Keywords: Biogas, Hydrogen sulfur, Reactive absorption, Adsorption on activated carbon, Aspen Plus®, Hydrodynamics, Low temperatures, BioGNVAL pilot plant, Vehicle fuel …

List of figures

List of tables

 22. Effect of temperature on the abatement of volatile organic silicon compounds.... Table 23. Comparison of the different biogas purification and upgrading technologies ........ Table 24. Performances, costs, advantages and disadvantages of separation processes [START_REF] Cloirec | Les Composés Organiques Volatils (COVs) dans l'Environnement[END_REF] Table 25. Conditions and composition of the raw biogas treated by BioGNVAL pilot plant ... Table 26. conditions of passage from one subsystem to another [START_REF][END_REF]

General introduction

Recent decades were accompanied by economic growth and prosperity for humanity. This growth gained from oil and natural gas production has been accompanied by an environmental pollution that may trigger irreversible changes in the environment with catastrophic consequences for humans. Moreover, issues related to the reduction of fossil reserves are still relevant, and the global primary energy demand is increasing, pushing the international community to pursue the development of renewable energies.

The law on energy transition plans to reduce the share of fossil fuel consumption in France to 30 % in 2030, while the share of renewable energy should be increased to 23 % in 2020 and 32 % in 2030 against [START_REF]Intergovernmental Panel on Climate Change[END_REF].1 % in 2012 as shown in the Fig. 1 [1].

Fig. 1: Distribution of final energy consumption in France [1]

Renewable energy is derived directly from natural phenomena. It takes many forms: sunlight, wind, wood heat, water power … The main renewable energies are: wind power, wave power, geothermal energy, solar energy, biomass, firewood and hydropower … According to the French Ministry of Energy [1], primary production of renewable energy rises to 22.4 Million tonnes of oil equivalent (Mtoe) in 2012. The distribution of renewabale energies production by sector is depicted in Fig. 2.

Among renewable energies, biogas is a possibility. It is storable, transportable, not intermittent and substitutable for fossil energies. These strengths justify the consolidation of this emerging sector and the preparation of its future development by ambitious public policies. The European primary energy production from biogas reached 5901 kilotonnes of oil equivalent (ktoe) in 2007, 7500 ktoe in 2008, 10086 ktoe in 2011 and 13379 ktoe in 2013.

Fig. 2: Distribution of production of renewable energies by sector [1]

Biogas production is at the heart of the policy priorities for 2020, the European Union aims for a production that will cover 50 % of transport sector needs. Biogas is a growing sector. Fig. 3 provides a quick review of biogas production in Europe in 2013 [START_REF] Eurobserv | Biogas barometer[END_REF]. Generally, this gas composed of methane and carbon dioxide, also contains other compounds as water, ammonia, volatile organic compounds and hydrogen sulfide.

Biogas can be valued in several applications such as production of heat and/or electricity, feed for fuel cells, injection into the natural gas grid and production of liquefied bio-methane. This last application presents an environmental and economic benefits. With hydrogen fuel, liquefied biomethane is one of the best fuel for the reduction of carbon dioxide emissions with a reduction potential up to 97 % compared to Diesel. It can effectively reduce emissions of greenhouse gases and the pollution, responsible for 42000 premature deaths annually in France [START_REF] Benkimoun | Diesel: 42000 morts prématurées chaque année en France[END_REF]. Biofuels (2.4) Hydropower ( 5) Firewood [START_REF] Rasi | Trace compounds of biogas from di_erent biogas production plants[END_REF] Heat pumps (1.4)

Wind power (1. Fig. 3: Biogas primary production in 2013 [START_REF] Eurobserv | Biogas barometer[END_REF] Liquefied bio-methane requires very low temperatures which may lead to solidification of impurities and thus facilities malfunctions. These impurities must be separated from the biogas. This implies implementation of a purification process to remove from the raw biogas all unwanted substances in order to maximize its methane content. In particular, the complete desulfurization of biogas is peremptory in order to ensure an optimal operation and a high purity of other compounds to valorize, such as carbon dioxide. Moreover, the presence of hydrogen sulfide in the wet biogas, is a poison for installations. In case of leakage, the presence of hydrogen sulphide in the biogas characterized by a rotten egg smell can be dangerous for operators working on the site. Hence, the need to remove all traces of this compound. Several technologies are used for the removal of H2S as adsorption on microporous solids, membrane technology, biological processes and absorption by means of liquid solvents.

The choice of the technique to implement is related to various parameters. The most important are the flow rate of biogas and hydrogen sulfide concentrations to treat. The solution must also respond to various economic, environmental and energetic imperatives as cost which must be reasonable, the threshold to reject which must be respected in an energy-efficient process. Therefore, there is no universal treatment method.

In this thesis, the choice fell on the use of a cryogenic method in order to combine biogas upgrading and biomethane liquefaction. The removal of H2S will be performed upstream of the process either by reactive absorption using an aqueous solution of sodium hydroxide (NaOH) in a structured packed column or by adsorption using activated carbon. These two technologies will be tested and compared in order to choose the most effective and the most suitable for the process set up.

Experiments were performed on a industrial pilot plant developed by Cryo Pur® company called "BioGNVAL". This pilot plant treats 85 Nm 3 /h of biogas from waste water treatment plant, which contains around 20 -100 ppm of H2S.

For absorption technology, the hydrodynamic of flows in structured packing columns was studied in order to develop a model able to predict realistically the key hydrodynamic parameters as pressure drop, liquid holdup and transition points but also effective interfacial area and global mass transfer coefficients.

The remainder of the study is based on simulations using Aspen Plus® V8.0 to study realistically the effectiveness of a structured packed column which uses sodium hydroxide as a chemical solvent for the selective removal of hydrogen sulfide from biogas. Results were compared with data from BioGNVAL pilot plant.

Finally, the dynamic aspect of the adsorption phenomenon is modeled, by predicting the breakthrough curve in the case of an adsorption column used for the removal of H2S.

Chapter 1: From biogas to biomethane

Résumé :

Après une introduction générale posant le contexte de la production de biogaz dans le paysage énergétique mondial, le premier chapitre est une introduction à la problématique de son obtention et de son usage sous une version raffinée en biométhane.

Le biogaz est produit par méthanisation lorsque des matières organiques sont décomposées dans des conditions bien définies et en absence d'oxygène. Il est constitué principalement de méthane et de dioxyde de carbone mais aussi de quantités variables de vapeur d'eau, de sulfure d'hydrogène et d'autres composés polluants. Afin de pouvoir utiliser le biogaz comme carburant pour véhicule, il doit être épuré en séparant le dioxyde de carbone et les autres composés contaminants du biogaz pour augmenter au maximum sa teneur en méthane.

Il existe différentes technologies utilisées dans le domaine du traitement des gaz. Les plus utilisées sont : l'absorption, l'adsorption, la technologie membranaire, la cryogénie, le traitement biologique et l'oxydation. Aujourd'hui, d'autres technologies sont développées telles que la biocatalyse, la photocatalyse et le plasma froid. Pour le moment, le manque d'informations à leur sujet limite leur application industrielle [START_REF] Biard | Contribution au développement d'un procédé de lavage chimique compact. Traitement du sulfure d'hydrogène par le chlore à l'échelle semi-industrielle et de COV odorants par oxydation avancée O3 / H2O2 à l'échelle du laboratoire[END_REF].

Introduction

Anaerobic digestion may be defined as the natural process of degradation by microorganisms under controlled conditions and in the absence of oxygen. This degradation results in the production of a gas mixture saturated with water outlet of the digester called biogas.

Anaerobic digestion is the result of four biochemical steps in which large carbon chains are converted into fatty acids and alcohols. These four steps are: Hydrolysis, acidogenesis, acetogenesis, methanogenesis [START_REF] De La Farge | Le biogaz, procédés de fermentation méthanique[END_REF].

 Hydrolysis

It takes place at the beginning of the fermentation and makes use of exo-enzymes in order to decompose the organic matter into simple substances.

 Acidogenesis

During this step, volatile fatty acids are formed. Carbon dioxide and hydrogen are also formed, they are used by microorganisms during the production of methane according to the chemical Reaction (R.1) shown below. The reaction enthalpy is about -567 kJ.mol -1 [START_REF] Zdanevitch | Etude de la composition du biogaz de méthanisation agricole et des émissions en sortie de moteur de valorisation[END_REF].

𝐶𝑂 2 + 4 𝐻 2 → 𝐶𝐻 4 + 2 𝐻 2 𝑂 (R.1)

 Acetogenesis

This stage involves the production of acetate, an indispensable substrate for the synthesis of methane.

 Methanogenesis

This last step results in the production of methane. It is ensured by the methanogenic bacteria, which can only use a limited number of carbon compounds, including acetate responsible for 70 % of methane production according to the chemical Reaction (R.2) shown below. The reaction enthalpy is about -130 kJ.mol -1 [START_REF] Zdanevitch | Etude de la composition du biogaz de méthanisation agricole et des émissions en sortie de moteur de valorisation[END_REF].

𝐶𝐻 3 𝐶𝑂𝑂𝐻 + 𝐻 2 𝑂 → 𝐶𝐻 4 + 𝐻 2 𝐶𝑂 3 (R.2)
The biogas needs to be purified and upgraded, which means that impurities are removed or valorized in order to produce a gas rich in methane called biomethane. There is a number of technologies available for this purpose as water scrubbing, membranes and pressure swing adsorption.

Biogas utilization

Biogas can be used in several applications. Sometimes it could be used raw, but almost always it has to be upgraded or as a minimum cleaned from its H2S content because the presence of this compound in the biogas even at very low concentrations could damage the installations.

Biogas can be used in all applications designed for natural gas such us production of heat and electricity known as combined heat and power (CHP), production of chemicals and/or proteins, fueling internal combustion engines and fuel cells, it may also be used as vehicle fuel or injected in the natural gas grids.

Direct combustion

The simplest method of biogas utilization is direct combustion. The biogas burner can be installed in heaters for production of hot water and hot air, in dryers for various materials, and in boilers for production of steam for process heat or power generation [START_REF] Walsh | Utilization of Biogas[END_REF]. Another limited application involves absorption heating and cooling to provide chilled water for refrigeration and hot water for industrial processes. These applications based on direct combustion do not require a high gas quality.

Biogas can also be used to fuel internal combustion engines to supply electric power for pumps, blowers, elevator and conveyors, heat pumps and air conditioners [START_REF] Walsh | Handbook on biogas utilization[END_REF].

Combined heat and power

Another application for biogas is CHP which involves the production of two or more forms of energy, generally electricity and thermal energy. This latter is generally the most used today but it could be a problem because the need of heat varies with season, and during summer for example, unused biogas is flared.

Injection into the natural gas grid

Upgrading of biogas to biomethane with a gas quality similar to natural gas and injecting it into the natural gas grid is an efficient way to integrate biogas into the energy sector. It allows the transport of large volume of biomethane and its utilization in wide areas where population is concentrated.

Vehicle fuel

Today, there is a big interest in using biogas as a vehicle fuel. But for this utilization, the raw biogas must be purified which means that contaminants are removed from biogas, and upgraded which means that CO2 is eliminated leading to a raise in the energy content of biogas. Finally, the biomethane needs to be liquefied by chilling it to very low temperatures (≈ -161 °C at atmospheric pressure). In order to prevent corrosion and solid formation, some requirements concerning component concentrations are needed before liquefaction. They are presented in Table 1. The H2S content must be lower than 4 ppm. This quantity is not in direct relation with corrosion or solidification aspects. This requirement ensures a high quality of biomethane and avoids odor problems caused by the presence of hydrogen sulfide.

Table 1. Concentration requirements before biogas liquefaction [8]

Compounds

Maximum concentration CO2 25 ppm H2S 4 ppm H2O 1 ppm Fig. 5 shows an example of use of biogas after its purification and upgrading using the Cryo Pur® system developed by Cryo Pur® Company. [START_REF][END_REF]

Fig. 5: Example of biogas utilization by Cryo Pur® Company

Biogas composition

The composition of biogas depends mainly on the type of substrates which are segmented according to their origin. The following classification is commonly used:  Household and industrial waste.  Sludge from sewage water treatment plants.  Agricultural and agro-industrial waste.

Household and industrial waste

To treat this type of waste, they are buried in landfill sites. The anaerobic conditions created by the landfill are sufficient to induce methanogenesis. Biogas production from this type of waste is characterized by variations of flow rates and composition due to the variation of the feedstock. This biogas production is also dependent on the progression of degradation of waste, moisture and temperature. These parameters are not the same on the entire degradation zone leading to variations in the composition of biogas produced from the same landfill. This variability phenomenon is also noticeable on the concentration of methane in biogas which varies around 15 % during a year [START_REF] Rasi | Trace compounds of biogas from di_erent biogas production plants[END_REF]. Two factors explain this: the high biological activity in summer and the rise of temperature. These conditions complicate the valorization of this type of biogas.

Household waste sorting improves the yield of biogas production. This process by which waste is separated into different elements achieves higher yields compared to unsorted waste. The difference may reach 100 m 3 /t [START_REF] Görisch | La production de biogaz[END_REF]. Household and industrial wastes are not all fermentable. The share of these fermentable wastes represents 45 % maximum [START_REF] Deublein | Biogas from waste and renewable resources : an introduction[END_REF]. Table 2 shows the principal compounds present in household waste.

Sludge from sewage water treatment plants

Biological treatment of urban wastewater is a widespread process which generates significant amounts of activated sludge. In order to stabilize these latter, an anaerobic degradation is used. It provides a solution to the storage and treatment of sludge. A small part of the produced biogas ensures 100 % of the energy needs of the wastewater treatment plant. The by-products of anaerobic digestion, as the digestate can be valorized via fertilization and amendment of farmland.

Today, the injection into the natural gas grid of biomethane issued from sludge of wastewater treatment plant is subject to strong demand from local authorities. According to the French Ministry of Ecology, Sustainable Development and Energy, by 2020, more than 60 wastewater treatment plants could be provided with the necessary facilities for energy recovery of waste to allow the injection of 500 GWh/year of biomethane into the national gas grid, which is equivalent to the annual consumption of more than 40000 households [1].

Unlike the treatment of household and industrial waste, anaerobic digestion of sludge from wastewater treatment plants is a controlled process conducted under optimal conditions for biogas generation.

Agricultural and agro-industrial waste

According to the European renewable energies Observatory [START_REF] Eurobserv | Biogas barometer[END_REF], France has about 60 million tons of organic material that can be valued in biogas. These agricultural wastes are divided into two groups: liquid effluents and solid waste. For liquid effluents, dry matter and dry organic matter rates influence the production of biogas, as shown in Table 3. Digesters used for biogas production from agricultural and agro-industrial wastes are optimized systems, regulated and stable. According to Boulinguiez [START_REF] Boulinguiez | Purification de biogaz -Elimination des COV et des siloxanes[END_REF], the composition of biogas from digesters varies with an amplitude of ± 5 % during the year. Table 3 shows that shortening (Alimentary fat) is the substrate with the highest methanogenic potential.

Assuming complete reaction without formation of by-products, Buswell Equation (1) shows that the theoretical yield of methane production may be estimated from the base elemental composition of a substrate [START_REF] Buswell | Laboratory studies of sludge digestion[END_REF].

𝐶 𝑐 𝐻 ℎ 𝑂 𝑜 𝑁 𝑛 𝑆 𝑠 + 𝑦 𝐻 2 𝑂 → 𝑥 𝐶𝐻 4 + (𝑐 -𝑥) 𝐶𝑂 2 + 𝑛 𝑁𝐻 3 + 𝑠 𝐻 2 𝑆 𝑥 = 1 8 (4 𝑐 + ℎ -2 𝑜 -3 𝑛 -2 𝑠) 𝑦 = 1 4 (4 𝑐 -ℎ -2 𝑜 + 3 𝑛 + 3 𝑠) (1)
The composition of biogas shown in Table 4 is defined according to the main types of substrates previously presented.

Environmental and economic issues

Biogas is a very interesting energy from an ecological and economic point of view.

The environmental interest of reducing CO2 emissions coincides with the approaches adopted to the production and consumption of renewable energies. The environmental impact of production and valorization of biogas is easily demonstrable.

From an economic point of view, digesters allow to value all the types of substrates discussed in the previous section at attractive cost. In addition, the digest that was recovered at the outlet of the digester could be valorized as fertilizer.

 Environmental issues

Anaerobic digestion is a natural phenomenon that rejects methane which is 23 times more potent as a greenhouse gas than carbon dioxide [START_REF]Intergovernmental Panel on Climate Change[END_REF]. The simple conversion by combustion of CH4 into CO2 reduced to 8 % the initial potential of greenhouse gas from biogas, emitted directly into the atmosphere. Carbon dioxide footprint could be further improved when the biogas is purified and upgraded. Moreover, the fossil energy used massively today releases large amounts of CO2 and it is not an inexhaustible resource. The valorization of biogas is therefore interesting for environmental protection by saving fossil fuels and avoiding methane emissions into the atmosphere. A medium agricultural digester allows the reduction of 1000 tons of CO2 each year [1].

 Economic issues

The economic feasibility of a biogas sector depends mainly on the composition of the biogas.

The methane concentration in a biogas produced from household and industrial wastes is low as seen in Table 4. The CH4 concentrations do not exceed 55 %. Moreover, the biogas produced can vary over time and it contains large amount of minor compounds, complicating biogas valorization. Biogas production from this type of waste is expected to be limited in future years in Europe. Restrictions and European standards are becoming more stringent, which makes the use of this source, unprofitable [START_REF] Deublein | Biogas from waste and renewable resources : an introduction[END_REF]. The most advanced biogas applications as vehicle fuel production or injection into the natural gas grid are hardly possible from this source, which requires advanced purification treatments. Considering an anaerobic digestion unit treating 50000 t/year of household waste, the total costs vary between 50 and 95 €/t while the incomes do not exceed 30 €/t [START_REF] Boulinguiez | Purification de biogaz -Elimination des COV et des siloxanes[END_REF].

The economic balance of biogas production within wastewater treatment plants depends on the size and savings made on the sludge. This stable and controlled anaerobic digestion process is financially viable. The wastewater treatment plant "Aquapol" situated in Grenoble, France treats 88 000 000 m 3 of wastewater each year which is equivalent to 8000 dry tons of sludge processed per year. Part of the produced biogas is used for internal energy needs of the plant (8 GWh/year). The other part (14 GWh/year) will be injected in the natural gas grid after purification and compression [START_REF] Leclerc | STEP d'Aquapole : Présentation de la méthanisation et de l'injection au réseau[END_REF].

The establishment of biogas purification process from agricultural waste is particularly attractive due to the reliability of the resource. The introduction of energy crops among the substrates improves the performance of anaerobic digestion giving a new economic aspect for the production of biogas.

From biogas to liquid biomethane

An advanced purification of biogas allows achieving an adequate quality threshold for use as vehicle fuel. The major step of this treatment is the separation of CO2, in addition to dehumidification, desulfurization, reduction of oxygen and the removal of trace compounds in biogas. Qualities and tolerances in impurities, for the production of liquid biomethane to be used as vehicle fuel are listed in Table 5.

Table 5. Tolerances in impurities for the use of liquid biomethane as vehicle fuel [START_REF] Boulinguiez | Purification de biogaz -Elimination des COV et des siloxanes[END_REF]

Compounds

Unit Liquid biomethane used as vehicle fuel

Methane -CH4 wt. % > 96

Carbone dioxide -CO2 wt. % < 3

Oxygen -O2 wt. % < 3

Water -H2O mg.m -3 < 30

Hydrogen sulfide -H2S mg.m -3 < 5

Total sulfur mg.m -3 < 120

Organosulfur mg.m -3 < 15

Hydrocarbons mg.m -3 < 200

Critical size of particles μm < 1

Liquid biogas can be produced using a cryogenic upgrading technology, based on differences in condensation temperature for different compounds. It can also be produced by mean of a conventional technology connected with a small-scale liquefaction plant. When using the first method, the carbon dioxide comes as a by-product which could be used in external applications bringing in extra income to the biogas upgrading unit.

The environmental benefits provided by the passage from the diesel to biomethane are impressive. Used as a fuel, Bio-LNG enables a considerable reduction of polluting emissions and represents a genuine alternative to diesel:  Zero emission of fine particles responsible for 42000 premature deaths annually in France.  -70 % NOx emissions.  -90 % CO2 emissions.  -99 % hydrocarbons emissions.  -50 % noise pollution.

Bio-LNG is a renewable energy produced from waste which makes it neutral in terms of greenhouse gas emissions. Its use allows the decarbonization of the energy mix. Bio-LNG is the only sustainable solution for long-distance haulage operated by heavy goods vehicles.

Conclusion

Biogas is produced when organic material is decomposed under anaerobic conditions. The main constituents are methane and carbon dioxide. To be able to use the raw biogas as a vehicle fuel it must be purified and upgraded, which means that impurities and CO2 respectively, are separated. There are a number of available upgrading technologies and the most commonly used are:

 Absorption  Adsorption  Membranes  Cryogenic technology
Other technologies exist such as oxidation and biological treatment. These techniques are known in the field of gas treatment. Today other technologies are being developed such as biocatalysis, photocatalysis and cold plasma. For the moment, the lack of information about them limits their industrial application [START_REF] Biard | Contribution au développement d'un procédé de lavage chimique compact. Traitement du sulfure d'hydrogène par le chlore à l'échelle semi-industrielle et de COV odorants par oxydation avancée O3 / H2O2 à l'échelle du laboratoire[END_REF].

The choice of the technology to be used to purify, upgrade and liquefy the biogas requires the knowledge of the thermodynamic properties of biogas and the representation of phase equilibrium. These thermodynamics aspects of biogas will be discussed in the next chapter.

Chapter 2: Thermodynamic aspects of biogas

Résumé :

Le choix de la technologie à utiliser pour purifier et liquéfier le biogaz exige la connaissance des propriétés thermodynamiques du biogaz.

Ce chapitre recense les propriétés thermodynamiques des fluides qui sont essentielles pour concevoir et optimiser les technologies de purification de biogaz. Il montre aussi que les propriétés thermophysiques du biogaz dépendent fortement de sa composition, en particulier des concentrations de méthane et de dioxyde de carbone. La présence de composés à faibles concentrations comme l'azote ou l'hydrogène sulfuré pourrait modifier les propriétés physiques du biogaz. Par exemple, les gaz d'enfouissement comprennent de petites quantités d'azote et d'oxygène qui affectent le comportement de phase du système CH4 -CO2.

Introduction

Thermodynamics can be used as a powerful tool for setting and evaluating processes used for purification and upgrading of biogas. Therefore, this study will focus on the thermodynamic investigation of biogas. It will present the thermodynamic properties of pure compounds and of the gas mixture (biogas) at a pressure of 1.103 bar. This pressure is considered because the biogas treated in the experimental part (Chapter 4) comes from the wastewater treatment plant at a pressure slightly above atmospheric pressure in order to avoid air infiltration into the biogas pipe.

Thermodynamic properties of pure component present in biogas

Biogas refers to a mixture of different molecules of gases as seen in Table 4. The thermodynamic aspects of the main components of biogas will be presented in this section.

Hydrogen sulfide

Hydrogen sulfide is a colorless gas with the characteristic foul odor of rotten eggs. It is very poisonous, corrosive, flammable and explosive. Its olfactory threshold varies between 0.7 and 200 g.m - 3 , depending on the sensitivity of each individual. The olfactory sensation is not proportional to the concentration of H2S in the air because it is possible that the smell felt at very low concentrations is attenuated or disappeared at high concentrations.

Hydrogen sulfide is created following the bacterial decomposition of organic matter in the absence of oxygen, such as in swamps and sewers. It also appears in volcanic gases and hot springs. Other sources of hydrogen sulfide are the industrial processes used in the oil and natural gas sectors, sewage treatment plants and factories producing pulp and paper … The thermos-physical properties of hydrogen sulfide are listed in Table 6.

Carbon dioxide

Carbon dioxide is a colorless and odorless gas which is naturally present in the Earth's atmosphere. The concentration of carbon dioxide in the atmosphere reached 405 ppm at the end of 2016, against only 283 ppm in 1839.

