Dylan Lorfing

Régis Olives

Quentin Falcoz

Emmanuel Guillot

Claude Le Men

Aras Ahmadi

Design and performance of a new type of boiler using concentrated solar flux

Keywords: Concentrated Solar Heating, Solar Boiling Modeling, Film Boiling

In order to reduce the amount of CO2 emissions associated with industrial boiling, the design of a new solar boiler operating with concentrated solar energy has been proposed and studied in the present work. A labscale solar boiler was built to show the feasibility of the concept through experiments, and a thermal model was then developed to study the efficiency of the solar boiler under different boiling conditions. Several peculiarities distinguish this new design from conventional boilers: (i) the use of vertical heating downwards from a receiver placed on top of the boiler, (ii) the conical shape of the receiver oriented upside down and submerged in the liquid, (iii) the solar receiver has a conical cavity on top where vertical solar flux is received, and its lower surface is submerged in boiling fluid, (iv) the direct use of solar energy with optical concentrating facilities for boiling without intermediate heat transfer fluids, and (v) the study of the unconventional filmboiling regime at elevated receiver temperatures which spaces the boiling fluid from the surface of the receiver. The model was validated by the data collected and showed good consistency between predictions and experimental results, with an average error of 3.79% for temperatures and 2.08% for vapor mass flow rate.

For the chosen operating conditions studied, the thermal efficiency of the solar boiler varied from 80% to 90%. Such a thermal model can play an important role for the study and design of new thermal processes involving concentrated solar heating for boiling such as desalination.

Introduction Energy-climate issues and the role of solar processes

In many industrial processes, boiling is required, for example, in the production of pressurized steam in power generation systems or for concentrating products in the food or chemical industry. A wide range of boilers operating at low pressure and high pressure already exist with proven efficiency and robustness where for the majority of applications however, they involve the use of fossil fuels (Steingress and Walker, 2012;Steingress et al., 2012). In the contemporary context where the reduction of CO2 emissions is an absolute necessity, it is essential to also explore alternative solutions mitigating the massive use of fossil boilers, when possible and relevant (IPCC, 2019;[START_REF] Schoeneberger | Solar for industrial process heat: A review of technologies, analysis approaches, and potential applications in the United States[END_REF].

Solar energy, and more particularly concentrated solar energy, is one of the solutions to be considered to replace part of the fossil fuels used for conventional boiling [START_REF] Kumar | Global advancement of solar thermal energy technologies for industrial process heat and its future prospects: A review[END_REF]. Indeed, the development of concentrated solar power (CSP) technologies and their application have technically proven the feasibility of using concentrated solar energy to provide a large heat flux which is a requirement in industrial boiling [START_REF] Achkari | Latest developments on TES and CSP technologies -Energy and environmental issues, applications and research trends[END_REF][START_REF] Fernández | Mainstreaming commercial CSP systems: A technology review[END_REF][START_REF] Irena | [END_REF]. The strong experience feedback in this technology makes it possible to deduce the strengths and weaknesses of concentrated solar heating when performing boiling. Desalination is a relevant example where the concentrated solar boiling could play an important role in reducing the use of fossil-based boilers in thermal distillations. Indeed, many desalination plants use thermal processes, such as Multi-Stage Flash distillation (MSF) and Multiple-effect distillation (MED) which represented in 2018, respectively 18 % and 7 % of the global desalination capacity [START_REF] Jones | The state of desalination and brine production: A global outlook -ScienceDirect[END_REF]. The concept of concentrated solar boiling can also be incorporated into the design and improvement of current thermal desalination systems which are supplied today, for the most part, by the energy released by the combustion of fossil fuels. In addition, the locations of areas where desalination is necessary for the supply of fresh water are mostly arid regions with high solar potential [START_REF] Pugsley | Global applicability of solar desalination[END_REF].

Presentation of solar boiler principles: from sun to steam Different types of solar receivers have been developed in the CSP sector. The design of these receivers strongly depends on the chosen solar concentration technology. As several forms of concentrators exist (Parabolic trough, Fresnel linear, solar tower, Dish-Stirling), various forms of receiver have been developed. For example, in tubular shape for solar towers [START_REF] Conroy | A review of steady-state thermal and mechanical modelling on tubular solar receivers[END_REF][START_REF] Ho | Advances in central receivers for concentrating solar applications[END_REF], or vacuum tube for parabolic trough [START_REF] Bilal Awan | Commercial parabolic trough CSP plants: Research trends and technological advancements[END_REF][START_REF] Sandeep | Solar parabolic trough collectors: A review on heat transfer augmentation techniques[END_REF]. But these receivers have been designed for power generation applications. In the majority of cases, boiling does not occur at the receiver; instead, an intermediary fluid is used to collect the energy in the form of sensible heat and bring it to the storage system or boiler; for this reason, the majority of solar receivers in the CSP sector are not solar boilers. This separation between the boiler and the receiver involves the use of thermal fluid, different heat transfer fluids can be used such as synthetic oils or molten salts for the most common ones [START_REF] Vignarooban | Heat transfer fluids for concentrating solar power systems -A review[END_REF]. The use of a heat transfer fluid has several advantages such as the practical ease of adding high capacity thermal energy storage (TES), which is one of the key elements for the viability of CSP technologies. But preventing the use of heat transfer fluid has also several advantages such as reduced environmental concerns, operating and maintenance costs, and increased overall efficiency by avoiding intermediate exchangers [START_REF] Hirsch | Advancements in the Field of Direct Steam Generation in Linear Solar Concentrators-A Review[END_REF]. For this reason, several studies were conducted on direct vaporization in the receiver [START_REF] Dinsenmeyer | Volume of fluid approach of boiling flows in concentrated solar plants[END_REF][START_REF] Ferchichi | Thermal and fluid dynamic analysis of Direct Steam Generation Parabolic Trough Collectors[END_REF][START_REF] Laing | Thermal energy storage for direct steam generation[END_REF][START_REF] Valenzuela | 12 -Thermal energy storage concepts for direct steam generation (DSG) solar plants[END_REF][START_REF] Yamada | Novel parabolic trough power system integrating direct steam generation and molten salt systems: Preliminary thermodynamic study[END_REF]. Note that today certain industrial installations use the direct steam generation [START_REF] Abengoa | Soluciones tecnológicas innovadoras para el desarrollo sostenible[END_REF][START_REF] Suncnim | Solar steam generators[END_REF], nevertheless these applications are only targeted in the context of power generation.

Outside the context of CSP and power generation, it is interesting to explore the technical feasibility of a direct boiling in solar receiver using concentrated solar flux and its overall efficiency at high receiver temperatures according to the scale of the boiler and the concentrated solar heat flux used. Additionally, to cope with the possibility of boiling corrosive or scaling liquids (e.g. seawater), boiling under film-boiling conditions using solar energy may be reinvestigated due to its potential protective effect [START_REF] Farrar | Film Boiling in a Scaling Liquid[END_REF]. Film-boiling describes a liquid-vapor flow heat transfer in which a vapor film separates the boiling liquid from the heating surface, where due to the lower conductivity of the vapor, the temperature of the heating surface can increase significantly (Jin and Shirvan, 2021). [START_REF] Bromley | Heat Transfer in Stable Film Boiling[END_REF] was the first to present an analytical approach for the calculation of heat transfer in this regime. Since then, many works have been carried out to study the phenomenon on simple geometries such as tube and plate [START_REF] Mohanta | Heat transfer correlation for film boiling in vertical upward flow[END_REF][START_REF] Wu | Solution of the boundary-layer equations for natural convection film boiling on vertical cylindrical surfaces[END_REF][START_REF] Yamada | Novel parabolic trough power system integrating direct steam generation and molten salt systems: Preliminary thermodynamic study[END_REF] and in different orientations, e.g. vertical, horizontal and downward [START_REF] Aursand | Inclined film boiling: Film stability and heat transfer[END_REF][START_REF] El-Genk | Transient boiling from inclined and downward-facing surfaces in a saturated pool[END_REF] where the resulting correlations are geometry specific.

