Hélène Tissoux

P Voinchet

C Lerouge

teaching and research institutions in France or abroad, or from public or private research centers.

Electron Spin Resonance (ESR) applied to optically bleached quartz grains extracted from sediment provides an age estimate for the last exposure of sediment to sunlight. This method has been increasingly used in archaeological, geological and geomorphological contexts for the last 30 years. However, its successful application is highly dependent on the geological context, the type and quality of the sampled material, but also its preparation.

In most of the ESR dating studies, Equivalent Dose calculation uses the multiple aliquot additive dose (MAAD) approach. Accuracy of the ED is the slightly correlated with the dose response of different aliquots of quartz grains to irradiation. In most of the measured quartz samples, a single saturating exponential dose response curve or a two components combining an exponential with a linear term are classically used in ESR dating to fit the experimental ESR data points derived from the aluminum (Al) or the titanium (Ti-Li, Ti-H) centers in quartz. However, large deviation and large associated errors may occasionally be observed. Heterogeneous bleaching (incomplete resetting) of the ESR dating signal can sometimes explain such deviations. Scattering of the dose response to irradiation may also be due to heterogeneous dose response of the different aliquots related to various nature and/or origin of the quartz grains constituting the dating sediment.

Quartz grains extracted from several sediments were measured by ESR and observed using cathodoluminescence (CL) technique. The CL colours of quartz are weak, compared with emission of other minerals such as carbonates or phosphates, but are highly variable and can be related to genetic conditions of quartz formation. Hence, luminescence microscopy can be used to reveal internal structures, growth zoning and lattice defects in quartz crystals not discernible by means of other analytical techniques, but also to reveal heterogeneity of the quartz grains within the sample, or even the presence of other minerals, not removed during the chemical process of quartz purification.

The cathodoluminescence gave evidence of several types of situation within our quartz samples. Some of them are made of monocrystalline quartz grains, emitting a uniform CL within the grain but with a light heterogeneity of the wavelength emission between the grains. Some are made of polycrystalline quartz grains, displaying a strong heterogeneity of the luminescence within and between the quartz grains. Finally we also observed some CL attributed to plagioclase and K-feldspar, which may contaminate, even strongly, the quartz samples.

The presentation will report this variability and discuss about its effect on the dose response curve when using the ESR dating of quartz grains.

1

  French geological survey, BRGM, 45060 Orléans cedex, France 2 Département de Préhistoire, Muséum National d'Histoire Naturelle, UMR 7194, Paris, France.


Tianyuan Wang

Elise El Ahmar

Christophe Coquelet
email: christophe.coquelet@mines-paristech.fr

Alkane solubilities in aqueous alkanolamine solutions with CPA EoS

Keywords: phase equilibria, data treatment, modelling, parameter determination

Absorption with aqueous alkanolamine solution is often used for acid gas removal processes. Thermodynamic model is very important to predict phase behaviour for designing and optimizing acid gas absorption units. To estimate alkanes losses in these processes, the knowledge of alkanes solubility in aqueous alkanolamine solutions is essential. In the present work, Cubic-Plus-Association Equation of State (CPA EoS) was applied to represent alkanes solubility in aqueous alkanolamine. The Average Relative Deviation on alkanes solubility in aqueous alkanolamine solution is less than 15%. Due to our model, the temperature of alkanes minimum solubility in pure water and alkanolamine solutions are correctly predicted. Water content is also well predicted at Vapour Liquid Equilibrium condition.

Introduction

Among fossil fuels, natural gas is the cleanest, in terms of CO 2 emission, burn efficiency and amount of air pollutant [START_REF]How much carbon dioxide is produced when different fuels are burned? -FAQ -U.S. Energy Information Administration (EIA)[END_REF]. Methane is the prevailing element of natural gas. However, there are a variety of impurities in the natural gas reservoir. Usually, natural gas contains considerable amounts of acid gas (CO 2 , H 2 S). With the presence of water, acid gas can lead to corrosion in equipments and pipelines. The presence of acid gas can also reduce the heating value of gas and make more air pollution. Acid gas is needed to be removed until acceptable standard is achieved: not greater than 4 ppm for H 2 S and less than 280 ppm for CO 2 [START_REF] Rojey | Le Gaz Naturel -Production[END_REF]. For this purpose, different acid gas removal processes have been developed in recent decades, such as absorption [START_REF] Kohl | Chapter 2 -Alkanolamines for Hydrogen Sulfide and Carbon Dioxide Removal[END_REF], adsorption [START_REF] Davison | Putting carbon back into the ground[END_REF], low temperature separation processes (CFZ [START_REF] Gale | Greenhouse Gas Control Technologies 9The CFZ TM process: A cryogenic method for handling high-CO2 and H2S gas reserves and facilitating geosequestration of CO2 and acid gases[END_REF] and Sprex [START_REF] Lallemand | Highly sour gas processing: H2s bulk removal with the sprex process[END_REF] processes) and membrane separation [START_REF] Sridhar | Separation of Carbon Dioxide from Natural Gas Mixtures through Polymeric Membranes-A Review[END_REF]. Chemical absorption with aqueous alkanolamines solutions is the most established and used technology for efficient removal of acid gas. As alkanolamines, we can cite monoethanolamine (MEA), diethanolamine (DEA), methyldiethanolamine (MDEA). After absorption step, solvent is regenerated into a second column by heating. During the operation, a part of alkanes can be absorbed as well. It is essential to estimate the losses of alkanes in the process design and simulation. For this reason, it is necessary to develop thermodynamic models in order to predict the solubilities of different alkanes in alkanolamine solutions.

Due to the presence of associating molecules like alkanolamine and water, it is better to consider molecular models or an Equation of State (EoS) containing an association term. In the beginning of 80s, Wertheim [START_REF] Wertheim | Fluids with highly directional attractive forces. I. Statistical thermodynamics[END_REF] has developed a model for association molecules. This model is used in the Statistical Associating Fluid Theory EoS [START_REF] Chapman | SAFT: Equation-of-state solution model for associating fluids[END_REF]. In 1996, Kontogeorgis et al. [START_REF] Kontogeorgis | An Equation of State for Associating Fluids[END_REF] has proposed to combine Wertheim's model and Cubic EoS. This model is called Cubic Plus Association EoS (CPA EoS). This EoS can easily consider the presence of hydrogen bonding. In this study, we have considered the Peng-Robinson (PR)-CPA EoS. It takes advantage of PR EoS for non-associating components as well. In 2015, Hajiw et al. [START_REF] Hajiw | Hydrocarbonswater phase equilibria using the CPA equation of state with a group contribution method[END_REF] have developed a Group Contribution (GC) version of this model. In previous studies, Avlund et al. [START_REF] Avlund | Modeling Systems Containing Alkanolamines with the CPA Equation of State[END_REF][START_REF] Avlund | Application of association models to mixtures containing alkanolamines, 20 Years SAFT Equ[END_REF] have applied CPA EoS to describe the phase equilibria of alkanolamine-water binary systems, and claimed that the results are satisfactory overall with an ARD less than 30%. Folas et al. [START_REF] Folas | Application of the Cubic-Plus-Association (CPA) Equation of State to Complex Mixtures with Aromatic Hydrocarbons[END_REF] and Oliveira et al. [START_REF] Oliveira | Mutual solubilities of hydrocarbons and water with the CPA EoS[END_REF] have used CPA EoS to study water-alkane binary systems. They have considered no temperature dependant Binary Interaction Parameters (BIPs). They claimed that the model has good performance for both alkane solubilities in water and water content in alkane. Since they found that the Average Relative Deviation is smaller than 30 % and 25 % respectively.

The aim of this study is to describe alkanes (methane, ethane, propane, n-butane, i-butane, npropane and hexane) solubility in aqueous MDEA, MEA and DEA solutions by using PR-CPA EoS. In the present work, pure compounds parameters are fitted with CPA EoS from experimental vapour pressure and liquid density. Then, for the water-alkane and wateralkanolamine binary systems, their BIPs are adjusted from experimental binary data. The above experimental data are taken from ThermoData Engine [START_REF] Frenkel | NIST ThermoData Engine[END_REF]. However, due to the lack of experimental data concerning alkane-alkanolamine binary systems, it is impossible to directly adjust their BIPs. We have determined the missing parameters from alkane-wateralkanolamine ternary systems experimental data. Because a large number of Vapour-Liquid-Equilibrium (VLE), Liquid-Liquid-Equilibrium (LLE) and Vapour-Liquid-Liquid-Equilibrium (VLLE) experimental data of these ternary systems are available in the open literature (as examples these two GPA research reports: Mokraoui et al. [START_REF] Mokraoui | Mutual solubility of hydrocarbons and amines[END_REF] and Valtz et al. [START_REF] Valtz | Amine BTEX solubility[END_REF]).

