. 4). Thus, heat flows are averaged by zone (i.e. a mean heat flow is estimated for the boiling area, one for the falling film are and one for the vapor area). The local temperature at the plate wall T w,i in front of a thermocouple (i) on the refrigerant side (Fig. 3) is calculated by extrapolation by Eq. [START_REF] Schnabel | Water as refrigerant -Experimental evaluation of boiling characteristics at low temperatures and pressures[END_REF].

T w,i = T 1,i - q̇i • l 2 λ p (14)
Thus, the average wall temperature for the considered area on the refrigerant side T w,z is then calculated by the Eq. 15.

T w,z = 1 n ∑ (T 1,i - l 2 • (T 2,i -T 1,i ) l 1 ) n i=1 (15)
Thereafter, only heat transfer phenomena in the boiling zone and the evaporation zone of the liquid film are considered.

When vaporizing water at low pressure in the VSPE, for a filling height H l , the periodic appearance of bubbles is observed at a distance H b below the free liquid surface [START_REF] Giraud | Experimental study of water vaporization occurring inside a channel of a smooth plate-type heat exchanger at subatmospheric pressure[END_REF] such that

For the liquid film zone, the saturation temperature T sat,z = T sat,f depends on the free surface pressure (P sat = P out,ref ) and is determined by Eq. 17.

T sat,f = T sat (P out,ref )

Thus, the heat transfer coefficient for each zone is determined by Eq. 18.

h z = q̇z T w,z -T sat,z (18)
The liquid column height (H l ) related to the feed mass flow rate, is calculated by the Eq. [START_REF] Bengt | Chapter Two -Advanced heat transfer topics in complex duct flows[END_REF].

H l = P in,ref -P out,ref ρ l g (19)
The distance between the nucleation site and the free interface (H b ) and the projection height of the droplets forming the liquid film, are determined manually by the software ImageJ.

The thermophysical properties of the fluid are calculated according to the saturation temperatures of each medium using Coolprop software.

Test of HTC correlations of literature

In order to evaluate the ability of these correlations to predict HTC in the experimental configuration used in this paper, i.e. for confined pool boiling and film liquid evaporation at subatmospheric pressures, a comparison is made between the HTC obtained following the data reduction introduced in 3.4 and the HTC predicted by correlations from the literature. The comparison conducted is: a graphical and statistical comparison conducted by calculating the Mean Absolute Percentage Error (MAE) (Eq. 20), the Mean Relative Percentage Error (MRE) (Eq. 21) and the correlation coefficient (𝑟) (Eq. 22). The MAE is used to check the accuracy of the predictions and the MRE is used to check whether a correlation is over or under predicted average. The percentage of predicted data within a range of ± 30%, noted τ 30 , is also calculated, to provide a quantitative indication of the correlations performance. This value, although it seems high, is consistent with the very complex phenomena of subatmospheric water boiling [START_REF] Giraud | Influences of confinement on subatmospheric water vaporization phenomena in a vertical rectangular channel[END_REF]. The correlation coefficient, always ranging between -1 and 1, measures the affinity between the experimental and predicted data:

-The closer r is to zero, the weaker the linear relationship.

-Positive values of r indicate a positive correlation when the values of the two variables tend to increase together.

-Negative values of r indicate a negative correlation when the values of one variable tend to increase and the values of the other variable decrease.

-The values 1 and -1 each represent "perfect" correlations, positive and negative respectively.

Thus, the relationship is said to be linear.

MAE = 100 N ∑ |h pred -h exp | h exp N i=1 (20)
MRE = 100 N ∑ ( h pred -h exp h exp ) N i=1 (21)
𝑟 = 𝑐𝑜𝑣 (h exp , h pred ) √𝑣𝑎𝑟(h exp ) • 𝑣𝑎𝑟(h pred ) (22)
To remain consistent, the constant (C) and coefficient (a and b) of the correlation Eq. ( 1) are fitted to the experimental database of Giraud et al. [START_REF] Giraud | Influences of confinement on subatmospheric water vaporization phenomena in a vertical rectangular channel[END_REF] (Table 3), minimising the sum of the differences between the experimental and theoretical values of the Nusselt number (Nu). The parameters constituting Eq. 1 are calculated as a function of the saturation temperatures of each medium considered. The definition of the carateritic length (δ) and velocity (U) depends on the area under consideration (table 7).

Pool boiling zone

The results of the comparisons between the experimental HTC of pool boiling and those predicted by the literature correlations are shown to the Fig. 5 (graphical comparison) and

h pred (W • m -2 • K -1 ) h exp (W • m -2 • K -1 )
Pool boiling zone Mc Nelly correlation (g) Figure 5 : Comparison between experimental HTC (hexp) of pool boiling zone and HTC predicted (hpred) by correlations of: (a) Dittus-Boelter [START_REF] Bengt | Chapter Two -Advanced heat transfer topics in complex duct flows[END_REF], (b) Labuntsov [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF], (c) Kruzylin [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF], (d) Stephan-Abdelsalam [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF], (e) Chang [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF], (f) Mc Nelly [START_REF] Baki | Boiling pure fluids at sub atmospheric pressures[END_REF] and (g) Yu [START_REF] Yu | Experimental research on water boiling heat transfer on horizontal copper rod surface at sub-Atmospheric pressure[END_REF] Table 4 : Statistical comparison between the HTC experimental of pool boiling zone and the HTC predicted by the correlations in the literature.

Pool boiling zone Correlations MAE [%] MRE [%] τ 30 [%] 𝑟 [-]

Dittus-Boelter [START_REF] Bengt | Chapter Two -Advanced heat transfer topics in complex duct flows[END_REF] 24 7 73 0.67 Kruzylin [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF] 28 -26 54 0.64 Labuntsov [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF] 69 -69 0 0.58 Stephan-Abdelsalam [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF] 77 -77 0 0.60 Chang [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF] 29 8 69 0.57 Mc Nelly [START_REF] Baki | Boiling pure fluids at sub atmospheric pressures[END_REF] 57 -57 3 0.55 Yu [START_REF] Yu | Experimental research on water boiling heat transfer on horizontal copper rod surface at sub-Atmospheric pressure[END_REF] 39 39 27 0.96 Table 4 indicates that the HTC experimental database for the pool boiling area did not accurately match the majority of the correlations considered.

Labuntsov (Eq.2) and Stephan-Abdelsalam (Eq. 4) correlations show the worst results by predicting 0% of the experimental database within an error band of ±30%. Mc Nelly (Eq. 9)

and Yu (Eq. 10) correlations, also poorly predict 3%, and 27% of the experimental database within an error band of ±30%, respectively. All these above-mentioned correlations give very poor precision and underestimate the experimental database. The Dittus-Boelter (Eq. 1), Kruzylin (Eq. 3) and Chang (Eq. 6) correlations, however, provided more accurate results.

h pred (W • m -2 • K -1 ) h exp (W • m -2 • K -1 )
Pool boiling zone Yu correlation satisfactory result and correctly predicts 69% of the database within an error band of ±30% with an accuracy of 29 %. It over-estimates the experimental values by 8 %.