Carbon dioxide is produced by all aerobic organisms when they metabolize carbohydrate and lipids to produce energy by respiration [START_REF] Bosak | Science is …: A source book of fascinating facts, projects and activities[END_REF]. It is also produced by burning fossil fuels such as coal, natural gas and oil. Significant amounts of CO2 are also released by volcanoes. The thermos-physical properties of carbon dioxide are listed in Table 7. Huge amounts of methane are buried in the earth's crust in the form of natural gas and on the ocean floor in the form of methane hydrates. Moreover, mud volcanoes, landfills, and animal digestion release methane. The thermos-physical properties of methane are listed in Table 8. The vapor pressure is the basis of all equilibrium calculation. The vapor pressure curves for the three molecules of interest are shown in Fig. 6.

Fig. 6: Vapor pressure of the main components present in biogas (■) H2S critical point ; (▲) CO2 critical point ; (•) CH4 critical point ; (□) H2S triple point ; (Δ) CO2 triple point ; (○) CH4 triple point ; (----) H2S ; ( ____ ) CO2 ; ( …… ) CH4

The vapor pressures are calculated using Antoine Equation [START_REF] Eurobserv | Biogas barometer[END_REF]. The carbon dioxide vapor pressures above -76.36 °C were retrieved from the works of Kidnay [START_REF] Kidnay | Vapor-Liquid Equilibria in the Nitrogen + Carbon Dioxide + Propane System from 240 to 330 K at Pressures to 15 MPa[END_REF], Yarym-Agaev [START_REF] Yarym-Agaev | Vapor-Liquid Equilibrium and Volumetric Properties of the Liquid Phase of the gamma-Butyrolactone-Carbon Dioxide System at Increasing Pressures[END_REF], Miller [START_REF] Miller | The Joule-Thomson Effect in Carbon Dioxide[END_REF] and Del Rio [START_REF] Del Rio | The vapour pressure of CO2 from 194 to 243 K[END_REF]. As shown in Fig. 6, the vapor pressure curve of CO2 continues at temperatures lower than the triple point temperature (-56.56 °C) where it passes from Vapor -Liquid Equilibrium to Vapor -Solid Equilibrium. The constants used by Antoine Equation ( 2) are listed in Table 9 for each component. 2.3. Thermodynamic properties of the gas mixture (biogas)

log 𝑃 = 𝐴 -( 𝐵 𝑇 + 𝐶 ) (2)

Phase equilibrium behavior of biogas

In this section, the gas mixture is assumed to consist of methane (1) and carbon dioxide (2) because all the other impurities are present at very low concentrations depending on the type of substrate as shown in Table 4. Further, the upgrading process is generally made after purification that consists in the elimination of all these pollutants such as H2S, H2O and siloxanes.

The liquefaction of biomethane requires very low temperatures, therefore a cryogenic technology for biogas upgrading could be envisaged. This technology requires correct design of heat exchangers to achieve maximal purity of biomethane, that's why the knowledge of the pressure -temperature behavior of the binary mixture methane -carbon dioxide is essential [START_REF] Riva | Solid -Liquid -Vapor Equilibrium Models for Cryogenic Biogas Upgrading[END_REF]. Fig. 7 presents the phase equilibrium behavior for the methane -carbon dioxide system. Fig. 7: Pressure -Temperature equilibrium behavior for the CH4 -CO2 system [START_REF] Riva | Solid -Liquid -Vapor Equilibrium Models for Cryogenic Biogas Upgrading[END_REF] (

■) CH4 triple point ; (•) CH4 critical point ; (□) CO2 triple point ; (○) CO2 critical point ; (▲) mixture quadruple point ; (--) vapor-liquid critical locus ; (---) three-phase loci ; (-) pure compound phase equilibria

In the context of CO2 separation by solidification of CO2, the authors [START_REF] Riva | Solid -Liquid -Vapor Equilibrium Models for Cryogenic Biogas Upgrading[END_REF] have focused on the solid phase because thanks to the high triple point of carbon dioxyde as seen in Table 7, CO2 could be separated from methane directly from its gaseous phase into a solid phase for pressures close to the atmospheric pressure.

In order to understand the phase equilibrium behavior of the system CH4 (1) -H2S (2), Langè et al. [START_REF] Langè | Phase Behavior of System Methane + Hydrogen Sulfide[END_REF] have studied this aspect for temperatures from 70 K up to the critical tempearture of H2S and pressures up to 250 MPa as seen in Fig. 8.

Density and dynamic viscosity of biogas

All the thermophysical properties discussed in this section are calculated using the software REFPROP V9.0 which calculates the thermodynamic and transport properties of industrially important fluids and their mixtures. The equation of state for calculating these properties is the GERG (European Gas Research Group) -2008 equation [START_REF] Kunz | The GERG -2008 Wide-Range Equation of State for Natural Gases and Other Mixtures: An Expansion of GERG -2004[END_REF].

Depending on its composition, biogas has characteristics that it is interesting to investigate such as density and viscosity.

At a pressure of 1,103 bar, slightly higher than the atmospheric pressure, the variation of density as a function of temperature is shown in Fig. 9.

Fig. 9: Biogas density as a function of temperature ( ____ ) Air ; ( …… ) Biogas with 40 mol% of CO2 ; (----) Biogas with 35 mol% of CO2

As seen in Fig. 9, biogas is lighter than air. Moreover, its density depends on the carbon dioxide content. The density of a biogas rich on carbon dioxide will be greater than that of a biogas containing an inferior CO2 concentration. This is due to the molecular weight of CO2 which is greater than that of methane as seen in Tables 7 and8.

At a pressure of 1,103 bar, the evolution of the viscosity of the biogas as a function of temperature is depicted in Fig. 10. As density, the biogas viscosity increases with the CO2 concentration. This is due to the higher viscosity of carbon dioxide compared to methane, at equal pressure and temperature as shown in Tables 7 and8.

Thermal conductivity of biogas

The thermal conductivity is a physical property which describes the ability of gases to conduct heat. The phenomenon of heat conduction in gases is explained by the kinetic gas theory which treats the collisions between the molecules. The thermal conductivity depends on thermal capacity at constant volume and viscosity [START_REF] Shao | Engineering Heat Transfer[END_REF]. At a pressure of 1,103 bar, the variation of thermal conductivity of biogas as a function of temperature is shown in Fig. 11. As shown in Fig. 11, the thermal conductivity increases with temperature. This is due to collisions between the molecules of biogas that increase with temperature, resulting a thermal energy transfer increase.

Thermal capacities

The constant pressure heat capacity and the constant volume heat capacity are respectively related by the following thermodynamic Equations ( 3) and [START_REF] De La Farge | Le biogaz, procédés de fermentation méthanique[END_REF].

𝐶 𝑃 = ( 𝜕𝐻 𝜕𝑇 ) 𝑃 (3)
𝐶 𝑉 = ( 𝜕𝑈 𝜕𝑇 ) 𝑉 (4)
At a pressure of 1,103 bar, the variations of heat capacities of biogas as a function of temperature are shown in Fig. 12 and13. Fig. 12 and 13 show that biogas specific heat capacities strongly depend on the methane concentration unlike the density, which is inversely proportional to the concentration of methane in the biogas. According to Vardanjan et al. [START_REF] Vardanjan | Biogas specific heat capacity variations during upgrading[END_REF], an increase of methane concentration from 50 % to 75 % will result in the increase of heat capacity by 17 %. In Fig. 12 and13, heat capacities increased by 5 % after the passage of the methane concentration from 60 to 65 %.

Energy content in biogas

An important property of biogas is the lower heating value (LHV) which measures its energy value. The LHV of biogas is proportional to its methane content. For example, a biogas containing 70 mol% of methane at 15 °C and at atmospheric pressure has a LHV equal to 6.6 kWh/m 3 [START_REF] Gélin | Complete oxidation of methane at low temperature over noble metal based catalysts: a review[END_REF].

The energy value of biogas can also be evaluated by the Wobbe Index. Compared to the calorific value of biogas which has been treated by a lot of authors, only few studies have been conducted on the Wobbe index of biogas. According to the Danish Technological Institute (DTI) [35], a biogas containing 60 mol% of methane, 38 mol% of carbon dioxide and 2 mol% of others has a Wobbe index of 19.5 MJ/m 3 .

Conclusion

This chapter has allowed identifying the thermophysical properties of biogas. These properties are essential to design and optimize technologies for biogas purification and upgrading.

The thermophysical properties of biogas discussed in this chapter strongly depend on the composition of biogas, especially the concentrations of methane and carbon dioxide.

The presence of minor compounds such as H2O, N2, O2 and H2S could change the physical properties of biogas. For example, landfill gas includes small amounts of nitrogen and oxygen which affect the phase behavior of the CH4 -CO2 system.

The technologies used for purification and upgrading of biogas will be discussed in the following section and a particular attention will be paid to the elimination of hydrogen sulfide.

Chapter 3: From molecules to the process

Résumé :

La valorisation du biogaz nécessite la mise en oeuvre d'un procédé de purification qui consiste à éliminer du biogaz brut toutes les substances indésirables afin d'augmenter au maximum sa teneur en méthane.

Ce chapitre présente une étude comparative des méthodes connues de séparation de H2S du biogaz. La méthode de séparation devra permettre de réduire la teneur résiduelle en H2S à moins de 1 ppm.

La première technique de séparation étudiée est l'absorption dans des contacteurs gaz -liquide. L'accent a été porté sur les principes fondamentaux basés sur l'absorption avec des solvants de type physique et chimique ainsi que leur efficacité dans l'élimination de l'H2S.

La deuxième technique de traitement de l'H2S étudiée est l'adsorption sur des solides microporeux comme le charbon actif et les zéolithes.

Le troisième procédé discuté est la séparation membranaire dont la performance dépend de deux paramètres qui sont : la perméabilité et la sélectivité membranaires. En général, les membranes de perméabilité élevée présentent une faible sélectivité et inversement. Cette méthode présente quelques inconvénients comme le risque de rupture dû au gradient de pression qui constitue la force motrice, l'exposition à certains solvants qui peut endommager ou boucher la membrane, le prix élevé des membranes ou encore les pertes en méthane qui peuvent être considérables.

La dernière technologie séparative étudiée lors de ce chapitre est la condensation cryogénique basée sur la thermodynamique des équilibres de phases.

Cette synthèse bibliographique a permis de comparer les différents procédés selon plusieurs critères comme l'efficacité de séparation, l'impact environnemental et les coûts d'investissement et d'exploitation.

Finalement, le choix s'est porté sur les deux technologies suivantes utilisées pour l'élimination du sulfure d'hydrogène:

• Absorption chimique dans une colonne à garnissage structuré utilisant l'hydroxyde de sodium (NaOH) comme solvant.

• Adsorption dans un lit fixe de charbon actif.

Introduction

Biogas purification requires the removal of minor compounds despite their low amount relative to methane. Among these compounds, the removal of water vapor and ammonia are sufficiently documented in the literature and do not constitute a technical difficulty. However, the treatment of hydrogen sulfide present at low concentration is a challenge. This gas is formed when organic material containing sulfur is decomposed under anaerobic conditions. It is very corrosive on most metals which negatively affects the operation and viability of equipment especially pumps, heat exchangers and pipes. In the following section, conventional purification and upgrading technologies will be described.

Absorption technology

In chemistry, absorption is an operation by which a substance combined in one state is transferred into another substance of a different state. The most frequent use of absorption is the separation of a gas mixture by the absorption of part of the mixture in a solvent. The two phases are brought into contact in an absorption column and are allowed to exchange mass and energy across their common interface, where the flux of H2S transferred is calculated using Equation ( 5) by applying the two-film theory presented in Fig. 14. This theory assumes that the mass transfer resistance is located on the boundary layer on the gas side and the liquid side respectively. The mass transfer between the liquid and the vapor phases heavily depends on the effective interfacial area.

𝑁 𝐻 2 𝑆, 𝑧 = 𝑘 𝐺 𝑅 𝑇 (𝑝 𝐻 2 𝑆 -𝑝 𝐻 2 𝑆 * ) = 𝑘 𝐿 (𝐶 𝐻 2 𝑆 * -𝐶 𝐻 2 𝑆 ) (5)
Where the gas-phase and liquid-phase mass transfer coefficients are calculated using Equations ( 6) and ( 7) respectively.

𝑘 𝐺 = 𝐷 𝐻 2 𝑆,𝐺 𝛿 𝐺 (6)
𝑘 𝐿 = 𝐷 𝐻 2 𝑆,𝐿 𝛿 𝐿 [START_REF] Walsh | Handbook on biogas utilization[END_REF] Where: This theory assumes also that the thermodynamic equilibrium is reached in the interface. Knowing that the H2S concentration is low, therefore the Henry's law presented in Equation ( 8) is applicable.

NH2S, z [mol.s -1 .m -
𝑝 𝐻 2 𝑆 * = 𝐻 𝐻 2 𝑆 𝑥 𝐻 2 𝑆 (8)
Fig. 14: Two-film theory [START_REF] Kohl | Gas purification[END_REF] There are two types of absorption processes: physical absorption and chemical absorption, depending on whether there is any chemical reaction between the pollutant and the absorbent.

For example, when water absorbs oxygen from the air, a mass of the gas moves into the liquid, and no significant chemical reactions occur between the solvent and the solute. In this case, the process is commonly referred to as physical absorption.

Chemical absorption occurs, when a chemical reaction is carried out in the liquid phase, to dissolve the compound to be removed, and thus enhance the efficiency of the process. An example of chemical absorption is the process for absorbing CO2 and/or H2S with aqueous solution of sodium hydroxide. Chemical solvents are favored over physical solvents when the concentration of pollutants at low partial pressure has to be reduced to a very low level [START_REF] Mokhatab | Handbook of natural gas transmission and processing[END_REF]. However, if the impurity is available in the feed gas at high partial pressure, physical solvents might be preferred to chemical solvents [START_REF] Burr | A comparison of physical solvents for acid gas removal[END_REF] In the presence of a chemical reaction, the rate of absorption increases. The flux of H2S transferred depends on an acceleration factor E, and is expressed in Equation [START_REF][END_REF].

𝑁 𝐻 2 𝑆 = 𝐸 𝑘 𝐿 𝐶 𝐻 2 𝑆 * (9)
The acceleration factor E is equal to the ratio between the flux of H2S transferred in the presence of a chemical reaction and the flux transferred in the absence of a chemical reaction. This factor characterizes the importance of the chemical reaction on the transfer compared to the diffusion process. The addition of a chemical solvent such as sodium hydroxide for the removal of hydrogen sulfide allows a significant increase of the acceleration factor which becomes controlled by the reaction rate. It will also cause an increase in temperature because of the exothermicity of the reaction. For low concentrations, the increase in temperature will be small.

Physical solvents

The first solvent studied and used in absorption process is water. Hydrophilic compounds present in the biogas such as CO2 and H2S are absorbed better in water than the hydrophobic and non-polar compounds as methane.

Table 10 shows the solubility of some compounds of biogas in water. In order to improve the effectiveness of absorption processes, other solvents were tested for purification and upgrading of biogas. They allow the reduction of the size of columns, the energy used for the treatment and the volumes of solvents involved.

Table 10. Solubility of the main compounds of biogas in water [39]

Compound

Selexol® is a physical absorption process developed by Allied Chemical Corporation, then improved by Norton. Today, it is owned by Universal Oil Products (UOP). It is made up of Dimethyl Ethers of Polyethylene Glycol (DMEPG) whose molecular weight is about 272 g.mol -1 .

Rectisol® is one of the older physical absorption processes. It was developed by Linde and Lurgi in order to separate the acid gases present in the syngas from gasification of coal. It has also been used for separation of CO2 in the syngas of units producing hydrogen and ammonia. This process uses a methanol-based solvent (MeOH) whose chemical formula is CH3OH.

These solvents exhibit a high selectivity for H2S removal compared to the other compounds, in particular CO2. Other processes exist such as Purisol® based on the use of N-Methyl-2-Pyrrolidone (NMP), and Morphysorb® destined for the separation of acid gases at high concentrations, using N-Formyl-Morpholine (NFM). The main physical solvents are summarized in Table 11 where selected physical properties are compared. The solubility of a compound in a physical solvent is often expressed as the volume of gas absorbed by the solvent volume. The absorption capacity of H2S in physical solvents is frequently higher than that of CO2. As shown in Table 12, the solubility of H2S in NMP is very important. Moreover, this physical solvent has a high selectivity for H2S. Table 12 also shows a very high solubility of water in some physical solvents. This can be harmful to these physical absorbents because the presence of water in the gas phase will cause its accumulation in the solvent and therefore the reduction of the solvent absorption capacity. Operating at ambient temperature or higher should be avoided during use of some physical solvents in absorption operations because it leads to solvent losses by volatility. A chiller is necessary because the solubility of acid gases in physical solvents is favored by low temperatures. The two most volatile physical solvents are methanol and NMP. They use absorption columns with respective temperatures of -30 °C and -5 °C to prevent evaporation of the product and improve the solubilization of the acid gases. However, the DMEPG shows minimal evaporation losses at ambient temperature due to its very low vapor pressure. Furthermore, it becomes viscous at low temperatures. The physical solvents such as DMEPG and NFM must be used at ambient or higher temperatures.

Chemical solvents

Alkanolamines are most commonly used in acid gas absorption processes. Their molecular structure contains at least both amino (-N) and hydroxyl (-OH) functional groups. The hydroxyl functional group increases the solubility of acid gases in water and reduces the solvent vapor pressure. The amino functional group provides the necessary alkalinity in aqueous solution to ensure the absorption of acid gases.

At equilibrium, in aqueous solution, the reactions between the alkanolamines (R1R2R3N) and acid gases, particularly CO2 and H2S are described by the following chemical equilibria [START_REF] Archane | Etude de l'absorption des gaz acides dans des solvants mixtes[END_REF]:

 Self ionization of water: 2𝐻 2 𝑂 ↔ 𝐻 3 𝑂 + + 𝑂𝐻 - (R.3)
 Protonation of alkanolamine:

𝑅 1 𝑅 2 𝑅 3 𝑁 + 𝐻 3 𝑂 + ↔ 𝑅 1 𝑅 2 𝑅 3 𝑁𝐻 + + 𝐻 2 𝑂 (R.4)
 Hydrolysis of hydrogen sulfide:

𝐻 2 𝑆 + 𝐻 2 𝑂 ↔ 𝐻𝑆 -+ 𝐻 3 𝑂 + (R.5)
 Bisulfide ion dissociation:

𝐻𝑆 -+ 𝐻 2 𝑂 ↔ 𝑆 2-+ 𝐻 3 𝑂 + (R.6)
 Carbon dioxide hydrolysis:

𝐶𝑂 2 + 2𝐻 2 𝑂 ↔ 𝐻𝐶𝑂 3 -+ 𝐻 3 𝑂 + (R.7)
 Bicarbonate ion dissociation:

𝐻𝐶𝑂 3 -+ 𝐻 2 𝑂 ↔ 𝐶𝑂 3 2-+ 𝐻 3 𝑂 + (R.8)
 Carbamate hydrolysis for primary and secondary amines:

𝑅 1 𝑅 2 𝑁𝐻 + 𝐻𝐶𝑂 3 -↔ 𝑅 1 𝑅 2 𝑁𝐶𝑂𝑂 -+ 𝐻 2 𝑂 (R.9)
The most used amines in industry are:

 Primary amines: monoethanolamine (MEA) and diglycolamine (DGA).  Secondary amines: diethanolamine (DEA) and diisopropanolamine (DIPA).  Tertiary amines: méthyldiethanolamine (MDEA) and triethanolamine (TEA).

The triethanolamine was the first used in gas processing industry. Table 13 lists the physical properties of some of these chemical solvents. Fig. 15 shows the chemical structure of some alkanolamines. The primary, secondary or tertiary amines are distinguished according to the degree of substitution of the nitrogen atom. Fig. 15: Chemical structure of some alkanolamines [START_REF] Lecomte | Le captage du CO2, des technologies pour réduire les émissions de gaz à effet de serre[END_REF] Primary amines such as monoethanolamine (MEA) are very reactive with H2S and CO2, but they react also with other impurities present in the biogas, which leads to a significant energy requirement for regeneration. They are also susceptible to degradation and corrosion.

Secondary and tertiary amines require less energy for regeneration and are less susceptible to degradation. However, they are less reactive than primary amines and are therefore used for less demanding objectives in terms of purity.

Bottoms [START_REF] Bottoms | Process for separating acidic gases[END_REF] was the first in 1930 to study the amines in the gas processing industry. He filed the first patent for the process of absorption of acid gases by ethanolamines.

In 1997, Pani et al. [START_REF] Pani | Absorption of H2S by Aqueous Methyldiethanolamine Solution at 296 and 343 K[END_REF] experimentally studied the absorption of hydrogen sulphide by a MDEA solution in a temperature range varying between 296 and 343 K. They developed a device to determine the kinetics of absorption of acid gases by alkanolamine solutions, The H2S concentrations used vary between 0 and 0.44 moles of gas per mole of amine. A mass transfer model incorporating a reversible reaction was used to test the experimental flow absorbed and to determine the diffusion coefficient of the MDEA.

In 1984, Blauwhoff et al. [START_REF] Blauwhoff | A Study on the Reaction between CO2, and Alkanolamines in Aqueous Solutions[END_REF] investigated the selective absorption of hydrogen sulphide and have shown that it significantly reduces the cost of gas treatment, by reducing the CO2 flux transferred.

It is important to note that the solvents presented in this section are more suitable for the elimination of high H2S concentrations present in natural gas. Other chemical solvents more suitable for the treatment of biogas are available for the absorption of hydrogen sulfide:  Aqueous solution of potassium carbonate (K2CO3), to which are added additives such as amines. This type of chemical solvent is used in many gas treatment processes such as Flexsorb HP and Catacarb processes, developed respectively by ExxonMobil® and Eickmeyer®.  Aqueous solution of sodium hydroxide (NaOH), also known as caustic soda.  Iron chelate solution Fe (III) according to the following reaction (R.10).

H / OH -CH2 -CH2 -N \ H HO \ CH2 \ CH2 \ N -H / CH2 / CH2 / HO HO \ CH2 \ CH2 \ N -CH2 -CH2 -OH / CH2 / CH2 / HO 𝐻 2 𝑆 + 𝐹𝑒 3+ → 𝑆 0 + 2 𝐹𝑒 2+ + 2 𝐻 + (R.10)
According Neumann and Lynn [START_REF] Neumann | Oxidative absorption of H2S and O2 by iron chelate solutions[END_REF], absorption of hydrogen sulfide by iron chelate solutions are advantageous for achieving high reaction rates (99.99 % H2S removal).

Hybrid solvents

There are processes that combine both physical and chemical absorption, implementing aqueous mixtures that contain water, amine and an organic solvent. This type of processes has been developed to treat gases containing significant fractions of acid gases. An example of such processes is the Hi-Pure process, where the gas to treat is put into contact, firstly with an aqueous solution of potassium carbonate, then with an aqueous solution of amine.

Patented by Shell, Sulfinol process uses an hybrid solvent containing a physical solvent called sulfolane, water and a chemical solvent. This latter determines the name of the mixture. It is called Sulfinol-D when DIPA is used as chemical solvent and Sulfinol-M when MDEA is used. Sulfinol-M is used for the selective removal of H2S in the presence of CO2.

Amisol is a process also used for the selective removal of H2S in the presence of CO2. It was developed by Bratzler and Doerges in 1974. The mixture used is composed of methanol as a physical solvent, and DEA or MEA as a chemical solvent.

Gas-liquid contactors

Application of the principle of absorption is based on contacting the gas and liquid phases in a gasliquid contactor. This latter, also called absorber, aims to achieve better mass exchange between the two phases in contact. The efficiency of a gas-liquid contactor is dependent on phenomena involved in the absorption process:

 Transfer laws in the vicinities of interfaces, in particular the transfer coefficients and the interfacial area.  Transport laws, in particular diffusion coefficients.  The thermodynamic equilibrium at the interface, especially the solubility of acid gases in the solvent.  The chemical reaction kinetics: the reaction schemes, the kinetic constants and orders of reactions.