To the best of our knowledge, only the paper of Carroll et al. [START_REF] Carroll | Phase equilibria in the system watermethyldiethanolamine-propane[END_REF] has allowed us to compare water content and MDEA vapour compositions. With optimized BIPs, solubilities of pure alkanes and mixtures of alkanes in aqueous alkanolamine solutions are predicted. These mixtures [START_REF] Mokraoui | Mutual solubility of hydrocarbons and amines[END_REF] are composed by ethane, propane, n-butane, i-butane, n-propane and hexane.

The PR-CPA EoS

The CPA EoS has an explicit part to account for hydrogen bonding, making it well suited to describe water-alkanolamines-alkanes systems, where water and alkanolamines molecules form hydrogen bonds between them and themselves (self and cross associations). PR-CPA EoS takes a cubic EoS as the basis and adds a correction for hydrogen bonding [START_REF] Kontogeorgis | An Equation of State for Associating Fluids[END_REF]. Here, the cubic part is the PR EoS [START_REF] Peng | A New Two-Constant Equation of State[END_REF] and the association part is from Wertheim [START_REF] Wertheim | Fluids with highly directional attractive forces. I. Statistical thermodynamics[END_REF] (Eq. 1).

 

                     i A A i r m m m m m i i X x g V RT b V b b V V T a b V RT P 1 ln 1 2 1 ) ( ) ( ) (   (1)
The model is detailed in the appendix A. For LLE and VLE, liquid and vapour compositions were calculated by two phase flash using algorithm of Michelsen et al. [START_REF] Michelsen | The isothermal flash problem. Part II. Phase-split calculation[END_REF]. For VLLE, all three phase compositions were calculated by multi-phases flash method (Michelsen [START_REF] Michelsen | Calculation of multiphase equilibrium[END_REF]) with stability analysis (Michelsen [23]).

Results

Pure components

Alkanes do not form any hydrogen bond between themselves, the parameters of association term (ε AiBj and β AiBj ) are set to zero. Cubic pure component parameters (a 0 , b, c 1 ) were calculated from critical pressure, critical temperature and acentric factor by using Eq A 9-12 from Appendix A. The values of pure components critical properties (Tc and Pc) and acentric factors (ω) are taken from Thermo Data Engine [START_REF] Frenkel | NIST ThermoData Engine[END_REF] (Table 1). The nomenclature of Huang and Radosz [START_REF] Huang | Equation of state for small, large, polydisperse, and associating molecules[END_REF] is adopted for water and all the alkanolamines. We consider as Kontogeorgis et al. [START_REF] Kontogeorgis | Multicomponent phase equilibrium calculations for water-methanol-alkane mixtures[END_REF] that 4C association scheme is the most suitable scheme for water. For alkanolamines, 4C association scheme is used to represent their association scheme. In previous work, Avlund et al. [START_REF] Avlund | Modeling Systems Containing Alkanolamines with the CPA Equation of State[END_REF][START_REF] Avlund | Application of association models to mixtures containing alkanolamines, 20 Years SAFT Equ[END_REF] mentioned that 4C association scheme is a natural choice for MEA. Also, they compared different association schemes for DEA and concluded that with 4C association scheme, pure component thermodynamic properties of DEA are correctly predicted. Moreover, as the molecule structure of MDEA is similar to the one of DEA, they have assumed that 4C association scheme is also suitable for MDEA. Consequently, the parameters of association term (ε AiBj and β AiBj ) of MDEA are assumed as the same as the ones of DEA.

Five parameters (a 0 , b, c

Where 𝑛 𝑣𝑎𝑝 and 𝑛 𝜌 are the number of vapour pressure, and liquid density. 𝑃 𝑠𝑎𝑡 is the vapour pressure. 𝜌 𝑙𝑖𝑞 is the liquid density. 𝑃 𝑠𝑎𝑡 𝑒𝑥𝑝 and 𝜌 𝑙𝑖𝑞 𝑒𝑥𝑝 were generated by using DIPPR correlations from Thermo Data Engine [START_REF] Frenkel | NIST ThermoData Engine[END_REF]. Pure component parameters are presented in Table 2.

Alkanolamine-water binary systems

Like Avlund et al. [START_REF] Avlund | Modeling Systems Containing Alkanolamines with the CPA Equation of State[END_REF][START_REF] Avlund | Application of association models to mixtures containing alkanolamines, 20 Years SAFT Equ[END_REF], we consider a constant value of BIP to represent alkanolaminewater binary systems. For each alkanolamine-water binary system, we have adjusted the corresponding BIP and a Flash type objective function (Eq. 3) has been chosen.

𝑓 𝑜𝑏𝑗 ,𝑎𝑚𝑖𝑛𝑒 -𝑤𝑎𝑡𝑒𝑟 = 100 × 𝑥 1 𝑐𝑎𝑙 -𝑥 1 𝑒𝑥𝑝 + 𝑦 1 𝑐𝑎𝑙 -𝑦 1 𝑒𝑥𝑝 𝑖 𝑛 𝑖=1 (3)
Where x 1 is the composition of alkanolamine in the liquid phase, y 1 is the composition of alkanolamine in the vapour phase.

Table 3 summarizes the BIPs and the references used in PR-CPA EoS for the VLE data treatment of alkanolamine-water systems. For MDEA-water and DEA-water binary systems, as usually in the vapour phase the concentration of alkanolamine is very low (order of magnitude of the amine mole fraction around 10 -6 ), only compositions of liquid phase are used to estimate BIPs. For MEA-water binary system, both compositions of liquid and vapour phases have been taken into account. From Figure 1, it can be seen that PR-CPA EoS with temperature independent BIPs is able to accurately represent phase behaviour of alkanolamine-water systems. [START_REF] Kim | Ebulliometric Determination of Vapor-Liquid Equilibria for Pure Water, Monoethanolamine, N-Methyldiethanolamine, 3-(Methylamino)-propylamine, and Their Binary and Ternary Solutions[END_REF], [START_REF] Xu | Vapor pressure measurements of aqueous N-methyldiethanolamine solutions[END_REF], [START_REF] Voutsas | Measurement and thermodynamic modeling of the phase equilibrium of aqueous N-methyldiethanolamine solutions[END_REF], [START_REF] Dell'era | Thermodynamics of aqueous solutions of methyldiethanolamine and diisopropanolamine[END_REF] DEA-water 311-473 6.93 -2.14 --0.114 [START_REF] Horstmann | Phase Equilibrium and Excess Enthalpy Data for the System Methanol + 2,2'-Diethanolamine + Water[END_REF], [START_REF] Cai | Binary Isobaric Vapor-Liquid Equilibria of Ethanolamines + Water[END_REF] MEA-water 283-373 9.30 1.73 1.74 0.23 -0.142 [START_REF] Kim | Ebulliometric Determination of Vapor-Liquid Equilibria for Pure Water, Monoethanolamine, N-Methyldiethanolamine, 3-(Methylamino)-propylamine, and Their Binary and Ternary Solutions[END_REF], [START_REF] Lenard | Vapor-Liquid Equilibria for Mixtures of 2-aminoethanol+ water[END_REF], [START_REF] Tochigi | Isothermal Vapor-Liquid Equilibria for Water + 2-Aminoethanol + Dimethyl Sulfoxide and Its Constituent Three Binary Systems[END_REF] a AAD = 1/NP× Σ| χi calc -χi exp |; b isotherm and isobar VLE data

Figure 1. Comparison of experimental data (symbols) and adjusted ones using PR-CPA EoS (solid lines) for MEA-water binary system; (×)=333 K:Lenard et al. [32], (□)=343 K:Kim et al. [26], (△)=353 K :[26], (○)= 363 K:Tochigi et al. [33], (◇)=373 K [26].3.3. Alkane-water binary systems

As explained by Hajiw [START_REF] Hajiw | Hydrate Mitigation in Sour and Acid Gases, phdthesis[END_REF], we also consider a second order polynomial equation with temperature (Eq .4) for the BIP to well describe the minimum solubility of alkane in aqueous solution. Consequently, we have considered a second order polynomial equation for the BIP (Eq. 4):

𝐵𝐼𝑃 = 𝑎 + 𝑏 • 𝑇 + 𝑐 • 𝑇 2 (4)
a, b, c were estimated by fitting alkane solubility data and using a Flash type objective function (Eq. 5).