Liquid film evaporation zone

For the liquid film evaporation zone, the comparison between the experimental HTC and those estimated by correlations from the litterature are shown in the Fig. 6 (graphical comparison) and Table 5 (statistical comparison).

h pred (W • m -2 • K -1 ) h exp (W • m -2 • K -1 )
Liquid film evaporation zone Chang correlation 0 1000 2000 3000 4000 5000

h pred (W • m -2 • K -1 ) h exp (W • m -2 • K -1 )

Liquid film evaporation zone Mc Nelly correlation

(n) Figure 6 : Comparison between experimental HTC (hexp) of liquid film evaporation zone and HTC predicted (hpred) by correlations of: (h) Dittus-Boelter [START_REF] Bengt | Chapter Two -Advanced heat transfer topics in complex duct flows[END_REF], (i) Labuntsov [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF], (j) Kruzylin [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF], (k) Stephan-Abdelsalam [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF], (l) Chang [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF], (m) Mc Nelly [START_REF] Baki | Boiling pure fluids at sub atmospheric pressures[END_REF] and (n) Yu [START_REF] Yu | Experimental research on water boiling heat transfer on horizontal copper rod surface at sub-Atmospheric pressure[END_REF] [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF] 36 -34 35 0.72 Labuntsov [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF] 73 -73 0 0.67 Stephan-Abdelsalam [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF] 80 -80 0 0.69 Chang [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF] 40 13 54 0.50 Mc Nelly [START_REF] Baki | Boiling pure fluids at sub atmospheric pressures[END_REF] 61 -61 1 0.64 Yu [START_REF] Yu | Experimental research on water boiling heat transfer on horizontal copper rod surface at sub-Atmospheric pressure[END_REF] 31 31 45 0.97

Table 5 shows that correlations from the literature do not accurately predict the experimental HTC database for the liquid film evaporation zone, despite their validity at lowpressure conditions and/or the amount and diversity of involved parameters in their development.

Labuntsov (Eq. 2) and Stephan-Abdelsalam (Eq. 4) correlations show the worst results by predicting 0 % of the experimental database within a ±30% error band. As for the Kruzylin (Eq. 3), Mc Nelly (Eq. 9) and Yu (Eq. 10) correlations, they also poorly perform by predicting 35%, 1% and 45% respectively of the experimental database within a ±30% error band.

However, Dittus-Boelter (Eq. 1) and Chang (Eq. 6) correlations provided satisfactory results.

Dittus-Boelter correlation (Eq. 1) correctly predicts 68 % of the database within an error band of ± 30% with an accuracy of 27%

h pred (W • m -2 • K -1 ) h exp (W • m -2 • K -1 )
Liquid film evaporation zone Yu correlation accuracy of 40%. The values calculated with the correlation (Eq. 7) are increased by 13% compared to the experimental values.

The poor results of the correlations from the litterature in accurately predicting the heat transfer coefficient may be due to the complexity of the coupled mechanisms involved during confined vaporization at subatmospheric pressure. Furthermore, with the exception of the correlations (Eq. 6) and (Eq. 7), the other correlations were originally designed to predict free boiling heat transfer coefficients and at higher pressures, although their validity at low pressures has been verified by [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF] [21] and [START_REF] Yu | Experimental research on water boiling heat transfer on horizontal copper rod surface at sub-Atmospheric pressure[END_REF]. All these correlations are also developed for higher heat flux densities than those observed in this study. From a physical point of view, the observations of Chang et al. [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF] are in perfect agreement with those observed by Giraud et al. [5] [16]. Therefore, the good performance of the correlations (Eq. 6) and (Eq. 7) was expected.

However, all these correlations in the literature (Eq. 1 to Eq. 10, except Eq. 5) do not take into account all the relevant parameters that influence the HTC at low pressure. For example, the liquid height is absent of these equations whereas it has been shown that this parameter strongly influence the heat transfer at low pressure [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF] [4] [START_REF] Schnabel | Water as refrigerant -Experimental evaluation of boiling characteristics at low temperatures and pressures[END_REF] [24] [START_REF] Giraud | Subatmospheric pressure boiling on a single nucleation site in narrow vertical spaces[END_REF]. Also, given the centimetric size of the bubbles during the vaporization of water at subatmospheric pressure [START_REF] Giraud | Vaporization of water at subatmospheric pressure:fundamentals of boiling phenomena and path towards the design of compact evaporators for sorption chillers[END_REF] , a parameter influencing their rise and their growth would obviously be determining. It should also be noted that the dimensional analysis adopted for the development of the correlations (Eq.

1 to Eq. 10, except Eq. 5), does not verify the physical independence of the physical quantities.

For the correlations of Labuntsov (Eq. 2), Kruzylin (Eq. 3), Stephan-Abdelsalam (Eq. 4), Chang (Eq. 6 and Eq. 7), Mc Nelly (Eq.9) and Yu (Eq. 10), the thermophysical properties of the fluid and the heat flux density (q̇), depend on the saturation conditions of the medium (P sat , T sat ). Eq.

5 also shows a dependence between the bubble diameter (d b ) and the thermophysical properties of the fluid. However, it is necessary to ensure that no physical quantity, apart from the variable of interest, depends on the variation of the other quantities [START_REF] Delaplace | Modélisation en génie des procédés par analyse dimensionnelle. Méthode et exemples résolus[END_REF]. This independence is a key factor for modeling by dimensional analysis since it makes possible to easily transpose results from one scale to another (similarity).

New pool boiling and liquid film evaporation HTC predictive correlations

Dimensional analysis

Based on the scientific knowledge generated by the literature, the rigorous dimensional analysis approach based on the Vaschy-Buckingham theorem requires the definition of the independent physical quantities governing the heat transfer phenomena occurring in the evaporator (Fig. 2a). These parameters, which are to be considered during boiling and evaporation at subatmospheric pressures in the evaporator, are the thermophysical properties of the liquid and vapour, the thermodynamic conditions (pressure and temperature) and boundary conditions.