Film thickness, residence time and flow regime also all have a vey important impact on the effeciency of the contactor.

The most common concept to evaluate the separation effeciency of packed columns is expressed in terms of Height Equivalent to a Theoretical Plate (HETP).

There is a large number of gas-liquid contactors in the industry, for mass and heat transfer between the two phases, as seen in Fig. 16. Generally, the gas and the liquid flow counter-currently in order to obtain significant concentration gradients and better absorption rate.

The gas-liquid contactors can be classified according to the dispersion mode of phases. Despite a few exceptions, the liquid phase is naturally the dispersed phase in gas treatment application. The choice of the absorber is mainly related to the physicochemical properties of the gas to be treated and to the chemical reactions involved, as well as gas and liquid flow rates implemented. Table 14 ranks the gas-liquid contactors according to the continuous phase, the fluid inclusion type and the main associated applications (See Fig. 16).

Manufacture of nitric acid

To promote the mass transfer, the absorbers are usually equipped with internal devices to generate the largest interfacial area in order to achieve better mass exchange between the two phases in contact.

In prior years, plate columns were heavily favored over packed columns. But, nowadays, packing columns are the most used in gas absorption applications. Only few specific applications with special design requirement can lead to different choices as in the case of very large flow rates or very soluble compounds where it is preferable to use plate or spray columns. Fig. 16: Main gas-liquid contactors [START_REF] Roquet | Modélisation et étude expérimentale de l'absorption réactive multiconstituants[END_REF] In a packed column, the gas and liquid normally flow counter currently as seen in Fig. 17. The liquid is sprayed from the top of the column to flow by gravity on the packing forming a large-area liquid film. The liquid enters in contact with the gas injected from the bottom of the column. Liquid flow must be sufficient to ensure uniform wetting of the packing and must not exceed a certain threshold in order to avoid flooding of the column.

The selection of the packing type and material is a very important issue in packed column design. The material should respect certain requirements as weight, pressure drop and especially corrosion resistance. It exists two types of packings: Those consisting of packing elements placed in a random disposition and those containing corrugated sheets arranged in an orderly manner. The first one is called random packing and the latter is called structured packing. Fig. 18 shows the two types of packing produced by Sulzer®.

Fig. 17: Schematic representation of a packed column

Today, structured packings are much more used than random packings. structured packings ensure a better transfer with a minimal pressure drop.

Adsorption technology

Adsorption is a growing process, that is increasingly used in biogas purification. It removes water vapor, odors and other impurities as hydrogen sulfide, from biogas streams. Adsorption is a surface phenomenon that occurs between a vapor or liquid phase and a solid. Molecules, ions or atoms forming a solid surface, are subjected to asymmetric forces that result in an attractive force field. This attractive force has a limited range, but enough to attract gas or liquid molecules located in the immediate vicinity of the interface. These are forces that cause fixing of molecules on the surface.

Adsorption is classified according to the nature of the interactions that allow the fixing of the adsorbate on the surface of the adsorbent. There are two types of adsorption, according to the nature of the interactions: physical and chemical adsorption.

Physical adsorption is a reversible phenomenon characterized by weak interaction forces as Van der Waals' forces, while chemical adsorption is usually irreversible involving strong binding energies.

In contrast to physical adsorption, chemisorption grows at high temperatures and causes the formation of a chemical compound on the surface of the adsorbent. Table 15 lists the criteria to differentiate between physical and chemical adsorption.

Mechanism of adsorption

During adsorption process, the fluid molecules bind to the surface of a solid following three steps, describing the mass transfer from the fluid phase to the solid surface.

During the first step called external diffusion, the molecules of the fluid phase migrate to the vicinity of the outer surface of the solid particles. To model the transfer of the fluid phase towards the outer surface of the solid phase, the Equation ( 10) is often used: The second step is called internal diffusion. It results from the transfer of fluid phase particles from the outside of the solid surface inside the pores. To simplify the problem to one spatial dimension, the pore is assumed spherical. The flow transferred into the pore is expressed by Equation [START_REF] Görisch | La production de biogaz[END_REF].

- 𝑑𝐶 𝑡 𝑑𝑡 = 𝑘 𝑓 ( 𝑎 𝑎𝑑𝑠 𝑉 ) (𝐶 𝑡 -𝐶 𝑒 ) (10) Where
𝐽 = -𝐷 𝑝 𝜀 𝑝 𝜏 𝜕𝐶 𝑝 𝜕𝑟 (11)
Where: The last step of the adsorption mechanism is the surface diffusion. It corresponds to the attachment of the vapor phase particles on the surface of the adsorbent. This step is very quick and independent of the overall process. The flux is then defined by Equation [START_REF] Deublein | Biogas from waste and renewable resources : an introduction[END_REF].

J [mol.m -2 .
𝐽 = -𝐷 𝑠 𝜀 𝑝 𝜏 𝜕𝑞 𝜕𝑟 (12)
Where Ds [m 2 .s -1 ] is the surface diffusivity and q [mol.kg -1 ] is the amount adsorbed.

Fig. 19 shows the different steps of the adsorption mechanism.

Materials used for H2S adsorption

The adsorbents are microporous solids, characterized by high surface area per unit weight, from 100 to over 2000 m 2 .g -1 . The classification of the International Union of Pure and Applied Chemistry (IUPAC) defines three kinds of pores by their size:  Microporous materials have pore diameters of less than 2 nm.  Mesoporous materials have pore diameters between 2 and 50 nm.  Macroporous materials have pore diameters of greater than 50 nm.

The most widely used adsorbents in industry remain: activated carbon, zeolite, silica gel and activated alumina. Other materials rarely used today as red mud may have good adsorption performance in H2S removal.

 Activated carbon

Activated carbon is characterized by a high degree of microporosity. This structure gives the activated carbon its very large surface area. This essential feature allows the activated carbon to be by far, the most widely used adsorbent in industry. Pores size between of 0.5 and 1 nm were found by Yan et al. [START_REF] Yan | Influence of Surface Properties on the Mechanism of H2S Removal by Alkaline Activated Carbons[END_REF] to have the best adsorption capacity. Activated carbon is characterized by a non-polar surface allowing it to preferentially adsorb nonpolar compounds. Activated carbon can be impregnated with potassium hydroxide (KOH) or sodium hydroxide (NaOH) which acts as catalysts to remove H2S, because non-impregnated activated carbon is a weak catalyst, and however, removes hydrogen sulfide at a much slower rate.

Bandosz [START_REF] Bandosz | On the adsorption/oxidation of hydrogen sulfide on activated carbons at ambient temperatures[END_REF] has shown that using low hydrogen sulfide concentrations, with a sufficient time in laboratory tests leads to comparable removal capacities of both impregnated and non-impregnated activated carbons. But in on-site applications, removal capacities vary greatly because of the presence of other constituents such as volatile organic compounds (VOCs) which may inhibit the removal capacity. According to Abatzoglou and Boivin [START_REF] Abatzoglou | A review of biogas purification processes[END_REF], the typical H2S adsorption capacities for respectively impregnated and non-impregnated activated carbons are 150 and 20 mg H2S/g of activated carbon.

Activated carbon is produced in different shapes and sizes depending on the application for which it is used:

 Extruded activated carbon (EAC).  Granular activated carbon (GAC).  Powder activated carbon (PAC). Fig. 20 shows the different shapes of commercial activated carbons.

Fig. 20: Shapes of commercial activated carbons [53]  Zeolites

Zeolites are microporous adsorbent materials. The size of the pores can be adjusted by ion exchange to catalyze selective reactions. According to the International Zeolite Association (IZA), in 2007, there are 176 crystal structures identified by a three letter code. The vast majority of these structures are synthetic while the rest exists in nature. The zeolites are particularly effective for removing polar compounds such as water and H2S, from non-polar gas streams, such as methane. Zeolites are low capacity adsorbents, with a surface area not exceeding 900 m 2 .g -1 but they have a good selectivity. Compared to activated carbon, they are less sensitive to heat.

The adsorption of H2S present in biogas is one of the application envisaged with zeolites. The measured adsorption capacities remain well below the purification yields obtainable with other systems. Yasyerli et al. [START_REF] Yasyerli | Removal of hydrogen sulfide by clinoptilolite in a fixed bed adsorber[END_REF] determined an adsorption capacity of 30 mg.g -1 with clinoptilolite, a natural zeolite during the treatment of a real biogas. Cosoli et al. [START_REF] Cosoli | Hydrogen sulphide removal from biogas by zeolite adsorption : Part i. GCMC molecular simulations[END_REF] reported an adsorption capacity of 40 mg.g -1 on synthetic zeolites, for H2S concentration of 1000 mg.m -3 .

Factors affecting the adsorption

Literature concerning the factors affecting adsorption dates back to 1914 [START_REF] Johns | The simulation of gold adsorption by carbon using a film diffusion model[END_REF]. The main factors affecting the adsorption rate are the temperature, the surface area and the porosity of the adsorbent, the competition between species, and the polarity of the adsorbent and the adsorbate.

 Temperature

Fig. 21 shows that during adsorption processes, the adsorbed amount increases as the temperature decreases. Moreover, physisorption releases heat. So, as any exothermic reaction, it is favored by low temperatures.

Fig. 21: Effect of temperature on some adsorbents [START_REF] Uop | UOP Molecular Sieves[END_REF]

Adsorbents: (───) Molecular sieves ; ( ─ ─ ─ ) Activated alumina ; ( ……. ) Silica gel

Unlike physical adsorption, chemical adsorption requires higher temperatures, because it is an endothermic phenomenon.

 Specific surface area

Adsorption performance increases with the specific surface area of the adsorbent. This proportionality was demonstrated by Bouchemal and Achour [START_REF] Bouchemal | Essais d'adsorption de la tyrosine sur charbon actif en poudre et en grain[END_REF] during adsorption study of tyrosine on activated carbon.  Selectivity

The concept of selectivity is crucial in the design of adsorption processes. The presence of competitive species at the surface of the adsorbent decreases the capacity of each species to be adsorbed. However, The higher the selectivity, the easier would be the separation.

 Pore size distribution

Adsorption is a surface phenomenon, hence the interest of porous structures. The porosity of the adsorbent material is therefore an important physical property. For example, the microporous activated carbon has a better adsorption capacity than the mesoporous activated carbon in the case of macromolecules. The thermal regeneration and impregnation could modify the pore volume of the adsorbent. The resolution of Equation ( 13) allows access to the porous distribution [START_REF] Evans | Capillary condensation and adsorption in cylindrical and slit-like pores[END_REF].

𝑁(𝑝 𝑝 0 ⁄ ) = ∫ 𝑁(𝑝 𝑝 0 , 𝑤 ⁄ ) 𝑓(𝑤) 𝑑𝑤 𝑤 𝑚𝑎𝑥 𝑤 𝑚𝑖𝑛 (13)
Where:

N (p/p0) [-], is the experimental adsorption isotherm. N (p/p0 , w) [-], is the local isotherm in a pore size of w. f (w) [-], is the pore distribution function. w [nm], is the opening dimension of a pore.

 Molecular weight and structure

If the molecular weight of particles is low, it means that they are light and move faster than those with high molecular weight. The probability of being adsorbed is therefore much greater.

If the molecular structure of particle is large, pores are filled rapidly with low yields to saturation, causing the decrease of free sites for other molecules.

 Polarity

For more affinity between the adsorbent and the adsorbate, they must have the same polarity [START_REF] Lesage | Etude d'un procédé hybride Adsorption / Bioréacteur à membranes pour le traitement des effluents industriels[END_REF]. For example, the structure of activated carbons is non-polar and therefore promotes the adsorption of nonpolar molecules. Hydrogen sulfide is a polar gas, it is adsorbed on the polar surfaces in the absence of water vapor. In the presence of water vapor in the gas, there is competitive adsorption to the advantage of water vapor which has a much higher partial pressure and which is much more polar than hydrogen sulfide.

Adsorption isotherms

In order to model the binding of a gas over a bed of adsorbent, it is necessary to choose a model to represent interactions between the gas and the solid. An adsorption isotherm is the curve presenting the static adsorption capacity of an adsorbate/adsorbent system at a given temperature. The curve presents the specific amount adsorbed, Na as a function of the relative pressure, P/P0 as seen in Fig. 22. According to the classification of IUPAC based on the one established by Brunauer [START_REF] Brunauer | [END_REF], there are six different isotherms profiles represented in Fig. 22.

The adsorption isotherm type I, is distinguished by the existence of a horizontal line, which results in the saturation of the adsorbent. This isotherm is characteristic of adsorbent having micropores, which are filled at low relative pressures. This is essentially a monolayer adsorption, often described by the Langmuir isotherm, where there may be strong interactions involved. The equation that describes the Langmuir isotherm is presented in Table 16.

The adsorption isotherm type II are widespread for non-porous or macroporous solids. The absence of a clearly identifiable inflection point on the curve corresponding to the filling of a monolayer and the absence of a continuous increase of the amount adsorbed are indicatives of energy heterogeneity of the surface regarding the interactions adsorbate / adsorbent.

Fig. 22: Classification of adsorption isotherms [IUPAC]

The adsorption isotherm type III corresponds to non-porous or macroporous solids. This isotherm is characterized by weak interactions adsorbate / adsorbent.

The adsorption isotherms type IV and V are characterized by a filling of mesopores, and a capillary condensation in the pores. The interactions adsorbate / adsorbent for type V isotherm are weaker than those of type IV.

The adsorption isotherm type VI is very rare. It is encountered in the case of very homogeneous surfaces.

 Modeling of adsorption isotherms

Several models have been proposed to describe the experimental adsorption isotherms. Despite their common interest, the assumptions defined for each model are different such as those concerning interactions that hold the fluid molecules on the surface of the adsorbent. Most of these models are described in the literature such as: Freundlich model, Elovich model, Temkin model, Toth model and Langmuir model. The latter is the best known, and probably the most widely used to describe the adsorption isotherm. It was developed by Irving Langmuir in 1916 [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids[END_REF], sixteen years before obtaining the Nobel Prize in chemistry. The Langmuir model assumes uniform energies of adsorption onto the surface and no transmigration of adsorbate in the plane of the surface [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids[END_REF].

The Freundlich model assumes that as the adsorbate concentration increases, the concentration of adsorbate on the adsorbent surface also increases [START_REF] Freundlich | Uber die adsorption in losungen[END_REF]. This proportionality is explained by the Freundlich expression which is an exponential equation.

The Temkin model assumes that in adsorption, the binding energies are distributed uniformly, and that due to interactions between the adsorbent and the adsorbate, the heat of adsorption of all the molecules in the layer decreases linearly with coverage [START_REF] Temkin | Adsorption equilibrium and the kinetics of processes on nonhomogeneous surfaces and in the interaction between adsorbed molecules[END_REF].

The Elovich model is derived from a kinetic principle assuming that the adsorption sites increase exponentially with adsorption, implying a multilayer adsorption [START_REF] Elovich | Theory of adsorption from solutions of non electrolytes on solid (I) equation adsorption from solutions and the analysis of its simplest form, (II) verification of the equation of adsorption isotherm from solutions[END_REF].

The Toth model [START_REF] Toth | Calculation of the BET-compatible surface area from any type I isotherms measured above the critical temperature[END_REF] was developed based on an improvement of the Langmuir model to reduce the error between experimental and predicted data. This model is applied in the case of multilayer adsorption.

The equations defining the main models are shown in Table 16. ] is the variation of adsorption energy. K0 and K1 [m 3 .g -1 ] are the Temkin and Hill-de Boer equilibrium constants respectively. K2 [kJ.mol -1 ] is the energetic constant of the interaction between adsorbed molecules. KFG [m 3 .mol -1 ] is the Fowler-Guggenheim equilibrium constant. W [kJ.mol -1 ] is the interaction energy between adsorbed molecules.

Adsorption processes

There are two major classes of adsorption processes: the temperature swing adsorption processes (TSA) and the pressure swing adsorption processes (PSA).

 Temperature swing adsorption processes

The temperature swing adsorption is the oldest cyclic adsorption process. It consists of two main phases: the adsorption phase and the desorption phase, during which the adsorber is heated. A precooling phase is therefore commonly added to bring the temperature to a level similar to that desired for adsorption.

The main advantage of temperature swing adsorption process compared to pressure swing adsorption is to desorb more easily species strongly adsorbed. For this reason, the temperature swing adsorption processes are used, for example, to capture volatile organic compounds present in many effluents.

Temperature swing adsorption is used whenever the energy required to regenerate a bed is sufficiently large that long high thermal cycles are needed given the strength of the bond between the adsorbate and adsorbent. For example, adsorption of H2O on zeolites.

However, significant time required to heat and cool the adsorber, prevent the use of temperature swing adsorption process in fast cycle. Moreover, the adsorption columns used for temperature swing adsorption cycles are large in size, which has an impact on the cost of the installation. However, unlike pressure swing adsorption processes, that use mechanical energy, temperature swing adsorption processes can use residual heat, which then reduce their operating cost.

 Pressure swing adsorption processes

The pressure swing adsorption process was initially introduced as an alternative to temperature swing adsorption process. It is mainly used in separation of some gas species from a mixture of gases, this is another option that complements the traditional separation processes as absorption and cryogenic distillation. In a pressure swing adsorption process, the feed pressure is generally greater than atmospheric pressure. The regeneration pressure may be less than the atmospheric pressure, in this case the process is called vacuum swing adsorption (VSA).

Since the adsorption step is performed at a higher pressure than the pressure of desorption step, intermediate steps are necessary: a compression is required to move from the low to the high pressure at the end of the regeneration step. A decompression step is also necessary to reduce the pressure at the end of the adsorption step. These four steps are the elements of a basic cycle called the Skarstrom cycle. Steps which constitute this cycle, and the variation of the pressure as a function of the different phases, are shown in Fig. 23.

Fig. 23: Skarstrom cycle stages and pressure variations [68]

To ensure continuous production, the pressure swing adsorption process must have at least two separation columns. These columns suffer the four steps mentioned above, but with a temporal phase shift leading to a cyclic operation of the pressure swing adsorption process, in which one of the columns is regenerated, while the gas mixture is separated in the other one.

Currently, there are several hundred thousand pressure swing adsorption processes installed worldwide. Their size varies from 6 l.hr -1 , often for the production of medical grade oxygen with a purity of 90 % to 2000 m 3 .h -1 typically for the production of pure hydrogen at 99.999 % [START_REF] Shivaji | Pressure Swing Adsorption[END_REF].

For purification, temperature swing adsorption is generally the process of choice. For bulk separation, pressure swing adsorption is more suitable.

Membranes technology

A membrane can be defined as a physical barrier for the selective transport of chemical species. As seen in Fig. 24, it allows the restricted passage of one or more constituents. The flux passing through the membrane is called permeate, while the retained is called retentate or concentrate. One or the other of these two flows may be advantageous according to the intended application, and can therefore be used as final product [START_REF] Soni | A general model for membranebased separation processes[END_REF].

The driving force in a membrane can be a pressure gradient, a concentration gradient, a temperature gradient, or an electrochemical gradient. Thus, the membranes include a wide variety of materials and structures. Table 17 shows the main materials used by membranes manufacturers.

Fig. 24: Schematic representation of a membrane [71]

The membrane performance is evaluated for its ability to separate different species from a gas mixture, and transport a maximum quantity of gas at high speed. These two criteria are generally in competition, a membrane is more efficient when it presents the best compromise: flow / selectivity.

A membrane can be gaseous, liquid, solid or a combination thereof [START_REF] Lakshminarayanaiah | Equations of Membrane[END_REF]. It covers a wide range of applications such as ultrafiltration, microfiltration, reverse osmosis, pervaporation, electrodialysis and gas separation. In addition, other medical applications such as blood oxygenators and artificial kidneys require the use of membrane technology. Of the applications listed in Table 18, reverse osmosis and ultafiltration are the most widely used industrially. The two essential parameters in the operation of a membrane are: permeability and selectivity. These parameters are used to provide informations on the membrane, and to describe its performance on the transfer of material through the barrier and its ability to separate one or more chemical species in a gas mixture.

The selectivity expressed most often by separation factor is defined as the ratio of the compositions of components i and j in the permeate relative to the composition ratio of these components in the retentate.

𝑆 𝑓 𝑖,𝑗 = ( 𝑥 𝑖 𝑥 𝑗 ) 𝑝𝑒𝑟𝑚𝑒𝑎𝑡𝑒 ( 𝑥 𝑖 𝑥 𝑗 ) 𝑟𝑒𝑡𝑒𝑛𝑡𝑎𝑡𝑒 ( 14
)
The permeability is used to indicate the ability of the membrane to feed the permeate. In order to ensure an attractive performance, the permeability ∏i [mol.m -2 .Pa -1 .s -1 ] must be high to ensure a large transmembrane flux.

𝛱 𝑖 = 𝐽 𝑖 𝛥𝑃 𝑖 (15)
Where Ji [mol.m -2 .s -1 ] is the transferred flux and ΔPi [Pa] is the partial pressure difference of the constituent i through the membrane.

Generally, the membranes with a high permeability have a low selectivity and vice versa.

 Membranes for gas separation

The membranes can be classified according to different viewpoints. They can be divided according to the nature: biological or synthetic, according to the morphology, or according to the structure.

Several scientific journals classify membranes into porous and non-porous membranes depending on the structure of the material. In general, a membrane may be thick or thin, and its structure may be homogeneous or heterogeneous with a transfer mechanism which may be active or passive [START_REF] Chen | Développement de nouvelles membranes à base de polyimide pour la séparation CO2 / CH4[END_REF]. Fig. 25 shows the main types of membrane. The figure shows a thin interface that forms the membrane. This interface can be homogeneous at the molecular level, that is to say, completely uniform in composition and structure. It can also be heterogeneous, comprising for example pores.

Fig. 25: Diagram of the main types of membranes [75]

The main types of membranes used today for gas separation are membranes with dense polymeric materials, where transfers follow a solute solubility and diffusion mechanism based on Fick's law.

𝐽 𝑖 = 𝐷 𝑖𝑗 𝜕𝑐 𝑖 𝜕𝑥 (16)
The structure of a porous membrane is like a sponge, it is also very similar to a conventional filter [START_REF] Chen | Développement de nouvelles membranes à base de polyimide pour la séparation CO2 / CH4[END_REF]. Most of the materials used are characterized by tortuous and interconnected pores, whose precise geometry is inaccessible. Table 19 shows the different categories of membranes according to the size of their pores provided by the International Union of Pure and Applied Chemistry.

 Geometric configuration of membranes

The geometric design is an essential step in any membrane process as this factor defines the active area of the module. The first membranes used are: flat sheet and tubular membranes. Today, these systems are still available, but their use has declined because of their low efficiency and high cost. They are mostly replaced by spiral wound, and hollow fibers membranes. Fig. 26 shows the four membrane contactors mentioned above.

The flat sheet membranes are the oldest and simplest to use. They have planar configuration and are mainly rectangular, though other geometries exist for membrane modules designed to rotate. Modules may be stacked to provide a double deck.

Tubular membranes consist of tubes having an inner diameter between 4 and 25 mm [START_REF] Boucif | Modélisation et simulation de contacteurs membranaires pour les procédés d'absorption de gaz acides par solvant chimique[END_REF]. They are based on a simple technology, easy to use and clean, but they are large energy consumers for a low exchange surface area per unit volume.

Spiral wound membranes consist of a flat sheet membrane coiled on itself around a perforated tube, which collects the residue. As seen in Fig. 26.c, the feed flows axially in the channels, while the permeate flows along a spiral path towards the porous tube [START_REF] Boucif | Modélisation et simulation de contacteurs membranaires pour les procédés d'absorption de gaz acides par solvant chimique[END_REF]. The hollow fiber modules consist of a bundle of hollow fibers of outer diameter less than 1 mm. The main advantage of hollow fiber membrane is their compactness, due to the membrane's high packing density. Also, They can be operated at very high pressures, due to the absence of membrane support. One disadvantage of hollw fiber membrane is the pressure drop. Hence the importance of fiber length criterion in the design of a separation unit.