𝑓 𝑜𝑏𝑗 ,ℎ𝑦𝑑𝑟𝑜𝑐𝑎𝑟𝑏𝑜𝑛 -𝑤𝑎𝑡𝑒𝑟 = 100 × 𝑥 1 𝑐𝑎𝑙 -𝑥 1 𝑒𝑥𝑝 𝑥 1 𝑒𝑥𝑝 𝑖 𝑛 𝑖=1
(5) Table 4 summarizes the adjusted BIPs and the references used for the VLE data treatment of alkane-water binary systems. The ARD on alkane solubility in water is less than 12 %. Satisfactory results are obtained with our model compared to the ARD (less than 30%) obtained with Oliveira et al. [START_REF] Oliveira | Mutual solubilities of hydrocarbons and water with the CPA EoS[END_REF]. We suspect that this higher deviation is due to their choice of using non temperature dependent BIPs. a LLE data were also considered; a, b and c are parameters in Eq. 4

Alkane-water-alkanolamine ternary systems

Previously, we have shown that using regressed BIPs for alkanolamine-water and alkanewater binary systems give satisfactory results. To the knowledge of the authors, there is no experimental VLE data concerning alkane-alkanolamine binary systems (only one set of data of methane-MDEA binary system is available in the open literature [START_REF] Jou | Solubility of Methane in Methyldiethanolamine[END_REF]). Consequently, BIPs of alkane-alkanolamine have been fitted using experimental alkane solubility in aqueous alkanolamine solutions. BIPs have the same expression as alkane-water binary systems (Eq. 4). A Flash type objective function is used and given by Equation 6.

𝑓 𝑜𝑏𝑗 = 100 × 𝑥 1 𝑐𝑎𝑙 -𝑥 1 𝑒𝑥𝑝 𝑥 1 𝑒𝑥𝑝 𝑖 𝑛 𝑖=1 (6)

Where

x 1 is alkane solubility in aqueous alkanolamine solutions Table 5 summarizes the obtained results. The model is generally able to describe different alkane solubilities in different aqueous alkanolamine solutions with ARD less than 10%. It is important to highlight, for each alkane-alkanolamine-water systems, thanks to a single temperature depend expression of BIPs, the model gives good accuracy (<10%) in all range of alkanolamine concentration. For examples, Figure 2a and 2b show propane solubility in 35 wt % aqueous MDEA solution. The ARD is 5.0% for all sets of experimental data at VLE and LLE conditions. Figure 3 represents propane solubility in aqueous DEA solution in function of concentration, PR-CPA EoS is in good agreement with experimental data. Nevertheless, our model has higher ARD (17%) on n-hexane solubility in aqueous MDEA solution. In fact, this system has been treated by using two sets of experimental data (Mokraoui et al. [START_REF] Mokraoui | Mutual solubility of hydrocarbons and amines[END_REF] and Alheseinat et al. [START_REF] Alhseinat | Solubility of Hexane in Aqueous Solutions of Methyldiethanolamine[END_REF]), the ARD are 8.8% and 32.8% respectively. Since Alheseinat et al. [START_REF] Alhseinat | Solubility of Hexane in Aqueous Solutions of Methyldiethanolamine[END_REF] mentioned that the uncertainty of their experimental data is more than 30%, this set of data is suspicious.

Discussion

Temperature of minimum solubility of methane

Hajiw [START_REF] Hajiw | Hydrate Mitigation in Sour and Acid Gases, phdthesis[END_REF] (2014) and Hajiw et al. [START_REF] Hajiw | Hydrocarbonswater phase equilibria using the CPA equation of state with a group contribution method[END_REF] (2015) have tested the capacity of PR-CPA EoS to predict the temperature of minimum of solubility for alkane-water binary systems. In this paper, the authors have also demonstrated that the prediction of this temperature is strongly dependant on the use of a temperature dependent BIP and recommended a second order polynomial expression. Using our model for methane-water binary system and at 5 MPa, we have predicted a temperature of 407 K. As shown in Figure 4, we have predicted the methane solubility as function of temperature (273-405 K) for 3 different aqueous MDEA solutions (25, 35 and 50 wt %) at 7.5 MPa (Since the solubility of methane do not have the same order of magnitude, their values are normalized). We observed that for each aqueous MDEA solution, it exists a temperature of minimum methane solubility. After, we also predict the temperature of minimum of solubility of methane in 3 MDEA aqueous solutions (25, 35 and 50 wt %) and pure water at 3 different pressures 5, 7.5 and 10 MPa, the result is summarized in Table 6. In fact, the temperature of methane minimum solubility is influenced by two factors: the concentration of alkanolamine and the pressure. As shown in Figure 5, the temperature of the minimum solubility decreases as the MDEA weight percent increases. It could be explained by the fact that the hydrogen bound between water is destroyed by introducing MDEA molecules. Less energy is needed to absorb the same quantity of methane in a higher MDEA concentration solution. Seen in Figure 6, the temperature of methane minimum solubility decreases while the pressure increases. In fact, higher pressure makes methane more soluble. Consequently, lower temperature is required.

The same approach can be applied to all alkanes.

Propane-MDEA-water alkane rich phase prediction

The ability of the model to predict water content was also evaluated. As seen in Figure 7, at VLE condition, predicted water contents are in good agreement with the only experimental data found in the literature and measured by Carroll et al. [START_REF] Carroll | Phase equilibria in the system watermethyldiethanolamine-propane[END_REF] in 1992. The ARDs are less than 20%. However, the model fails to predict water content in LLE conditions. After comparison with experimental data of Carroll et al. [START_REF] Carroll | Phase equilibria in the system watermethyldiethanolamine-propane[END_REF]. The ARDs are less than 124%. The biggest deviations are observed at low temperatures. It is not surprising that EoS fails to predict both VLE and LLE without mixing rules. We suggest as an alternative solution, to modify the expression of the co-volume b, by introducing a binary interaction parameter l ij . MDEA composition has been also measured by Carroll et al. [START_REF] Carroll | Phase equilibria in the system watermethyldiethanolamine-propane[END_REF] in propane rich phase, the composition of MDEA corresponds to few ppm at low temperature. MDEA composition is poorly predicted by our model. In fact, predicted MDEA composition is 10-100 times lower than experimental data at LLE conditions. The only agreement between model and experimental data is at temperature above 373 K and pressure under 5 MPa. Mokraoui et al. [START_REF] Mokraoui | Mutual solubility of hydrocarbons and amines[END_REF] did not manage to measure the same system as Carroll et al. [START_REF] Carroll | Phase equilibria in the system watermethyldiethanolamine-propane[END_REF]. Meanwhile, for pentane solubility in aqueous MDEA solutions, Mokraoui et al. [START_REF] Mokraoui | Mutual solubility of hydrocarbons and amines[END_REF] showed that the concentration of MDEA is less than 10 ppm in the liquid alkane rich phase. We conclude that more studies concerning the measurement of MDEA composition in different alkanes rich phase data are needed for the model validation.

We have also evaluated the model performance to predict the water content of this ternary system without BIPs for propane-water and propane-MDEA systems, the ARD on water content is 21% in VLE region. The water content value concerning propane-water-MDEA ternary system at given T and P should be lower but very close to the one of propane-water binary system in the same condition of T and P because the molar composition of water is 64.4 % in the aqueous solution. Therefore, we have predicted the water content without BIP for propane-water binary system, the ARD is 8% compared to the experimental data published by Kobayashi et al. [START_REF] Kobayashi | Vapor-Liquid Equilibria For Binary Hydrocarbon-Water Systems[END_REF] and Blanco et al. [START_REF] Blanco | Water dew points of binary nitrogen+water and propane+water mixtures. Measurement and correlation[END_REF]. We conclude that the vapour phase data published by Carroll et al. [START_REF] Carroll | Phase equilibria in the system watermethyldiethanolamine-propane[END_REF] are probably suspicious.

Alkane-alkanolamine-water ternary systems prediction at VLLE conditions

The experimental VLLE data of ethane-MDEA-water, propane-MDEA-water, and n-butane-MDEA-water ternary systems were not used for data adjustment, since for these systems we have already enough VLE or LLE data. Therefore, VLLE data is used for model validation; they are compared with those predicted with PR-CPA EoS, see Figures 8, 9 and 10. For ethane-MDEA-water ternary system, the overall ARD for ethane solubility is less than 6% in MDEA with three concentrations (0% wt%, 25 wt % and 50 wt %). The largest ARDs, 14 % are observed for ethane solubility in pure water. For propane-MDEA-water ternary system, the ARD for all concentrations of MDEA are less than 2 %. For n-butane-MDEA-water ternary system, PR-CPA EoS can accurately predict n-butane solubility in pure water, 25 wt % MDEA, 35 wt % MDEA and 50 wt % MDEA. The ARDs are less than 8 %. Moreover, the slope of each curve represents the enthalpy of absorption of solute i (Δh i abs ) given by Equation 7.

𝜕𝑙𝑛 𝑥 𝑖 𝜕1/𝑇 𝑃 = - 𝛥ℎ 𝑖 𝑎𝑏𝑠 𝑅 (7)
The absorption is endothermic, if the sign is negative, otherwise exothermic. For propane and n-butane, the absorption is always endothermic because once need energy to destroy the hydrogen bound between water-water, water-MDEA and MDEA-MDEA once alkanes molecules are introduced. However, for ethane, it seems that we have a particular behaviour since there is no tendency of the enthalpy of absorption (exothermic, athermic and endothermic for 0, 25 and 50 wt % MDEA respectively).