In this work, the proof of concept of a solar boiler design to directly generate the steam under a solar receiver was carried out using the medium-power solar furnace at the Odeillo laboratory in France (Figure 1). The medium-power solar furnace provides a substantial concentrated heat flux on a focal zone (point D in Figure 1).

Even though the solar collection and concentration systems are not the focus of this study, it is important to have a global view of the system. On this device, a heliostat (Figure 1A) permits the direct solar irradiation to be redirected onto a second reflector (Figure 1C); this reflector then concentrates the flux and orients it vertically. More details concerning this installation are provided in Section 2.1.

The vertical descending concentration presented is similar to that of "Beam-Down" solar concentrators [START_REF] Grange | Preliminary Optical, Thermal and Structural Design of a 100 kWth CSPonD Beam-down On-sun Demonstration Plant[END_REF]. As the solar fields of solar towers, the "Beam-down" system allows a point focusing of solar energy, to work at high power density and flexible scales of installation. But unlike the solar tower, where the beam is directed upwards to the top of a tower where the receiver is located, the "beam down" technology has a second reflector which allows the beam to be conveyed to the ground level. This helps reducing construction costs compared to a solar tower. In addition, focusing solar radiation at ground level seems to be more practical to integrate it into various processes. These systems have been tested on laboratory scale, such as in Masdar city (Marwan Mokhtar et al., 2014) or in Miyazaki [START_REF] Kodama | Flux Measurement of a New Beam-down Solar Concentrating System in Miyazaki for Demonstration of Thermochemical Water Splitting Reactors -ScienceDirect[END_REF] and industrial facilities are currently under development such as in the Yumen Xinneng factory [START_REF] Bcp | BCP solar technology[END_REF] or the San Filippo del Mela project [START_REF] Group | Performance of a 100 kWth Concentrated Solar Beam-Down Optical Experiment[END_REF].

Because the solar furnaces have a point focus, a conically shaped receiver was chosen (Figure 2E). The advantage of this shape is that it traps the solar flux in a cavity to avoid both radiation and reflection losses, conical shapes have already been studied for Dish-Stirling installation and have shown good optical performances [START_REF] Daabo | The optical efficiency of three different geometries of a small scale cavity receiver for concentrated solar applications[END_REF][START_REF] Zhang | Combined optics and heat transfer numerical model of a solar conical receiver with built-in helical pipe[END_REF]. The conical cavity in vertical descending flow has also been studied and confirms the good optical results of this geometry [START_REF] Li | Performance investigation on solar thermal conversion of a conical cavity receiver employing a beam-down solar tower concentrator[END_REF]. Moreover, because boiling takes place directly in contact with the receiver, on the fluid side, the conical shape permits a larger contact surface for heat transfer.

Goals of the present work

Several objectives are targeted by this study; the first objective is to provide experimental results and a proof of concept for a solar boiler where concentrated solar radiation is received on a main solar receiver to operate in boiling mode at the level of solar receiver without intermediate heat transfer fluids. The second objective is to present a thermal model, fitted to the specific conical geometry of the solar receiver, to estimate the efficiency of such a boiler and its potentials when using the concentrated downward vertical solar flux and operating under different boiling conditions. Note that in this design, the conical receiver is submerged in the boiling tank and therefore operation under film boiling conditions may space the surface of the receiver from that of the liquid to prevent scaling or corrosion. Consequently, the experimental data are used to estimate the precision of the developed model and to adjust the correlation of the heat transfer under film boiling for the particular conical geometry of the studied solar receiver.

Materials and methods

Experimental setup

A lab-scale experimental device was constructed in the Procédés, Matériaux et Energie Solaire (PROMES) laboratory and was supplied with concentrated solar heat from the Odeillo solar furnace. The solar boiler module allowed boiling under concentrated solar heating to be studied. The optical facilities of the Odeillo solar furnace enabled the collection and concentration of solar energy with a downward vertical flux (Figure 1).

A first reflector, made of float low-iron glass with a silver coating, collects solar radiation (Figure 1A) and directs it through a trap door. The beams pass through adjustable carbon shutters that permit the input power to be regulated (Figure 1B) and are finally reflected again by a low-iron glass parabolic mirror with a silver coating.

The parabolic mirror has an apparent diameter of 1.5 m (Figure 1C) and allows the solar energy to be concentrated at the focal point (Figure 1D). In the laboratory installation used, the heliostats mirror is oversized compared to the parabolic reflector. Thanks to this, even when the angle of incidence is large, the redirected solar energy irradiated the entire parabolic mirror. In practice, this implies that concentrated solar energy will not depend on the sun's position, but only on direct normal irradiance (DNI).

Figure 2 shows the solar module subject to concentrated solar heating. The boiler is composed of two polymethyl methacrylate tanks: (i) the main tank where boiling takes place (Figure 2A) and (ii) the secondary tank where the overflow is collected (Figure 2B).

The boiler also includes a metallic receiver (Figure 2E), which receives the concentrated solar heat and transfers it to the main tank. The receiver shape is defined by four geometric parameters: the external diameter (Drec,out); the internal diameter of the receiver (Drec,in), which designates the base diameter of the conical cavity; the opening half-angle of the cone (αrec,cone); and the receiver thickness (trec), as shown in Figure 3A. To complete the geometric consideration, the receiver is defined by its emissivity (εrec) and absorptance (αrec). The receiver is assumed to have an opaque gray surface. Given this and using Kirchhoff's law, absorptance and emissivity coefficient can be linked via a strict equality (εrec = αrec). The receiver used in this study is made of copper, and its characteristics are summarized in Table 1.

A metal support disk (Figure 2C), which uses metallic sticks to hold the solar receiver, is placed on top of the main tank. A rail system enables the entire module to be mobile, allowing the boiler to be placed at the focal point under the parabolic concentrator. Regarding the emissivity (εrec), the value used here is an approximation from the literature for black oxidized copper (OMEGA (TM), 2021). Indeed, even if the receiver were originally made of non-oxidized copper, because of ageing at high temperatures, a layer of oxidized copper would quickly appear and modify the optical properties.

The water supply was provided by a Watson Marlow 313S peristaltic pump. The pump extracts the water from a feed tank and conveys it to the main boiler tank. The non-vaporized water collected in the secondary tank is evacuated at a constant flow rate. The liquid mass flow rates are measured at the inlet and outlet of the system by measuring the mass variation of the reservoirs using weighing balances with data acquisition. Balances are OHAUS brand Explorer with a definition of 0.1 g and a capacity of 12 kg. The mass flow rate of the exiting vapor is then calculated as the difference between these two liquid flow rates. To estimate the error propagations for the estimation of mass flow rates, a minimum measurement period of 10 s was chosen, which gives an accuracy of 2.0 × 10 -5 kg s -1 .

Temperatures are measured using six K-type thermocouples. K-type thermocouples provide temperature measurements with an accuracy of 1°C for temperature between 0 and 1,100°C. Measurements are carried out for the surroundings (Tamb), the inlet water (Tl,in), the outlet liquid and vapor water (Tl,out and Tv, respectively), and the upper and lower surfaces of the receiver (Trec,sup and Trec,inf, respectively). The data logger used for the acquisition of the temperature measurements was a Midi LOGGER GL240 from Graphtec©.

An empirical relation developed in the laboratory for the used equipment was employed to calculate the concentrated power at the top of the receiver ( ). This empirical relation takes DNI measurements from a pyrheliometer (Kipp & Zonen CHP1) as input and accounts for the opening of the carbon shutter (Figure 1B).