Overall, the selection of a given configuaration should be addressed individually based on membrane properties and the throughput rates desired [START_REF] Ohya | Polyimide membranes: applications, fabrications and properties[END_REF]. Table 20 shows the characteristics of different geometries of membranes.

 Biogas purification by membrane processes

In the case of biogas, separation of compounds is often limited to three species: methane, carbon dioxide and hydrogen sulfide. Polyurethane membranes show significant selectivity between methane and hydrogen sulfide compared to that of methane and carbon dioxide. The other types of membranes such as polyimide, polyamide, polysulfone and cellulose acetate membranes show a significant selectivity between methane and carbon dioxide, at the expense of selectivity between methane and hydrogen sulfide. Today, despite the limited studies and few results, it is possible to obtain methane concentrations above 95 % in the retentate. However, this residue may contain significant hydrogen sulfide concentrations. But the main disadvantage of the membrane process remains the methane loss, with methane weight percentages which can reach 15 % in the permeate. In addition, the membrane processes require treatment upstream, to separate the volatile organic compounds and water vapor. The membrane's resistance to breaking due to the pressure gradient is one important technical limitation. Exposure to certain solvents and materials causes the membrane to get either damaged or blocked up. These limitations are of great importance since membranes usually are expensive.

Cryogenic technology

The term cryogenic refers to the science of very low temperatures. The cryogenic separation process consists on passing the pollutant from the gas phase to the liquid or solid phase by lowering the temperature, in order to separate it from the carrier gas which is the methane. The pollutant is recovered, and will then be destroyed or valorized for possible use. This technology is based on the thermodynamics of phase equilibria. The thermodynamic equilibrium between the different phases results in a graph called phase diagram, which generally uses pressure and temperature as variables.

To purify and upgrade biogas with the cryogenic technology, the gas is chilled and the differences in condensation or solidification temperatures for different compounds are used to separate impurities and carbon dioxide from biogas, which can be seen in Table 21. The technology can be used to upgrade biogas by cooling it at atmospheric pressure in order to seperate carbon dioxide at temperatures related to the CO2 partial pressures upstream and downstream the refrigeration unit, typically from -90 °C to -120 °C. Then, the biogas is chilled to produce liquid biogas (LBG) at temperatures which depend on the pressure: -120 °C at 1.5 MPa to -162 °C at atmospheric pressure.

To ensure a high purity of the products and an optimal operation, all traces of hydrogen sulfide should be removed upstream of the process using one of the conventional technologies presented above.

Depending on the temperature of the process, different purity grades can be reached. A lower temperature results in a higher removal efficiency. A study was performed to condense the volatile organic silicon compounds (VOSiC) contained in biogas. This process involves cooling the biogas at different temperatures, in order to evaluate the effect of cold on volatile organic silicon compounds removal. Table 22 shows the results obtained by different authors using this process at different temperatures. Production of liquid biogas is a suitable upgrading technology for landfill gas, which usually consists of significant amount of nitrogen, hard to separate from methane with conventional technologies. However, when the methane is liquefied, nitrogen can be separated due to its lower condensation temperature [START_REF] Benjaminsson | Nya renings -och uppgraderingstekniker för biogas[END_REF].

Table 22. Effect of temperature on the abatement of volatile organic silicon compounds

Condensation temperature [°C] Abatement rate of VOSiC [%] References

Cooling biogas to very low temperatures is energy intensive but in some occasions the product is more valuable. If the biogas production plant is situated on the countryside, far from the end users, it is more space efficient to transport biogas in its liquid state. Today pressurised (200 bar) gas is delivered in gas vessels stored on a mobile compressed biogas (CBG) storage, leading to transportation of a huge share of steel, compared to gas [START_REF] Pettersson | LCNG-stuidemöjligheter med LNG i fordonsgasförsörjningen i Sverige[END_REF].

Producing liquid biogas also leads to a renewable fuel available for heavy duty vehicles. The fuel can be stored as liquid biogas on the vehicle, which increase the driving distance per tank. The requirement is that the vehicle is running frequently, otherwise liquid biogas will vapourize and the methane will be vented to the atmosphere [START_REF] Johansson | Production of liquid biogas, LBG, with cryogenic and conventional upgrading technology. Description of systems and evaluations of energy balances[END_REF].

An advantage of the cryogenic technology is that it does not need any water or solvent to function, although it requires external cooling equipment such as a refrigeration system.

Choice of the separation process

Various technologies are available in order to purify and upgrade biogas. Water scrubbing and pressure swing adsorption dominated the market until 2008. But lately, membrane separation units and chemical scrubbers have increased their market share as seen in Fig. 27.

Fig. 27: Most technologies used for the purification and upgrading of biogas [International Energy Agency]

The choice of the technology to be used depends on multiple parameters, such as the final use, the incentives, the flow rate, the nature and diversity of species present, and the concentrations to treat. The solution chosen has to meet different requirements, both technical and economic. Other criteria can sometimes be decisive: environment, maintenance, temperature and pressure … There is therefore, no universal treatment technology.

Table 23 provides an overview of the different biogas purification and upgrading techniques. It indicates the methane concentration in the purified gas, loss of methane and the substances used in the process such as water and chemicals solvents.

The most important criterion for environmental impact of technology is methane losses. The portion of methane which slips away from raw biogas because of the separation technology itself, contributes to global warming, so regulations of most European countries require that the methane slip has to be burnt. The two main selling points of a biogas treatment unit, are its efficiency and cost. Table 24 compares the main purification and upgrading technologies according to these two criteria. Operating and investment costs are very variable depending on the technology, for abatement performance, often above 90 %.

Generally, the more expensive technologies, both in investment and in operation are those of oxidation, but during the work of this thesis, only anaerobic processes are studied. -Possible pollutant recovery.

-High operating costs, related to the liquid phase in general.

-Generating a polluted aqueous effluent.

-Can generate additional separation operations.

Adsorption on activated carbon 80 -90 7 -55 0.7 -2.4

-Very easy to use.

-Tolerates flow variations.

-Possible pollutant recovery.

-Add an operating cost, associated with the regeneration of the adsorbent.

Condensati on 50 -90 5 -37 1.4 -8.2 -Possible pollutant recovery.

-Icing Possibility

The investment costs of the different technologies do not differ greatly, especially at high flow rates. They are presented in Fig. 28. [START_REF] Bauer | Biogas upgrading -Review of commercial technologies[END_REF] Separation technologies: ( __ Δ __ ) Chemical absorption (Amines) ; (--□--) Water wash ; (-. ○-. ) Membranes ; ( … + … ) Adsorption (PSA)

Fig. 28: Comparison of investment costs of different biogas purification and upgrading technologies

Chemical absorption using amines as chemical solvent is slightly more expensive in terms of investment, and membrane separation process is less costly for low flow rates. This investment cost criterion begins to converge for all technologies at higher flow rates.

Conclusion

This chapter has introduced the various technologies used for the purification and upgrading of biogas. These separation methods were compared according to several criteria such as separation efficiency, the environmental impact and investment and operating costs.

The work of this thesis is part of the project led by the company Cryo Pur®, which aims to create an innovative biogas cryogenic purification and upgrading process for the production of a renewable fuel and liquid carbon dioxide.

For carbon dioxide capture, Cryo Pur® company has developed a new technology which consists of anti-sublimating the carbon dioxide on a low temperature surface (from -90 °C to -120 °C), thus transforming CO2 directly from its gaseous phase into a solid phase frosted on the cold surface [START_REF] Clodic | CO2 capture by anti-sublimation -Thermo-economic process evaluation[END_REF]. Having regard to the need to liquefy the biogas at low temperatures to be used as vehicle fuel, the best alternative to the conventional technologies is therefore to upgrade biogas with cryogenic technology.

Biogas upgrading depends on the concentrations of hydrogen sulfide present. Indeed, this compound must be completely eliminated upstream of the process to ensure high quality products and to prevent corrosion of equipment such as heat exchangers used in the cryogenic process. Hence the need to use a process with a very high efficiency, to be able to eliminate all the hydrogen sulfide present in the biogas. On the whole, the lowest methane losses are indicated for chemical absorption and adsorption processes, and the highest one relates to membranes and water wash. Finally, the choice was focused on the following two technologies used for the removal of hydrogen sulfide:

 Chemical absorption in a structured packed column using sodium hydroxide (NaOH) as solvent.

 Adsorption in a fixed bed using activated carbon.

The next chapter will present the technology developed by Cryo Pur® Company for purification and upgrading of biogas.

Introduction

Experiments were performed on the industrial demonstrator "BioGNVAL" treating 85 Nm 3 /h of biogas from the Valenton water treatment plant, the second biggest in France run by the SIAAP (Public society serving the Paris region). The demonstrator shown in Fig. 29 was developed by the Cryo Pur® Company. It was built in partnership with SUEZ as part of the BioGNVAL project, and partially funded by the 'Invest in the Future' program run by the ADEME (French Environment and Energy Management Agency). GNVert (Engie) and IVECO are also partners in the BioGNVal project, providing the Bio-LNG distribution station and the heavy goods vehicle Flex Fuel gas / Bio-LNG respectively.

Fig. 29: BioGNVAL demonstrator located at Valenton water treatment plant [9]

The BioGNVAL pilot plant uses a cryogenic method to purify and liquefy biogas efficiently without loss of methane and without emitting greenhouse gases. The system generates two products from biogas: liquid bio-methane and bioCO2 at purity level greater than 99.995 % respecting EIGA (European Industrial Gases Association) specifications [START_REF]Carbon Dioxide Source Certification, Qualify Standards and Verification[END_REF].

The general principle of operation of the BioGNVAL pilot plant is depicted in Fig. 30. It takes place in three main stages:  Pretreatment or purification which is to remove trace compounds present in the biogas as hydrogen sulfide, water vapor and siloxanes.

 CO2 capture or biogas upgrading which consists in separating carbon dioxide from biogas. The content of carbon dioxide in biogas is typically greater than 30 %.  Liquefaction of the biomethane after purification and upgrading biogas.

Fig. 30: Schematic representation of Cryo Pur® system [9]

This chapter is divided into two sections: The first part consists of a general presentation of the pilot plant. The second part is dedicated to the operating principle of the subsystems. The raw biogas, whose conditions and composition are presented in Table 25, comes from the anaerobic digester through the line of biogas. The latter distributes the biogas in all sub-systems, starting with the desulfurization subsystem, until the liquefaction subsystem. Conditions at the outputs of each sub-system in terms of composition and temperature are defined. If one of the conditions is not met, the biogas is routed through the biogas treatment line to the flare. These conditions are shown in Table 26. Once the full treatment is performed, the liquefied biogas is stored in a mobile container presented in Fig. 31. The operating principle of reactive absorption is simple. The system mainly comprises a structured packed column, a water circuit and two heat exchangers, used to cool biogas and the liquid phase. Fig. 32 shows the apparatus setup for the desulfurization process.

General presentation of the BioGNVAL pilot plant

Table 25. Conditions and composition of the raw biogas treated by BioGNVAL pilot plant

Conditions

Table 26. conditions of passage from one subsystem to another [9]

Passage

The biogas is saturated with water vapor at the input of the pilot plant. It firstly passes through a heat exchanger (Green line) to cool the gas phase and condense a portion of the water vapor contained therein. A phase separator recovers the condensed water vapor and sends it to a drainage tank. Thereafter, the biogas enters the bottom of the absorption column where it is contacted in a countercurrent with the washing water sprayed from the top (Violet line). This water is neutralized by an aqueous solution of sodium hydroxide. The injected quantity is controlled by a pH meter placed on the tank TK-230-04. This tank showed in Fig. 32 provides the column with the liquid phase. The liquid phase is recirculated by a pump and is cooled by an exchanger to a temperature slightly higher than 2 °C to prevent freezing.

Biogas finally exits from the top of the column at a temperature of about 5 ° C. The cooling duty is provided by the liquid phase on the surface of the packing in direct contact within the column. High efficient mass and heat transfer between the liquid and the biogas are achieved thanks to the packing surface. In order to test the adsorption technology, the absorption process could be bypassed. Therefore, the biogas passesthrough two fixed bed adsorption columns placed in series, as seen in Fig. 33.

Fig. 33: Piping and instrumentation diagram of adsorption subsystem for the removal of hydrogen sulfide [9] Equipment: CV (Control Valve) ; FV (Flow Valve) ; MV (Manual Valve) ; PSV (Pressure Safety Valve) ; BL (Blower) ; F (Filter) ; HX (Heat Exchanger) ; SP (Separator) ; CU (Condensing Unit) ; TK (Tank). Instrumentation: FT (Flow Transmitter) ; TE (Temperature Element) ; AT (Analyzer Transmitter) ; XY (Limit Switch)

These two columns packed with impregnated activated carbon, provide a continuous treatment of hydrogen sulphide. To further improve the functioning of activated carbon, a small amount of oxygen is added to oxidize the hydrogen sulfide, which will make larger molecules and allow to block them into the pores.

Dehumidification and siloxanes icing subsystem

The biogas leaves the desulfurization subsystem at 5 °C and with a H2S content lower than 1 ppm. Dehumidification and cooling the biogas continue until -40 °C. At this temperature, only 125 ppm of water vapor remains in the biogas and the heaviest Siloxanes are removed.

Biogas cooling continues until a temperature of -87 ° C. This eliminates the siloxanes, as well as water vapor, whose content is reduced to less than 1 ppm at the outlet of the subsystem.

To ensure a continuous operation and to prevent the accumulation of ice that could block the passage of biogas, the chillers used contain two evaporators placed in parallel to alternate operation in frosting and defrosting mode.

Carbon dioxide capture subsystem

After having been purified of hydrogen sulphide, water vapor and siloxane, the biogas is fed into the CO2 capture subsystem to be upgraded. The biogas is now composed of methane and carbon dioxide at atmospheric pressure, that's to say a carbon dioxide partial pressure below its triple pressure. After cooling the biogas to -120 ° C in this subsystem, the carbon dioxide thus undergoes the phenomenon of antisublimation, which means that it is transformed directly from its gaseous phase into a solid phase frosted on the cold surface of the heat exchanger as seen in Fig. 34.

To allow continuous operation of the upgrading system, two heat exchangers are used. When the first is frosting the carbon dioxyde, the second operates in defrost mode. Thus carbon dioxide is recovered in the liquid phase and then stored in a cryogenic vessel. This strategy allows recovery of CO2 at a very high purity level (99.995 %) which could be used for industrial and food applications.

Biogas liquefaction subsystem

After purification and upgrading of biogas, the biomethane is sent to the liqufaction subsystem which is composed of a compression unit, a cooling and liquefaction unit and a storage unit for the liquefied biomethane.

At the outlet of the upgrading unit, a gas analyser ensures that the biomethane produced is composed of less than 2.5 % of residual carbon dioxide. Once the required biomethane quality reached, it is supplied to the liquefaction subsystem to be compressed and liquefied. Compression of biomethane increases its liquid-vapor saturation temperature, which means its liquefaction at a high enough temperature level reducing electrical consumption of the refrigeration machine. Fig. 34: Carbon dioxide antisublimation [START_REF][END_REF]

Experimental results concerning the removal of hydrogen sulfide by chemical absorption using sodium hydroxide

The main objectives of the experiments on the absorption column is to maintain the concentration of hydrogen sulfide between 0 and 1 ppm at the outlet of the column throughout the testing period, for a content at the entrance equal to 20 ppm. The biogas temperature at the outlet of the column should be slightly greater than 2 ° C whatever its temperature at the inlet.

These objectives have been reached on the demonstrator. The hydrogen sulfide content at the outlet of the absorption column was maintained between 0 and 1 ppm as seen in Fig. 35. The biogas temperature at the outlet of the packing column was maintained between 5 and 6 °C even when the biogas temperature at the inlet reaches 35 °C as shown in Fig. 36.

The concentrations measurement are made using a biogas analyser, model "Gas 3200 R Biogas" bought from "Gas Engineering and Instrumentation Technologies Europe" GEIT® company. The measurements of H2S concentration are made with 3-electrodes electrochemical cell designed for biogas applications with several measuring range: from 0 -50 ppm to 0 -9999 ppm. The sensitivities of measurement tools used in the experiments are shown in Table 27. Sodium hydroxide was used as chemical solvant for the removal of hydrogen sulfide. The weight percentage of NaOH in water is equal to 30.5 wt%. Averaged over all experiments, sodium hydroxide consumption was assessed at 6 l/h (solution containing 30.5 wt% of NaOH and 69.5 wt% of H2O). This consumption was calculated with an indicator allowing the NaOH level measurement in the tank TK-240-02 (See Fig. 32). This consumption is very excessive relative to the theoretical consumption of 1.35 l/hr of commercial caustic soda. This overconsumption is explained by desorption of hydrogen sulfide, because the reaction between hydrogen sulphide and sodium hydroxide is reversible (R.11). It leads to the formation of sodium sulfide (Na2S) that is unstable in water. Fig. 37 shows the desorption phenomenon observed during experiments. Indeed, stopping the injection of sodium hydroxide at 13:45:00 (See Fig. 37) shows that the hydrogen sulfide content at the outlet of the column is greater than its content at the inlet. This explains the overconsumption of the solvent. To overcome this problem, the injection of sodium hypochlorite (NaOCl) has been proposed to prevent the regeneration of hydrogen sulfide, and to obtain soluble by-products in water (Na2SO4 and NaCl) as seen in Reaction 2. The injections of sodium hydroxide and sodium hypochlorite will respectively be controlled by the pH and conductivity measurements. This solution will reduce the consumption of the solvents.

𝑁𝑎𝑂𝐻 + 𝐻

Conclusion

This section has presented two different processes (absorption and adsorption) used for the removal of H2S in order to reduce its content to less than 1 ppm throughout the operation.

The other undesirable components present in the biogas such as H2O, siloxanes and CO2 are captured through cryo-condensation, freezing each component.

The CO2 is retrieved in liquid form at a high level of purity enabling revalorization. Once purified, the biomethane fulfills the characteristics necessary to be used as fuel for heavy goods vehicles (HGV).

To allow the absorption column to operate at full capacity, an hydrodynamic study is necessary. This study will optimise the flow rates involved and ensure a better mass transfer in the packing column, with a lower consumption of the liquid phase and a lower pressure drop. This study which will be presented in the next chapter, is also of great interest for the design of packed columns. Ce chapitre compare trois modèles existants utilisés pour la prédiction des paramètres hydrodynamiques dans des colonnes à garnissage structuré.

Ces modèles sont utilisés pour évaluer la perte de pression, la rétention liquide, l'aire interfaciale effective, les coefficients de transfert de masse et les points de transition. Les résultats de ces modèles sont comparés à des données expérimentales afin de choisir celui avec le meilleur ajustement.

Les comparaisons ont été effectuées en utilisant deux systèmes : Air -Eau et Air -Kerosol 200 et un garnissage structuré de type Flexipac 350Y.

Le modèle choisi est basé sur des corrélations semi-empiriques contenant des constantes et des exposants définis selon des mesures expérimentales. Pour rendre le modèle plus représentatif du système d'intérêt (biogaz contenant du H2S / solution aqueuse d'hydroxyde de sodium), ces constantes ont été modifiées et certains exposants ont été ajustés en fonction de la vitesse superficielle du liquide et la densité.

Une fois le modèle modifié, les résultats de la perte de pression ont été comparés aux données expérimentales obtenues sur le démonstrateur BioGNVAL. Les résultats obtenus sont en bon accord mais il est judicieux de noter que ce modèle peut perdre de sa précision en variant les applications.

Par conséquent, ce modèle est idéal pour prédire avec précision les trois régions opérationnelles d'une colonne à garnissage structuré à petite échelle utilisée pour des applications de biogaz ou de gaz naturel.

Introduction

Today, in modern absorption columns, structured packings are widely used, thanks to their higher capacity and lower pressure drop compared to random packings. Structured packings were used for the first time in 1950 [START_REF] Strigle | Packed tower design and applications, Random and structured packings[END_REF]. They are in continuous development to expand their use and improve their efficiency. They provide a large surface area for the liquid and gas phases to be in direct contact within the column. High efficient mass transfer between the two phases is achieved thanks to the packing surface. This work compares three existing models used for the prediction of hydrodynamic parameters in structured packing columns. These models are used to evaluate pressure drop, liquid holdup, effective interfacial area, mass transfer coefficients and transition points. The results obtained with these models are compared to experimental data in order to choose the one with the best fit. Comparisons were made using Flexipac 350Y structured packing and two systems: Air -Water and Air -Kerosol 200. The model chosen is based on semi-empirical correlations using constants and exponents defined according to experimental measurements. To adapt the model to biogas application and to make it more representative of the system of interest, these constants were optimized and some exponents have been adjusted. Once the model modified, the results of pressure drop were compared to data from BioGNVAL pilot plant.

Theoretical principles

In a packed column, hydrodynamics and mass transfer processes occur simultaneously. They are correlated and the link parameter is liquid holdup hL defined as the volume of the liquid per unit volume of the column. Equation ( 17) defined by Chan and Fair [START_REF] Chan | Industrial and Engineering Chemistry Process Design and Development[END_REF] for sieve trays illustrated the relation between the two processes.

𝑘 𝑉 𝑎 𝑒 = 316 𝐷 𝑉 0,5 (1030 𝑓 + 867 𝑓 2 ) ℎ 𝐿 0,5 (17)
Regarding the hydrodynamic analysis, increasing the velocity of liquid and gas, results in an increase of the liquid holdup and the thickness of the liquid film which leads to an increase in pressure drop.

About mass transfer analysis, increasing liquid holdup causes the enlargement of the interfacial area leading to higher mass transfer rates.

The curve which represents the evolution of the pressure drop or the liquid holdup as a function of the gas capacity factor Fc is divided by two points (loading and flooding points) into three operating regions as seen in Fig. 38.

The liquid flow rate is not influenced by the counter-current flow of the gas in the preloading region, as can be seen in Fig. 38.b which presents the evolution of the liquid holdup as a function of the gas capacity factor defined by Equation [START_REF] Biard | Contribution au développement d'un procédé de lavage chimique compact. Traitement du sulfure d'hydrogène par le chlore à l'échelle semi-industrielle et de COV odorants par oxydation avancée O3 / H2O2 à l'échelle du laboratoire[END_REF].

𝐹 𝑐 = 𝑢 𝑉 × √𝜌 𝑉 ( 18
)
The loading point represented by the line AA in Fig. 38 is reached when the slope of the liquid holdup curve starts to increase, or when the wet pressure drop curve starts to deviate from the pressure drop in a dry column. The flooding point is represented by the line BB in Fig. 38. It is the point where the slope of pressure drop and liquid holdup curves tends toward infinity. Therefore, it is necessary to predict accurately the transition points because they characterize the capacity of a packing column. According to Paquet [START_REF] Friedrich | Establishing a facility to measure the efficiency of structured packing under total reflux[END_REF], under-predicting the flooding point will prevent the column to operate at its optimal conditions, and its capacity could be very low. However, over-predicting the flooding point may lead to higher pressure drop which could be problematic.

As reported by Paquet [START_REF] Friedrich | Establishing a facility to measure the efficiency of structured packing under total reflux[END_REF], the main disadvantage of this model is that it requires six specific constants for each type of packing. The ones needed for Flexipac 350Y and for some other types of packing, are presented in Table 28. 20) and ( 21) illustrated in Table 29 are used by Billet and Schultes to calculate the effective interfacial area respectively at loading point, in the loading region and at the flooding point.

Table 29. Effective interfacial area in packing columns using Billet and Schultes model [98]

Parameter Correlation

Effective interfacial area at loading point

( 𝑎 𝑒 𝑎 ) 𝑙𝑝 = 1.5 (𝑎 𝑑 ℎ ) -0.
The Billet and Schultes model is composed of several correlations that describe liquid holdup and pressure drop in the preloading, loading and flooding regions. Velocities and liquid holdup at loading and flooding points are calculated using the equations listed in Table 30.