Multi-component alkanes solubilities prediction in aqueous amine solutions

As a final test, we have used our model to predict multi-component alkanes solubilities in alkanolamine solutions. Few experimental data for alkane mixtures solubility in aqueous alkanolamine solutions are available. Mokraoui et al. [START_REF] Mokraoui | Mutual solubility of hydrocarbons and amines[END_REF] measured the solubility of two mixtures (MIX1 and MIX2, Table 7) in 35 wt % DEA, 25 wt % and 50 wt % MDEA, within temperature range from 298 to 333 K and pressure range from 0.6 to 4 MPa. The reported data concern VLE and VLLE conditions. As for non-associative molecules, the PR-CPA EoS is reduced to PR EoS, the BIP between alkanes one taken from the work of Gao et al. [START_REF] Gao | A simple correlation to evaluate binary interaction parameters of the Peng-Robinson equation of state: binary light hydrocarbon systems[END_REF]. From Table 8 once can notice that the model has good predictability for the majority compounds, i.e. ethane and propane for MIX1; propane and n-butane for MIX 2. The ARDs are less than 30% for ethane and propane. Meanwhile, for C4 to C6, ARDs are more important.

Conclusion

In this work, the PR-CPA EoS was applied to describe the solubility of alkanes in aqueous MDEA, DEA and MEA solutions. Pure compounds parameters of associating compounds were determined by regression from experimental data. The model describes both pure component liquid density and vapour pressure within ARD lower than 3%. Then, PR-CPA EoS was applied to model phase equilibria of alkanolamine-water and alkane-water binary systems. It showed good agreement for these binaries studied in this work. Experimental data of alkanes in aqueous alkanolamine solutions (solubility) were used to correlate BIPs of corresponding alkane-alkanolamine binary systems. With optimized parameters, alkane solubilities in aqueous alkanolamine solutions were successfully described by the model; the ARDs are under 10%. Furthermore, the temperature of minimum solubility of methane was well described. Water content was also accurately predicted in VLE conditions (ARDs are less than 12%) but not in LLE conditions, PR-CPA EoS must be improved. Solubility of two alkanes mixtures in aqueous alkanolamine solutions were studied in this work. The model was able to predict main components solubility (The ARDs are less than 30%). This study showed that PR-CPA EoS is accurate for predicting alkane solubility in aqueous alkanolamine solutions. Our future objective is to improve our model by taking into account the impact of acid gases. The introduction of H 2 S and CO 2 leads to chemical reaction taking place with alkanolamine and water and different electrolyte species will be formed.

Nomenclature

List of abbreviations

Greek letters j i B A

 =association volume parameter between site A in molecule I and site B in molecule j ∆=association strength (L/mol) j i B A  =association energy parameter between site A in molecule I and site B in molecule j (bar L mol -1 ) η=reduced density ρ=density (mol/L)

APPRENDIX A: PR-CPA description

Where V m is the molar volume, b is the molar co volume parameter, a(T) the temperature dependent energy parameter of the equation of state, ρ the molar density (ρ=1/ V m ), g r the radial distribution function, and X Ai the fraction of sites A on molecule i (hence the subscript A i ) that do not form bonds with other hydrogen bonding sights. X Ai is dependent on the association strength Δ AiBj between it and sights belonging to other molecules of the same or different substance, like for example sight B on molecule j named B j (hence the superscript A i B j on Δ). X Ai is given as

     j B B A B j A j j i j i X x X  1 1
Eq (A. 1)

X Bj is the fraction of sites B on molecule j that don't form hydrogen bonds. The association strength Δ AiBj is dependent on the radial distribution function g r , the association energy ε AiBj , and the association volume β AiBj between sights A i and B j . The relation is given by equation:

j i j i j i B A ij B A r B A b RT g                      1 exp
Eq (A.

2)

The cross-association between two associating molecules is modelled by using the CR1 combining rule (Eq. A 3) and (Eq. A 4) for the cross-association energy and volume. Where k ij is the binary interaction parameter. While a i and a j are the energy parameters of the equation of state for pure substances i and j respectively. For substance i, a i is defined as: Where a 0,i and c 1,i are parameters of the equation of state for substance i. For water and alkanolamines they were fit to vapour pressure and liquid density experimental data. For alkanes they were calculated from critical properties and the acentric factor: Eq (A. 12)

Figure 2 .Figure 3 .

 23 Figure 2. (b. is the zoom of a.) Comparison between experimental data from Jou et al.[21] for propane solubility in 35 wt % aqueous MDEA solution and adjusted data using PR-CPA EoS (solid lines). (×)=273 K, (□)=298 K, (△)=313 K (○)=323 K, (*)=348 K, (■)=398 K, (▲)=398 K, (•)=423 K, dashed line: VLLE interface.

Figure 4 .

 4 Figure 4. PR-CPA EoS prediction of the ratio of methane solubility and the minimum solubility as function of temperature at 75 bar. solide line=25 wt % MDEA, dashed line=35 wt % MDEA, dotted line=50 wt % MDEA.

Figure 5 .

 5 Figure 5. PR-CPA EoS prediction of the temperature of minimum solubility of methane as function of MDEA concentration at different pressure. (×)=5 MPa, (□)=7.5 MPa, (○)=10 MPa.

Figure 6 .

 6 Figure 6. PR-CPA EoS prediction of the temperature of minimum solubility of methane as function of pressure in different aqueous MDEA solutions (□)=25 wt % MDEA, (×)=35 wt % MDEA, (○)=50 wt % MDEA and (△)=pure water.

Figure 7 .

 7 Figure 7. Comparison between experimental data from Caroll et al. [19] for water content in propane rich phase of propane-MDEA(35 wt %)-water ternary system and predicted data using PR-CPA EoS (solid lines). (◆)=273K, (■)=298K, (▲)=323K, (×)=348K, (*)=373K, (•)=398K, (+)=423K.

Figure 8 .

 8 Figure 8. Comparison between experimental data from Mokraoui et al. [17] for solubility of ethane in aqueous MDEA solution and predicted data using PR-CPA EoS (solid lines). (△)=Pure water, (×)=25 wt % MDEA, (□)=50 wt % MDEA.

Figure 9 . 1 Figure 10 .

 9110 Figure 9. Comparison between experimental data from Mokraoui et al. [17] for solubility of propane in aqueous MDEA solution and predicted data using PR-CPA EoS (solid lines). (△)=Pure water, (×)=25 wt % MDEA, (□)=50 wt % MDEA.

  0.0026 0.0027 0.0028 0.0029 0.003 0.0031 0.0032 0.0033 0.0034 0

6 ) 8 )

 68 values of ε and β were found by fitting to pure compound vapour pressure and liquid density experimental data. The equation used to find the radial distribution function is:The values of a(T) and b for the mixture are founded using classical van der Waals mixing rules:Where b i is the molar co volume parameter of the equation of state for pure substance i. For associative molecules, it is determined by fitting to pure compound vapour pressure and liquid density experimental data. For alkanes b i was calculated from critical properties acentric factor:

Table 1 . Pure components critical properties and acentric factors from Thermo Data Engine [16].

 1

	Compound Tc /K Pc /Mpa	ω
	Methane	190.56	0.546	0.0011548
	Ethane	305.32	4.872	0.099493
	propane	369.82	4.248	0.152291
	n-butane	425.15	3.794	0.19955
	i-butane	407.84	3.639	0.18444
	n-pentane	469.7	3.370	0.251506
	n-hexane 507.53	3.028	0.30044

Table 2 . PR CPA parameters for compounds for association compounds considered in this work.

 2

	Compound scheme	a 0 /bar L 2 mol -2	b /L mol -1	c 1	ε AiBj /bar.L.mol -1	β AiBj	Tc /K	Range of Tr	ARD a
									For P sat	For ρ L	P sat ρ L
	MEA	4C	1.333	5.467	0.763	168.23	0.0142 671.4 0.42-0.92 0.43-0.61 1.8 0.6
	DEA	4C	3.065	9.246	1.02	201.76	0.0083	768	0.45-0.88 0.41-0.48 0.3 1.3
	MDEA	4C	3.339	11.346	0.695	201.76	0.0083 741.9 0.39-0.9 0.38-0.63 0.9 2
	water	4C	0.123	1.445	0.674	170.48	0.0698 647.29 0.43-0.95 0.43-0.95	1 1.6

a ARD = 1/np× Σ|1 -χi calc /χi exp | × 100%.

Table 3 . Average Absolute Deviation (AAD) in liquid and vapor composition between PR-CPA adjusted data and experimental ones of water with MEA, DEA or MDEA binary system.

 3

	T /K	No BIP			Adjusted BIP	references b
		AAD a	AAD a	AAD a x 1 ×100 AAD a y 1 ×100	k ij
		x 1 ×100	y 1 ×100
	MDEA-water 313-450	12.26	1.26	1.20	0.005	-0.190

Table 4 . Comparison between alkane solubility data water with the calculated ones.