The HTC (h) on the refrigerant side, the target variable or variable of interest, depends mainly on : the pressure (P sat ), the vapor mass velocity (G v ), the heat flow (q̇), the enthalpy of change of state from liquid to vapour (∆h lv ) , the difference between the refrigerant side wall temperature T w and the temperature of the saturated liquid medium T sat noted (∆T) (maximum temperature difference for a bubble to develop), the densities of the liquid and vapour (ρ l and ρ v ), the vapour density at liquid temperature (ρ b ), the specific heat (or mass heat capacity) of the liquid (c p l ), the thermal conductivity of the liquid (λ l ), the dynamic viscosities of the liquid and vapor (μ l and μ v ), the surface tension at the liquid-vapour interface (σ), the thickness of the plate containment (e), the liquid column height (H l ), and the acceleration of gravity (g).

The crucial point for a rigorous dimensional analysis is the verification of the physical independence of the listed physical quantities. It is then necessary to make sure that no physical quantity, except the variable of interest h, is influenced by the variation of the other quantities [START_REF] Delaplace | Modélisation en génie des procédés par analyse dimensionnelle. Méthode et exemples résolus[END_REF]. All the thermophysical properties of the refrigerant, listed here, except the vapour density at liquid temperature (ρ b ), are physically dependent on the saturation conditions of the medium (P sat , T sat ). The vapor mass velocity is defined by Eq. 23 which shows that G v , ∆h lv , and q̇(∆T)

do not have to be listed simultaneously. It is then sufficient to take one of these parameters (for example: G v ).

G v = q̇z ∆h lv (23)
Thus, the problem can be set as follows:

h = h(H l , μ v , ρ v , ρ b , σ, g, λ l , G v , e, μ l , ρ l , c p l ) (24)
The mathematical model of the physical law studied (Eq. 24) indicates that the HTC (h) depends on 12 independent dimensional parameters. Thus, the Vaschy-Buckingham theorem allows us to express Eq. 24 in dimensionless form by Eq. 25 (see details in Appendix A).

Nu = Nu(F * , μ * , ρ * , ρ v * , Bo, Ca, Pr, We) (25)
Thus, Eq. 25 shows that the Nusselt number (Nu) depends on 8 dimensionless groups.

Since the heat transfer phenomena in the pool boiling zone and the evaporation zone of the liquid film observed (Fig. 3), are considered, the thermophysical properties of the fluid vary according to each medium as they are calculated according to the saturation temperatures defined by Eq. 16 and Eq.17. Although the physical meaning and/or definition of the dimensionless numbers introduced in Eq. 25 may depend on the area considered (pool boiling or liquid film evaporation), the formulation for each area remains identical. The definition of these and their meaning are described in Table 6.

Table 6 : Dimensionless numbers

Notations Formulas Physical definitions and meaning

Nu h δ λ l

Nusselt number compares the effective thermal energy transport through the boundary layer between the refrigerant and the core plate wall (convective transfer) to purely diffusive conduction (conduction in the medium).

F * e H l
The form factor is the ratio between the plate confining thickness (e) and the water column height (H l ) .It allows to find the right filling ratio and channel thickness for better system performance and is thus of great importance for the evaporator sizing.

ρ v * ρ b ρ v
The ratio between the density of vapour at liquid temperature (ρ b ) and the density of vapour at saturation temperature (ρ v ) is a determining parameter that influences the rise and growth of bubbles at low pressure.

F * * e δ
The shape factor is the ratio between the containment thickness of the plate (e) and the characteristic length (δ).

Ca U μ l σ

Capillary number compares the force of viscosity to the force of interfacial tension (e.g., two-phase microchannel flow).

We ρ l U 2 δ σ

Weber number characterizes the flow of fluids at the interface of a multiphase system. It corresponds to the ratio of inertial forces to surface tension forces.

The definition of the characteristic length (δ) and velocitie (U) depends on the area considered. The physical quantities that constitute them are different whether the pool boiling zone or the falling film zone is considered. These numbers are defined in table 7.

Table 7 : characteristic lengths (δ) and velocities (U) Pool boiling

Notations Formulas Physical definitions and meaning Zones

δ = δ f ( μ l 2 ρ l 2 g ) 1 3
δ f is the visco-gravity length. Liquid film evaporation

U = U f q̇f ρ v ∆h lv
U f is the characteristic average evaporation velocity of the liquid film, determined on the basis of the total heat flux at the wall.

Liquid film evaporation

Thus, for the pool boiling zone, F * * (Table 6) is the Bond number (Bo). It represents the ratio of buoyancy forces to surface tension forces. It is commonly used in confined boiling studies and can be seen as an indicator of bubble "squeezing" [START_REF] Bonjour | Etude phénoménologique de l'ébullition en espace confiné à partir d'un site de nucléation isolé[END_REF].

For the falling film zone, F * * (Table 6) is the shape factor between the containment thickness of the plate (e) and visco-gravity length (δ f ) (noted Bo f ) since the flow of the liquid film falling on the vertical smooth plate (Fig. 2) is induced by the gravity force (gravity flow).

The thickness of the liquid film necessarily depends on the visco-gravity length and the flow velocity. The HTC during evaporation of a liquid film running down a wall depends on the thickness of the film, which depends on the flow regime.

Development of news HTC correlations

The monomial form (Eq. 26) was preferred to simplify the identification of the unknowns. Indeed, the relatively general form, i.e., the one based on a sum of monomials where each monomial is a product of internal measurements at different powers, more accurate, makes the identification of the unknowns very complex and requires a lot of experimental data.

Moreover, in the search for the process relationship via a monomial form, it is necessary to rely on an experimental program where each of the dimensionless numbers varies within a sufficiently wide range. Otherwise, the value of the exponents a i∈ [1;8] is not guaranteed and may bias the value of the constant C [START_REF] Delaplace | Modélisation en génie des procédés par analyse dimensionnelle. Méthode et exemples résolus[END_REF].

Nu = C • (F * ) a 1 • (μ * ) a 2 • (ρ * ) a 3 • (ρ v * ) a 4 • (Bo) a 5 • (Ca) a 6 • (Pr) a 7 • (We) a 8 (26)
For the range of operating points tested in this study, the dimensionless numbers μ * , ρ * and Pr show small variations that bias the values of coefficients and the constant C of Eq. 26. As a result, their elimination from the list of thermophysical properties is possible and this does not change the appearance of the scatterplot cloud formed by Eq. 26. This elimination is in line with the suggestion of Kew and Corwell (1997) quoted by [START_REF] Amalfi | Flow boiling and frictional pressure gradients in plate heat exchangers: part 2, comparison of literature methods to database and new prediction methods[END_REF], who encourages to eliminate any non-dimensional group that does not improve the accuracy of the prediction model. However, there is no evidence that the non-dimensional group eliminated do not improve the accuracy of the model prediction in another range of operating conditions and/or with another fluid. Thus, in this paper Eq. 26 is simplified by becoming the Eq. 27 but it is highly recommended to use Eq. 26 for higher variation of μ * , ρ * and Pr.