Table 31 presents the correlations used by Billet and Schultes to calculate the liquid holdup in the loading region. This property depends on the liquid holdup in the preloading region and at the flooding point. The first one is theoretically derived from a force balance, while the second is purely empirical. The liquid holdup in the preloading region does not depend on the gas properties. It is only a function of the liquid properties and its velocity, as seen in Equation [START_REF] Kunz | The GERG -2008 Wide-Range Equation of State for Natural Gases and Other Mixtures: An Expansion of GERG -2004[END_REF]. As stated in the thesis of Paquet [START_REF] Friedrich | Establishing a facility to measure the efficiency of structured packing under total reflux[END_REF], the hydraulic area of the packing accounts for the surfaces that were not completely wetted by the liquid flow.

The equations used to calculate pressure drop are listed in Table 32.

f s 𝑓(𝑠) = ( ℎ 𝐿 ℎ 𝐿,𝑙𝑝 ) 0,3 exp ( 𝑅𝑒 𝐿 200 ) (44)
The expression of dry pressure drop is obtained by applying a force balance. The wall factor K is used to take into account the free spaces more available at the wall. The constant Cp used to calculate the resistance coefficient ψ0 characterizes the geometry of the packing.

For the wetted packing column, Equation ( 43) used to calculate pressure drop replaces the void fraction (ε) by an effective void fraction (ε -hL) which depends on liquid holdup, reducing the volume available for the gas flow. This equation introduces a wetting factor fw to account for any change in the surface of the packing caused by the wetting action [START_REF] Friedrich | Establishing a facility to measure the efficiency of structured packing under total reflux[END_REF].

SRP model

The SRP (Separations Research Program) model [START_REF] Fair | A comprehensive model for the performance of columns containing structured packings[END_REF] was developed at the University of Texas [100]. The latest version of this model was published in the work of Fair et al. in 2000 [96]. According to Paquet [START_REF] Friedrich | Establishing a facility to measure the efficiency of structured packing under total reflux[END_REF], the SRP model considers the void fraction as a series of wet columns where the gas flow passes through. Unlike the Billet and Schultes model, the geometry depends on the angle and dimensions of corrugations.

To calculate liquid holdup and effective interfacial area, the SRP model uses a correction factor that takes into account the packing surface that is not completely wetted by the liquid flow.

The prediction of the effective interfacial area is based on a simple equation that depends on the liquid holdup correction factor and a surface enhancement factor as seen in Table 33. The surface enhancement factor is equal to 0.35 for stainless steel sheet metal packing [100].

Table 33. Effective interfacial area in packing columns using SRP model [START_REF] Fair | A comprehensive model for the performance of columns containing structured packings[END_REF] Parameter Correlation

Reynolds number

𝑅𝑒 𝐿 = 𝑠 𝑢 𝐿 𝜌 𝐿 𝜇 𝐿 (46)
Froude number

𝐹𝑟 𝐿 = 𝑢 𝐿 2 𝑠 𝑔 (47)
Weber number

𝑊𝑒 𝐿 = 𝑠 𝜌 𝐿 𝑢 𝐿 2 𝑔 𝜎 𝐿 (48)
Solid -liquid film contact angle 𝐹𝑜𝑟 𝜎 𝐿 ≤ 0,055 𝑁. 𝑚 -1 𝑐𝑜𝑠𝛾 = 0,9 𝐹𝑜𝑟 𝜎 𝐿 > 0,055 𝑁. 𝑚 -1 𝑐𝑜𝑠𝛾 = 5,211 × 10 -16,835 𝜎 𝐿 (

Correction factor 𝐹 𝑡 = 29,12 𝑠 0,359 (𝑊𝑒 𝐿 𝐹𝑟 𝐿 ) 0,15 𝜀 0,6 𝑅𝑒 𝐿 0,2 (𝑠𝑖𝑛𝛳) 0,3 (1 -0,93 𝑐𝑜𝑠𝛾)

Effective interfacial area

𝑎 𝑒 𝑎 = 𝐹 𝑡 𝐹 𝑆𝐸 (51)
The SRP model uses the effective gravity which takes into account forces that oppose the flow of the liquid film over the packing. These forces are caused by the pressure gradient, buoyancy and shear stress in the gas phase [START_REF] Friedrich | Establishing a facility to measure the efficiency of structured packing under total reflux[END_REF]. An iterative approach exploiting this effective gravity is used to calculate liquid holdup. The calculation steps followed for predicting liquid holdup in a packing column are shown in Table 34.

Initial condition for the iterative approach

( 𝛥𝑃 𝛥𝑧 ) 𝑖𝑡𝑒𝑟 = ( 𝛥𝑃 𝛥𝑧 ) 𝑑 (53)
Iterative approach

ℎ 𝐿 = ( 4 𝐹 𝑡 𝑠 ) 2 3 [ 3 𝜇 𝐿 𝑢 𝐿 𝜌 𝐿 𝜀 𝑔 𝑒𝑓𝑓 𝑠𝑖𝑛𝛳 ] 1 3 𝛥𝑃 𝛥𝑧 = ( 𝛥𝑃 𝛥𝑧 ) 𝑑 [1 -ℎ 𝐿 (71,35 𝑠 + 0,614)] 5 (54)
Convergence

𝐼𝑓 𝛥𝑃 𝛥𝑧 ≠ ( 𝛥𝑃 𝛥𝑧 ) 𝑖𝑡𝑒𝑟 → ( 𝛥𝑃 𝛥𝑧 ) 𝑖𝑡𝑒𝑟 = 𝛥𝑃 𝛥𝑧
and restart from ( 52)

𝐼𝑓 𝛥𝑃 𝛥𝑧 ≈ ( 𝛥𝑃 𝛥𝑧 ) 𝑖𝑡𝑒𝑟 → 𝐶𝑜𝑛𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒 (55)
The constants A and B used to calculate the pressure drop in a dry column depend on the type of the packing. For metal structured packings, A and B are equal to 0.177 and 88.77 respectively [101]. Table 35 presents the equations used for the prediction of pressure drop in preloading and loading regions.

Table 35. Pressure drop in packing columns using SRP model [START_REF] Fair | A comprehensive model for the performance of columns containing structured packings[END_REF] Parameter Correlation

Liquid film thickness 𝛿 = ( 3 𝜇 𝐿 𝑢 𝐿 𝑎 𝑔 𝜌 𝐿 𝑠𝑖𝑛𝛳 ) 1 3 (56)
Gas flow channel diameter

𝑑 ℎ𝑉 = (𝑏 ℎ -2 𝑠 𝛿 ) 2 𝑏 ℎ [( 𝑏 ℎ -2 𝑠 𝛿 2 ℎ ) 2 + ( 𝑏 ℎ -2 𝑠 𝛿 𝑏 ) 2 ] 0,5 + 𝑏 ℎ -2 𝑠 𝛿 2 ℎ (57)
Gas capacity factor at loading point

𝐹 𝑐,𝑙𝑝 = [0,053 𝑔 𝑑 ℎ𝑉 𝜀 2 (𝑠𝑖𝑛𝛳) 1,15 (𝜌 𝐿 -𝜌 𝑉 ) ( 𝑢 𝐿 𝑢 𝑉 √ 𝜌 𝐿 𝜌 𝑉 ) -0,25 ] 0,5 (58)
Pressure drop enhancement factor

𝐹 𝑙 = 3,8 ( 𝐹 𝑐 𝐹 𝑐,𝑙𝑝 ) 2 𝑠𝑖𝑛𝛳 ( 𝑢 𝐿 2 𝑔 𝑑 ℎ𝑉 𝜀 2 ) 0,13 (59)
Pressure drop in preloading region

( 𝛥𝑃 𝛥𝑧 ) 𝑝𝑙 = ( 𝛥𝑃 𝛥𝑧 ) 𝑑 ( 1 [1 -ℎ 𝐿 (71,35 𝑠 + 0,614)] ) (60)
Pressure drop in loading region

( 𝛥𝑃 𝛥𝑧 ) = 𝐹 𝑙 ( 𝛥𝑃 𝛥𝑧 ) 𝑝𝑙 (61)

Delft model

The Delft model [START_REF] Olujić | Predicting the efficiency of corrugated sheet structured packings with large specific surface area[END_REF] was developed in a joint academic project between Montz Company and Delft University of Technology. The Delft model considers that all the packing surface area is wetted by the liquid film [START_REF] Friedrich | Establishing a facility to measure the efficiency of structured packing under total reflux[END_REF]. The prediction of the effective interfacial area with the Delft model is based on an empirical correlation presented in Equation [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids[END_REF].

𝑎 𝑒 = 𝑎 (1 -𝛺) (1 + 𝐴 𝑢 𝐿 𝐵 ) (62)
According to Paquet [START_REF] Friedrich | Establishing a facility to measure the efficiency of structured packing under total reflux[END_REF], Ω is equal to 0.1 for Montz Packing and for most packing with holes as Flexipac and Mellapak. A and B are constants specific to the type and size of the packing. For example, these two constants are respectively equal to 2.143 x 10 -6 and 1.5 for Montz® Packing B1-250 [100].

The Delft model introduces a new expression to define the effective liquid flow angle as seen in Equation [START_REF] Freundlich | Uber die adsorption in losungen[END_REF].

𝛼 𝐿 = 𝑎𝑟𝑐𝑡𝑎𝑛 [ 𝑐𝑜𝑠(90 -𝜃) 𝑠𝑖𝑛(90 -𝜃) 𝑐𝑜𝑠 [𝑎𝑟𝑐𝑡𝑎𝑛 ( 𝑏 2 ℎ )] ] (63)
This model uses a simple function for predicting liquid holdup consisting on the product of the specific surface of the packing and the thickness of the liquid film.

ℎ 𝐿 = 𝛿 𝑎 [START_REF] Temkin | Adsorption equilibrium and the kinetics of processes on nonhomogeneous surfaces and in the interaction between adsorbed molecules[END_REF] The expression of the liquid film thickness is the same adapted by the SRP model except that it uses the effective liquid flow angle.

For the prediction of the pressure drop, the Delft model uses the same equations as the SRP model. The only difference is situated in the preloading region. As reported by Paquet [START_REF] Friedrich | Establishing a facility to measure the efficiency of structured packing under total reflux[END_REF], the Delft model assumes that the gas flows in a regular zigzag pattern through the packed column. It uses three parameters which contribute to the calculation of the pressure drop in the preloading region. The details of calculation of pressure drop in the preloading region are summarized in Table 36.

Table 36. Pressure drop in preloading region using Delft model [97]

Parameter Correlation

Effective gas velocity

𝑢 𝑉,𝑒 = 𝑢 𝑉 𝜀 (1 -ℎ 𝐿 ) 𝑠𝑖𝑛𝜃 (65)
Effective liquid velocity 𝑢 𝐿,𝑒 = 𝑢 𝐿 ℎ 𝐿 𝜀 𝑠𝑖𝑛𝛼 𝐿 [START_REF] Toth | Calculation of the BET-compatible surface area from any type I isotherms measured above the critical temperature[END_REF] Relative Reynolds number for gas phase 𝑅𝑒 𝑉𝑟 = 𝜌 𝑉 𝑑 ℎ𝑉 (𝑢 𝑉,𝑒 + 𝑢 𝐿,𝑒 ) 𝜇 𝑉

Effective Reynolds number for gas phase

𝑅𝑒 𝑉𝑒 = 𝜌 𝑉 𝑑 ℎ𝑉 𝑢 𝑉,𝑒 𝜇 𝑉 (68)
Fraction of the flow channel occupied by the liquid phase

𝜑 = 2 𝑠 2 𝑠 + 𝑏 (69)
Fraction of the channels ending at the column wall

𝜓 = 2 𝛱 𝑎𝑟𝑐𝑠𝑖𝑛 ( ℎ 𝑝𝑒 𝑑 𝑐 𝑡𝑎𝑛𝜃 ) + 2 ℎ 𝑝𝑒 𝛱 𝑑 𝑐 2 𝑡𝑎𝑛𝜃 (𝑑 𝑐 2 - ℎ 𝑝𝑒 2 𝑡𝑎𝑛 2 𝜃 ) 0,5 (70)
Gas/Liquid friction coefficient

𝜉 𝐺𝐿 = [-2 𝑙𝑜𝑔 10 [ 𝛿 3,7 𝑑 ℎ𝑉 - 5,02 𝑅𝑒 𝑉𝑟 𝑙𝑜𝑔 10 ( 𝛿 3,7 𝑑 ℎ𝑉 + 14,5 𝑅𝑒 𝑉𝑟 )]] -2 (71)
Gas/Gas friction coefficient 𝜉 𝐺𝐺 = 0,722 (𝑐𝑜𝑠𝜃) 3,14

Direction change factor for bulk zone

𝜉 𝑏𝑢𝑙𝑘 = 1,76 (𝑐𝑜𝑠𝜃) 1,63 (73)
Direction change coefficient for wall zone 𝜉 𝑤𝑎𝑙𝑙 = 34,19 𝑢 𝐿 0,44 (𝑐𝑜𝑠𝜃) 0,779 + 4092 𝑢 𝐿 0,31 + 4715 (𝑐𝑜𝑠𝜃) 0,445 𝑅𝑒 𝑉𝑒 [START_REF] Chen | Développement de nouvelles membranes à base de polyimide pour la séparation CO2 / CH4[END_REF] Coefficient for gas/liquid friction losses

𝜍 𝐺𝐿 = 𝜉 𝐺𝐿 𝜑 ℎ 𝑝𝑏 𝑑 ℎ𝑉 𝑠𝑖𝑛𝜃 (75)
Coefficient for gas/gas friction losses

𝜍 𝐺𝐺 = 𝜉 𝐺𝐺 ℎ 𝑝𝑏 𝑑 ℎ𝑉 𝑠𝑖𝑛𝜃 (1 -𝜑) (76)
Coefficient for losses caused by direction change

𝜍 𝐷𝐶 = ℎ 𝑝𝑏 ℎ 𝑝𝑒 (𝜉 𝑏𝑢𝑙𝑘 + 𝜓 𝜉 𝑤𝑎𝑙𝑙 ) (77)
Pressure drop in preloading region

∆𝑃 𝑝𝑙 = ∆𝑃 𝐺𝐺 + ∆𝑃 𝐺𝐿 + ∆𝑃 𝐷𝐶 = 1 2 𝜌 𝑉 𝑢 𝑉,𝑒 2 (𝜍 𝐺𝐺 + 𝜍 𝐺𝐿 + 𝜍 𝐷𝐶 ) (78)

Models evaluation

The three models introduced in the previous section are evaluated and compared in order to choose the most effective in the prediction of hydrodynamic properties. To achieve this, the models are compared using two systems: Air / Water and Air / Kerosol 200. These systems have been chosen because of the lack of experimental data in the open literature concerning the system of interest (biogas with H2S / aqueous solution of sodium hydroxide). Kerosol is a paraffin, characterized by a low surface tension and high viscosity as seen in Table 37. "200" refers to its boiling point (200 °C).

The differences in liquid surface tension, density and viscosity between water and Kerosol 200 allow comparison of models for different conditions, highlighting the effects on pressure drop and liquid holdup. The experimental data were retrieved from the work of Erasmus [100]. The type of packing used for this comparison is Flexipac® 350Y. This packing is different with respect to the one used in the BioGNVAL pilot plant (Montz® B1-420), but no literature data are available for this last. The dimensions for the packing Flexipac® 350Y are outlined in Table 38 and the relative constants used by Billet and Schultes are shown in Table 28.

Pressure drop and liquid holdup

In Fig. 39, the experimentally determined pressure drop and liquid holdup over Flexipac® 350Y [100] are compared to the results obtained with the models using an Air -Water system. Fig. 39.a shows that SRP and Billet and Schultes models are accurate in predicting the pressure drop in preloading region (Fc < 1.9). The Delft model predicts the correct shape of the pressure drop curve, but compared to experimental data, the results obtained are not realistic.

Although the results are not accurate, Fig. 39.b shows that the model by Billet and Schultes is the best in predicting liquid holdup in a structured packed column. The Delft model assumes that the liquid holdup is not influenced by the gas velocity, which explains the constant shape of the curve. The modified Billet and Schultes model shown in Fig. 39 will be presented in section 5.4. The average absolute deviations between predictive models (Billet & Schultes, SRP and Delft) and experimental results for pressure drop and liquid holdup are shown in Table 39.

Table 39. Deviation between predictive models and experimental data

Model

Effective interfacial area

In a packed column, the gas and the liquid phases are brought into contact and exchange mass and energy across their common interfacial area. The effective interfacial area accounts for the dead area that does not actively take part in the mass transfer process.

Fig. 40 shows the results of the effective interfacial area obtained with the three models, and compared to experimental data. Fig. 40 shows that most models overpredict the effective interfacial area. The Delft model assumes that the liquid load does not influence the effective interfacial area which presents 90 % of the overall specific area of Flexipac® 350 Y as seen in Fig. 40.c.

Compared to the Delft model, the SRP model predicts the right slope of the curve. However, for liquid loads above 16 m.h -1 , the predicted effective interfacial area becomes larger than the packing specific surface.

The Billet and Schultes model [START_REF] Billet | A physical model for the prediction of liquid hold-up in two-phase counter-current columns[END_REF] is accurate in predicting the effective interfacial area.

The evaluation of the three models shows that the Billet and Schultes model predicts hydrodynamic parameters more accurately than SRP and Delft models. Therefore, the model by Billet and Schultes is retained for the further study.

Changes made to Billet and Schultes model and results

The Billet and Schultes model was developed for random packings, then it was extended to cover a limited number of commercially available structured packings.

To make this model more realistic and more accurate in predicting hydrodynamic parameters for structured packings, some constants and exponents defined according to experimental observations and used in the correlations were modified as function of liquid load and density. The constants and exponents to modify were selected following a sensitivity analysis. The values of the constants and exponents have been optimized by minimization of an objective function based on the deviations between modelling and experimental results. The modifications made to Equations ( 31), ( 36), ( 43) and ( 44) are shown in Tables 40, 41 and 42 for liquid holdup and pressure drop. These equations are reminded below by highlighting the modified constants.

ℎ 𝐿,𝑝𝑙 = 𝑪 𝟏 ( 𝜇 𝐿 𝑎 2 𝑢 𝐿 𝜌 𝐿 𝑔 ) 1 3 ( 𝑎 ℎ 𝑎 ) 2 3 (79)
𝑑𝑃 𝑑𝑧 = 𝑪 𝟑 𝜓 𝐿 𝑓 𝑤 𝑎 (𝜀 -ℎ 𝐿 ) 3 𝐹 𝑐 2 2 1 𝐾 (81)
In order to improve predictions, Equations ( 31), ( 36), ( 43) and ( 44) were slightly modified based on the experimental results of Erasmus [100], but using only three values of liquid load (uL = 6 m/h, uL = 20.5 m/h and uL = 35.5 m/h) for regression set. The modifications made to equations are colored in red. Statistical deviation between experimental data and the modified model results are presented in Table 43 for both systems. After validation of the modified model, it was used to predict pressure drop on a real structured packing column used for the removal of H2S from biogas. The results between experimental data obtained from BioGNVAL pilot plant and the refined model are shown in Table 44. The specific constant Cp for pressure drop over Montz B1-420 packing was set to 0.14 by fitting it on experimental data. The difference between the two results could be explained by pressure drop in the piping which does not contain packing.

Conclusion

This chapter evaluated three semi-empirical models for prediction of hydrodynamic parameters used for an industrial application concerning biogas purification: Billet and Schultes, SRP and Delft. Flexipac® 350Y structured packing was considered here. Its capacity is closely related to hydrodynamics and mass transfer characteristics. That is why, the performances of these hydrodynamic models were investigated and compared based on existing experimental data, and the choice was finally made on the model of Billet and Schultes. The correlations of this model were improved in order to develop an accurate prediction of hydrodynamic parameters in a structured packing column. This model allows to precisely predicting the key hydrodynamic parameters: liquid holdup, pressure drop, effective interfacial area and especially the two transition points: loading and flooding points.

The results of pressure drop using the modified model were compared to those obtained on BioGNVAL pilot plant. Good agreement was obtained with experimental data. It is wise to note that this model may lose generality with varying applications but for the activities of interest, it wins precision. Therefore, this predictive model is ideal to predict accurately the three operating regions of a small scale structured packing column used for biogas or natural gas applications. It would allow the design of structured packing columns without the need of experimental data collected on a pilot plant. The operative conditions of the existing columns could also be optimized using the modified model to operate at full capacity.

After the hydrodynamic study, it will be interesting to investigate the thermodynamic properties in order to accurately predict the efficiencies of H2S separation from biogas. This study based on simulations using the chemical process optimization software Aspen Plus® V8.0 will be presented in the following section.

Chapter 6: Comparison of experimental and simulation results for the removal of H2S from biogas by means of sodium hydroxide in structured packed columns

Résumé :

Ce travail est basé sur des simulations utilisant le simulateur de procédés Aspen Plus V8.0 et les résultats sont comparés à des données expérimentales issues de la littérature et du démonstrateur « BioGNVAL ». Le modèle « Rate-based » a été considéré lors des simulations afin de déterminer l'efficacité de séparation pour différentes conditions opératoires. Ce modèle a été adopté car les résultats expérimentaux ont montré une concentration constante de NaOH dans la phase liquide en fonction du temps. Une approche γ/φ est employée pour décrire l'équilibre vapeur -liquide : le modèle Electrolyte Non-Random Two-Liquid « ENRTL » est utilisé pour représenter les non-idéalités de la phase liquide, alors que l'équation d'état de Redlich-Kwong est utilisée pour le calcul des propriétés de la phase vapeur. Afin d'étudier de façon réaliste l'efficacité des colonnes à garnissage utilisant de la soude comme solvant pour l'élimination sélective de l'H2S, le modèle thermodynamique a été vérifié et validé en se basant sur différentes données expérimentales disponibles. Pour une estimation rigoureuse de la solubilité des gaz dans le solvant, les constantes de Henry obtenues ont été comparées aux données expérimentales issues du rapport de recherche RR-48 du Gas Processors Association [105] ainsi qu'à une équation semi-empirique proposée par Harvey [START_REF] Harvey | Semiempirical correlation for Henry's constants over large temperature ranges[END_REF]. Des propriétés physiques comme les capacités thermiques des composants purs, utilisées dans les calculs de transfert de matière et de chaleur ont été aussi vérifiées. L'équation développée par Aly et Lee [START_REF] Aly | Fluid Phase Equil[END_REF] a été adoptée pour calculer les capacités calorifiques et les résultats obtenus ont été comparés aux données expérimentales fournies par Elliott et Lira [START_REF] Elliott | Introductory chemical engineering thermodynamics[END_REF]. Le calcul de la masse volumique de la phase liquide dans Aspen Plus a été effectué en fonction de la composition massique de NaOH dans la soude à 25 °C. Les résultats obtenus ont été vérifiés grâce aux données expérimentales de Herrington et al. [109]. Concernant la viscosité de la phase liquide, un modèle correctif adapté pour les électrolytes appelé modèle de « Jones-Dole » [110] a été appliqué dans Aspen Plus. Les paramètres de ce modèle correctif ont été optimisés pour les ions HCO3 -, Na + et CO3 2-en utilisant respectivement les données expérimentales de viscosité des systèmes KHCO3 -H2O [START_REF] Palaty | Viscosity of diluted aqueous K2CO3 / KHCO3 solutions[END_REF], NaOH -H2O [START_REF] Vargaftik | Dictionary of thermophysical properties of gases and liquids[END_REF] et K2CO3 -H2O [114]. Ce modèle est remplacé par celui de Breslau et Miller [111] dans le cas où la concentration des électrolytes dépasse 0,1 mol/l. Les résultats de ces modèles ont été comparés aux données expérimentales de Klochko et Godneva [START_REF] Klochko | Electrical conductivity and viscosity of aqueous solutions of NaOH and KOH[END_REF]. Pour le calcul de la tension superficielle de la phase liquide, le modèle de Onsager-Samaras a été considéré dans Aspen Plus et les résultats de ce dernier ont été vérifiés en se basant sur les données expérimentales présentées dans le travail de Gel'perin et al. [START_REF] Gel'perin | The relation between the surface tension of aqueous solutions of inorganic substances and concentration and temperature[END_REF]. Les réactions chimiques impliquées ont été spécifiées. Elles sont toutes à l'équilibre chimique, à l'exception de la réaction entre le CO2 et l'ion hydroxyde (OH -) qui est cinétiquement contrôlée. Pour les réactions d'équilibre, les constantes d'équilibres ont été calculées en s'appuyant sur des données d' Edwards et al. [118] sauf pour les réactions entre H2S et OH -et entre HS -et OH -où les constantes d'équilibres relatives ont été calculées en utilisant l'énergie libre de Gibbs faute de données dans la littérature. Pour la réaction cinétique entre CO2 et OH -, l'équation d'Arrhenius a été considérée dans Aspen Plus où le facteur de fréquence et l'énergie d'activation ont été extraits du travail de Pinsent et al. [START_REF] Pinsent | The kinetics of combination of carbon dioxide with hydroxide ions[END_REF]. Les résultats des constantes de vitesse obtenues en fonction de la température ont été vérifiées à l'aide des résultats expérimentaux de Faurholt [120]. Après la validation des paramètres physico-chimiques, les simulations ont été réalisées en utilisant les mêmes conditions employées dans le démonstrateur « BioGNVAL » comme présenté dans le tableau suivant.