The DNI measurement system is maintained by the laboratory's technical team and allows for an accuracy of 1 W.m-2 for the entire range of solar irradiance naturally available on site. The shutter opening can be controlled during experiments by the operator to regulate the concentrated solar power ( ) entering the boiler. The empirical relationship implicitly takes into account the optical performance of the global solar concentration system before the receiver which is estimated 48%. These performances include the optical properties of the heliostat mirror and the parabolic mirror. The maximum concentration rate that can be obtained by the concentrator is close to 15000. This high concentration factor allows working with a high power density.

Finally, the imaging system was based on a Basler acA1920-150um CMOS camera paired with a 35-mm focal length lens to observe the fully submerged cone. The Shadowgraph method is used to allow visualization of the receiver. Regarding the lighting conditions required for the experiment, natural light is sufficient to apply the method. In the experiment presented, the acquisition speed of the camera is fixed at 25 frames per second.

The camera was operated using free Basler Pylon software that allows for a user-friendly configuration, viewing, and the acquisition of images. The purpose of the imaging apparatus was to visualize the boiling process in real time. By observing the cone, one can see the transition between the different stages of the boiling process and estimate when the vapor film forms between the receiver and the liquid. As illustrated in Figure 3B, the receiver is placed at the top of a tank. This principal tank, where boiling takes place, is fed with water via a bottom input (point I in Figure 3B); the water mass flow rate at this inlet is denoted , and the inlet temperature is denoted Tl,in. At the top of the tank, the overflow is evacuated via a secondary tank and is then ejected as the outlet liquid water ( , ) at a temperature of Tl,out (point O in Figure 3B). The flow of steam below the receiver is denoted and leaves the boiler at a temperature Tv. Finally, losses via the tank wall ( , ) are included; these losses are shown in Figure 3B by the orange arrow. Table 2 summarizes all the heat flows implied in the solar boiler. To give an overview of the system, the thermal network diagram illustrating the heat transfers and the link with the temperatures involved is also given in The overall design of the receiver is given in Figure 3A. A conical shape was chosen for the receiver to increase the solar efficiency on its upper surface and at its lower contact surface between the receiver and the water.

The characteristics used to define the receiver were given in Section 2.1.

Similar to the receiver, the tank is characterized by several geometric parameters. Four parameters allow the tank geometry to be defined. These parameters are the tank width, length, height, and wall thickness denoted Wtank, Ltank, Htank, and ttank, respectively, as shown in Figure 3B. Table 3 summarizes the above-mentioned parameters involved in the design of the solar boiler.

Table 3: The solar boiler design model consists of the mass balance for water under the liquid-vapor equilibrium condition (Equations ( 1) and ( 2)) in the main boiler tank (Figure 3B). The heat balance and heat transfer equations are given for the different parts of the overall module: (i) the heat balance in the upper area of the receiver and the heat transfer from the receiver to the water (Equations ( 3) and ( 4), respectively) and (ii) the heat balance equation for the water in the main boiler tank (Equation ( 5)). The heat transfer terms mentioned in the model are expressed using their theoretical formulations given in Section 2.4. Note that, by convention, the heat transfer terms in Equations ( 3) -( 5) are positive for inputs and negative for outputs. Because this is the design stage and there is a need for future experimental validations of the principal heat transfer results, all balance equations are given under steady-state conditions in this study.

Boiling equations under liquid-vapor equilibrium conditions

Knowing the conditions of the water supply, the mass balance equation is given for water (Equation ( 1)) at a given temperature and pressure (T and P, respectively). In addition, the liquid-vapor equilibrium condition for water (Equation ( 2)) allows us to relate the boiling temperature (Tb) to the operating pressure in the module.

, ,

! (1) ,
Here, , , , and , are the mass flow rate of the liquid entering the boiler, the vapor flow rate, and the mass flow rate of the liquid leaving the boiler, respectively. These flows are represented in Figure 3B. In

Equation (2), ,
is the saturated temperature of pure water at a given pressure P.

Upper area of the receiver and heat transfer to the boiling fluid

The overall energy balance in the upper zone located on the top surface of the solar receiver is given by the following equation specifying all the heat flows adjacent to this zone (Figure 3A):

, , , 0 (3)
where is the concentrated solar energy reaching the receiver and , in W, is the conductive heat flow that is transmitted from the upper surface of the receiver at Trec,sup to its lower surface at Trec,inf as a result of the temperature difference in the solid material of the receiver (Trec,sup > Trec,inf).

Calculating the overall energy balance in the lower zone on the bottom surface of the receiver and neglecting the transfer by the lateral part of the receiver, the heat transfer via conduction through the receiver is equal to the heat transferred to the fluid for boiling in the film-boiling regime:

0 (4)

Heat balance for fluid in the main boiler tank

Under steady-state conditions, the heat balance for the water inside the tank below the surface of the receiver is

#$ % & $ , % ' , #$ , % & $ , % ' & # , ' 0 (5)
where , and , are the total mass flow rates of outgoing vapor and liquid, respectively; hl,in' and hl,out' are the mass enthalpies for the incoming and outgoing liquids, respectively; hv' is the mass enthalpy of the outgoing vapor; and

, represents the heat loss in the surroundings of the boiler tank. In this study, the liquid at the inlet is assumed to be pure water, the thermodynamic properties of which are evaluated using the MIT model [START_REF] Sharqawy | Thermophysical properties of seawater: a review of existing correlations and data[END_REF], and the vapor mass enthalpy is evaluated for pure water.

Formulation of the heat transfer phenomena

To complete the governing equations of the model (Section 2.3), the heat transfers defined in Equations ( 1) -

(5) need to be quantified using relevant theoretical formulations depending on the nature of the transfer (radiative, conductive, or convective) and the specificities of the geometry chosen in this study for the design of the solar boiler.

Receiver heat losses by reflection

As the concentrated solar power is supplied inside the cavity from above (the focal point is located inside the cavity) thanks to the second parabolic reflector, it is assumed here that all solar radiative heat concentrated ( ) is transmitted to the conical part of the top surface of the receiver (Green line in Figure 5) and solar radiation on the ring part of the receiver (Blue line in Figure 5) is neglected. To estimate the heat reflected by and leaving the conical cavity of the receiver, several assumptions are made. First, the receiver is considered to have an opaque gray surface. This assumption is commonly made for metal and permits the transmittance to be set equal to zero. The reflectance, absorptance, and emissivity are then constant, only depending on the receiver materials and the temperature. Under these assumptions and using Kirchhoff's law, the absorptance and the emissivity coefficient can be linked via a strict equality. The absorptances used for the different materials considered in this study are listed in Appendix 1.

The second assumption is that the upper surface of the receiver is a diffuse emitter, absorber, and reflector.

This assumption is also commonly used for unpolished metal when the external radiation arrives on the surface at a reasonable incident angle (i < 60°) [START_REF] Modest | Radiative Heat Transfer[END_REF]. The input radiation is also considered as a uniform vertical flux for simplicity. In addition, in this context, the assumption of a uniform vertical flux is necessary because the actual distribution of the radiation is not known precisely.

The ring surface of the receiver (Ssup,ring) (the blue line in Figure 5) is a flat surface delimited by the diameter of the conical base of the receiver (Drec,in) and the receiver diameter (Drec,out). Because it is assumed that all the concentrated solar power ( ) reaches the conical cavity, the reflection losses of this surface are not considered ( ), but the radiative losses ( To determine the reflective losses, the method proposed by [START_REF] Modest | Radiative Heat Transfer[END_REF] is used. This method estimates the energy reflected by a given surface, here Ssup,cone, in a closed cavity. As the presented cavity is open from above, and according to the literature, an imaginary surface is defined, which is the conical base surface (the black dash dotted line in Figure 5) for artificially enclosing the cavity. The details of the calculation are given in Appendix 1.

The power reflected by the internal cone and leaving the receiver ( , ) is obtained using Equation ( 6):

, & . )1 & + 1 & 1 & + !. 1 & ,! - (6)
Where,

, and are the heat loss by reflection at the upper surface of the receiver and the heat concentrated at the receiver, respectively; εrec is the receiver emissivity; and F is the receiver cavity view factor.