Nu = C • (F * ) a 1 • (ρ v * ) a 2 • (F * * ) a 3 • (Ca) a 4 • (We) a 5 (27)
Based on the regression analysis of the experimental database, using the least squares method (minimizing the sum of the differences between the experimental and theoretical values of the Nusselt number Nu), following HTC correlations is obtained:

-Pool boiling zone

h δ b λ l = 2.664 • 10 -6 • ( e H l ) 0.410 • ( ρ b ρ v ) 0.157 • ( e δ b ) -0.382 • ( U b μ l σ ) -2.812 • ( ρ l U b 2 δ s σ ) 1.630 (28)
-Liquid film evaporation zone

h δ f λ l = 1.195 • 10 -8 • ( e H l ) -0.411 • ( ρ b ρ v ) 0.231 • ( e δ f ) 0.767 • ( U f μ l σ ) -3.247 • ( ρ l U v 2 δ f σ ) 1.929 (29)
The domain of correlations application is the important factor that limits its performance. The correlations Eq. (28) and Eq. ( 29) are developed according to the range covered by the experimental test (Table 1). Thus, for working pressures ranging from 0.7 to 1.8 kPa, and for dimensionless numbers which vary according to the ranges listed in Table 8.

The heat fluxes and superheats that allowed the determination of the experimental HTC (Eq. 18) are between 1352-14507 W. m -2 and 1.12-7.14 K respectively. on the other hand (negative sign for the pool boiling zone, positive sign for the liquid film evaporation zone). Although the obtained values of the exponents have no physical meaning since they are obtained by a mathematical fitting, the obtained opposite sign could be expected.

Indeed, for a given thickness of the narrow channel, low H l leads to relative high bubble frequency. Since the bubble frequency is high, the liquid film created by these bubbles is often regenerated leading to thicker falling film. On the contrary, high H l leads to low bubble event and the falling film could get thinner before being regenerated again. Thus, the estimated HTC is expected to be higher in the pool boiling zone for low H l than for high H l values, whereas in the falling film evaporation zone, the reverse trend is expected: a higher HTC is expected for high H l than low H l . Regarding the dimensionless group e δ f/b

, the physical meaning is different regarding the considered zone. Indeed, in the pool boiling zone, δ b refers to the capillary length whereas in the liquid film zone δ f refers to the visco-gravity length. Nevertheless, the obtained sign could be explained by the fact that, for a given running conditions, a lower e leads to a warmer environment fostering boiling in on hand whereas it increases friction and refrains the falling of the liquid film in the other hand. From a physical point of view, the new models involve different dimensionless numbers of great importance for the sizing of two-phase heat exchangers (Table 6). For example, the form

Evaluation and validation of the developed correlations

-2 • K -1 ) h exp (W • m -2 • K -1 )
Liquid film evaporation zone New correlation

factor (e/H l ) should help in finding the right filling ratio and channel thickness to obtain the best system performance. They are then made up of dimensionless numbers that have a physical meaning and relevance to the phenomenon being studied. However, a predictive correlation must not only have reasonable statistical similarity to the experimental results, but it must also capture trends in the experimental data. Thus, Fig. 8 represents the variations of the calculated HTC (hpred) by the newly proposed correlations as a function of the heat fluxes ( q̇), at fixed pressures (Psat). According to Fig. 8, at each given Psat, hpred increases with heat flux (q). That is, when q and/or Psat increase, hpred increases accordingly. In other words, at subatmospheric pressure, when the heat flux is constant, the water pool boiling or liquid film evaporation HTC increases with pressure. This trend is consistent with studies in the literature done at subatmospheric or atmospheric pressure [13] [11] [20] [START_REF] Kiyomura | A predictive model for confined and unconfined nucleate boiling heat transfer coefficient[END_REF]. For given conditions (constant Psat), these said trends are easily analyzed with (Eq. 18) which allows to determine the experimental heat transfer coefficients. Thus, the proposed correlations correctly capture the relationships between HTC (h), working pressure (Psat) and heat flux (q̇) verified extensively in the literature, for different pressure and/or temperature conditions.

The constants and coefficients of the correlations (Eq. 28) and (Eq. 29) are specific to the evaporator studied (Fig. 2) and to the domain covered by the experimental program described in subsection 2.3. In order to extend their areas of validity, they are evaluated by fitting their constants and coefficients with different experimental data obtained at low pressure by other authors (Table 9 ). According to Fig. 9, the developed pool boiling correlation predicts 71% (with MRE =-1%, MAE=5%) and 85% (with MRE = -2%, MAE = 4%) of the respective Zajaczkowski et al. [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF] and Chang et al. [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF] experimental databases. As for the liquid film evaporation correlation, it predicts 92% (with MRE =2%, MAE = 2%) of the experimental data from Chang et al. [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF] (Intermittent region). Thus, from these results, it can be inferred that the dimensionless numbers found in this work are relevant for correlating the pool boiling and liquid film evaporation HTC at low pressure.

It is believed that the proposed correlations could then be used more widely. By keeping the dimensionless numbers and fitting them with more data one would be able to predict all the HTC of water at subatmospheric pressure and at higher pressure since these numbers consider many relevant parameters. As a result, these new correlations and/or the dimensionless numbers used should allow the design of various two-phase evaporators.

Conclusion

Raw experimental data, obtained in order to study heat transfer phenomena during the vaporization of water at pressures ranging from 0.7 to 1.7 kPa in a vertical evaporator channel with smooth plates of standard size (0.2 m wide x 0.5 m high) included inside a thermosyphon loop mimicking a sorption system are processed and exploited.

A graphical and statistical comparison between the HTC from the experimental database and the HTC estimated by empirical correlations from the literature is made. Among the correlations used, only those of Dittus-Boelter (Eq.1), Kruzylin (Eq. 3) and Chang (Eq. 6 and Eq. 7) gave good statistical results but still, show poor performance (Table 10).

Table 10: Results obtained from the statistical comparison between the experimental HTC and the HTC predicted by Dittus-Boelter (Eq.1), Kruzylin (Eq. 3) and Chang (Eq. 6 and Eq. 7) correlations. Based on a rigorous dimensional analysis, two news methodical correlations to predict the pool boiling and liquid film evaporation HTC are developed by fitting the experimental database by mean of the least squares method. Unlike the empirical correlations in the literature, these new models consider among others the thermophysical properties of the fluid (saturation pressure, hydrostatic pressure), the heat flow, the parietal superheating, the filling height of the channel and the geometrical parameters (boundary conditions). They therefore involve various dimensionless numbers of great importance for the dimensioning of two-phase heat exchangers and the optimization of their design like for example, the form factor (e H l ⁄ ), which it is believed to help finding the correct filling ratio and channel thickness to obtain the best system performance. Moreover, it is believed that the proposed correlations could then be used more widely by keeping the dimensionless numbers and fitting them with more data. Currently, the new correlations predict 86% (with MAE = 19%, MRE = 3%, r = 0.79) and 83% (with MAE = 18%, MRE = 2%, r = 0.84) of the experimental data in τ 30 respectively.