Introduction

This chapter was treated at an international stay conducted within the "Group on Advanced Separation Processes & GAS Processing" at Politecnico di Milano. This group is led by the Professor Laura Pellegrini who is an expert internationally recognized in process simulation and using the software Aspen Plus®.

Purification of biogas particularly requires the removal of hydrogen sulfide, which negatively affects the operation and viability of equipment especially pumps, heat exchangers and pipes, causing their corrosion. Several methods described in chapter 2 are available to eliminate hydrogen sulfide from biogas. Herein, reactive absorption in structured packed column by means of chemical absorption in aqueous sodium hydroxide solutions is considered. This study is based on simulations using Aspen Plus™ V8.0, and comparisons are done with data from BioGNVAL pilot plant treating 85 Nm 3 /h of biogas which contains about 30 ppm of hydrogen sulfide. The rate-based model approach has been used for simulations in order to determine the efficiencies of separation for different operating conditions. To describe vapor-liquid equilibrium, a γ/ϕ approach has been considered: the Electrolyte Non-Random Two-Liquid (ENRTL) model has been adopted to represent non-idealities in the liquid phase, while the Redlich-Kwong equation of state has been used for the vapor phase. In order to validate the thermodynamic model, Henry's law constants of each compound in water have been verified against experimental data. Default values available in Aspen Plus™ V8.0 for the properties of pure components as heat capacity, density, viscosity and surface tension have also been verified. Reactions involved in the process have been studied rigorously. Equilibrium constants for equilibrium reactions and the reaction rate constant for the kinetically controlled reaction between carbon dioxide and the hydroxide ion have been checked. Results of simulations of the pilot plant purification section show the influence of low temperatures, concentration of sodium hydroxide and hydrodynamic parameters on the selective absorption of hydrogen sulfide.

Aspen Plus® simulations

The aim of these simulations in Aspen Plus™ V8.0 is to study realistically the effectiveness of a structured packed column which uses sodium hydroxide as a chemical solvent for the selective removal of hydrogen sulfide.

Fig. 43: Flowsheet of the absorption process simulated using Aspen Plus®

Unlike amines, sodium hydroxide is not regenerable but it is very effective in removing low contents of H2S [START_REF] Kohl | Gas purification[END_REF].

Although the liquid solution is recycled, experimental data showed that the NaOH consumption is quite constant. NaOH concentration can be assumed to be constant over time in the liquid phase. This assumption justifies the use of the model "Rate-based" for the study.

The rate-based modeling approach is realistic compared to the traditional equilibrium-stage modeling approach that has been employed extensively in the process industries over the decades. The rate-based models assume that separation is caused by mass transfer between the contacting phases, and use the Maxwell-Stefan theory to calculate mass transfer rates [START_REF] Taylor | Real-World Modeling of Distillation[END_REF]. Conversely, the equilibrium-stage models assume that the contacting phases are in equilibrium with each other, which is an inherent approximation because the contacting phases are never in equilibrium in a real column. The rate-based modeling approach has many advantages over the equilibrium-stage modeling approach. The rate-based models represent a higher fidelity, more realistic modeling approach and the simulation results are more accurate than those attainable from the equilibrium-stage models [START_REF] Chen | A Rate-Based Process Modeling Study of CO2 Capture with Aqueous Amine Solutions using aspenONE Process Engineering[END_REF].

The Electrolyte Non-Random Two-Liquid model proposed by Chen and Song [START_REF] Chen | Generalized electrolyte-NRTL model for mixed-solvent electrolyte systems[END_REF] is used for calculating the liquid phase properties, while the Redlich-Kwong equation of state is used to calculate the vapor phase properties. This model is verified and validated using various experimental data from literature.

Coefficients aij, bij, cij, dij and eij are summarized in Table 45 for each system.

Where Ps,j is the vapor pressure of the component j, and T * is the reduced temperature. They are calculated respectively using the expressions [START_REF] Harasimowicz | Application of polyimide membranes for biogas purification and enrichment[END_REF]

Where Tc,water is the critical temperature of water.

Coefficients aij, bij and cij used by Harvey [START_REF] Harvey | Semiempirical correlation for Henry's constants over large temperature ranges[END_REF] in Equation ( 84) are summarized in Table 46 for each system. The obtained results are similar, with no major differences. Fig. 44, 45 and 46 show the adequacy of the results for the systems CH4 -H2O, CO2 -H2O and H2S -H2O respectively. The average absolute deviation is equal to 1.2 % for CH4 -H2O system, 1.9 for CO2 -H2O system and 7.8 % for H2S -H2O system.

Validation of heat capacity for carbon dioxide

Some physical properties as heat capacities of pure components used for heat and mass transfer modelling were also checked. The ideal gas heat capacity equation [START_REF] Cloirec | Les Composés Organiques Volatils (COVs) dans l'Environnement[END_REF] developed by Aly and Lee [START_REF] Aly | Fluid Phase Equil[END_REF] is used for Aspen Plus® simulations. Fig. 47 shows for example the comparison of results obtained with the adopted model [START_REF] Aly | Fluid Phase Equil[END_REF] in Aspen Plus® with experimental data [START_REF] Elliott | Introductory chemical engineering thermodynamics[END_REF].

𝐶 𝑝 * 𝑖𝑔 = 𝐶 1𝑖 + 𝐶 2𝑖 ( 𝐶 3𝑖 𝑇 ⁄ sinh(𝐶 3𝑖 𝑇 ⁄ ) ) 2 + 𝐶 4𝑖 ( 𝐶 5𝑖 𝑇 ⁄ cosh(𝐶 5𝑖 𝑇 ⁄ ) ) 2 (87)
The values of the constants of equation ( 87) used to calculate the heat capacities of carbon dioxide are listed in Table 47. [START_REF] Elliott | Introductory chemical engineering thermodynamics[END_REF] The deviations between experimental and calculated results of heat capacity of carbon dioxide are presented in Fig. 48.

NaOH mass fraction

The deviations between experimental and calculated results of liquid density are depicted in Fig. 50.

Fig. 50: Deviation between experimental and calculated results of liquid density

Validation of liquid viscosity of NaOH -H2O

For the liquid viscosity, a corrective model for electrolytes called "Jones-Dole" is applied in Aspen Plus®. This model uses the mass fraction of the solvent in the liquid phase. This model is presented in Equation ( 88) [110].

𝜇 𝑙 = 𝜇 𝑠𝑜𝑙𝑣 (1 + ∑ ∆𝜇 𝑐𝑎 𝑙 𝑐𝑎 ) (88)
Where μsolv is the viscosity of the liquid solvent mixture, calculated using the Andrade model and Δμ l ca is the contribution to the viscosity correction due to apparent electrolyte ca (cation-anion).

In Aspen Plus®, the ENRTL model calculates the viscosity of the liquid solvent mixture by default using the modified Andrade Equation [START_REF] Clodic | CO2 capture by anti-sublimation -Thermo-economic process evaluation[END_REF].

ln 𝜇 𝑙 = ∑ 𝑓 𝑖 ln 𝜇 𝑖 * 𝑙 + ∑ ∑(𝑘 𝑖𝑗 𝑓 𝑖 𝑓 𝑗 + 𝑚 𝑖𝑗 𝑓 𝑖 2 𝑓 𝑗 2 ) 𝑗 𝑖 𝑖 ( 89
)
Where fi is by default the mole fraction of the component i. kij and mij are binary parameters, they allow accurate representation of complex liquid mixture viscosity. kij and mij are given respectively by Equations ( 90) and [START_REF] Strigle | Packed tower design and applications, Random and structured packings[END_REF]. When the electrolyte concentration exceeds 0.1 M, Aspen Plus® uses the Equation (92) of Breslau and Miller instead of that of Jones and Dole [111].

𝑘 𝑖𝑗 = 𝑎 𝑖𝑗 + 𝑏 𝑖𝑗 𝑇 (90)
𝑚 𝑖𝑗 = 𝑐 𝑖𝑗 + 𝑑 𝑖𝑗 𝑇 (91)
∆𝜇 𝑐𝑎 𝑙 = 2.
Where Ve is the effective volume. It is given by Equation [START_REF] Chan | Industrial and Engineering Chemistry Process Design and Development[END_REF].

𝑉 𝑒 = 𝐵 𝑐𝑎 -0.002 2.6
For salts involving univalent ions (93.a)

𝑉 𝑒 = 𝐵 𝑐𝑎 -0.011 5.06

For other salts (93.b)

Where Bca is calculated using Equation [START_REF] Schultes | Research on mass transfer columns "Old hat or still relevant ?[END_REF].

𝐵 𝑐𝑎 = (𝑏 𝑐,1 + 𝑏 𝑐,2 𝑇) + (𝑏 𝑎,1 + 𝑏 𝑎,2 𝑇) (94)
c a ca is the concentration of apparent electrolyte ca. It is calculated using Equation [START_REF] Billet | A physical model for the prediction of liquid hold-up in two-phase counter-current columns[END_REF].

𝑐 𝑐𝑎 𝑎 = 𝑥 𝑐𝑎 𝑎 𝑉 𝑚 𝑙 (95)
Where x a ca is the mole fraction of the apparent electrolyte ca and V l m is the molar volume of the liquid mixture calculated by the Clarke model.

The electrolyte correction model parameters were improved for the ion HCO 3-using KHCO3 -H2O viscosity data [START_REF] Palaty | Viscosity of diluted aqueous K2CO3 / KHCO3 solutions[END_REF]. The regression of parameters for Na + ion was performed with viscosity data of NaOH -H2O system [START_REF] Vargaftik | Dictionary of thermophysical properties of gases and liquids[END_REF]. For ion CO3 2-, parameters have been optimized considering experimental data for the K2CO3 -H2O system [114]. For other ions, values provided by Aspen Plus® database were used. Fig. 51 shows the fit between experimental data and Aspen Plus® results for the viscosity of the liquid phase as a function of the mass fraction of sodium hydroxide. Data of this comparison are carried out at a temperature of about 25 °C. [START_REF] Klochko | Electrical conductivity and viscosity of aqueous solutions of NaOH and KOH[END_REF] The deviations between experimental and calculated results of liquid viscosity are shown in Fig. 52.

Fig. 51: Comparison between model and experimental data for liquid viscosity of NaOH -H2O ( ____ ) Aspen Plus® ; (♦) Experimental values

Validation of surface tension of NaOH -H2O

To calculate the liquid phase surface tension, Aspen Plus® uses the model of Onsager-Samaras presented by Equation [START_REF] Fair | A comprehensive model for the performance of columns containing structured packings[END_REF]. Results obtained were compared to experimental data found in literature, as shown in Fig. 53 [START_REF] Gel'perin | The relation between the surface tension of aqueous solutions of inorganic substances and concentration and temperature[END_REF].

𝜎 = 𝜎 𝑠𝑜𝑙𝑣 + ∑ 𝑥 𝑐𝑎 𝑎 𝑐𝑎 ∆𝜎 𝑐𝑎 ( 96
)
Where σsolv is the surface tension of the solvent mixture calculated using the General Pure Component Liquid Surface Tension Model [START_REF] Horvath | Handbook of Aqueous Electrolyte Solutions[END_REF] and Δσca is the contribution to the surface tension correction due to apparent electrolyte ca, calculated using Equation [START_REF] Olujić | Predicting the efficiency of corrugated sheet structured packings with large specific surface area[END_REF].

∆𝜎 𝑐𝑎 = 80 𝜀 𝑠𝑜𝑙𝑣 𝑐 𝑐𝑎 𝑎 𝑙𝑜𝑔 [ 1.13 × 10 -13 (𝜀 𝑠𝑜𝑙𝑣 𝑇) 3 𝑐 𝑐𝑎 𝑎 ] ( 97
)
Where ɛsolv is the dielectric constant of the solvent mixture. [START_REF] Gel'perin | The relation between the surface tension of aqueous solutions of inorganic substances and concentration and temperature[END_REF] The deviations between experimental and calculated results of surface tension are presented in Fig. Fig. 54: Deviation between experimental and calculated results of surface tension

Fig. 53: Comparison between model and experimental data for liquid phase surface tension of 5 wt% NaOH aqueous solution ( ____ ) Aspen Plus® ; (♦) Experimental values

Validation of chemical parameters

All the reactions involved in the process have been specified. These reactions are assumed to be in chemical equilibrium. Only the reaction between carbon dioxide and hydroxyl ion is kinetically controlled. The reactions defined in Aspen Plus® are presented in the following expressions.

The calculation of the temperature-dependent equilibrium constants requires the knowledge of coefficients A, B and C, which were taken from the work of Edwards et al [118]. Coefficients used for Reactions (R.13) to (R.17) are presented in Table 48.

A temperature-dependent expression was proposed in order to define coefficients A, B and C for Reaction (R.18). The values of the defined coefficients are presented in Table 49. The same method has been adopted in order to validate the equilibrium constant for Reaction (R.19).

For kinetic-controlled reactions (R.24) and (R.25), the power law expression (100) is adopted by Aspen Plus®. k and E parameters are given in Table 50 for reactions (R.24) and (R.25), knowing that the concentration is based on the molarity. The reaction rate constants for the kinetic reaction between CO2 and OH -have been verified against experimental data [120]. Fig. 56 shows the good agreement between the results. [120] The deviations between Arrhenius equation ( 88) and experimental results are depicted in Fig. 57.

Simulation results

After validation of physicochemical parameters, Aspen Plus® simulations of the packed absorption column of the pilot plant have been performed using the Rate-based model. Details of the column used in the experiments, as well as the description of the given gas and liquid inlets are defined in Table 51. The type of packing used for this comparison is Flexipac® 350Y. This packing is different with respect to the one used in the pilot BioGNVAL (Montz® B1-420), but no literature data are available for this last. When designing a packed column, it is desired to minimize the flow of liquid to reduce the consumption of water and the energy needed by the pump for its circulation. However, the flow must allow the absorption of H2S and reduce its content to less than 1 ppm. This is a very important parameter since it has an influence on the thermodynamic and hydrodynamic conditions.

In 2007, Sanchez et al. showed that with a fixed air flow, an increase in liquid flow rate will improve the velocity of transfer [START_REF] Sanchez | Hydrodynamic and mass transfer in a new concurrent two-phase flow gas-liquid contactor[END_REF]. Fig. 58 obtained using Aspen Plus® demonstrates that increasing the liquid flow improves the absorption of H2S. However, a mass flow rate of 240 kg/h is sufficient to eliminate the totality of hydrogen sulfide with a minimal pressure loss.

As shown in Fig. 58, chemical conditions strongly influence the transfer percentage. The absorption rate increases with the concentration of sodium hydroxide in the liquid phase.

Fig. 58: Influence of the liquid flow on the absorption of hydrogen sulfide

Sodium hydroxide concentrations: ( ____ ) 0.5 g/l ; (----) 1.9 g/l Table 52 shows that the increase in liquid flow causes a rise in pressure drop. This increase is limited, and shows that this parameter does not depend too much on the liquid flow.

The results obtained on the BioGNVAL pilot plant were compared to those from the modified Billet and Schultes model. These results confirm the precision of this model for activities of interest. The dependence between the pressure drop and the gas flow rate is much more important as shown in Fig. 59, because the gas has the continuous phase in the packed column. [BioGNVAL, 2015] The deviations between experimental and calculated results of pressure drop are presented in Fig. 60. When hydrogen sulfide is absorbed into a sodium hydroxide solution, it can react directly with hydroxyl ions by a proton transfer reaction as seen in Reaction R. [START_REF] Biard | Contribution au développement d'un procédé de lavage chimique compact. Traitement du sulfure d'hydrogène par le chlore à l'échelle semi-industrielle et de COV odorants par oxydation avancée O3 / H2O2 à l'échelle du laboratoire[END_REF]. Compared to the diffusion phenomena, this reaction is extremely rapid and can be considered instantaneous. Since hydrogen sulfide is absorbed more rapidly than carbon dioxide by aqueous sodium hydroxide solutions, partial selectivity can be attained when both gases are present as seen in Fig. 61. Selectivity is favored by short gas-liquid contact times and low temperatures [START_REF] Kohl | Gas purification[END_REF].

Fig. 59: Influence of gas flow rate on the pressure drop ( ____ ) Modified Billet and Schultes model ; (♦) Experimental values

Fig. 61: Influence of the concentration of NaOH in the removal of H2S and CO2 Compound: ( ____ ) CO2 ; (----) H2S

A key parameter affecting the overall performances of the absorption unit is the temperature, since it affects physicochemical properties (such as the solubility of acidic compounds in the aqueous phase, according to the Henry's law) and the chemical reactions in the liquid phase. Fig. 62 shows a good agreement between the results of Aspen Plus® simulations and those obtained from the pilot plant. The deviations between experimental and calculated results of hydrogen sulfide concentration at the outlet of the column are depicted in Fig. 63.

Conclusion

The thermodynamic model used in the simulations (Electrolyte NRTL) was validated with experimental data from the literature. Simulations were performed in order to study the influence of temperatures, chemical and hydrodynamic parameters on H2S absorption. The simulation results were compared to experimental data obtained on the BioGNVAL pilot plant. The comparison was successful and shows that the two results are in good agreement. The model allows predicting realistically the separation efficiencies of H2S in biogas.

The simulation results confirm the observations made on the demonstrator. The NaOH aqueous solution is effective for the removal of H2S in a packing column. The removal efficiency reaches values higher than 99.5 % throughout the operation period of the demonstrator. The use of NaOCl is important to prevent the accumulation of H2S in the aqueous solution by creating an irreversible reaction.

From a practical point of view, the use of hazardous substances (NaOH and NaOCl) requires an operator to manipulate them, which complicates the commercialization of the technology. Furthermore, salt precipitation may occur and can cause blockage of pumps and heat exchangers.

Despite the advantages of absorption technology, this system will be bypassed in order to test another promising technology that requires less financial means and which does not use hazardous chemical products. This technology is adsorption using activated carbon. It will be discussed in the next chapter.

Chapter 7: Modeling hydrogen sulfide adsorption onto activated carbon

Résumé :

Le dernier chapitre concerne la modélisation du procédé d'adsorption du H2S sur charbon actif. Un modèle dynamique a été développé pour modéliser la courbe de percée du système H2S -charbon actif.

La courbe de percée est utilisée pour décrire l'évolution spatio-temporelle de la concentration de H2S en phase gazeuse. Ce type de modélisation de la dynamique de la colonne d'adsorption est basé sur la définition des bilans de matière : un bilan de masse de la phase gazeuse où le transfert par convection domine et un bilan massique des particules adsorbantes où le transfert par diffusion domine.

Au cours de cette étude, les différents coefficients de transfert de masse impliqués dans le processus ont été estimés dans les mêmes conditions dans lesquelles les expérimentations sur le démonstrateur "BioGNVAL" ont eu lieu.

Les simulations ont été effectuées seulement pour des concentrations élevées en H2S car une unité d'adsorption industrielle qui traite de faibles concentrations en H2S atteint la saturation dans une période de quelques mois (≈ 3 mois). La simulation de cette période avec un pas de temps de 0,01 s exige des temps de calculs énormes qui ne sont pas supportés par la machine de travail. De plus, l'augmentation du pas de temps entraînera la divergence des calculs.

Les résultats obtenus avec le modèle développé devraient être comparés aux données expérimentales afin d'ajuster le coefficient global de transfert de masse et les paramètres d'équilibre donnés par l'isotherme d'adsorption.

Introduction

This chapter deals with modeling the breakthrough curve in the case of hydrogen sulfide adsorption onto activated carbon. Physical adsorption using activated carbon is a traditional technology widely used for the removal of H2S. The adsorption efficiency depends on several factors, such as relative humidity, temperature, concentration of H2S in biogas and characteristics of the activated carbon. To improve efficiency, the activated carbon may be impregnated with sodium hydroxide (NaOH), potassium hydroxide (KOH), sodium carbonate (Na2CO3), sodium bicarbonate (NaHCO3), potassium iodide (KI) or potassium permanganate (KMnO4).

Operating conditions of the adsorption column for the removal of hydrogen sulfide

The adsorbent chosen for this study is activated carbon impregnated with a base (NaOH or KOH) dedicated to the elimination of hydrogen sulfide. The selected commercial adsorbent called "Airpel Ultra DS" is provided by Desotec® Company. Its properties are shown in Table 53. The extruded activated carbon was chosen because it causes a lower pressure drop than granular and powder activated carbons.

The micropore volume is an important parameter for the kinetics of hydrogen sulfide adsorption. Compared to the other commercial adsorbents, the "Airpel Ultra DS" contains a high volume of micropores.

The hydrogen sulfide adsorption capacity is also influenced by other operating conditions. One of the important parameters is the relative humidity. As seen in Fig. 64 presenting the breakthrough curve of H2S, the best hydrogen sulfide adsorption capacities are obtained for values of relative humidity between 55 % and 100 % with an optimal value of 85 %. The breakthrough curve presents the evolution of concentration of the pollutant to be removed (H2S) as a function of time. It predicts the time required for the saturation of the activated carbon in the adsorption bed.

The adsorption using impregnated activated carbon is improved by injection of a small quantity of oxygen. The amount injected is generally of the order of 4 times the amount of hydrogen sulfide present in the biogas. The oxidation of hydrogen sulfide will make larger molecules to block them in the micropores. 𝑃𝑒 𝑝 = 𝑑 𝑝 𝑢 𝑉 𝐷 𝐻 𝑠 𝑆-𝐵𝑖𝑜𝑔𝑎𝑠 [START_REF] Gel'perin | The relation between the surface tension of aqueous solutions of inorganic substances and concentration and temperature[END_REF] The H2S molecular diffusion coefficient is calculated using The Equation [START_REF] Horvath | Handbook of Aqueous Electrolyte Solutions[END_REF] proposed by Wilke and Fairbanks [START_REF] Wilke | Diffusion coefficients in multicomponent gas mixtures[END_REF]. This equation is derived from the theories of Maxwell and Stefan. [START_REF] Horvath | Handbook of Aqueous Electrolyte Solutions[END_REF] Equations ( 118), ( 119), ( 120) and ( 121) are used to estimate the diffusivities for the different binary systems at low pressure. These equations developed by Slattery and Bird [START_REF] Slattery | Calculation of the diffusion coefficient of dilute gases and of the self-diffusion coefficient of dense gases[END_REF] from a kinetic theory are shown in Table 56.

𝐷

 Internal mass transfer coefficient

The hydrogen sulfide molecules are now located on the surface of the activated carbon. The diffusion mechanisms govern the transport of hydrogen sulfide molecules into the pores of activated carbon. In porous materials, diffusion mechanisms are of four types: molecular diffusion caused by collisions between molecules, Knudsen diffusion caused by collisions of the molecules with the walls of the pore, surface diffusion caused by the electrostatic forces exerted by the walls on the molecules and Poiseuille diffusion caused by the difference in total pressure across a particle. The equations used to estimate these diffusivities are listed in Table 57.