 4

		T /K	ARD x 1	a	b×10 3 /K -1	c×10 6 /K -2	References
	methane-water	274-423	6	-1.597	8.398	-8.29	[37,38]
	ethane-water	259-444	7	-1.517	8.198	-9.236	[39,40]
	propane-water	274-422	9	-1.114	6.256	-7.370	[41,42]
	butane-water a	273-423	7	-0.751	4.074	-4.467	[43,44]
	i-butane-water a	278-363	10	0.198	-2.152	5.205	[45]
	pentane-water a	273-477	12	-0.704	3.932	-4.749	[35-37]
	hexane-water a	273-425	12	-1.026	5.438	-6.565	[46,48]

Table 5 . Comparison between experimental data of alkane solubility in aqueous alkanolamine solutions and the adjusted ones obtained with PR-CPA EoS.

 5

	T independent BIP	T dependent BIP

Table 6 . Methane minimum solubility temperature in aqueous MDEA solution and in pure water predicted by PR-CPA EoS.

 6

	P /MPa		T /K
	5	407	340	317	286
	7.5	402	335	312	280
	10	395	330	306	277
	wt % MDEA	0	25	35	50

Table 7 . Alkane mixture composition from Mokraoui et al. [17].

 7

		Mole Composition
	Alkanes	MIX1	MIX2
	ethane	0.5	0.02
	propane	0.3	0.5
	n-butane	0.1	0.23
	i-butane	0.02	0.1
	n-pentane	0.05	0.1
	n-hexane	0.03	0.05

Table 8 . PR-CPA EoS prediction of alkane mixture solubility in aqueous alkanolamine solutions.

 8

						ARD a %
			T /K	P /MPa	Ethane	Propane
		50 wt% MDEA	298-333	0.6-4.2	22	13
	MIX1	25 wt% MDEA	298-333	0.7-4.0	16	11
		35 wt% DEA	298-333	0.6-4.3	29	17
		50 wt% MDEA	298-333	0.5-1.3	-	30
	MIX2 b	25 wt% MDEA	298-333	0.5-1.3	-	19
		35 wt% DEA	298-333	0.5-1.4	-	9
	a: ARD is average relative deviation determined on solubility of main conponents
	b: propane is the main component of MIX 2


Guillaume Bertrand
email: g.bertrand@brgm.fr

Daniel Cassard

Mario Billa

Jean-Michel Angel

François Tertre

Bruno Tourliere

Predictive assessment of rare earth occurrences

come

Introduction

Rare earth elements (REE) are a group of 17 metallic chemical elements that have relatively similar chemical and physical properties. They comprise the 15 lanthanides, plus scandium (sometimes excluded) and yttrium. REE are used in a wide range of manufactured products, especially in green energy and digital technologies, and are considered as critical raw materials by the European Commission [START_REF]Report on critical raw materials for the EU, Report of the ad-hoc working group on defining critical raw materials[END_REF] . China has the most abundant resources and production of REE in the world, although significant resources, either known or inferred, exist in other countries. The goal of the present study is to assess REE occurrences in Europe, using a predictivity methodology developed by BRGM to evaluate the favorability of known mineralization to host a targeted commodity, the DataBase Querying (or DBQ) method 2 . Its purpose is not to evaluate the REE endowment of Europe, but to pinpoint mineral occurrences that may contain REE, in order to identify favorable areas and help guiding further exploration works.

Data sources and preparation

Data that was used for this work came from both the EURare IKMS (see presentation from Cassard et al. in this conference) and the ProMine Mineral Deposit (MD) database [START_REF] Cassard | ProMine Mineral Databases: New Tools to Assess Primary and Secondary Mineral Resources in Europe[END_REF] . Datasets required by the DBQ method have to include a large number of occurrences or deposits, and to list their type and the presence/absence of a wide panel of commodities. The EURare IKMS dataset was extracted and completed to list all contained commodities (deduced from the mineralogy mostly) and to classify deposits in 17 distinct metallogenic families (16 from the ProMine project, plus placers, even though it is not a metallogenic type sensu stricto), in order to apply subsequent processing to coherent deposit populations. Additional information from publication [START_REF] Al-Ani | REE minerals in carbonatite, alkaline and hydrothermal rocks, northern and central Finland[END_REF][START_REF] Paulick | Rare Earth Element (REE) exploration potential and projects in Greenland[END_REF][START_REF] Thrane | REE deposits and occurrences in Greenland[END_REF][START_REF] Tukiainen | The Motzfeldt Centre of the Igaliko Nepheline Syenite complex, south Greenland -a major resource of REE elements[END_REF] was also included. The resulting 'augmented' dataset contained 158 REE deposits, amongst which 104 were sufficiently documented (in terms of deposit type and contained commodities) to be processed with the DBQ method. The ProMine Mineral Deposit database was also queried for all deposits belonging to the identified metallogenic families, with their type and contained commodities. This ProMine dataset was merged with the 104 IKMS deposits. After removal of duplicates, the final dataset contained 8326 occurrences and deposits, classified in 17 metallogenic families and described for their content in 52 commodities. 244 deposits out of the 8326 contain REE.

Predictive assessment

The DBQ method was developed within the ProMine project 2 , to perform predictive assessment on datasets that are not big enough to provide reliable results with "usual" methods (e.g., Weight of Evidence), which is often the case when targeted commodities are "rare" elements and/or by-products. The first step of DBQ is to calculate, for each metallogenic family, an enrichment ratio (ER, or ratio of frequency of REE in the metallogenic family vs. frequency of REE in the whole dataset). ER>1 indicates a metallogenic family enriched in REE, while ER<1 indicates a depleted one. Enriched metallogenic families are (by order of decreasing ER) alkaline/peralkaline intrusions (ER=26.25), pegmatites (ER=7.36), IOCG (ER=2.59), placers (ER=2.57), igneous replacement deposits (ER=1.77), residual deposits (ER=1.20), sandstone-and shale-hosted deposits (ER=1.19) and igneous intermediate deposits (ER=1.11). These 8 families contain a total of 2074 occurrences or deposits, amongst which 193 contain REE. For each enriched family, a frequency of occurrence in all deposits containing REE was calculated, per commodity (Table 1). This spectrum of commodity frequencies per metallogenic family constitutes the signatures that will be searched for. Note that the "igneous intermediate" and "IOCG" families (in red in Table 1) have only 4 and 5 deposits, respectively, containing REE, therefore their signatures have little statistical value and must be considered cautiously. All deposits in the dataset are compared to the characteristic signature of the metallogenic family they belong to and ranked according to their proximity (or similarity) to this signature, using the following formula:

where "binary presence value" is 1 if the ranked deposit contains the commodity, or 0 if not. In order to compare ranks between different enriched metallogenic families, ranks were ponderated by the enrichments ratios ER. We have then calculated a ponderated rank for each of the 2074 occurrences belonging to the 8 enriched metallogenic families.

Results and Conclusion

The DBQ method is not spatial in its principle (i.e., the location of a mineralization has no impact on its rank). However, as we know the geographic coordinates of deposits, it is possible to spatialize the results, for instance by drawing maps of rank density. Figure 1 is a map of kernel density of ponderated ranks in Europe. It highlights areas of high density that are a priori favorable for REE. Note that the distribution of results also depends on the spatial heterogeneity of records in the ProMine database (e.g., higher density of occurrences in France that therefore appears more favorable, compared to other countries with less data). As alkaline/peralkaline intrusions is by far the most enriched metallogenic family, the map closely mimics their spatial distribution in Europe, highlighting potentially favorable areas, such as the Svidnya-Seslavtsi region in Bulgaria, or the Laacher See intrusion in western Germany. In addition, it shows favorable areas related to other metallogenic families. Examples are for instances IOCG deposits in the northern Svecofennian belt (northern Sweden), placers in French Britany and Normandy, pegmatites in southern Austria or the Iberian massif (northern Portugal), or numerous bauxite occurrences in the Delfi region (Greece). Although the DBQ method is not dedicated to discovering new deposits, it is appropriate to assess favorability for by-products commodities in known occurrences where they were not necessarily searched for. The present study confirms that although REE are not mined in the EU, potential resources exist from a wide range of metallogenic provinces and geodynamic settings [START_REF] Goodenough | Europe's rare earth element resource potential: An overview of REE metallogenetic provinces and their geodynamic setting[END_REF] . It highlights mineral occurrences and areas in Europe that are favorable for REE, according to their DBQ ranking, and could therefore be interesting targets for future exploration works.