Pool boiling zone

To extend their validity domains, the new correlations are evaluated by fitting the constants and coefficients with experimental data from Zajaczkowski et al. [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF] and Chang et al. [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF] (Table 9 ). The pool boiling correlation correctly predicts 71% (with MRE = -1%, MAE = 5%) and 85% (with MRE = -2%, MAE = 4%) of the databases of respective authors. As for the liquid film evaporation correlation, it predicts 92% (with MRE = 2%, MAE = 2%) of the experimental data from [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF] (Intermittent region). Thus, from these satisfactory results, the new proposed models constitute a first step towards the development of tools to assist in the sizing of two-phase heat exchangers and the optimization of their operation.

In order to allow a generalization and to better understand the impact of all the parameters taken into account, experimental test campaigns with a wider range of pressures are envisaged in order to make the correlations more robust and to allow the insertion of dimensionless numbers removed from (Eq. 26) due to their small variation. Research on the decisive effect that non-condensable gases such as air can have on heat transfer by vaporization at sub-atmospheric pressure also seems necessary. Tests will be also performed with other fluids.

M 1 0 1 1 1 1 1 0 1 0 1 1 0 L 0 1 -1 -3 -3 -2 0 1 1 1 -1 -3 2 T -3 0 -1 0 0 -1 -2 -2 -3 0 -1 0 -2 Θ -1 0 0 0 0 0 0 0 -1 0 0 0 1
In order to find the dimensionless groupings, linear combinations between the rows of the matrix with dimensions D (Table A.1) to transform the central matrix C into an identity matrix are made. These operations lead to the matrix with modified dimensions Dm (Table A.2) an identity matrix (modified central matrix Cm), the dimensionless numbers characterizing the physical phenomenon can be written, based on the coefficients contained in the columns of the new residual matrix R (modified residual matrix Rm). Each non-repeated variable represented by G i (i.e. each column of the modified residual matrix) forms a dimensionless number π i (Eq.

A.1) [START_REF] Delaplace | Modélisation en génie des procédés par analyse dimensionnelle. Méthode et exemples résolus[END_REF]. The process relationship (Eq.24) can now be expressed in dimensionless form by the Eq. (A.2). π G v = G v e μ l ; π σ = σ e ρ l μ l 2 ;

π g = ρ l 2 g e 3 μ l 2 ; π λ l = λ l μ l c p l )

(A.2)
The space of the dimensionless groupings (π i ) obtained Eq. (A.2) is modified in order to simplify the interpretation of the dimensionless numbers and to try to understand the physical phenomena involved in the process of boiling and evaporation at subatmospheric pressure in a PHE. These rearrangements, which consist of raising some to different powers, multiplying or dividing them, adding or subtracting them, make it possible to go from Eq. (A.2) to (Eq.25).

  1,i represents the temperature measured by the thermocouple (i) on the refrigerant (primary fluid) side and T 2,i the temperature measured by a thermocouple (i) on the secondary fluid side. Due to the driving pressure imposed in the evaporator by the condenser (pressure set by the thermostatic bath B2) and the evaporator (pressure set by the thermostatic bath B1), vaporization (boiling) of the refrigerant occurs. The PMMA plate on the refrigerant side being transparent, the visualization of boiling phenomena occurring in the channel concerned is allowed and recorded by a high-speed camera (Phantom VEO410). Acquisitions are performed at 3000 images per second with a resolution of 1280 x 800 and an exposure time of 191.8 µs. (a) (b) Figure 2 : (a) Vertical smooth plate evaporator (VSPE); (b) Location of thermocouples located in the center plate.

Figure 3 :

 3 Figure 3: Configuration of a pair of thermocouples

Figure 4 :

 4 Figure 4: Schematic diagram of the temperature development of the secondary fluid and refrigerant as a function of evaporator height [17]

  H l > H b . And H b varies according to the operating conditions of each test point. So, the saturation temperature in the boiling zone T sat,z = T sat,b depends on the hydrostatic pressure (P static = ρ l g d s ) and the free surface pressure (P sat = P out,ref ). This saturation temperature is determined by Eq. 16. T sat,b = T sat (P out,ref +ρ l gH b )

Dittus-

  Boelter correlation shows the best performance by satisfactory predicts 73 % of the database within an error band of ±30% with an accuracy of 24%. For the pool boiling zone, this correlation over-estimates the experimental values by 7%. Kruzylin correlation correctly predicts 54 % of the database within a ±30% error band with an accuracy of 28%. The correlation (Eq. 3) underpredicts 26% of the experimental data. Chang correlation offers a

  the modified capillary length. It is the characteristic distance below which capillary phenomena are predominant over gravitational phenomena. The modified capillary length characterizes the relationship between the surface tension forces and the gravitational forces acting on the interface. Pool boiling U = U b q̇b ρ v ∆h lv U b is the superficial velocity of the vapour bubble.

  for the pool boiling zone, negative sign for the liquid film evaporation zone) and e δ f/b

Fig. 7 a

 7 Fig.7a and b reflect the graphical comparisons between the experimental HTC and the HTC calculated by the correlations (Eq. 28) and (Eq. 29) (named thereafter "new correlation").

Figure 7 :

 7 Figure 7 : Comparison between the experimental HTC (hexp) and the HTC calculated (hpred) by the new correlations: (a) pool boiling zone, (b) liquid film evaporation zone.

Figure 8 :

 8 Figure 8 : Variations of HTC predicted by the proposed new correlations as a function of heat flux

2 )

 2 Psat = 0.775 kPa (Pool boiling) Psat = 1.837 kPa (Pool boiling) Psat = 0.755 kPa (Liquid film evaporation) Psat = 1.712 kPa (Liquid film evaporation)

Figure 9 :

 9 Figure 9 : Comparison between experimental HTC (hexp) from the literature and HTC calculated (hpred) by the new correlations

Table 1 :

 1 Specification of the different measuring devices and standard data uncertainties associated.

	Devices	Type	Accuracy of	Range	Data
			the device		uncertainties
	Thermometer	T-type	0.1 K	-10 to 85 °C	0.06 K
		thermocouple
	Refrigerant flow	Thermal mass	± 1%	0.04 -2 kg h -1	0.01 kg h -1
	meter	flow meter	of full scale
	Coolant flow	Electromagnetic	± 0.5%	0.017 -0.034 kg s -1 0.00006 kg s -1
	meter	flow meter	of reading
	Pressure	Pressure	± 0.075%
	measurement	transmitter with	of reading	0 -5 kPa	0.0009 kPa
		ceramic sensor

Table 2 :

 2 Domain covered by the experimental test Pressures, temperatures, flow rates etc. are recorded by a LabVIEW program, once the desired experimental conditions are ensured and steady-stated conditions are reached.