Figure 5: Representation of the different surfaces that are relevant to the calculation of the radiative phenomena.

Receiver heat losses by radiation

The radiative heat loss ( ) as well as those from the ring part of the receiver ( , ,

).

Regarding the conical cavity, the same assumptions as made in Section 2.4.1 following the same method proposed by [START_REF] Modest | Radiative Heat Transfer[END_REF] lead to Equation (8). Details of the calculation are presented in Appendix 1.

The ring part of the upper side of the receiver can be also be treated as a gray surface. Assumption is made that no concentrated solar power irradiates this part of the receiver (all the concentrated power arrives in the cavity). In addition, because the ring is flat (with a view factor equal to 1), Equation ( 9) can be used to estimate the heat losses by radiative emission at this surface:

, , , , , (7)
, , &. /, ,+ 1 & 1 & + ! 1 & ,! 0 12 , / 3 (8) , , &. /, + 0 12 # , / 3 & 4 3 ' (9)
where Ssup,ring and Ssup,cone are the surfaces of the upper ring of the receiver and the conical lateral part, respectively, in m 2 ; εrec is the receiver emissivity; F is the receiver view factor; σSB is the Stefan-Boltzmann constant; and Trec,sup and Tamb are the temperatures of the upper side of the receiver and the atmosphere, respectively. Here, the thermal loss that takes place at the lateral surface is neglected. This assumption can be justified if the thickness of the receiver is small compared to the outer diameter. In this case, the lateral surface is very small in comparison to the upper and lower sides of the receiver. To account for the radiation losses by the lateral part of the receiver, Equation ( 9) can be modified to replace the surface of the upper ring (Ssup,ring)

with a new surface by adding Ssup,ring to the lateral surface (Srec,lat).

Receiver heat losses by convection

Because of the high temperature of the upper surface of the receiver, convection is taken into account at this level such that

, &$ 5 4 . /, # , / & 4 ' ( 10
)
where

,
is the heat loss to the environment by convection from the upper surface of the receiver, hrecamb is the global convective heat transfer coefficient, and Ssup,rec is the surface of the upper side of the receiver.

In this study, the air velocity is assumed to be weak enough that free convection is the dominant convective phenomena and forced convection can be neglected.

Because of the geometry, two correlations need to be incorporated. The first correlation characterizes the heat transfer coefficient for free convection on the upper surface of a hot plate (hring-amb), and the second correlation determines the heat transfer coefficient for free convection on an inclined wall (hcone-amb). Here, the upper internal surface of the cone is assumed to be an inclined plane. The correlations used for the calculation of these heat transfer coefficient are taken from the literature [START_REF] Rohsenow | Handbook of Heat transfer[END_REF]. These correlation are given in details in appendix 2.

Consequently, the expression of the global convective heat transfer coefficient can be written as

$ 5 4 $ 5 4 . /, $ 5 4 . /, . , / (11)
Accordingly, two assumptions are made here. First, the interaction between the conical part and the flat part is considered insignificant, so that the two heat transfer coefficients add together and result in a global convective heat transfer coefficient, and second, the conical shape of the surface inside the cone is assumed to be an inclined plane with the same area.

Conductive heat transfer inside the receiver

The conductive heat transfer describes the heat transmitted from the upper surface of the receiver at Trec,sup to its lower surface at Trec,inf and is formulated as

6 . ,4 7 # , / & , ' (12)
where λrec is the conductivity of the receiver material in W m -1 K -1 , Srec,m is the mean surface of the receiver subject to a flow of conductive heat in m 2 , and trec is the receiver thickness in m.

The conductivity is assumed to be constant; its value is taken at the receiver mean temperature (Trec,m) and is calculated using a polynomial correlation of the thermal conductivity for the considered receiver material. The correlation coefficients are determined by fitting the experimental data of the conductivity versus the temperature from the literature [START_REF] Graves | The thermal conductivity of AISI 304L stainless steel[END_REF][START_REF] Holmgren | Review of thermal conductivity of cast iron[END_REF].

Heat transfer to liquid under film boiling

With respect to the heat transfer between the receiver and the liquid, where the heat transfer coefficient is determined for the specific case of boiling with the Leidenfrost effect (hboiling), the global heat transfer is

$ . , # , & ' (13)
Where, Srec,inf is the inferior receiver surface and Trec,inf and Tb are the receiver inferior surface temperature and the boiling temperature, respectively.

hboiling is composed of two components: convective transfer and radiation. The thickness of the film is variable and increases slightly with the height of the cone, which implies an interdependence between the radiative and convective components of the heat transfer. [START_REF] Bromley | Heat Transfer in Stable Film Boiling[END_REF] demonstrated that this dependence can be described by

$ $ , 8 9 $ , 8 $ : ; < $ , (14)
The heat transfer coefficient for the radiative transfer (hboiling,rad) is given by Equation ( 15), while that for the convective heat transfer in the film-boiling regime (hboiling,conv0) is given at the limit hboiling,rad → 0 by [START_REF] Rohsenow | Handbook of Heat transfer[END_REF] and is formulated in Equation ( 16).

$ , + 0 12 # , 3 & 3 ' # , & ' (15)
$ , 8 = > ?@ @ & @ !∆$ °6 < B # , & 'C D ; 3 EFFFFFFGFFFFFFH "J 4" (16)
Here, ρl and ρv, the liquid and vapor densities, respectively, in kg m -3 , were taken at the boiling point (Tb) for the liquid and at the arithmetic mean between the upper surface temperature of the receiver (Trec,inf) and the boiling point (Tb) for the steam. ∆$ / ° is the efficient latent heat, and Lc is the characteristic length corresponding to the length of the submerged wall. μv and λv, the viscosity and the thermal conductivity of the vapor, respectively, are taken at the arithmetic mean between the lower surface temperature of the receiver (Trec,inf) and the boiling point (Tb).

The efficient enthalpy of vaporization, ∆$ / °, is

∆$ ° ∆$ 91 0.34 M / # , & ' ∆$ : (17)
Where, cpv is the thermal heat capacity of the vapor in J kg -1 K -1 and Δhv is the enthalpy of vaporization of the water.

Beyond the Leidenfrost point at the film pool boiling, there is no contact between the solid surface and the liquid. In this situation, there is an uncanny similarity between the conditions in the stable vapor film and the condensation of a laminar film because in the latter case, the external liquid is not easily set into motion, and therefore the film boiling correlation is historically obtained from the theory of condensation. The "D term" in Equation ( 16),

N O P O Q 5O P !∆R P °SP T U P #V WXY,Z[\ 5V ] '^Y
_ à, is inspired by the expression of the Nusselt number in the condensation theory on a horizontal cylinder, as used and assimilated by [START_REF] Bromley | Heat Transfer in Stable Film Boiling[END_REF] for the description of film boiling with some modifications. For condensation on a horizontal cylinder, the original expression implied the density, viscosity and conductivity of the liquid instead of vapor in the case of film boiling, and the C coefficient in Equation ( 16) was 0.728 instead of 0.62 for film boiling [START_REF] Lienhard | A Heat Transfer Textbook[END_REF]. The results for other situations were also reported by [START_REF] Lienhard | A Heat Transfer Textbook[END_REF]: the correction coefficient C was reported 0.828 for the sphere and 0.874 for the vertical cone with the apex on top and the base insulated where in the latter case, the gravity g in the expression has been rectified by also considering the effect of the cone half-angle.

Finally, the = coefficient accounts for the geometry of the receiver and the type of flow involved. Several values, summarized in Table 4, have been proposed by different authors for different geometries and flows. In Table 4, stagnant interface and dynamic interface refer to the condition of the liquid-vapor interface during boiling. In the case of a stagnant phase interface, a no-slip boundary is used, while in the case of a dynamic interface, zero shear stress is considered. A recent study [START_REF] Aursand | Inclined film boiling: Film stability and heat transfer[END_REF] proposed a = coefficient for an inclined plane that was determined according to the inclination, Reynolds number, capillary number, and vapor-thrust number.