Table 1:

Frequencies (in percent) of occurrence of commodities in deposits containing REE, or characteristic signatures per enriched metallogenic family; in yellow, 10≤frequency<40; in orange, frequency≥40. * note that the "all metallogenic families" column includes deposits from families that are not enriched in REE (ER≤1). 2,50 0,00 0,00 0,00 0,00 0,00 55,56 0,00 Li 5,74 5,00 0,00 0,00 0,00 9,46 0,00 5,56 0,00 Sn 5,33 0,00 0,00 10,00 0,00 5,41 0,00 5,56 0,00 Zn 5,33 15,00 25,00 0,00 0,00 0,00 0,00 5,56 0,00 Pb 4,92 7,50 25,00 0,00 0,00 0,00 0,00 5,56 0,00 Ag 4,51 0,00 0,00 5,00 0,00 0,00 0,00 5,56 0,00 Ni 4,51 0,00 0,00 0,00 0,00 0,00 0,00 50,00 0,00 Mn 4,10 10,00 0,00 0,00 20,00 0,00 0,00 11,11 0,00 Ba 3,69 12,50 0,00 0,00 0,00 0,00 0,00 0,00 0,00 Co 3,69 0,00 0,00 5,00 0,00 0,00 0,00 11,11 0,00 Mg 3,28 5,00 0,00 0,00 0,00 1,35 9,09 11,11 0,00 W 3,28 0,00 0,00 0,00 0,00 1,35 0,00 5,56 0,00 Sr 2,87 15,00 25,00 0,00 0,00 0,00 0,00 0,00 0,00 Bi 2,46 0,00 0,00 10,00 0,00 2,70 0,00 0,00 0,00 As 2,05 0,00 0,00 0,00 0,00 0,00 0,00 11,11 0,00 Sb 1,64 0,00 0,00 0,00 0,00 0,00 0,00 5,56 0,00 Cd 1,23 0,00 0,00 0,00 0,00 0,00 0,00 5,56 0,00 V 1,23 0,00 25,00 0,00 0,00 0,00 0,00 5,56 0,00 Rb 0,82 0,00 25,00 0,00 0,00 0,00 0,00 0,00 0,00 Se 0,82 0,00 0,00 0,00 0,00 0,00 0,00 5,56 0,00

Figure 1 :

 1 Figure 1: Map of kernel density of ponderated REE scores in Europe, obtained by the DBQ method.


Philippe Jacquet
email: philippe.jacquet@nokia-bell-labs.com

Dalia Popescu

Self-similarity in urban wireless networks: Hyperfractals

We introduce a model of Poisson patterns of fixed and mobile nodes on lines designed for urban wireless networks. The pattern obeys to "Hyperfractal" rules of dimension larger than 2. The hyperfractal pattern is best suitable for capturing the traffic over the streets and highways in a city. We show that the network capacity under ad hoc routing algorithms scales much better than with the classic uniform Poisson shot model. The scaling effect depends on the hyperfractal dimensions. We show this results in two different routing models: nearest neighbor routing with no collision, minimum delay routing model assuming slotted Aloha and signal to interference ratio (SIR) capture condition, power-path loss and Rayleigh fading. The novelty of the model is that, in addition to capturing the irregularity and variability of the node configuration, it exploits self-similarity, a characteristic of urban wireless networks.

I. INTRODUCTION

Future networks require challenging and diverse communication scenarios with topologies designed to fit specific usecases [START_REF] Blaszczyszyn | Random linear multihop relaying in a general field of interferers using spatial aloha[END_REF], [START_REF] Herculea | Straight: Stochastic geometry and user history based mobility estimation[END_REF]. The explosion of the Internet of Things (IoT), devices employing ad-hoc communication has brought back in the attention the ad-hoc networks. Traditionally, for the modeling of ad-hoc topologies, uniform Poisson spatial models are employed. These models have been successfully applied to analyze wireless networks that exhibit a high degree of randomness. The characteristics and network metrics have been extensively studied in scientific literature [START_REF] Baccelli | Stochastic Geometry and Wireless Networks[END_REF] and results about the scaling laws are well-known [START_REF] Gupta | The capacity of wireless networks[END_REF].

Urban infrastructures display a certain degree of regularity that has been modeled by the Manhattan grid [START_REF] Karabacak | Mobility performance of macrocell-assisted small cells in manhattan model[END_REF], used as a lattice with users positioned in its corners. In reality, users have random positions on streets and the density of the users on each street depends on the level of importance of the respective street in the city road map (boulevard, street, alley, etc).

Recently, models of fractal repartition have been introduced [START_REF] Jacquet | Optimized outage capacity in random wireless networks in uniform and fractal maps[END_REF], [START_REF]Capacity of simple multiple-input-single-output wireless networks over uniform or fractal maps[END_REF] and have shown that environment displays selfsimilarity characteristics. For example, a department has rural areas, with low density of population, and urban areas, with high density of population, namely cities and towns. The towns are split in neighborhoods, each neighborhood is organized in blocks separated by streets. Blocks are made of buildings that are themselves split in apartments and so on. This description is very close to a fractal object that is based on self-similarity. Figure 1 represents Indianapolis downtown road map and is a perfect illustration of a map exhibiting self-similarity properties. Results that exploit self-similarity are very promising. In [START_REF] Jacquet | Optimized outage capacity in random wireless networks in uniform and fractal maps[END_REF], the authors showed that a limit of the capacity in a network with a non-collaborative protocol is inversely proportional to the fractal dimension of the spatial repartition of terminals. In their model the nodes have locations defined as a Poisson shot inside a fractal subset, for example a cantor set.

By definition a fractal dimension is smaller than the Euclidean dimension; it can be arbitrary smaller. In this work we propose a new model, which we call "Hyperfractal", for the ad-hoc urban wireless networks. This model captures not only the irregularity and variability of the node configuration but the self-similarity of the topology as well. The hyperfractal model is not a Poisson shot model in a fractal support but rather is a Poisson shot model which has support a measure which has scaling properties. It is a kind of generalization of fractal Poisson shot models, and in some cases, in fact in every case of our urban traffic models, it will have a dimension that is larger than the Euclidean dimension and this dimension can be arbitrarily large.

The radio model comprises urban-specific phenomenons such as the "urban canyon" propagation effect which is the most realistic in urban context.

Using insights from stochastic geometry and fractal geometry, we derive scaling laws of information routing metrics as well of throughput capacity and we prove by numerical analysis and simulations the accuracy of our expressions.

II. SYSTEM MODEL AND GEOMETRY

A. Urban Geometry: hierarchical grid street model

Let us assume that map is the unit square and it is divided into a grid of streets, horizontal (or West-East oriented) streets and vertical (or North-South oriented) streets, similar to a Manhattan grid. The horizontal (resp vertical streets) streets have abscissas (resp. ordinates) which are integer multiple of inverse power of two. The number of binary digits after the coma minus indicates the level of the street, starting with the street with abscissas (resp ordinate) 1/2 being at level 0.

We assume that the streets of level 0 are the highways of the city which support the most traffic, the level 1 are the main streets, the other levels are secondary streets, the traffic decreasing with level number.

The street grid can, of course, be based on on the powers of a different number, for example 3 or 4. This particular model is realistic for a modern US city. Figure 1 shows a map of Indianapolis as an example. It could also model the pattern of older cities in the ancient world. In this case, the model would display a similar hierarchical street distribution but plugged into a more chaotic geometric pattern instead that of the grid pattern.

B. Hyperfractal Mobile nodes distribution

It is assumed that the density of mobiles on streets decays as a decreasing function of the street. The process of assigning points to the streets is performed recursively, in iterations, similar to the process for obtaining the Cantor Dust [START_REF] Wikipedia | Cantor set -wikipedia, the free encyclopedia[END_REF].

We notice that the two streets of level 0 form a central cross which splits the map in exactly 4 quadrants. Let us assume a probability p and denote by q the complementary probability 1 -p . A mobile node is dispatched on a street according to the following procedure:

• with probability p the mobile node is located on the cross according to a uniform distribution. • otherwise, with probability q /4, it is located in one the four quadrants where the assignation procedure continues recursively. The procedure stops when the mobile node is assigned to a cross of a level m ≥ 0. A cross of level m consists of two intersecting segments of streets of level m. An example of a decreasing density in the street assignment process performed in L = 4 steps is given in Figure 2.

Taking the unit density for the initial map, the density of mobile nodes in a quadrant is q /4. Let µ H be the density of mobile nodes assigned on a street of level H. It satisfies:

µ H = (p /2)(q /2) H (1)
The measure (understood in the Lebesgue meaning) which represents the actual density of mobile nodes in the map has strong scaling properties. The most important one is that the map as a whole is identically reproduced in each of the four quadrants but with a weight q /4 instead of 1. Thus the measure has a structure which recalls the structure of a fractal set, such as the Cantor map. A crucial difference lies in the fact that its dimension, d m , is in fact greater than 2, the Euclidean dimension. Indeed, considering the map in only half of its length consists into considering the same map but with a reduced weight by a factor q /4. Thus, one obtains:

1 2 dm = q /4 (2)
Thus

d m = log( 4 q ) log 2 > 2 (3)
This property can only be explained via the concept of measure. We coin here the term hyperfractal to refer to this new kind of self scaling object in euclidean space. Notice that when p → 0 then d m → 2 and the measure tends to the uniform measure in the unit square.