	Parameters	Field
	Filling ratio	[10 -50] %
		(equivalent in [5 -25] cm)
	Temperature set point at the condenser	[2 -15] °C
	(bath B2 -T fs )	(equivalent in an operating pressure of [0.7 -1.7] kPa)
	Temperature set point at the thermostatic	[30 -45] °C
	bath B3	(equivalent in a mass vapor
		quality inlet of [1.7 -3.8] %)
	Temperature set point at the evaporator	[10 -25] °C
	(bath B1 -T in,fs )	(equivalent in a driving pressure of [0.3 -2.1] kPa)
	Channel thick (refrigerant side -e)	2 mm, 4 mm, 6 mm

Table 3 :

 3 Constants and coefficients fitted to experimental database of Giraud et al.[START_REF] Giraud | Influences of confinement on subatmospheric water vaporization phenomena in a vertical rectangular channel[END_REF]

	Constants and coefficients fitted
	C	a	b	Zones
	38.405	0.484	-1.976	Pooling boiling
	0.067	0.742	-1.541	Liquid film evaporation

Table 4

 4 (statistical comparison).

Table 5 :

 5 Statistical comparison between the experimental HTC of liquid film evaporation zone and the HTC predicted by the correlations from the literature

	Liquid film evaporation zone

Table 8 :

 8 Dimensionless numbers range of variation

		(a) Pool boiling
	Dimensionless numbers	Validity range
	Nusselt number, Nu	2.751 ≤ Nu ≤ 21.118
	Form factor, F *	0.007≤ F * ≤ 0.106
	Vapor density ratio, ρ v *	1.259 ≤ ρ v * ≤ 2.507
	Capillary number, Ca	0.001 ≤ Ca ≤ 0.013
	Bond number, Bo	0.722 ≤ Bo ≤ 2.192
	Weber number, We	0.164 ≤ We ≤ 16.948
	(b) Liquid film evaporation
	Nusselt number, Nu	0.064 ≤ Nu ≤ 0.530
	Form factor, F *	0.007 ≤ F * ≤ 0.106
	Vapor density ratio, ρ v *	1.332 ≤ ρ v * ≤ 2.524
	Capillary number, Ca	0.001 ≤ Ca ≤ 0.016
	Modified Bond number, Bo f	31.111 ≤ Bo f ≤ 117.917
	Weber number, We	0.002 ≤ We ≤ 0.479
	Eq. (28) and (29) show opposite sign exponents for the dimensionless group

Table A .

 A

	1 : Matrix with dimensions D.
	h	H l	μ v	ρ v	ρ b	G v	σ	g	λ l	e	μ l	ρ l	c p l

Table A .

 A

		2 : Matrix with modified dimensions Dm
	ℎ	H l	μ v	ρ v	ρ b	G v	σ	g	λ l	e	μ l	ρ l	c p l
	-1	1	0	0	0	-1	-1	-3	0	1	0	0	0
	1	0	1	0	0	1	2	2	1	0	1	0	0
	0	0	0	1	1	0	-1	-2	0	0	0	1	0
	1	0	0	0	0	0	0	0	1	0	0	0	1
					π i =	e a i1 μ l a i2 ρ l a i3 c pl a i4 G i					(A.1)

Acknowledgements :

The authors wish to thank the "Ministère des Affaires Etrangères de la France" (French Ministry of Foreign Affairs) for funding research stays in Lafset through a cooperation grant with Senegal (Service de Coopération et d'Action Culturelle (SCAC) of the French Embassy in Dakar). This project focuses on compact plate evaporators using water as refrigerant in order to properly size and design them.

Intermittent region

The experimental data from Zajaczkowski et al. [START_REF] Zajaczkowski | Experimental verification of heat transfer coefficient for nucleate boiling at sub-atmospheric pressure and small heat fluxes[END_REF] and Chang et al. [START_REF] Chang | Sub-atmospheric boiling heat transfer and thermal performance of two-phase loop thermosyphon[END_REF] are compared with the corresponding values estimated by the new correlations (Table 9 ), as shown in Fig. 9.

Appendix A: Dimensional Analysis Approach

The main step of the dimensional analysis is the choice of fundamental quantities, i.e., the quantities according to which the dimensions of the independent physical parameters governing the studied phenomena are expressed. In the international system, the fundamental quantities are the mass M, the length L and the time T. In heat transfer problems, as in the case of this study, it is necessary to add a fourth dimension Θ. To go back to the dimensionless numbers that characterize the physical problem under study, this requires the free choice of the repeated variables, provided that they are dimensionally independent and that their dimensions cover all the fundamental dimensions (the determinant of the matrix is non-zero) [START_REF] Delaplace | Modélisation en génie des procédés par analyse dimensionnelle. Méthode et exemples résolus[END_REF]. The repeated variables (e, μ l , ρ l , c p l ) and the non-repeated variables (H l , μ v , ρ v , ρ b , ∆h lv , σ, g, λ l ) form the matrix with dimensions D (Table A.1). The repeated variables form the central matrix C (blue background) and the unrepeated variables form the residual matrix R (grey background).


Nils Ferrand
email: nils.ferrand@inrae.fr

Julie Latune

Mariana Machado-Rios

Eliza Crouzet

Mariana Rios

Elisa Crouzet Inrae

Exploring the complexity of socio-environmental management for transitions, through participatory modeling and games

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

a. Resilience has pros and cons. Do "we" want to change or not ?

2. Top-down management for adaptation has shown its limits through decades. Real, multi-level participation is an option to "co-change".