The values obtained for the dimensionless numbers and the diffusion coefficients are presented in Table 58. As seen in Table 58, the molecular diffusion is the dominant transport mechanism. The Poiseuille diffusivity can be neglected because the pressure drop over a particle is very small [START_REF] Ruthven | Principles of adsorption and adsorption processes[END_REF].

Knudsen diffusion and surface diffusion contribute to global internal diffusion. This contribution is very small compared to the contribution of external diffusion. Therefore, only external diffusion will be considered. The external mass transfer coefficient calculated using Equation ( 101) is equal to 0.021 m.s -1 .

Modeling of breakthrough curves

The breakthrough curve is used to describe the spatiotemporal evolution of the concentration of H2S in the gas phase. This type of modeling the dynamics of the adsorption column is based on the definition of mass transfer balances: a mass balance of the gas phase where the transfer by convection dominates and a mass balance of the adsorbent particles where the diffusion transfer dominates [START_REF] Sigot | Epuration fine des biogaz en vue d'une valorisation énergétique en pile à combustible de type SOFC : Adsorption de l'octaméthylcyclotétrasloxane et du sulfure d'hydrogène[END_REF].

Several hypotheses govern the flow of the biogas in the absorption column: The properties and the superficial velocity of the biogas are assumed constant throughout the adsorption column and the porosity of the adsorption column is considered uniform [START_REF] Boulinguiez | Procédé d'adsorption et régénération électrothermique sur textile de carbone activé : Une solution pour la problématique des COV dans des gaz à fort potentiel énergétique[END_REF]. Based on the assumptions outlined, the mass balance of the adsorption column can be equated as seen in Equation [START_REF] Wilke | Mass Transfer in the Flow of gases through Granular Solids extended to Low Modified Reynolds Numbers[END_REF]. This equation has been used by several authors as Ruthven [START_REF] Ruthven | Principles of adsorption and adsorption processes[END_REF], Suzuki [START_REF] Suzuki | Adsorption Engineering[END_REF], Hwang et al. [START_REF] Hwang | Adsorption and thermal regeneration of methylene chloride vapor on an activated carbon bed[END_REF], Brosillon et al. [START_REF] Brosillon | Mass Transfer in VOC Adsorption on Zeolite: Experimental and Theoretical Breakthrough Curves[END_REF], Yang [START_REF] Yang | Adsorbents: Fundamentals and Applications[END_REF], Boulinguiez [START_REF] Boulinguiez | Procédé d'adsorption et régénération électrothermique sur textile de carbone activé : Une solution pour la problématique des COV dans des gaz à fort potentiel énergétique[END_REF] and Sigot [START_REF] Sigot | Epuration fine des biogaz en vue d'une valorisation énergétique en pile à combustible de type SOFC : Adsorption de l'octaméthylcyclotétrasloxane et du sulfure d'hydrogène[END_REF]. -3 ] is the density of activated carbon. ɛ [-] is the void fraction of the adsorption column. q [g.kg -1 ] is the amount of H2S adsorbed per kg of activated carbon.

-
The mass balance Equation [START_REF] Wilke | Mass Transfer in the Flow of gases through Granular Solids extended to Low Modified Reynolds Numbers[END_REF] takes into account the axial dispersion, convection, the accumulation of the gas phase and the overall mass transfer by adsorption [START_REF] Boulinguiez | Procédé d'adsorption et régénération électrothermique sur textile de carbone activé : Une solution pour la problématique des COV dans des gaz à fort potentiel énergétique[END_REF]. To simplify the problem, initially the axial diffusion coefficient is neglected. The mass balance presented by Equation ( 126) is simplified.

𝑢 𝐿 𝜕𝐶 𝜕𝑧 + 𝜕𝐶 𝜕𝑡 + 𝜌 𝜀 𝜕𝑞 𝜕𝑡 = 0 (127)
Considering a single transfer resistance material represented by the linear driving force model, the flux transferred may then be expressed relative to the solid phase concentration as shown in Equation [START_REF] Kataoka | Mass Transfer in Laminar Region between Liquid and Packing Materials Surface in the Packed-Bed[END_REF], or relative to the gas phase concentration as shown in Equation [START_REF] Doytchava | Mass Transfer from Solid Particles to Fluid in Granular Bed[END_REF]. By combining the equation ( 127) and ( 129), the expression ( 136) is obtained [START_REF] Sigot | Epuration fine des biogaz en vue d'une valorisation énergétique en pile à combustible de type SOFC : Adsorption de l'octaméthylcyclotétrasloxane et du sulfure d'hydrogène[END_REF].

𝑢
To simplify the system of equations ( 136) and [START_REF] Suzuki | Adsorption Engineering[END_REF], three time constants associated to mass transfer τ1, τ2 and τ3 [s -1 ] are highlighted. The system of equation becomes [START_REF] Sigot | Epuration fine des biogaz en vue d'une valorisation énergétique en pile à combustible de type SOFC : Adsorption de l'octaméthylcyclotétrasloxane et du sulfure d'hydrogène[END_REF]:

1
Where:

1 𝜏 1 = 𝑢 𝐿 (
In order to solve the system of equations numerically, it is necessary to discretize it by means of Euler method which uses the finite difference quotient. The system of equations thus obtained is given by the Expressions (143) and (144). The boundary and initial conditions are given respectively by Equations ( 145) and (146) [START_REF] Sigot | Epuration fine des biogaz en vue d'une valorisation énergétique en pile à combustible de type SOFC : Adsorption de l'octaméthylcyclotétrasloxane et du sulfure d'hydrogène[END_REF]. After the construction of the discrete form of the analytical mass balance equations, the input parameters for the simulation of the breakthrough curve are listed in Table 60. The simulation was performed for high hydrogen sulfide concentrations (5, 7.5 and 10 mol%) as seen in Table 60. The breakthrough curves obtained are presented respectively in Fig. 67, 68 and 69. An industrial adsorption column with such dimensions (0.8 m in height and 0.8 m in diameter) and treating low concentrations of H2S (< 100 ppm) reaches saturation after few months (≈ 3 months).

The simulation of such period with a time step of 0.01 s demands enormous time calculations that are not supported by the working machine. In addition, increasing the time step will cause the divergence of calculations.

Conclusion

Based on the mass balance equations, a dynamic model has been developed to simulate the breakthrough curve for the system H2S -Activated carbon. To simulate the dynamic behavior of an adsorption column for the removal of low concentrations of H2S, a powerful calculation tool is necessary.

The results obtained with the model developed should be compared to experimental data in order to adjust the overall mass transfer coefficient and equilibrium parameters given by the Langmuir isotherm. The axial dispersion coefficient should be added in the mass balance equation to evaluate its contribution on the overall mass transfer coefficient.

This study has highlighted the analytical difficulties due to low concentrations of H2S. It has also showed the importance of the experimental work to develop an accurate model which could be a reliable design tool for biogas purification units.

Conclusions and perspectives

This thesis is part of the innovative project led by the Cryo Pur® society that aims to develop a technology to purify and upgrade biogas in order to be used as a liquid fuel or injected into the natural gas grid. The purpose of this work is to ensure a continuous removal of hydrogen sulfide upstream of the process because of the risks of toxicity, corrosion and odors. This thesis has allowed drawing scientific and technical conclusions.

From a technical point of view, two desulfurization technologies were tested: chemical absorption in a packing column using sodium hydroxide as an aqueous solvent and adsorption using impregnated activated carbon in a fixed bed. Both techniques have been experimented on a demonstrator "BioGNVAL" developed by Cryo Pur® Society which treats 85 Nm 3 /h of real biogas from the sewage treatment plant of Valenton. The general process is characterized by the following conditions:

 The concentrations of methane and carbon dioxide are about ≈ 60 mol% of CH4 and ≈ 35 mol% of CO2.  The process operates at a pressure slightly higher than the atmospheric pressure to prevent the infiltration of oxygen.  The desulfurization of biogas must be complete to ensure a high quality of CO2 and biomethane.  The concentration of H2S in the biogas is low and generally varies between 10 and 100 ppm.

Both technologies have shown satisfactory separation efficiency greater than 99.5 %. The bibliographic study has allowed choosing an efficient absorption process that uses sodium hydroxide as a chemical solvent for the selective separation of hydrogen sulfide. This method is based on contacting the biogas to be treated with the aqueous NaOH in a structured packing column.

Despite the advantages of this process, it also has some drawbacks. For example, in the presence of CO2, the aqueous solution of sodium hydroxide cannot be regenerated because there is formation of sodium carbonate (Na2CO3) precipitate. Moreover, when the tanks of NaOH and NaOCl reach their lowest level, they must be filled by an operator who has to handle hazardous products. This operation is an obstacle to the commercialization of the process which must be automated without the repetitive intervention of an operator. Nevertheless for high H2S concentrations for processes which are continuously operated with personnel, this absorption process can be adopted.

To overcome these drawbacks for fully automated biogas purification equipment, the adsorption technology using impregnated activated carbon was tested. It has achieved similar separation efficiencies to the absorption technology without the use of hazardous products and without the need for an operator to ensure continuous operation of the process. Furthermore, the investment and operating costs of the adsorption process are lower than those of the absorption process.

From a scientific point of view, experimental data collected on the BioGNVAL pilot plant were used to develop a new hydrodynamic model which predicts accurately the key hydrodynamic parameters in a structured packing column: liquid holdup, pressure drop, effective interfacial area and the two transition points: loading and flooding points. It also allows the design and optimization of structured packing columns to operate at full capacity. This model was developed based on an existing model: Billet and Schultes. Some constants and exponents present in the correlations of this last model were developed following a sensitivity analysis and then optimized based on experimental data to finally implement a new model adapted to predict precisely the hydrodynamic parameters of a small scale structured packing column used for biogas or natural gas applications.

After the hydrodynamic study, simulations using the chemical process optimization software Aspen Plus® V8.0 have been performed in order to determine the efficiencies of separation for different operating conditions. The rate-based model approach has been used. To describe vapor-liquid equilibrium, a γ/ϕ approach has been considered: the Electrolyte Non-Random Two-Liquid (ENRTL) model has been adopted to represent non-idealities in the liquid phase, while the Redlich-Kwong equation of state has been used for the vapor phase.

In order to improve the simulation of absorption, Henry's law constants of the main biogas components (CH4, CO2 and H2S) in water, the properties of pure components, equilibrium constants for the equilibrium reactions and the reaction rate constant for the kinetically controlled reaction between carbon dioxide and the hydroxide ion have been verified against experimental data. After the verification and modification of the thermodynamic model, the results of the simulations were compared to experimental data obtained on the demonstrator "BioGNVAL". The comparison shows that the two results are in good agreement. In connection with this work, it would be interesting to study the perspective of incorporating the hydrodynamic model developed, in the database of Aspen Plus.

For adsorption technology, a dynamic model has been developed to simulate the breakthrough curve for the system H2S -Activated carbon. During this study, the different mass transfer coefficients involved in the process have been estimated under the same conditions in which the experiments on "BioGNVAL" pilot plant were held.

The results obtained with the model developed should be compared to experimental data in order to adjust the overall mass transfer coefficient and equilibrium parameters given by the adsorption isotherm. Another perspective is to consider the reaction phenomena due to the impregnation of the activated carbon in order to have a prediction tool able to manage the design of adsorption columns. It will be also interesting to reinsert the axial dispersion coefficient in the mass balance equation to evaluate its contribution.

Fig. 1 :Fig. 2 :Fig. 3 :Fig. 4 :Fig. 5 :Fig. 6 :Fig. 9 :Fig. 10 :Fig. 11 :Fig. 12 :Fig. 13 :Fig. 20 :Fig. 22 :Fig. 23 :Fig. 24 :Fig. 25 :Fig. 26 :Fig. 27 :Fig. 30 :Fig. 31 :Fig. 32 :Fig. 33 :Fig. 34 :Fig. 36 :

 12345691011121320222324252627303132333436 Fig. 1: Distribution of final energy consumption in France [1] ................................................ Fig. 2: Distribution of production of renewable energies by sector [1] ................................... Fig. 3: Biogas primary production in 2013 [2] .......................................................................... Fig. 4: Simplified diagram of production of biomethane ......................................................... Fig. 5: Example of biogas utilization by Cryo Pur® Company [9].............................................. Fig. 6: Vapor pressure of the main components present in biogas ......................................... Fig. 7: Pressure -Temperature equilibrium behavior for the CH4 -CO2 system [29] ............. Fig. 8: Pressure -Temperature equilibrium behavior for the CH4 -H2S system [30] ............. Fig. 9: Biogas density as a function of temperature ................................................................ Fig. 10: Biogas viscosity as a function of temperature ............................................................. Fig. 11: Thermal conductivity of biogas as a function of temperature .................................... Fig. 12: Heat capacity at constant pressure of biogas as a function of temperature .............. Fig. 13: Heat capacity at constant volume of biogas as a function of temperature ................ Fig. 14: Two-film theory [36] .................................................................................................... Fig. 15: Chemical structure of some alkanolamines [43] ......................................................... Fig. 16: Main gas-liquid contactors [48] ................................................................................... Fig. 17: Schematic representation of a packed column ........................................................... Fig. 18.a: Random packing Nutter Ring [Sulzer] ....................................................................... Fig. 18.b : Structured packing Mellapak [Sulzer] ...................................................................... Fig. 19: Transport mechanism of the adsorbate molecules on the adsorbent surface ........... Fig. 20: Shapes of commercial activated carbons [53]............................................................. Fig. 21: Effect of temperature on some adsorbents [57] ......................................................... Fig. 22: Classification of adsorption isotherms [IUPAC] ........................................................... Fig. 23: Skarstrom cycle stages and pressure variations [68] .................................................. Fig. 24: Schematic representation of a membrane [71] .......................................................... Fig. 25: Diagram of the main types of membranes [75] .......................................................... Fig. 26: The geometric configurations of membrane contactors [76] ..................................... Fig. 27: Most technologies used for the purification and upgrading of biogas [International Energy Agency] ......................................................................................................................... Fig. 28: Comparison of investment costs of different biogas purification and upgrading technologies [88] ...................................................................................................................... Fig. 29: BioGNVAL demonstrator located at Valenton water treatment plant [9] .................. Fig. 30: Schematic representation of Cryo Pur® system [9] ..................................................... Fig. 31: Simplified flowsheet of the BioGNVAL pilot plant [9] ................................................. Fig. 32: Schematic diagram of the absorption subsystem for elimination of hydrogen sulfide [9] ............................................................................................................................................. Fig. 33: Piping and instrumentation diagram of adsorption subsystem for the removal of hydrogen sulfide [9] ................................................................................................................ Fig. 34: Carbon dioxide antisublimation [9] ............................................................................. Fig. 35: Variation of the H2S content at the inlet and at the outlet of the absorption column .................................................................................................................................................. Fig. 36: Variation of the biogas temperature at the inlet of the demonstrator and at the outlet of the absorption column .............................................................................................. Fig. 37: Observation of the desorption phenomenon of H2S ................................................... Fig. 38.a: Pressure drop evolution in a packing column .......................................................... Fig. 38.b: Liquid holdup evolution in a packing column ...........................................................

Fig. 39 .Fig. 43 :Fig. 44 :Fig. 45 :Fig. 49 :Fig. 50 :Fig. 51 :Fig. 52 :Fig. 54 :Fig. 59 :Fig. 60 :Fig. 61 :Fig. 62 :Fig. 63 :Fig. 64 :Fig. 65 :

 39434445495051525459606162636465 Fig. 39.a. Pressure drop evaluation for liquid load uL = 20.5 m/h ........................................... Fig. 39.b. Liquid holdup evaluation for liquid load uL = 20.5 m/h ............................................ Fig. 40.a. Prediction of effective interfacial area by Billet and Schultes model for the system Air / Kerosol 200 ....................................................................................................................... Fig. 40.b. Prediction of effective interfacial area by SRP model for the system Air / Kerosol 200 ............................................................................................................................................ Fig. 40.c. Prediction of effective interfacial area by Delft model for the system Air / Kerosol 200 ............................................................................................................................................ Fig. 41. Liquid holdup and pressure drop with an Air -Water system using Flexipac® 350Y packing ................................................................................................................................... Fig. 42. Liquid holdup and pressure drop with an Air -Kerosol 200 system using Flexipac® 350Y packing ........................................................................................................................... Fig. 43: Flowsheet of the absorption process simulated using Aspen Plus® ......................... Fig. 44: Henry coefficients for CH4 -H2O system ................................................................... Fig. 45: Henry coefficients for CO2 -H2O system ................................................................... Fig. 46: Henry coefficients for H2S -H2O system ................................................................... Fig. 47: Comparison between model and experimental data for heat capacity for carbon dioxide .................................................................................................................................... Fig. 48: Deviation between experimental and calculated results of carbon dioxide heat capacity................................................................................................................................... Fig. 49: Comparison between model and experimental data for liquid density of NaOH -H2O ................................................................................................................................................ Fig. 50: Deviation between experimental and calculated results of liquid density ............... Fig. 51: Comparison between model and experimental data for liquid viscosity of NaOH -H2O ......................................................................................................................................... Fig. 52: Deviation between experimental and calculated results of liquid viscosity ............. Fig. 53: Comparison between model and experimental data for liquid phase surface tension of 5 wt% NaOH aqueous solution .......................................................................................... Fig. 54: Deviation between experimental and calculated results of surface tension ............ Fig. 55: Comparison of results of equilibrium constant for reaction R.18 ............................. Fig. 56: Comparison of results of reaction rate constant for reaction R.24 .......................... Fig. 57: Deviation between experimental and calculated results of reaction rate constant (R.24) ...................................................................................................................................... Fig. 58: Influence of the liquid flow on the absorption of hydrogen sulfide ......................... Fig. 59: Influence of gas flow rate on the pressure drop ....................................................... Fig. 60: Deviation between experimental and calculated results of pressure drop .............. Fig. 61: Influence of the concentration of NaOH in the removal of H2S and CO2 .................. Fig. 62: Influence of the temperature of liquid in the absorption of hydrogen sulfide ......... Fig. 63: Deviation between experimental and calculated results of H2S concentration leaving the packing column ................................................................................................................ Fig. 64: Influence of relative humidity in the adsorption of hydrogen sulfide using Airpel Ultra DS [53] .................................................................................................................................... Fig. 65: Schematic representation of an adsorption column [122] ....................................... Fig. 66: Influence of Langmuir equilibrium parameters on adsoption sites occupied .......... Fig. 67: Breakthrough curve simulated for H2S molar percentage of 5 % ............................. Fig. 68: Breakthrough curve simulated for H2S molar percentage of 7.5 % .......................... Fig. 69: Breakthrough curve simulated for H2S molar percentage of 10 % ...........................

Fig. 4 :

 4 Fig. 4: Simplified diagram of production of biomethane

  and C [-]: Component specific constants.

Fig. 8 :

 8 Fig. 8: Pressure -Temperature equilibrium behavior for the CH4 -H2S system [30] (─) three-phase equilibrium boundaries ; (•••) critical curves ; (--) SVE, VLE and SLE of CH4 and H2S ; (■) quadruple point QP2 ; (▲) quadruple point QP1 ; (Δ) Upper Critical EndPoint UCEP1 ; (□) Upper Critical EndPoint UCEP2

Fig. 10 :

 10 Fig. 10: Biogas viscosity as a function of temperature ( ____ ) Air ; ( …… ) Biogas with 40 mol% of CO2 ; (----) Biogas with 35 mol% of CO2

Fig. 11 :

 11 Fig. 11: Thermal conductivity of biogas as a function of temperature ( ____ ) Air ; ( …… ) Biogas with 40 mol% of CO2 ; (----) Biogas with 35 mol% of CO2

Fig. 12 :Fig. 13 :

 1213 Fig. 12: Heat capacity at constant pressure of biogas as a function of temperature ( …… ) Biogas with 60 mol% of CH4 ; (----) Biogas with 65 mol% of CH4

  2 ]: flux of H2S transferred by unit area. pH2S [Pa]: partial pressure of H2S in the gas phase. p * H2S [Pa]: partial pressure of H2S at the interface. CH2S [mol.m -3 ]: concentration of H2S in the liquid phase. C * H2S [mol.m -3 ]: concentration of H2S at the interface. δG and δL [m]: thickness of the stagnant film on the gas side and the liquid side respectively. DH2S,G and DH2S,L [m 2 .s -1 ]: H2S diffusion coefficients in gas phase and liquid phase respectively.

Fig. 18

 18 Fig. 18.a: Random packing Nutter Ring [Sulzer]

  : Ct [mol.m -3 ]: the concentration of the compound in the fluid phase. Ce [mol.m -3 ]: the concentration of the compound at the surface of the adsorbent. kf [m.s -1 ]: the external mass transfer coefficient. aads [m 2 ]: the useful surface area for external transfer. V [m 3 ]: Volume of the adsorption bed.

  s -1 ]: the transferred flux. Dp [m 2 .s -1 ]: the pore diffusion coefficient. ɛp [-]: the porosity. τ [-]: the tortuosity. Cp [mol.m -3 ]: the compound concentration in the pore. r [m]: radius of the pore.

Fig. 19 :

 19 Fig. 19: Transport mechanism of the adsorbate molecules on the adsorbent surface

  (a) Flat sheet membrane (b) Tubular membrane (c) Spiral wound membrane (d) Hollow fiber membrane

Fig. 26 :

 26 Fig.26:The geometric configurations of membrane contactors[START_REF] Boucif | Modélisation et simulation de contacteurs membranaires pour les procédés d'absorption de gaz acides par solvant chimique[END_REF]

Fig. 31

 31 Fig. 31 shows the general operating principle of the BioGNVAL demonstrator. It consists of 7 subsystems:  Chiller.  Desulfurization subsystem, either by chemical absorption in a structured packing column using sodium hydroxide, or by adsorption onto activated carbon in a fixed bed respectively.  Biogas dehumidification and siloxanes icing subsystem.  Subsystem for the capture of carbon dioxide.  Biogas liquefaction subsystem.  Biogas treatment line with flaring output.

Fig. 31 :

 31 Fig.31: Simplified flowsheet of the BioGNVAL pilot plant[START_REF][END_REF]

Fig. 32 :

 32 Fig. 32: Schematic diagram of the absorption subsystem for elimination of hydrogen sulfide [9] Equipment: CV (Control Valve) ; HX (Heat Exchanger) ; CU (Condensing Unit) ; TK (Tank) ; P (Pump) ; VP (Vaccum Pump) ; Instrumentation: FT (Flow Transmitter) ; TT (Temperature Transmitter) ; TE (Temperature Element) ; PT (Pressure Transmitter) ; PDT (Pressure Difference Transmitter) ; PHT (pH Analyzer Transmitter) ; LT (Level Transmitter) ; GD (Gas Detector)

Fig. 36 :

 36 Fig. 36: Variation of the biogas temperature at the inlet of the demonstrator and at the outlet of the absorption column

Fig. 37 :

 37 Fig. 37: Observation of the desorption phenomenon of H2S

  :24 13:55:12 14:24:00 14:52:48 15:21:36 15:50:24 16:19:12 16:48:00 17:16:48

Fig. 39 .

 39 Fig. 39.a. Pressure drop evaluation for liquid load uL = 20.5 m/h Models: ( __ Δ __ ) Billet & Schultes ; (--□--) SRP ; (-. ○-. ) Delft ; ( … + … ) Billet & Schultes modified "Section 5" ; Experimental values: (♦) [100]

Fig. 40 .Fig. 40 .

 4040 Fig. 40.a. Prediction of effective interfacial area by Billet and Schultes model for the system Air / Kerosol 200 Models: ( __ Δ __ ) Billet & Schultes ; (♦) Experimental values [100]

Fig. 41 .Fig. 42 .

 4142 Fig.[START_REF] Descamps | Etude de la capture du CO2 par absorption physique dans les systèmes de production d'électricité basés sur la gazéification du charbon intégrée à un cycle combiné[END_REF]. Liquid holdup and pressure drop with an Air -Water system using Flexipac® 350Y packing

6. 2 . 1 .