Despite the fact that the analogy between film condensation and boiling is not suitable nor yet fully verified for complex geometries, high curvatures and vertical shapes due to the large disparity between vapor film thickness and that of liquid for both processes, satisfactory predictions can be obtained for limited conditions and like most dimensional analyzes and hydrodynamic stability analyzes postulated in the literature [START_REF] Incropera | Fundamentals of Heat and Mass Transfer[END_REF][START_REF] Lienhard | A Heat Transfer Textbook[END_REF][START_REF] Dhir | Laminar film condensation on plane and axi-symmetric bodies in non-uniform gravity[END_REF]), the "D term" remains preponderant in the expression of the convective heat transfer coefficient for film boiling and the C coefficient can be identified for varying geometries on the basis of an appropriate choice of characteristic length, as the order of magnitude of the C coefficient is the same with respect to well-defined geometries such as cylinder, sphere and plate.

As no expression of the = coefficient is available in the literature for an inverted cone (oriented upside down)

under stagnant conditions, and as a first approach, the value for an inclined plate under pool boiling conditions is used in this study. To justify this approximation, it should be noted that for a large part of the surface of the cone, the radius of its cross section is sufficiently large compared to the expected vapor film thickness.

According to the recommendations of the literature [START_REF] Ghiaasiaan | Two-Phase Flow, Boiling, and Condensation: In Conventional and Miniature Systems[END_REF], for an inclined plate the = value is taken equal to that of the vertical surface (here, 0.663). To account for the inclination, the gravity constant was replaced by the apparent gravity g.cos(αrec,cone) in Equation ( 16).

Table 4: Different values of the C coefficient used for the calculation of the heat transfer coefficient in the filmboiling regime.

Condition Value Reference

Vertical surface (with stagnant liquid vapor interface) 0.663 [START_REF] Ghiaasiaan | Two-Phase Flow, Boiling, and Condensation: In Conventional and Miniature Systems[END_REF] Vertical surface (with dynamic interface) 0.943 [START_REF] Ghiaasiaan | Two-Phase Flow, Boiling, and Condensation: In Conventional and Miniature Systems[END_REF] Horizontal cylinder 0.620 [START_REF] Bromley | Heat Transfer in Stable Film Boiling[END_REF] Horizontal surface (with CCl4 or pentane) 0.425 [START_REF] Berenson | Film-Boiling Heat Transfer From a Horizontal Surface[END_REF] The model was programmed and implemented in Python 2.7, where the CoolProp library [START_REF] Bell | Pure and Pseudo-pure Fluid Thermophysical Property Evaluation and the Open-Source Thermophysical Property Library CoolProp[END_REF] was employed to calculate of the fluid properties and the phase equilibrium conditions. The model used the damped least-squares method for the system of nonlinear equations and employed the Levenberg-Marquardt modification in the SciPy.Optimize package [START_REF] Pauli | Emissitivity of Common Materials[END_REF] as implemented in MINPACK [START_REF] Moré | User guide for minpack-1[END_REF].

Results and discussion

Boiler experiments under concentrated solar heating

Figure 6 shows the conical receiver for a single experiment under different boiling conditions. In this experiment, once the solar boiler had been adjusted, regulation shutters were fully open. These images were taken at key moments in the experiment. Regarding these images, it should be noted that they provide a qualitative view of the boiling phenomena and are not intended for quantitative measurements. Indeed, due to the conical geometry and the large variation in the optical coefficient between the vapor and the liquid, deformation effects appear. They can be seen by comparing Figures 6A and6B.

Figure 6A corresponds to the beginning of the experiment; in this picture, the shutter is fully closed and no concentrated power is arriving at the receiver. The second picture (Figure 6B) was taken after opening the shutter. The DNI value during the acquisition of this picture was approximately 890 W m -2 , which corresponds to a concentrated power of 760 W according to the laboratory empirical correlation established for the solar concentrator used in these experiments. Next, an increase in the power input from 760 W to 850 W provoked the start of the boiling crisis shown in Figure 6C. This increase in power input was due to the actual increase of DNI during the day of the experiment.

It is observed that the film started from the top of the immersed part of the cone and then extended downward (Figure 6C). When a stable film-boiling regime is established, it can be seen that the vapor film thickness decreases from top to bottom, having its smallest thickness at the lowest part of the immersed receiver. The time needed to establish a relatively stable vapor film depends on the meteorological conditions (DNI) and the material and shape of the receiver. Nevertheless, according to the experiment with the small labscale receiver shown in Figure 6, for an operation under a maximum solar power of 850 W, the shift from the nucleate phase to the film-boiling phase (the shift from Figure 6B to Figure 6D) generally takes 10-20 s. During this short period, the receiver temperature rises sharply from a temperature close to the bubble temperature of water to temperatures above 800 °C.

Under the experimental conditions, after a relatively stable film-boiling regime is established, the total solar energy input can be divided by five, from 850 W to 180 W, without exiting the film-boiling regime. As expected, to reach film-boiling conditions, it is first necessary to impose a strong solar heat flux to reach a high surface temperature on the receiver and to pass through the boiling crisis.

As discussed earlier, for the same solar power input, two different boiling conditions can occur: (i) film boiling and (ii) nucleate boiling. Under film-boiling conditions, as shown in Figure 6D, the vapor encompasses the cone so that direct contact between the liquid and the metallic receiver is prevented or reduced, as opposed to nucleate boiling (Figure 6B), where a large part of the receiver is in contact with the liquid phase. Vapor formation in the latter case takes place at discrete nucleation sites, and the bubbles formed at these sites regularly leave the receiver surface.

The measurements were taken after the film boiling was established. The experimental setup was used to study the solar boiler behavior under the film-boiling regime at different solar heating powers. The measurements took place at steady-state conditions over several stable DNI conditions. Table 6 summarizes the main operating conditions and associated measurements for these five representative steady-state conditions. The values presented correspond to average values measured over periods of operation at stable regime. The DNI values were recorded during the experiments, and the amount of concentrated solar power at the receiver ( ) was obtained on-site. As previously described in Section 2.1, the vapor water flow rates ( ) were calculated from the measured mass flow rates of the liquid streams entering and leaving the solar boiler.

Table 6: Characteristics and measurements for the solar boiling experiments at the Odeillo facility in France. (1) Accuracy: ±2 × 10 -5 kg s -1 ;

(2) calculated value, (3) accuracy: ±1 °C (4) Low boiling temperature due to Odeillo facility altitude (2,500 m above sea level).

The experimental results listed in Table 6 enable the different heat flow terms to be calculated to verify the energy balance closure in Section 3.2.1 and to compare the model predictions with the experimental results in Section 3.2.2.

The results obtained from the lab-scale experiments in Table 6, as well as the observation of established film boiling (Figure 6D), demonstrate the feasibility of a solar boiler using concentrated solar power and operating under film-boiling conditions.

Comparison between the model and experiments and adjustment in the film-boiling regime

Energy balance closure

To validate the coherence of the experimental results presented in Table 6, heat balance closure was verified.

Equations ( 3) -( 5) were combined into Equation ( 19) to formulate the heat balance for the entire solar boiler:

, , , & #$ % & $ , % ' & , #$ , % & $ , % ' 0 (19)
Considering the DNI averages received on the five experimental points presented in Table 6, the concentrated solar energy ( ) is approximated thanks to the empirical laboratory correlation established for the solar concentrator used in these experiments. Moreover, knowing the concentrated solar energy and the material properties of the receiver, the reflected power ( ) is calculated using Equation ( 6). Next, the temperatures measured in Table 6 are used as inputs for the calculation of the heat transfer terms (Equation 7-12), therefore the optical losses are all determined as shown in Table 7. Table 7 summarizes all the heat flows approximated from the experimental results and gives the absolute and relative errors (in percent) of the energy balance closure with respect to the input solar energy ( ). The global energy balance closure presents an average error of 9.36% over the five experiments. Given the complexity of the system, using a concentrated solar system coupled with boiling phenomena, and considering the large number of sources of errors (such as the estimation of the concentrated power, the calculation of losses, and the errors related to the measurement of temperatures and mass flow rates), a global average error of less than 10 % in the energy balance seems to be a promising result. These results also demonstrate the consistency of the experimental data.