3 .P👉

 3 Participatory simulation, aka role-playing games, are efficient to a. get them to explore and tackle complexity b. get participants to explore and challenge their own social & political constraints c. imagine new options (technical and social) and endorse them ! 4. Participatory modeling trusts stakeholders' (all) capacity to : a. elicit, share, structure knowledge about their environment and actions b. build actionable models that they can trust and use for change Background & design context for this workshop • French public research unit (Montpellier), multi-disciplinary, specialized in water management & governance, working internationally ○ Focus on engineering participatory methods • Supporting public action with governments, NGOs, CBOs, agencies • A 25y action-research on complex systems management with / by / for their stakeholders, incl. citizens, through participatory modeling • A focus on autonomous adaptation • With robuts physical and digital methods • An ethical perspective on action-research nils.ferrand@inrae.fr Steps & tools in decision-for-change 5 justice nils.ferrand@inrae.fr Less modeling experts, more autonomy ⇒ Wat-A-Game (WAG): A toolkit for (self) participatory modeling & gaming (cc IRSTEA-CIRAD, 2009 http://watagame.info) → « Let-them » (= all actors) model their own territory → Include their own roles, resources, activities, events !!! Get a shared model (& playable) of their socio-environmental system → Use it to explore and test together possible change Actions & trajectories based game dedicated to coastal resilience and transitions Planissim Sénégal + EPURE unit & ACTED (& Ths R Lombard-Latune) Participatory design of waste-water management strategies (Rufisque & Ranérou) • Kits & methods to supports CBOs (on WASTE-WAG) • 40 training workshops, 533 persons Source : R Lombard-Latune, Irstea EPURE III-Instance of Results Planning : building a sanitation system at local and communal level Other info on technologies : -Costs : self construction, maintenance -Les conséquences et gains sont réévalués sur le plateau, ainsi que les enjeux.Un débat est organisé qui peut conduire à ajouter, enlever, créer des actions nouvelles.Plusieurs stratégies sont ainsi comparables.

nils.ferrand@inrae.fr

nils.ferrand@inrae.fr Online models / games example


Quelques principes à partager pour préparer votre processus Preparer, planifier la participation

Objectif de cette réunion ⇒

 réunion Suite aux 35 entretiens réalisés, il s'agit de la réunion prévue pour dessiner le cadre d'une possible future concertation territoriale en 2023 et après • La recherche est ici pour aider à mettre en oeuvre une démarche nouvelle, utilisant selon les besoins des modèles et méthodes participatives dédiés • 3 types de compétences sont associées : ○ Modèles et modélisation territoriale sur les enjeux de soutenabilité et transition ○ Ingénierie de la participation pour l'action publique ○ at io n & ré su lt at s an né e 1 Formation, transfert, pérennisation institutionnelle cc.by.nc.sa INRIA STEEP & INRAE G-EAU, 2022 cc.by.nc.sa INRIA STEEP & INRAE G-EAU, 2022

1.

  Vous avez devant vous un tableau avec les principes principaux issus des entretiens, donc de vos points de vue. (re-)Lisez-les svp 2. Vous avez 5 minutes pour ajouter éventuellement un ou deux principes ou objectifs (sur post-it) 3. Puis venez ensemble marquer des --/-/+/++ sur les principes selon que vous les considérez à rejeter totalement / peu souhaitable / souhaitable / prioritaire 4. Discussion & 1ère synthèse Principes et pratiques de participation ((Des formations plus détaillées pourront être proposées ultérieurement dans le projet)) • Quelques définitions • Quelques idées de cadrage • Comment préparer et organiser une participation ? • Rôle de la modélisation (participative) • Comment savoir où l'on en est, et où l'on va ? Pilotage réflexif • Pour aller plus loin • Discussion La participation ? "Pratique qui consiste à consulter et impliquer les acteurs pertinents dans la définition des programmes, dans la prise de décision et dans les processus de conception des politiques, des organisations et des institutions en charge de ces politiques." (traduit de Rowe & Frewer, 2004) ET dans la mise en oeuvre des actions ! = redonner des pouvoirs (influencer, décider, agir) A n im a t io n , m é d ia t io n , d y n a m iq u e d e g r o u p e s o n t d e s t e c h n iq u e s r e la t io n n e ll e s ! ! ! Réunions publiques Débats "tous publics" sur réseaux sociaux

• 2

 2 niveaux de participation : directe (citoyen.ne.s) et intermédiaire (représentants) • La participation ne remplace pas la décision et la gestion par les élus et institutions -elle l'accompagne et l'enrichit par des apports et engagements • La participation "amont" est (théoriquement) préconisée par la loi (Ord. 2016) -elle évite (théoriquement) les contestations ultérieures • Toutes les étapes de décision (ci-dessus) sont utiles et complémentaires • Une méthode (p.e. jury citoyen, budget participatif, focus group) n'est pas une étape • "ça" prend du temps, mais "ça" évite l'imposition, l'acceptologie et les recours • Les participation commence chez et pour les gens, pas en réunion publique… • Il faut apprendre à participer et faire participer (élus, citoyen.ne.s, gestionnaires…) -il y a des principes et pratiques non évidentes • Il faut décider l'organisation et les règles de participation, et les formaliser pour avoir un cadre de procédure qui rassure et engage TOU.TE.S les participant.e.s ⇒ garantir le respect mutuel et que la participation ait une réelle influence Préparer, planifier la participation : Élaborer une stratégie de concertation Objectif Parvenir à un plan de participation co-construit avec les participants. Pour chaque étape de la décision, définir qui participe et selon quelles méthodologies. et écrire une charte de participation Méthode 1-Quels sont les objectifs du processus participatif ? 2-Quelles sont les différentes parties prenantes ? 3-Quelles sont les différentes étapes de la décision ? 4-Lister les actions de participation 5-Spécifier le rôle de chaque partie prenante à chaque étape, 6-Préciser quels outils seront mobilisés à chaque étape 7-Ecrire et partager une charte et un "contrat" de participation 8-Choisir éventuellement un.e garant.e de la paricipation (Ferrand et al., 2021) (cc)by.sa Cooplage@watagame.info UMR G-eau 2021 la Drôme : Faire émerger des propositions d'actions de la part des citoyens sur la rivière et sa gestion pour nourrir le SAGE en amont de sa révision. (cc)by.sa Cooplage@watagame.info UMR G-eau 2021 GP : groupe pilote (10 pers.) GDE : groupe Débat sur l'Eau (46 pers.) (cc)by.sa Cooplage@watagame.info UMR G-eau 2021

…

  du pain consommé en PACA … de la farine pour faire le pain consommé en PACA Un exemple à l'échelle locale : flux de biodéchets sur le territoire de Grenoble Alpes Métropole 6 Source : Quentin Desvaux (2019). Stage Inria / GAM.