One could extend the map in the whole quarter of the plane. In this case the map is made of half infinite horizontal and vertical streets. If a vertical (resp. horizontal) street has level H, it contains mobile nodes with uniform density µ H = (p /2)(q /2) H .

C. Canyon effect and relays

Due to the presence of buildings, the radio wave can hardly propagate beyond the streets borders. The buildings are made of concrete, glass and steal which generate a formidable obstacle for propagation. Therefore, we adopt the canyon propagation model where the signal emitted by a mobile node propagates only on the axis where it stands on. Considering the given construction process, the probability that a mobile node is placed in an intersection tends to zero and mobiles positioned on two different streets will never be able to communicate. Therefore, one needs to add relays in some street crossings in oder to guarantee connectivity and packet delivery.

A relay consists of two connected wireless devices: a first one transmitting and receiving on the North-South axis, and the second one, transmitting and receiving on West-East axis. We again make use of a hyperfractal process to select the intersections which will contain a relay.

) 2 / (q p 2 ) 2 / (q 2 ) 2 / (q 2 ) 2 / (q 2 ) 2 / (q ) 2 / (q p ) 2 / (q p ) 2 / (q p
Figure 3: Relays placement process Denote by p a fixed probability and q = 1 -p the complementary probability. A run for selecting a street crossing requires two processes: the in-quadrant process and the insegment process. The selection starts with the in-quadrant process as follows:

• with probability p 2 , the selection is the central crossing of the two streets of level 0; • with probability p(q/2), the relay is placed in one of the four street segments of level 0 starting at this point: North, South, West or East, and the process continues on the segment with the in-segment process • with probability (q/2) 2 , the relay is placed in one of the four quadrants delimited by the central cross and the inquadrant process continues recursively. The process of placing the relays is illustrated in Fig. 3. We perform M independent runs of selection. If one crossing is selected several times (e.g. the central crossing), only one relay will be installed in the respective crossing. This reduction will mean that the number of actually placed relays will be much smaller than M .

Following a reasoning similar to the mobile placement, the relay placement is hyperfractal with a hyperfractal dimension d r :

d r = 2 log(2/q) log 2 . ( 4
)
Let p(H, V ) be the probability that the run selects a crossing of two streets, one horizontal street of level H and one vertical street of level V . There are 2 H+V of such crossings. We have:

p(H, V ) = p 2 (q/2) H+V . (5)
Thus the probability that such crossing is selected to host a relay is

1-(1 -p(H, V )) M . When M is large, the probability is approximately 1 -exp(-M p(H, V )). If
L H (ρ) = V ≥0 2 V (1 -exp(-p(H, V )ρ)) . (6)
We notice that L H (ρ) = L 0 ((q/2) H ρ) and that L 0 (ρ) satisfies the functional equation: It is known from [START_REF] Flajolet | Mellin transforms and asymptotics: Harmonic sums[END_REF], [START_REF] Jacquet | Analytical depoissonization and its applications[END_REF] that this classic equation has a solution such as L 0 (ρ) = O(ρ 2/dr ).

L 0 (ρ) = 1 -exp(-p 2 ρ) + 2L 0 ((q/2)ρ). (7)
The average total number of relays in the city, R(ρ), has the expression:

R(ρ) = H,V ≥0 2 H+V (1 -exp(-p(H, V )ρ)) (8)
and satisfies the functional equation

R(ρ) = 1 -exp(-p 2 ρ) + 4L 0 (p(q/2)ρ) + 4R((q/2) 2 ρ). (9)
From the same reference, [START_REF] Flajolet | Mellin transforms and asymptotics: Harmonic sums[END_REF], [START_REF] Jacquet | Analytical depoissonization and its applications[END_REF], one gets

R(ρ) = O(ρ 2/dr log ρ) (10)
Since 2/d r < 1 the number of relays is much smaller than ρ. In the following we will take ρ = O(N ). Due to lack of space, complete proofs will be given in a extended version of the paper.

Given the process of construction, the probability of existence of mobile nodes is independent of the probability of existence of relays. Furthermore, the propagation limited to the axes does not ensure full connectivity. The connectivity graph presents a giant component. A complete Hyperfractal map containing both mobile nodes and relays is presented in Fig. 4.

III. ROUTING AND CAPACITY

Here it is considered that the nodes in the network, both mobile nodes and relays, communicate like in a mobile ad hoc network where packets are routed forward from their sources to their destinations. We consider a table driven routing where each nodes looks into a routing table to determine the next relay to send the packet. The distributed protocols needed to construct the routing tables in every node is not taken into account. To simplify, we consider that the routing tables are built by a central entity which has a full knowledge of the topology of the network as well as the precise locations of the nodes.

The routing table will be computed according to a minimum cost path over a cost matrix [t ij ] where t ij represents the cost of directly transmitting a packet from node i to node j. The min cost path from node i to node j which optimizes the relaying nodes (either mobile nodes or fixed relays) is denoted m ij and satisfies:

m ij = min k {m ik + t kj } , ∀(i, j), (11)
Due to the canyon effect some nodes can be disconnected from the rest of the network, several connected components may appear and some routes may not exist. In the case node i and node j cannot communicate m ij = ∞. We restrict our analysis to the giant component of the network which contains the central node [ 1 2 , 1 2 ]. We know that the size of this giant component is strictly of order N .

In this paper two routing strategies are considered:

• the nearest neighbor routing;

• the minimum delay routing.

To simplify, it is assumed that the channel is noiseless, i.e. every node is reachable given it is aligned (i.e. on direction North-South or West-East) regardless of the distance towards the transmitter. However, due to interference from other transmitters, the quality of the connection may significantly drop with respect to the relative distance between the nodes.

1) The nearest neighbor routing: In this strategy the next relay is always a next neighbor on an axis, i.e. there exist no other nodes between the transmitter and the receiver. Thus

     t ij = 1 if nodes i and j are aligned and k such that d(i, j) = d(i, k) + d(k, j) t ij = ∞ otherwise
This formula implies that, although the farther nodes on the axis might be reachable, the interference created by the nearest neighbors gives the farther node an infinite cost.

2) The minimum delay routing: In this model, the underlying medium access control is slotted Aloha with per slot and per node transmission probability p A . Considering interference, required SIR, and attenuation factors, we denote p ij the probability that node j correctly receives a packet transmission from node i at a given slot. Clearly, p ij ≤ p A (1 -p A ), since a required condition is that node i transmits and node j does not. Therefore, the average delay required for node i to successfully transmit a packet to node j is t ij = 1/p ij . The quantity m ij will be the cost of the minimum average path delay .

One should notice that the alternative strategy where we consider the minimum average number of retransmission on the path to the destination will provide the same shortest path. Indeed it will consist into multiplying the coefficients t ij and m ij by the factor p A .

A. Capacity

In this section a known result [START_REF] Malik | On the throughput capacity of wireless multi-hop networks with aloha, node coloring and csma[END_REF] for the throughput capacity of the ad-hoc networks is reminded and extended. Let us remind the following notations and results. For the hyperfractal distribution of nodes we assume N mobile nodes and that the relay nodes distribution satisfies ρ = N . It is already known that R(N ) = o(N ). Denote by G N the giant component of the network, its size

|G N | is Θ(N ).
Let ζ(N ) be the throughput capacity, defined as the expected number of packets delivered to their destinations per slot. It is a metric that depends on the number of nodes, N , SIR threshold, K, attenuation coefficient. α, the medium access scheme parameters and the expected transmission rate of each node, Ω i (N ).

Theorem 1. The throughput capacity of random wireless networks is of order [START_REF] Malik | On the throughput capacity of wireless multi-hop networks with aloha, node coloring and csma[END_REF] :

ζ(N ) = Θ N 2 i∈G N Ω i (N ) i,j∈G N m ij . ( 12
)
We intentionally keep vague the question whether or not the relay nodes should be mentioned or not in the giant component or if those relays generate traffic or limit their action to forwarding packets generated by mobile nodes. In any case this does not change the order of magnitude of the global throughput estimate.

One can notice that the quantity:

D N = i,j∈G N m ij |G N |(|G N -1) (13)
is the average path cost in the giant component.

In the nearest neighbor routing it is assumed that all nodes require the same quantity of bandwidth β: ∀i : Ω i = β. With Aloha and with the min path cost we have exactly ∀i : Ω i = p A slot bandwidth per node. To simplify we, it is assumed that the nodes transmit something even if they have no packet in their buffer. A complete proof of this results can be found in [START_REF] Malik | On the throughput capacity of wireless multi-hop networks with aloha, node coloring and csma[END_REF].