Based on the propagated error in the vapor mass flow rate measurement (±2 × 10 -5 kg s -1 ), the resulting error for the estimate of the thermal power transmitted to the vapor is 50 W. Admitting a high precision DNI acquisition system, the use of the empirical correlation of the laboratory induces the same order of magnitude for the uncertainty of the reported DNI values. A temperature error will have less impact on the heat flow estimates. Indeed an error of 10°C on the receiver temperature will generate only a discrepancy of 4.2 W in the estimation of the various losses, by radiation and convection. For fluid temperature, an error of 1°C generates an error of less than 1 W on the calculation of heat transfer to vapor and liquid.

Experiment versus model

To validate the capacity of the model to predict the boiler performance, a comparison was made between the model and the main experimental results, i.e., the mass flow rate of the steam ( ) and the upper and lower surface temperatures of the receiver (Table 8). These variables were chosen because the mass flow rate of the steam is the most important information available to assess the boiling power of the module and the high temperatures of the upper and lower sides of the receiver impact the internal heat transfer flows and heat losses in the boiler.

The Model results were calculated using the equations and methods presented in Section 2. All inputs and parameters used in the calculation, i.e. the concentrated solar power, the mass flow rates, the temperatures, the dimensions and materials used in the solar boiler and the external conditions, were taken identical to the conditions of each experiment. The comparative results between the model and the experiments are listed in Table 8. (1) Accuracy: ±2 × 10 -5 kg s -1 ; (2) accuracy: ±1 °C.

The average error using the model is 4.77% with respect to the upper and lower surface temperatures of the receiver and 6.49% with respect to the mass flow rate of the steam. These results show that the model fits the experimental results well for the main operating parameters of the boiler and therefore validates the model.

Several assumptions were made to build the model presented in this work. Among them, the main assumptions are the uniformity of temperatures on the upper and lower surface of the receiver and the use of inclined plate correlation for the modeling of heat transfer by film boiling. Given the difficulty of choosing suitable correlations for film boiling and on an atypical inverted conical geometry, these simplifying assumptions were made to first check the consistency of a 0d thermal representation of the system before involving surface heterogeneity, spatial temperature distribution and dynamics. Nevertheless, the predictions made by the model are already close to the observed experimental values. Despite the prospect of further improvements to the experimental setup and the thermal model, as shown by the results and the level of uncertainty reported in Table 8 and later in Table 9, this model already allows to evaluate the thermal performance of the solar boiler according to its geometry, the dimensions and the choice of materials used as well as the operating conditions with low requirements in terms of numerical simulation and resolution.

Moreover, it should be reminded that these predictions have been also validated with experimental data.

About the film boiling heat transfer calculation, the results given in Table 8 were obtained using the following value of the C coefficient in Equation ( 16), C = 0.663; this is a general value from the literature for a vertical surface in a stagnant liquid [START_REF] Ghiaasiaan | Two-Phase Flow, Boiling, and Condensation: In Conventional and Miniature Systems[END_REF]. Even though the model is valid using this general value of the C coefficient, it is reasonable to adjust this value because the film boiling in the experiments conducted in this study occurred on a specific geometry of solar receiver. The reason for such an adjustment is twofold. First, as explained in Section 2.4.5 and shown in Table 4, different values have been chosen by different authors for this coefficient in the literature depending on the geometric configuration and the type of flow in the vapor film.

The experiments in this study were performed using a new conical receiver, oriented upside down, with a downward vertical heat transfer, and moreover, the flow was impacted by the presence of potential waves near the liquid-vapor interface. These two points indicate the need to adjust the C coefficient for the particular receiver studied here. The appropriate value of the C coefficient (C = 1.069) was obtained by fitting the data using least-squares minimization between the predictions of model and the experiments for the parameters summarized in Table 9. (1) Accuracy: ±2 × 10 -5 kg s -1 ;

(2) accuracy: ±1 °C;

(3) calculated with adjusted C = 1.069.

Comparing Table 8 and9, an improvement in the model predictions appears when adjusting the C coefficient.

Indeed, with respect to the upper and lower surface temperatures of the receiver, the average error decreased from 4.77% to 2.08% from the initial model to the adjusted model. For the mass flow rate of the steam, the average error decreased from 6.49% to 3.79% when using the model with the adjusted value of the C coefficient. Further calculations and analyses in the following sections were performed using the adjusted model presented in this section.

Thermal efficiency under nucleate and film boiling conditions

To better illustrate the efficiency of the solar boiler, a reference case was defined with the design configuration and operating conditions indicated in Table 10. The concentrated solar power on the reference case was set to 1,000 W to remain in the order of magnitude of the solar power studied through the experiments (Tables 6 and7). The receiver size was calculated to allow a considerable increase in solar flux over the receiver to achieve boiling at film boiling regime under high surface temperatures. Finally, the inlet mass flow rate was adjusted to achieve a vaporization rate of 0.8. The performance indicator used here is the thermal efficiency, which is defined by the following equation:

i 1 & ∑ , (20)
Where, i stands for rad, refl, conv and wall which are respectively the radiative, reflective and convective losses on the top of the receiver, as well as the thermal losses through the tank wall. All these losses are calculated using the corresponding equations presented in section 2.

The model predictions for the different heat flows are shown in Table 11 and Figure 7, where the main losses and the corresponding thermal efficiencies can be identified. nucleate-boiling conditions.

The greatest loss was due to the radiosity of the receiver, that is, the combination of radiation and reflection from the upper side of the conical part of the receiver. For the reference case in the film-boiling regime, these losses were approximately 15%. There are methods to reduce these losses, e.g., changing the geometry of the receiver or changing the absorptance and emissivity of the receiver.

The losses due to the convection and radiation of the ring part of the receiver had a smaller impact on the performance, 3.99% and 5.23%, respectively, with respect to the concentrated solar power ( ). The tank losses were low compared to the other losses. A small part of the heat was also removed by the liquid leaving the boiler, i.e., 2.32% of the solar energy input. This proportion depends on the desired vaporization rate of the system. Here, this ratio was set close to 0.8. Controlling the inlet mass flow rate will permit the vaporization rate to be increased or decreased.

According to Equation ( 20), the thermal efficiency of the reference case boiler is 82%. In this configuration, the predicted upper and lower surface temperatures of the receiver are equal to 817 K and 814 K, respectively.

These temperatures are high enough to maintain the film-boiling regime according to the literature [START_REF] Ghiaasiaan | Two-Phase Flow, Boiling, and Condensation: In Conventional and Miniature Systems[END_REF] but are sufficiently low to ensure operation away from the melting point of the receiver material.

As a reminder, two main features distinguish the solar boiler proposed from conventional boilers used in industry. The First is the use of concentrated solar energy as the primary heat source for boiling directly under the solar receiver, and the second is to operate in the film boiling conditions. When comparing the reference case with the thermal performances of a conventional boiler running on fossil fuels, the typical boiler efficiencies range from 88% for low-performance coal boilers to 95% for high-performance gas boilers (Esa Kari Vakkilainen, 2016). Accordingly, two points should be emphasized. Firstly, the reference case here is not yet optimized, and secondly, the solar boiler does not require any fuel combustion and avoids CO2 emissions so that even at lower performance, the solar boiler remains a promising renewable choice of heat supply for industrial boiling.