  Tour de table Merci de vous présenter rapidement en indiquant : • prénom, nom • si vous représentez une ou plusieurs organisations (et vous-même !) • de quelle partie du territoire vous venez • 3 mots-clés qui selon vous représentent la concertation et la participation cc.by.nc.sa INRIA STEEP & INRAE G-EAU, 2022 Premiers résultats des enquêtes préalables

	Enjeux du territoire Des connaissances manquantes sur le territoire
	Enquêtes sur
	• Les enjeux du territoire et ses priorités • des élu.e.s engagé.e.s sur le territoire
	• Les connaissances manquantes sur le territoire • des représentants de l'État
	• Les expériences de participation et les besoins, principes, étapes qui en • des socioprofessionnels du tourisme
	découlent • des personnes travaillant dans les parcs
	• Les souhaits et les craintes à propos d'un processus de concertation sur le • des personnes engagées sur les enjeux
	territoire ○ d'énergie
	○ de biodiversité
	○ d'eau et rivières
	○ sociaux et territoriaux

et gestionnaires Citoyen.ne.s Scientifiques et experts

	Tout processus participatif engageant les participant.e.s dans des Politiques Décision technocratique "imposée", à base de sciences & techniques + étapes décisionnelles "acceptologie" successives conduisant à des décisions partagées 🡪 résistance citoyenne	Construction de décisions socialement acceptables -manque de bases scientifiques et techniques
	et des réalisations,
	fortement influencées par
	cette participation. Partage des connaissances vers des solutions
	scientifiquement et socialement acceptables
	≠ information,
	acceptologie

cc.by.nc.sa INRIA STEEP & INRAE G-EAU, 2022 Quels apports des acteurs ? -risque de manque de capacité de mise en oeuvre Adapté de Thomas (2004) cc.by.nc.sa INRIA STEEP & INRAE G-EAU, 2022

Ex2. Politique de l'eau partagée (NC) participative, guide des outils pour agir

			Planification de
			la participation
	Ce qui a été produit
	▪ 1 plan de participation
	▪ des règles de
	participation (charte)		Méthodes participatives. Un guide pour l'utilisateur
	les relations entre le ▪ une charte précisant	Hulot, 2015 LIEN Fondation Nicolas	Fondation Roi Baudoin, 2006 LIEN
	processus participatif
	et la Commission
	Locale de l'Eau (institution officielle)	La concertation, coeur du développement durable CERTU, Conseil général de
		l'Isère, Conseil régional
		Rhône-Alpes, DIREN,
		Rhônalpénergie-Environne
		ment, 2005
		LIEN

Comment impliquer les citoyens dans la gestion de l'eau ? Retours d'expériences de Parcs Naturels Régionaux

	Pour aller plus loin : PAUSE (10 min) C'est quoi un "modèle" et la "modélisation" ?
	des "webinaires" disponibles
	sur http://watagame.info/ingepart ((Retour en groupes après la pause))
		AERMC, 2016 LIEN
	Guide de	Démarches participatives et
	concertation	dialogue territorial
	territoriale et de	URCPIE Rhône-Alpes, 2017
	facilitation	LIEN
	Lisode, 2017
	LIEN
	cc.by.nc.sa INRIA STEEP & INRAE G-EAU, 2022

cc.by.nc.sa INRIA STEEP & INRAE G-EAU, 2022 cc.by.nc.sa INRIA STEEP & INRAE G-EAU, 2022 ★ Un "

modèle" est une image de la réalité qui permet d'en explorer et comprendre des propriétés, des fonctionnements ; et donc aussi d'en débattre et de décider

  . ○ Les modèles sont souvent mathématiques et construits avec des hypothèses et des données sur le "fonctionnement du monde" (p.e. la météo, l'économie, l'hydrologie).

Il y a aussi des modèles sans chiffres (en graphiques ou textes). ■ La carte d'un GPS, un schéma technique, un compte-rendu… des modèles ! ○ Certains modèles aident à envisager des futurs possibles (prédiction), pas tous… ○ Ils respectent des règles pour "bien" représenter la réalité et être utilisables ○ Un modèle est un choix de représentation.

Il n'est jamais "neutre".

	ARMENDARIZ & al, 2017, Construire des politiques	alimentaires urbaines, Editions Quae, 160p.

BECU, N, https://lienss.univ-larochelle.fr/Becu-Nicolas Chamas & al, 2021. https://doi.org/10.1016/j.jclepro.2020.123894 p.e. votre "modèle" des enjeux croisés du territoire cc.by.nc.sa INRIA STEEP & INRAE G-EAU, 2022 C'est quoi un "modèle" et la "modélisation" ? ★ Un "

modèle" est une image de la réalité qui permet d'en explorer et comprendre des propriétés, des fonctionnements ; et donc aussi d'en débattre et de décider.

	○ Les modèles sont souvent mathématiques et construits avec des hypothèses et des
	données sur le "fonctionnement du monde" (p.e. la météo, l'économie, l'hydrologie). Il
	y a aussi des modèles sans chiffres (en graphiques ou textes).
	■ La carte d'un GPS, un schéma technique, un compte-rendu… des modèles !
	○ Certains modèles aident à envisager des futurs possibles (prédiction), pas tous…
	○ Ils respectent des règles pour "bien" représenter la réalité et être utilisables
	○ Un modèle est un choix de représentation.

Il n'est jamais "neutre". ★ La modélisation est

  le processus pour bâtir un modèle. Souvent une affaire de spécialistes, elle peut aussi faire participer d'autres pour permettre d'intégrer leurs connaissances et de mieux partager les choix et principes du modèle. ○

Modéliser ensemble c'est donc partager et organiser des idées sur le monde, et en construire ainsi un modèle autour duquel discuter

  . ○ Un modélisateur aide à ne pas faire et exprimer n'importe quoi…

Ma, & al, 2021, https://doi.org/10.1016/j.jclepro.2021.127368 González-Rosell, & al, 2020. "Integrating Stakeholder Views and System Dynamics to Assess the Water-Energy-Food Nexus in Andalusia" Water 12, no. 11: 3172. https://doi.org/10.3390/w12113172

Source : Bevione, M. (2021) L'analyse des interactions entre flux et acteurs pour la compréhension des enjeux socio-écologiques à l'échelle d'un territoire : application à la production du fromage AOP Beaufort dans la vallée de la Maurienne

Source : Bevione, M. (2021) L'analyse des interactions entre flux et acteurs pour la compréhension des enjeux socio-écologiques à l'échelle d'un territoire : application à la production du fromage AOP Beaufort dans la vallée de la Maurienne

Source : Krieger, E. 2021. Describing and designing socio-technical organisations through the biophysical description of territorialized supply chains. Master thesis at STEEP / Mines ParisTech.

Modélisation "experte" vs. participative : que choisir ?

Un même sujet (Eau / énergie / alimentation)

• des maths et un modèle "expert simple" • des schémas et un modèle participatif complexe, intégrant de (plus) nombreux points de vue et enjeux, et "conçu avec des gens"

Modélisation participative : comment ? quels résultats ? -d'une amélioration de la communication sur le territoire -d'un processus de concertation pérenne, effectif et pertinent -que le processus impacte la prise de décision -que le processus impacte l'action Craintes :

-que le processus soit inutile, frustrant, non transparent -que le processus ajoute des filtres dans la décision, qu'on accentue encore la mise en oeuvre de projets -que les messages à destination du grand public soient déformés -que les gens pensent qu'il y a une confiscation du pouvoir -qu'on ait trop d'entre-soi -qu'on n'aborde pas tous les enjeux

Réconciliation de données…

Données collectées lacunaires et/ou incohérentes résultats du modèle