B. Average path cost and capacity estimate

In the context of nearest neighbor routing strategy, we prove the following result: Theorem 2. The average number of hops in a Hyperfractal is:

D N = O N 1- 2 (1+1/dm)dr (14)
where N is the number of mobile nodes and d m and d r are respectively the hyperfractal dimensions of mobile nodes and relays.

We conjecture (supported by the simulations) that with the minimum delay routing strategy using Aloha we have the same scaling.

Conjecture 2.1. The average path cost in minimum delay routing in a Hyperfractal scales as:

D N = O N 1- 2 (1+1/dm)dr (15)
Notice that when d r → 2 (i.e. when the relay distribution tends to be uniform Poisson) then D N tends to be in N

1 1+dm
which is o(N 1/3 ) much smaller than the average cost with uniform Poisson shot model, where the average path cost is O(N 1/2 ) in the plane [START_REF] Gupta | The capacity of wireless networks[END_REF]. In fact N 1/3 would be the order of magnitude of the average path length in uniform Poisson in a cube. The estimate D N = o(N 1/(1+dm) ) suggests the idea that the hyperfractal distribution operates as if the distribution of mobile nodes was a uniform distribution in an hypercube of dimension 1 + d m .

Corollary 2.1. The capacity in a Hyperfractal with N mobile nodes scales as:

ζ(N ) = Θ N 2 (1+1/dm)dr (16)
In consequence, when d r → 2 and d m → ∞ the capacity tends to scale linearly.

IV. PROOFS OF THEOREM 2

Proof. The proof will be left sketchy due to the lack of room. Mobile node mH situated on a line of level H wants to send a packet to its destination, mV , situated on a line of level V . The largest order of magnitude for path cost occurs when mV and m H are on non parallel streets as illustrated in Figure 5,a). Furthermore, the dominant case is V = H = 0, the other cases will just introduce extra factors which will not change the obtained orders of magnitude.

Given that the densities of the population on the support street of mobile nodes are high, the game will consist into diverting the packet by following a vertical line of level x > 0 with a much lower density. A similar phenomenon happens towards mobile mV , the packet will return on the support street of mV as "close" as possible to node mV , after following an horizontal street of level y > 0. Since it is considered V = H, by symmetry we will only consider that x = y. In order for the packet to change direction in its route, it is mandatory that a relay exists at the crossing. Let L(x, y) be the average distance between a random mobile node on a street of level y to the first relay to a street of level x. Every crossing between streets of level x and y is independent and holds a relay with probability 1 -exp(-ρp(x, y)). Since such crossings are regularly spaced by interval 2 -x we get:

L(x, y) ≤ 2 -x 1 -exp(-ρp(x, y)) . ( 17
)
Our aim is now to count the number of nodes traversed by the packet on its route. There is no room to prove that the main contribution in the count comes from the number of traversed mobile nodes. It is first assumed that the two streets of level x have a relay at their intersection. In this case, the average number of traversed nodes is upper bounded by 2N µ 0 L(x, 0) + 2N µ x . If x = α log ρ log(2/q) for 1/2 < α < 1, then

L(x, 0) = ρ -2α/dr 1 -exp(-p 2 ρ 1-α ) = O(ρ -2α/dr ). (18)
We also have µ x = O(ρ -α log(2/q ) log(2/q) ) = o(ρ -2α/dr ). The probability that there exists a valid relay at level x street intersection is 1 -exp(-ρp(x, x))) which tends to zero when α > 1/2 since ρp(x, x) = p 2 ρ 1-2α . Let us assume this holds (to be verified afterward).

Following this observation, an intermediate level, 0 < y < x, is added to the route with:

y = β log ρ log(2/q) (19)
with β = 1 -α. We consider the first relays on the streets of level y counted from the intersection of the streets, they are at average distance L(y, x). The probability that the intersection two streets of level y has a relay is 1 -exp(-ρp(y, y)) which tends to 1 when ρ → ∞ since ρp(y, y) = p 2 ρ 1-2β . The average distance between the intersection (x, x) with the routes of levels y which have a valid relay at their intersection is

L(y,x) 1-exp(-ρp(y,y)) = O(ρ -2β/dr ).
Level y adds a number of hops of an average quantity 2N µ y L(x,y) 1-exp(-ρp(y,y)) thus the number of extra hops if of order: O(N ρ -2β dm dr ). The total number of hops is now O(N ρ -2α/dr ) + O(N ρ -2β dm dr ). This quantity is minimized for α = dm 1+dm and β = 1 1+dm . We confirm that α > 1/2 The minimized value of the number of hops is, thus:

D N = O(N ρ - 2 (1+1/dm )dr ) (20)
V. NUMERICAL RESULTS

A. Aloha model with Rayleigh fading

We consider slotted, synchronized Aloha scheme and that all nodes are backlogged. Each transmitting node uses the same nominal transmit power. Path-loss between node i and node j is modeled by the power-law function l(i, j) = (Ar) α , when the two nodes i and j are aligned, where A, α are some constants and r is the distance between transmitter and receiver, otherwise l(i, j) = 0. The reception undergoes Rayleigh fading F ij independent over nodes and time, the signal received by receiver j from transmitter i at time slot n becomes F ij l(i, j).

By assuming the background noise power negligible, and that node i is in transmit mode, the successful reception of a signal transmitted from node i to a node j at a given time slot occurs when: where K is the SIR threshold related to the bit-rate when a particular modulation plus coding scheme is considered. B is the subset of nodes transmitting at the considered time slot. In the particular case of a relay we have to separate the signals coming from the vertical street with the signals received from the horizontal streets In order to avoid computational expensive and timeconsuming simulations of ALOHA protocol, use the following results computing the probabilities of successful reception, p ij when independent Rayleigh fading is applied:

F ij l(i, j) > K k∈B-{i} F kj l(k, j) (21)
p ij = p A (1 -p A ) k =i,j w kj (K/l(i, j)) . (22)
where w kj (θ) is the Laplace transform of the signal produced by node k over node j, these quantities are simple rational functions of θ. We will take α = 4 and K = 1. Now that we have this result on the probability of successful reception, we apply Dijkstra algorithm on the cost matrix t ij = 1/p ij . Two configurations are studied, one where the fractal dimension of mobiles is equal to the ones of relays, d m = d r and the second one, where the two values differ, d m > d r .

B. Simulation results

Figure 6 validates the scaling law for the average number of equipped intersection introduced in equation ( 10) for both evaluation setups. Figure 7 validates the Conjecture 2.1 by simulations and computation using minimum path cost algorithm. In fact it shows that the proposed scaling law is rather pessimistic, offering improvement for future work.

Another simulation is performed on the throughout capacity. Thanks to the given closed form expressions, computations can be done fast and conveniently. We compute the throughput capacity using Theorem 1 where the quantities m i,j are obtained by using a minimum path cost computation algorithm, where the values in the cost matrix are the inverse of the expression [22].

Figure 8 illustrates the results of this computation. Again, the conjectured scaling law is pessimistic.

VI. CONCLUSION

This work introduced a new model, which called "Hyperfractal", for the ad hoc urban wireless networks. The model

Figure 1 :

 1 Figure 1: lndianapolis downtown road map

Figure 2 :

 2 Figure 2: Example of mobile density Map obtained in L = 4 iterations. The lines are thicker for the level obtained first and become thinner with the increase of the level.

Figure 4 :

 4 Figure 4: Hyperfractal map of level 10 with mobiles and relays, d m = 3, d r = 3

Figure 5 :

 5 Figure 5: Routing in a Hyperfractal a) intermediate levels x and y, b)extra intermediate levelsIn order for the packet to change direction in its route, it is mandatory that a relay exists at the crossing. Let L(x, y) be the average distance between a random mobile node on a street of level y to the first relay to a street of level x. Every crossing between streets of level x and y is independent and holds a relay with probability 1 -exp(-ρp(x, y)). Since such crossings are regularly spaced by interval 2 -x we get:

Figure 6 :

 6 Figure 6: Average number of equipped intersections for the two configurations, a)d m = d r = 3, b)d m = 3, d r = 2.2

Figure 7 :

 7 Figure 7: Average path cost scaling for the two configurations, a)d m = d r = 3, b)d m = 3, d r = 2.2

Figure 8 :

 8 Figure 8: Throughput capacity scaling for the two configurations, a)d m = d r = 3, b)d m = 3, d r = 2.2 captures the irregularity and variability of the node configuration and, in addition to previous works, the self-similarity of the topology. The hyperfractal model is a Poisson shot model which has support a measure with scaling properties. We showed here the scaling of of interest like throughput capacity and the number of hops and path cost under two routing algorithms. The results show that the scaling is much better than with the classic Poisson shot model.

  The relay Poisson model is interesting as it generates independent crossings. For this reason, we keep the relay Poisson model from now on.The average number of relays on a streets of level H is denoted by L H (ρ) and satisfies the identity:

the number of crossing selection run is a Poisson random variable of mean ρ, then the probability that a crossing hosts a relay is exactly 1 -exp(-M p(H, V )).