Also, to compare these results in a more conventional situation, the calculation was made for the same reference case under nucleate-boiling conditions, replacing the calculation of the film-boiling heat transfer coefficient in the model (Equation ( 13)) with the correlation used in nucleate boiling [START_REF] Delhaye | Transferts de chaleur : ébullition ou condensation des corps purs[END_REF].

The results under nucleate-boiling conditions are listed in Table 11 and shown in Figure 7B. As expected, lower losses are recorded under these conditions. This is due to the lower temperature of the receiver. The temperature of the upper side of the receiver under nucleate boiling is 376.13 K, i.e., close to the boiling point.

For this reason, the convective and radiative losses at the ring level are negligible. In Table 11, a slight decrease in the radiosity losses can be observed when changing from film boiling (151 W) to nucleate boiling (131 W).

When comparing film boiling with nucleate boiling, it is obvious that, for the same operating conditions, the thermal efficiency is higher for nucleate boiling as a result of its lower heat losses. The lower temperature in nucleate boiling can be explained by the high efficiency of this type of heat transfer, where a small temperature difference allows the transfer of a large amount of energy.

However, for corrosive or scaling fluids (e.g., in the case of desalination), all the advantages and the drawbacks of the two boiling regimes need to be examined. Increasing the material lifetime and avoiding gradual performance losses due to fouling can be reasons to choose film boiling over nucleate boiling.

Conclusions

This study introduced a new boiler module operating with concentrated solar energy. Such a module can be designed at different scales and for different purposes, including for the desalination of seawater. The most important features distinguishing this new boiler and its modeling from conventional boilers are first, the use of vertical heating downward from a receiver placed on top of the boiler where the conical receiver is oriented upside down and it is directly in contact with the boiling liquid, and second, the direct use of solar energy with optical concentrating facilities for boiling without intermediate heat transfer fluids for which the unconventional film-boiling regime at elevated temperatures in the receiver is particularly investigated to avoid direct contact between the boiling fluid and the solar receiver. Nowadays when the solar energy is used in CSP to perform boiling, it is not a direct use at the level of solar receiver and a secondary heating circuit and an intermediary fluid are necessary, where in this case, the boiling takes place in a secondary heat exchanger.

Below, the main outcomes of the present work are given as follows:

The experimental setup permitted the feasibility of this new type of direct concentrated solar boiler at the level of solar receiver to be demonstrated on a lab scale. The proof of concept nevertheless requires further investigations. Indeed, the potential applicability of such a boiler must be studied in-depth and experimentally

for corrosive fluids such as in the case of seawater desalination. It should be noted that the experiments using corrosive and scaling fluids have not yet been carried out to verify the protective behavior of the vapor film during the boiling under film boiling. This aspect is one of the most important perspectives of this work.

Another important outcome of this work is the validation of a 0d thermal model which allowed a fast and precise prediction of the thermal performances of the solar boiler studied. Indeed, the recorded experimental data enabled a validation of the theoretical model developed to describe the solar boiler. In this first study, the model rigorously describes the heat transfer phenomena and the efficiency of the solar boiler in different operational regimes without considering the process dynamics. The comparison between the experiments and the predictions of the model showed a very satisfying consistency between the theoretical representation of thermal phenomena and the physical reality of solar boiling under film boiling. However, in order to better understand the thermal phenomena involved during the film boiling such as the presence of waves and turbulence near the boiling zone, the spatial distribution of temperatures on the receiver surface, the nonuniformity of heat losses and the influence of process dynamics, the experimental setup and the corresponding measurements and imaging system could be extended, for example by increasing the number of temperature measurements on the surface of the receiver and by re-evaluating more precisely and dynamically the distribution of incoming solar power to the receiver.

The experimental data were also used to determine the geometric coefficient C used in the calculation of the heat transmitted by film boiling; this coefficient was adapted to the used conical geometry and proved more accurate for the presented cases. However, more experiments with different conical shapes and using different receiver materials may provide a better view of the sensitivity of the proposed C coefficient value to these design parameters.

Figure 1 :

 1 Figure 1: Layout of the solar medium-power vertical concentrator installation at the Odeillo solar furnace in

Figure 2 :

 2 Figure 2: Pictures of the solar boiling module, which is composed of tanks, a support disk, and a copper solar

  Figure3A. Depending on the geometry, the surface temperature, and the properties of the receiver material,

Figure 4. 10 Figure 3 :

 103 Figure 4.

  Description of the design parameters of the solar boiler.Modeling the solar boiler at liquid-vapor equilibrium

  due to radiation, convection, and reflection, respectively, on the upper side of the receiver in W.

,

,

  ) can be divided into two parts depending on the part of the receiver considered, as shown in Equation (7). The total radiative losses ( , ) include the radiative losses from the conical part of the receiver ( , ,

Figure 6 :

 6 Figure 6: Boiling conditions on a conical receiver heated from above: (A) no boiling, (B) nucleate boiling, (C) beginning of film boiling, and (D) fully established film boiling.

Figure 7 :

 7 Figure 7: Distribution of the heat flows in the solar boiler module for the reference case under (A) film-boiling and (B)

  constant (W m -2 K -4 ) λThermal conductivity (W m -2 K -1 )

Table 1 :

 1 Characteristics of the copper conical receiver used for the lab-scale experiments.

	Parameter	Value	Unit
	αrec,cone	30	°
	Drec,in	0.021	m
	Drec,out	0.040	m
	trec	0.005	m
	εrec (=αrec)	0.77

Table 5 :

 5 Synthesis of the different variables used in the numerical resolution

	Inputs	Parameters	Model variables	Heat calculations
		e ,	, /	,
	,

Table 7 :

 7 Heat flow estimation from the experimental data (in absolute values).

	Measure						,		,	Energy balance
	point	(W)			(W)	(W)	(W)	(W)	(W)	closure (absolute
										error)
	N°1	611	79.43	86.61	21.97	382.21	3.76	255.28	122.03	6.9% (41.92 W)
	N°2	671	87.20	93.40	23.31	408.46	3.94	277.18	123.70	9.3% (62.27 W)
	N°3	651	84.55	81.91	22.32	377.14	4.06	277.25	123.99	8.7% (56.92 W)
	N°4	703	91.29	88.70	23.13	388.86	4.05	286.38	123.40	12.2% (85.05 W)
	N°5	704	91.48	98.30	24.36	412.08	4.11	292.91	124.44	9.7% (68.40 W)

, (W) , (W) ,

Table 8 :

 8 Comparison between the experimental results and the model predictions for the mass flow rate and

		the receiver temperatures
		Vapor mass flow	Receiver upper surface temperature	Receiver lower surface temperature
		( ) (kg s -1 )	(Trec,sup) (K)	(Trec,inf) (K)
	points	Experimental (1)	Model (% error)	Experimental (2)	Model (% error)	Experimental (2)	Model (% error)
	N°1	1.02 × 10 -4	0.90 × 10 -4 (12.4%)	1008.4 1015.6 (0.7%)	983.9	1012.1 (2.9%)
	N°2	1.11 × 10 -4	1.05 × 10 -4 (5.3%)	1027.5 1056.6 (2.8%)	1011.1	1052.8 (4.1%)
	N°3	1.11 × 10 -4	1.00 × 10 -4 (10.1%)	994.5 1043.0 (4.9%)	969.4	1039.2 (7.2%)
	N°4	1.15 × 10 -4	1.13 × 10 -4 (1.4%)	1014.4 1077.1 (6.2%)	983.8	1073.2 (9.1%)
	N°5	1.17 × 10 -4	1.13 × 10 -4 (3.2%)	1040.7 1078.1 (3.6%)	1010.9	1074.1 (6.2%)

Table 9 :

 9 Comparison between the experimental results and the model predictions for the mass flow rate and the receiver temperatures with the C value adjustment.

		Vapor mass flow		Receiver upper surface temperature	Receiver lower surface temperature
			( ) (kg s -1 )		(Trec,sup) (K)			(Trec,inf) (K)
		Experimental	Adjusted	(%	Experimental	Adjusted	(% error)	Experimental	Adjusted	(% error)
	points	(1)	Model (3)	error)	(2)	Model (3)		(2)	Model (3)
	N°1	1.02 × 10 -4	0.95 × 10 -4 (6.6%)	1008.4	980.2 (2.8%)	983.9	976.6 (0.8%)
	N°2	1.11 × 10 -4	1.12 × 10 -4 (0.6%)	1027.5	1021.4 (0.6%)	1011.1	1017.4 (0.6%)
	N°3	1.11 × 10 -4	1.06 × 10 -4 (4.4%)	994.5	1007.7 (1.3%)	969.4	1004.0 (3.5%)
	N°4	1.15 × 10 -4	1.20 × 10 -4 (4.7%)	1014.4	1042.0 (2.7%)	983.8	1038.0 (5.5%)
	N°5	1.17 × 10 -4	1.20 × 10 -4 (2.7%)	1040.7	1042.9 (0.2%)	1010.9	1039.0 (2.8%)

Table 10 :

 10 Boiler configuration and operating conditions for the reference case.

	Parameters	Values	Units
	Boiler configuration
	αrec,cone	20	°
	Drec,in	0.04	m
	Drec,out	0.06	m
	trec	0.006	m
	εrec	0.77
	Operating conditions
		1000	W
	,	3.5 × 10 -4	kg s -1
	Tl,in	20.0	°C
	Pl,in	1.0	atm

Table 11 :

 11 Prediction of the heat flows in the solar boiler for the reference case in the film-boiling and nucleate-

	boiling regimes.
	Boiling regime			Heat (W)
		Solar energy at	Radiosity	Ring radiation	Convection	Tank	Thermal
		receiver	losses	losses	losses	losses	efficiency
	Film Boiling	1000	110	30	36	6	82%
	Nucleate Boiling	1000	94	1	5	6	89%

2.4.6

Heat losses from the boiler tank Heat losses in the tank ( , ) are divided into two components: losses through the side walls ( , ,

) and losses through the bottom ( , ,

). These two types of losses are combined in Equation ( 18). Here, the temperature in the tank is considered uniform and equal to the boiling point.

Here, i represents lat for the lateral losses and bot for the bottom losses, htank,in,i is the convection coefficient between the water in the tank and the inside wall, and htank,out,i is the convection coefficient between the outside surface of the wall and the environment.

The calculation of the heat transfer coefficients $ , , and $ , , was performed using the correlations given by [START_REF] Rohsenow | Handbook of Heat transfer[END_REF]. Correlations for (1) vertical plates under natural convection for the interior and exterior lateral side walls, (2) horizontal plates downward facing for the exterior of the lower surface of the tank, and (3) horizontal plates upward facing for the inside of the bottom surface of the tank, were chosen. The thermal conductivity was calculated for Plexiglas at the average temperature inside the wall.

Numerical strategy

Given the model inputs, including the operating pressure (P), the feedwater temperature (Tl,in) and flow rate ( , ), and the concentrated power ( ) (Inputs in Table 5), as well as the dimensions for the design of the solar boiler module (Parameters in Table 5), the overall system to solve consists of five equations ( 1) -( 5), i.e., the mass and heat balances, the phase equilibrium condition for water, and the relationships between the different heat transfer terms. The heat transfer terms are explicitly determined by their formulations, as stipulated in Section 2.4. Consequently, there are five independent variables in the main governing equation system of the model; these are the temperatures on the upper and lower surfaces of the receiver (Trec,sup and Trec,inf, respectively), the boiling temperature (Tb), the water vapor flow rate ( ), and the outlet liquid water flow rate ( , ). Table 5 summarizes the different elements provided in the model.

Finally, the model was modified to compare the performance of the solar boiler with that of a boiler operating in the widely used nucleate regime. It is generally known that the use of film boiling involves less thermal efficiency; however, as discussed in this study, the effect of the spacing between the receiver and the boiling fluid due to the formation of a vapor film while operating in the film-boiling regime can potentially protect the receiver from scaling or chemical reactions in the case of desalination or the heating of corrosive fluids.

Therefore, the choice of boiling regime and the corresponding temperature of the receiver need to account not only for the thermal efficiency of the boiler but also for the lifetime and persistency of the thermal properties of the receiver over time.

The following steps need to be taken to demonstrate the potential of this new solar boiler. First, the main design parameters that impact the performance of the boiler need to be established and the potential performance that can be achieved for an optimized solar boiler needs to be determined. As mentioned in the introduction, the model is intended for general applications, particularly for seawater desalination. Therefore, the impact of salinity on the results needs to be investigated. Finally, the integration of this type of boiler in a real industrial process needs to be studied to estimate the potential of this technology, in particular in terms of reducing CO2 emissions. Note that the performance of solar concentration installations before the solar receiver was not included in the discussions as this is a large research topic in itself and is considered outside the scope of this work. Nevertheless, "beam-down" technology is a very dynamic field of research and improvements in their performance are eagerly awaited in the years to come. The energy balance of the infinitesimal elements for a cavity with an isothermal diffusive gray surface follows the equation proposed by [START_REF] Modest | Radiative Heat Transfer[END_REF]:

where k is the surface power absorbed by an infinitesimal element of the surface r, + is the emissivity of the surface r, , 5m is the view factor between the surface r and the surface s, g p is the external contribution to the irradiation on the surface r, and q is the black body emissive power of the surface r.

In the present situation, the cavity is composed of two surfaces. The first, i = 1, is the conical lateral surface of the receiver at a temperature of Trec,sup with an emissivity of + . The second (i = 2) surface is a virtual surface that permits the cavity to be enclosed and corresponds to the conical base surface. Because each beam arriving from surface 1 to surface 2 leaves the system and does not return, this surface can be treated as a perfect absorber (+ t 1) that does not emit power (T2 = 0 K).

Given these considerations and applying Equation (A1) for i = 1, the following expression is obtained:

The calculation of the view factor is made using the formula proposed by [START_REF] Bien | Configuration factors for thermal radiation from isothermal inner walls of cones and cylinders[END_REF]:

where

To simplify the notation, the view factor F1-2 is denoted F.

The total absorbed power by the conical part of the receiver ( ) can be expressed using the power flux absorbed:

. k ; . (A1.5)

Finally, by taking the heat balance at the conical cavity level, the following equation can be written

The combination of Equations (A1.2), (A1.5), and (A1.6) gives, ,

,

The two loss terms can be expressed as

Calculation of natural convection heat transfer coefficients

In Section 2.4.3, the following heat transfer coefficients are used: hring-amb and hcone-amb. The calculation of these coefficients is done by using the approach proposed by Rohsenow et al. [START_REF] Rohsenow | Handbook of Heat transfer[END_REF] for an upwardfacing plate and a plate at an arbitrary angle of tilt, respectively.

Free convection on the upper surface of a hot plate

In the ring part, Equations (A2.1 -A2.4) are used for estimating the Nusselt number.

Free convection on an inclined plate

For the conical part, the heat transfer is estimated by applying the calculation method proposed for a plate of arbitrary angle. The tilt angle ξ is equal to 90° for vertical plates, 0° for a horizontal surface facing upwards, and 180° for a horizontal surface facing downwards. In the current geometry, it can be related to the cone halfangle (αrec,cone) by the following equation :

Here, the determination of the heat transfer coefficient requires performing three different parallel calculation methods and to take the largest value obtained. The first method consists in following the calculation of the upward-facing plate as presented previously in Equations (A2.1 -A2.6) through which the gravity (g) is replaced by the maximum between 0 and g.cos(ξ). The second method is the same as the method proposed for a vertical plate where g is replaced by g.sin(ξ). Equations used in this method to calculate the Nusselt are the same as for upward facing plate (Equation A2.1) where m is taken equal to 6 and the laminar and turbulent Nusselt are calculated using Equations (A2.9 -A2.10):