 21 Validation of the Temperature-Dependent Henry's constant for CH4 -H2O, CO2 -H2O and H2S -H2O systems For a rigorous estimation of gas solubility in the solvent, the Henry's constants obtained with Aspen Plus® should be verified by experimental data. The Henry's constants based on the mole fraction scale are taken from Aspen Plus® databanks for the gaseous components (CH4, CO2 and H2S) with water. The temperature dependence of the Henry's constants used by Aspen Plus® is represented by: ln 𝐻 𝑖𝑗 = 𝑎 𝑖𝑗 + 𝑏 𝑖𝑗 𝑇 + 𝑐 𝑖𝑗 ln 𝑇 + 𝑑 𝑖𝑗 𝑇 + 𝑒 𝑖𝑗 𝑇 2

Fig. 44 :Fig. 45 :Fig. 46 :

 444546 Fig. 44: Henry coefficients for CH4 -H2O system ( __ □ __ ) Aspen Plus® ; (--Δ--) Harvey equation [106] ; (♦) Experimental values[105]

Fig. 48 : 6 . 2 . 3 .

 48623 Fig. 48: Deviation between experimental and calculated results of carbon dioxide heat capacity6.2.3. Validation of liquid density of NaOH -H2OThe calculations of the density of the liquid phase have been verified depending on the mass fraction of sodium hydroxide at 25 ° C. The results obtained with Aspen Plus ™ are in good agreement with experimental data [109] as shown in Fig.49.

Fig. 49 :

 49 Fig. 49: Comparison between model and experimental data for liquid density of NaOH -H2O ( ____ ) Aspen Plus® ; (♦) Experimental values [109]

Fig. 52 :

 52 Fig. 52: Deviation between experimental and calculated results of liquid viscosity

Fig. 55 :

 55 Fig. 55: Comparison of results of equilibrium constant for reaction R.18 ( ____ ) Aspen Plus® ; (----) Equation (99) ; ( …… ) Proposed equation

Fig. 57 :

 57 Fig.57: Deviation between experimental and calculated results of reaction rate constant (R.[START_REF] Miller | The Joule-Thomson Effect in Carbon Dioxide[END_REF]

Fig. 60 :

 60 Fig. 60: Deviation between experimental and calculated results of pressure drop

Fig. 62 :

 62 Fig. 62: Influence of the temperature of liquid in the absorption of hydrogen sulfide ( ____ ) Aspen Plus® ; (♦) Experimental values[BioGNVAL, 2015]

Fig. 63 : 2 S

 632 Fig. 63: Deviation between experimental and calculated results of H2S concentration leaving the packing column

Fig. 64 :

 64 Fig. 64: Influence of relative humidity in the adsorption of hydrogen sulfide using Airpel Ultra DS [53] Relative humidity: ( …… ) 0 % ; (-.. -..) 33 % ; (----) 58 % ; (-. -. -) 85 % ; ( ____ ) 100 %

Fig. 67 :Fig. 68 :

 6768 Fig. 67: Breakthrough curve simulated for H2S molar percentage of 5 %

Table of contents

 of Abstract .............................................................................................

Erreur ! Signet non défini.

		Sc			Schmidt number	-
	Sh Nomenclature T	Sherwood number Temperature	-K
		a TB A Tc ae uL b uL,e B uL,lp CFl uV uV,e Ch uV,Fl CL uV,lp Clp V WeL Cp z		Specific geometric packing surface area Boiling temperature Constant used for the calculation of pressure drop Critical temperature Effective interfacial area Superficial liquid velocity Length of the corrugation base Effective liquid velocity Constant used for the calculation of pressure drop Superficial liquid velocity at loading point Superficial gas velocity Specific packing constant for calculation of hydrodynamic parameters at Effective gas velocity flooding point Specific packing constant for hydraulic area Superficial gas velocity at flooding point Specific packing constant for mass transfer calculation in liquid phase Superficial gas velocity at loading point Gas mass flow Specific packing constant for calculation of hydrodynamic parameters at Liquid Weber number loading point Specific packing constant for pressure drop calculation Unit length	K m 2 /m 3 K -m/s m 2 m/s m m/s -m/s -m/s m/s -m/s -kg/h ---m
		CV DV 𝑑𝑃 𝑑𝑧			Specific packing constant for mass transfer calculation in gas phase Gas-phase diffusion coefficient Pressure drop	-m 2 /s Pa/m
	d dh dhV 𝑑𝑃 ( 𝑑𝑧 )	𝑑		Packing diameter Hydraulic diameter Diameter of the gas flow channel Dry pressure drop	m m m Pa/m
	(	dp f Fc 𝑑𝑃 ) 𝑑𝑧 𝑝𝑙 Fc,lp αL FrL γ Fl δ Ft ε FSE ζDC fw ζGG g ζGL geff θ h µL hL µV hL,Fl νL hL,lp ξbulk hL,pl ξGG K ξGL kG ξwall kL ρL L ρV M σL nFl σW nlp φ P Ψ0 Pc ΨFl Pe ΨL ReL Ψ'L ReV ψlp ReV,e Ω	Particle diameter Approach to flood Gas capacity factor Pressure drop in preloading region Gas capacity factor at loading point Liquid flow angle Liquid Froude number Solid -liquid film contact angle Enhancement factor for pressure drop calculation Liquid film thickness Correction factor Void fraction Surface enhancement factor for effective interfacial area calculation Coefficient for losses caused by direction change Wetting factor Coefficient for gas/gas friction losses Gravitational constant Coefficient for gas/liquid friction losses Effective gravitational constant Corrugation angle Corrugation height Dynamic viscosity of the liquid phase Liquid holdup Dynamic viscosity of the gas phase Liquid holdup at flooding point Kinematic viscosity of the liquid phase Liquid holdup at loading point Direction change coefficient in the bulk zone Liquid holdup in preloading region Gas/Gas friction coefficient Wall factor Gas/Liquid friction coefficient Gas-phase mass transfer coefficient Direction change coefficient near the wall Liquid-phase mass transfer coefficient Liquid density Liquid mass flow Gas density Molar mass Surface tension of the liquid phase Exponent for calculation of liquid holdup at flooding point Surface tension of water Exponent for calculation of liquid holdup at loading point Fraction of the flow channel occupied by the liquid phase Pressure Resistance coefficient for dry pressure drop calculation Critical pressure Resistance coefficient for pressure drop calculation at flooding point Peclet number Resistance coefficient for wet pressure drop calculation Liquid Reynolds number Resistance coefficient for wet pressure drop calculation Gas Reynolds number Resistance coefficient for pressure drop calculation at loading point Effective Reynolds number for gas phase Fraction of the packing surface occupied by holes	m -Pa/m (m/s) (kg/m 3 ) 0.5 ° (m/s) (kg/m 3 ) 0.5 ° -m --------m/s 2 ° m/s 2 kg/m.s m kg/m.s m 3 /m 3 m 2 /s m 3 /m 3 -m 3 /m 3 -m 3 /m 3 ---m/s kg/m 3 m/s kg/m 3 kg/h N/m kg/mol N/m ----Pa -Pa -------
		ReV,r		Relative Reynolds number for gas phase
		s			Length of the corrugation side	m

Table 1 .

 1 Concentration requirements before biogas liquefaction [8] .....................................

Table 2 .

 2 Composition and characterization of household waste [13] .....................................

Table 3 .

 3 Production yields for agricultural and agro-industrial substrates[START_REF] Boulinguiez | Purification de biogaz -Elimination des COV et des siloxanes[END_REF] ......................

Table 4 .

 4 Biogas composition depending on the type of substrate [14] ...................................

Table 5 .

 5 Tolerances in impurities for the use of liquid biomethane as vehicle fuel[START_REF] Boulinguiez | Purification de biogaz -Elimination des COV et des siloxanes[END_REF] .........

Table 6 .

 6 Thermo-physical properties of hydrogen sulfide [19] ................................................

Table 7 .

 7 Thermo-physical properties of carbon dioxide [19] ...................................................

Table 8 .

 8 Thermo-physical properties of methane [19] ............................................................

Table 9 .

 9 Constants used by Antoine Equation for the calculation of H2S, CO2 and CH4 vapor pressures ..................................................................................................................................

Table 10 .

 10 Solubility of the main compounds of biogas in water[START_REF] Lide | Handbook Chemistry and physics[END_REF] ...................................... Table 11. Properties of physical solvents [40] ......................................................................... Table 12. Solubility of gases in physical solvents at 25 °C and 0.1 MPa [41] ...........................

Table 13 .

 13 Physical properties of some chemical solvents [43] ................................................

Table 14 .

 14 Classification of main gas-liquid contactors [48]......................................................

Table 15 .

 15 Criteria to differentiate between physical and chemical adsorption[START_REF] Ruthven | Principles of adsorption and adsorption processes[END_REF] ...............

Table 16 .

 16 Equations governing adsorption isotherm models, and their linear forms[START_REF] Hamdaoui | Modeling of adsorption isotherms of phenol and chlorophenols onto granular activated carbon. Part I. Two-parameter models and equations allowing determination of thermodynamic parameters[END_REF] .....

Table 17

 17

. The membrane materials used by manufacturer [71] ............................................. Table 18. Classification of membrane separation processes [73] ........................................... Table 19. Distribution of membranes according to pore size [IUPAC] .................................... Table 20. Characteristics of the different geometries of membranes [76] ............................. Table 21. Condensation or solidification temperatures, at atmospheric pressure, for the different compounds present in biogas ................................................................................... Table

Table 31 .

 31 Liquid holdup in preloading and loading regions [98] ..............................................

.....................................

Table 27. Sensitivities of measurement tools .......................................................................... Table 28. Constants for the Billet and Schultes model [98] ..................................................... Table 29. Effective interfacial area in packing columns using Billet and Schultes model [98] Table 30. Liquid holdup and velocities at loading and flooding point [99] ..............................

Table 32 .

 32 Pressure drop in packing columns using Billet and Schultes model [98] .................

Table 33 .

 33 Effective interfacial area in packing columns using SRP model [96] ........................

Table 34 .

 34 Liquid holdup in packing columns using SRP model [101] ....................................... Table 35. Pressure drop in packing columns using SRP model [96] .........................................

Table 36 .

 36 Pressure drop in preloading region using Delft model [97] .....................................

Table 37 .

 37 Physical properties of the systems tested [100] ......................................................

Table 38 .

 38 Dimensions of Flexipac® 350Y [100] .........................................................................

Table 39 .

 39 Deviation between predictive models and experimental data ................................

Table 40 .

 40 Changes made to calculate liquid holdup ................................................................

Table 41

. Changes made to calculate Pressure drop for liquid density less than 900 kg.m -3 . Table

42

. Changes made to calculate Pressure drop for liquid density higher than 900 kg.m -3 ................................................................................................................................................

Table 43 .

 43 Statistical deviation between the modified model and experimental data for pressure drop and liquid holdup predictions .........................................................................

Table 44 .

 44 Comparison between modified correlations and experimental data for the prediction of pressure drop in a structured packing column ................................................

Table 45 .

 45 Coefficients used by Aspen Plus® to calculate Henry's constant ...........................

Table 46 .

 46 Coefficients used by Harvey to calculate Henry's constants ..................................

Table 47 .

 47 Values of the constants used by Equation (87) to calculate the heat capacity of carbon dioxide ........................................................................................................................

Table 48 .

 48 Coefficients used in the calculation of the equilibrium constant ..........................

Table 49 .

 49 Coefficients used in the calculation of the equilibrium constant of Reaction (R.6) ................................................................................................................................................ Table 50. Parameters k and E for kinetic-controlled reactions [119] ....................................

Table 51 .

 51 Details of the simulated process ............................................................................ Table 52. Influence of liquid flow rate on the pressure drop ................................................ Table 53. Activated carbon properties used for this study [53] ............................................

Table 54

 54

. Biogas composition and operating conditions of the adsorption process ............. Table 55. Sherwood number estimation [123] ...................................................................... Table 56. Estimation of the binary diffusion coefficients [135] ............................................. Table 57. Equations used to estimate internal diffusion coefficients ................................... Table 58. Estimation of dimensionless numbers and diffusion coefficients ......................... Table 59. Boundary and initial conditions of the system ....................................................... Table 60. Input parameters for the simulation of the breakthrough curve ..........................

Table 2 .

 2 Composition and characterization of household waste[START_REF] Suez | Etude biogaz -etat des lieux et potentiel du biométhane carburant[END_REF]

	Compounds	Composition [wt. %] Dry matter [wt. %] Dry organic matter [wt. %]
	Putrescible	33.0	44	77
	Papers	11.7	68	80
	Cardboards	12.0	70	80
	Various incombustible	8.5	90	1
	Tetra brik	8.5	70	60
	Glasses	5.4	98	2
	Plastic	4.9	85	90
	Green waste	4.5	50	79
	Textiles	4.2	74	92
	Metals	3.7	90	1
	Special waste	2.0	90	1
	Various fuels	1.6	85	75

Table 3 .

 3 Production yields for agricultural and agro-industrial substrates [

14] Substrates Dry matter [wt. %] Dry organic matter [wt. %] Production yield [m 3 /t of dry organic matter] Production yield [m 3 /t of substrate]

	Pig manure	6 to 25	75 to 80	300 to 450	22 to 60
	Cow manure	8 to 30	80 to 82	350 to 700	21 to 168
	Chicken manure	10 to 60	67 to 77	300 to 800	20 to 180
	Horse manure	28	25	500	35
	sheep manure	15 to 25	80 to 85	350	50 to 74
	Grass silage	26 to 80	67 to 98	500 to 600	87 to 440
	Hay	86 to 93	83 to 93	500	356 to 432
	corn straw	86	72	500	310
	Foliage	85	82	400	279
	Sorghum	25	93	700	162
	Helianthus annuus	35	88	750	231
	Distillation residues	12	90	430	77
	Brewery waste	15 to 21	66 to 95	500	50 to 100
	Vegetable waste	5 to 20	76 to 90	600	23 to 108
	oleaginous residues	92	97	600	536
	Fruit residues	40 to 50	30 to 93	450 to 500	60 to 232
	press cake	88	93	5550	450
	slaughterhouse waste	15	80 to 90	450	58
	bakery waste	50	80 to 95	450	665
	Shortening waste	99	99	1200	1117

Table 4 .

 4 Biogas composition depending on the type of substrate [

14] Compounds Unit Biogas from household and industrial waste Biogas from wastewater treatment plants Agricultural biogas

	CH4	% mol.	40 -55	65 -75	45 -75
	CO2	% mol.	25 -30	20 -35	25 -55
	N2	% mol.	10	0 -5	0 -5
	O2	% mol.	1 -5	0.5	0 -2
	NH3	% mol.	Traces	Traces	0 -3
	COV	mg.Nm -3	< 2500	< 3000	< 1500
	H2S	mg.Nm -3	< 3000	< 4000	< 10000

Table 6 .

 6 Thermo-physical properties of hydrogen sulfide[19]

	Properties	Unit	Value
	Molar mass	g.mol -1	34.08
	Auto-ignition temperature	°C	270
	Solubility in water (1.013 bar and 0 °C)	vol/vol	4.67
		Solid phase
	Melting point	°C	-85.7
	Latent heat of fusion (1.013 bar at melting point)	kJ.kg -1	69.73
		Liquid phase
	Boiling point at 1.013 bar	°C	-60.3
	Vapor pressure at 20 °C	bar	17.81
	Liquid phase density (1.013 bar at boiling point)	kg.m -3	949.2
	Latent heat of vaporization (1.013 bar at boiling point)	kJ.kg -1	546.41
		Gas phase
	Gas phase density (1.013 bar and 15 °C)	kg.m -3	1.45
	Viscosity (1.013 bar and 0 °C)	Pa.s	1.13 x 10 -5
	Thermal conductivity (1.013 bar and 0 °C)	mW.m -1 .K -1	15.61
	Specific volume (1.013 bar and 25 °C)	m 3 .kg -1	0.7126
	Heat capacity at constant pressure (1.013 bar and 25 °C)	kJ.mol -1 .K -1	0.0346
	Heat capacity at constant volume (1.013 bar and 25 °C)	kJ.mol -1 .K -1	0.026
		Critical point
	Critical temperature	°C	99.95
	Critical pressure	bar	90
	Critical density	kg.m -3	347.28
		Triple point
	Triple point temperature	°C	-85.45
	Triple point pressure	bar	0.232

Table 7 .

 7 Thermo-physical properties of carbon dioxide[19]

	Properties	Unit	Value
	Molar mass	g.mol -1	44.01
	Concentration in air	Vol %	0.0405
	Solubility in water (1.013 bar and 0 °C)	vol/vol	1.7163
		Solid phase
	Melting point	°C	-56.57
	Latent heat of fusion (1.013 bar at melting point)	kJ.kg -1	204.93
	Solid phase density	kg.m -3	1562
		Liquid phase
	Boiling point	°C	-78.45
	Vapor pressure at 20 °C	bar	57.291
	Liquid phase density (19.7 bar at -20 °C)	kg.m -3	1256.74
		Gas phase
	Gas phase density (1.013 bar and 15 °C)	kg.m -3	1.87
	Viscosity (1.013 bar and 0 °C)	Pa.s	1.37 x 10 -5
	Thermal conductivity (1.013 bar and 0 °C)	mW.m -1 .K -1	14.67
	Specific volume (1.013 bar and 25 °C)	m 3 .kg -1	0.5532
	Heat capacity at constant pressure (1.013 bar and 25 °C)	kJ.mol -1 .K -1	0.0374
	Heat capacity at constant volume (1.013 bar and 25 °C)	kJ.mol -1 .K -1	0.0289
		Critical point
	Critical temperature	°C	30.98
	Critical pressure	bar	73.77
	Critical density	kg.m -3	467.6
		Triple point
	Triple point temperature	°C	-56.56
	Triple point pressure	bar	5.187
	2.2.3. Methane
	Methane is a hydrocarbon which is naturally present in the Earth's atmosphere at very low
	concentrations (1.82 ppm in 2012) [21].

Table 8 .

 8 Thermo-physical properties of methane[19]

	Properties	Unit	Value
	Molar mass	g.mol -1	16.043
	Auto-ignition temperature	°C	595
	Solubility in water (1.013 bar and 2 °C)	vol/vol	0.054
		Solid phase
	Melting point	°C	-182.46
	Latent heat of fusion (1.013 bar at melting point)	kJ.kg -1	58.682
		Liquid phase
	Boiling point at 1.013 bar	°C	-161.48
	Liquid phase density (1.013 bar at boiling point)	kg.m -3	422.36
		Gas phase
	Gas phase density (1.013 bar at boiling point)	kg.m -3	1.816
	Viscosity (1.013 bar and 0 °C)	Pa.s	1.0245 x 10 -5
	Thermal conductivity (1.013 bar and 0 °C)	mW.m -1 .K -1	30.57
	Specific volume (1.013 bar and 25 °C)	m 3 .kg -1	1.5227
	Heat capacity at constant pressure (1.013 bar and 25 °C)	kJ.mol -1 .K -1	0.0358
	Heat capacity at constant volume (1.013 bar and 25 °C)	kJ.mol -1 .K -1	0.0274
		Critical point
	Critical temperature	°C	-82.59
	Critical pressure	bar	45.99
	Critical density	kg.m -3	162.7
		Triple point
	Triple point temperature	°C	-182.46
	Triple point pressure	bar	0.117

Table 9

 9

	. Constants used by Antoine Equation for the calculation of H2S, CO2 and CH4 vapor pressures
	Components	A	B	C	Temperature range [K]	References
	H2S	4.43681 829.439 -25.412	138.8 -212.8	[26]
	H2S	4.52887 958.587	-0.539	212.8 -349.5	[26]
	CO2	6.81228 1301.679 -3.494	154.26 -195.89	[27]
	CH4	3.9895	443.028	-0.49	90.99 -189.99	[28]

Table 11 .

 11 Properties of physical solvents[START_REF] Bucklin | Comparison of Fluor Solvent and Selexol Processes[END_REF]

		Selexol®	Purisol®	Rectisol® Morphysorb®	Fluor solvent®
	Solvent	Dimethyl Ethers of Polyethylene Glycol	N-Methyl-2-Pyrrolidone	Methanol	N-Formyl-Morpholine	Propylene Carbonate
	Chemical formula	(CH3O(CH2CH2O)xCH3) 3 ≤ x ≤ 9	C5H9NO	CH3OH	C5H9NO2	C4H6O3
	Maximum temperature	175				65
	[°C]
	Vapor pressure at 25 °C [kPa]	0.097	32	1.3 (at 20 °C)	9	3
	Viscosity at 25 °C [Pa.s]	5.8	1.65	0.6	9.5	3
	Boiling point [°C]	240	202	110	242	240
	Melting point [°C]	-23 / -29	-24	-98	21	-48
	Molar mass [g.mol -1 ]	280	99	32	115.3	102

Table 12 .

 12 Solubility of gases in physical solvents at 25 °C and 0.1 MPa[START_REF] Descamps | Etude de la capture du CO2 par absorption physique dans les systèmes de production d'électricité basés sur la gazéification du charbon intégrée à un cycle combiné[END_REF]

	vol.vol -1	DMEPG	PC	NMP	MeOH (at -25° C)
	H2	0.013	0.0078	0.0064	0.0054
	N2	0.02	0.0084		0.012
	O2		0.026	0.035	0.02
	CO	0.028	0.021	0.021	0.02
	CH4	0.066	0.038	0.072	0.051
	CO2	1	1	1	1
	NH3	4.8			23.2
	H2S	8.82	3.2	10.2	7.06
	SO2	92.1	68.6
	H2O	730	300	4000

Table 13 .

 13 Physical properties of some chemical solvents[START_REF] Lecomte | Le captage du CO2, des technologies pour réduire les émissions de gaz à effet de serre[END_REF]

		MEA	DGA	DEA	DIPA	TEA
	Chemical formula	HOC2H4NH2	H(OC2H4)2NH2 (HOC2H4)2NH (HOC3H6)2NH (HOC2H4)3N
	Molecular weight	61.08	105.14	105.14	133.19	148.19
	Boiling point [°C]	170.50	221.11	269.00	248.72	360.00
	Melting point [°C]	10.5	-12.5	28.0	42.0	22.4
	Viscosity [cP]	24 (20 °C)	40 (15.6 °C)	350 (20 °C)	870 (30 °C)	1013 (20 °C)

Table 14 .

 14 Classification of main gas-liquid contactors[START_REF] Roquet | Modélisation et étude expérimentale de l'absorption réactive multiconstituants[END_REF]

	Gas-liquid contactors	Continuous phase	Fluid inclusion type	Main applications
	Bubble column	Liquid	Bubbles	Oxidation / Chlorination
	Gas-liquid agitated vessel	Liquid	Bubbles	Oxidation / Fermentation
	Spray tower	Gas	Drops and liquid films	Gas scrubbing dust-laden
	Packed column	Gas	Drops and liquid films	Gas scrubbing
	Venturi tube	Gas	Drops and liquid films	Gas scrubbing dust-laden
			the flow is stratified,
			with entrained
	Plate column	Liquid	bubbles in the liquid and a mist or spray of
			liquid droplets in the
			vapour

Table 15 .

 15 Criteria to differentiate between physical and chemical adsorption[START_REF] Ruthven | Principles of adsorption and adsorption processes[END_REF]

	Criteria	Physical adsorption	Chemical adsorption
	Process temperature	Low	High
	Type of bond	Van der Waals	Chemical
	Interaction forces	30 to 40 kJ.mol -1	80 to 800 kJ.mol -1
	Layers	Monolayers or multilayers	Monolayers only
	Kinetic	Fast and reversible	Slow and irreversible
	Desorption	Easy	Difficult

Table 16 .

 16 Equations governing adsorption isotherm models, and their linear forms[START_REF] Hamdaoui | Modeling of adsorption isotherms of phenol and chlorophenols onto granular activated carbon. Part I. Two-parameter models and equations allowing determination of thermodynamic parameters[END_REF] [g 1-(1/n) .m(3/n) .kg -1 ] is the Freundlich constant indicative of the relative adsorption capacity of the adsorbent. KT [-] is the Toth equilibrium constant. n [-] is the Freundlich constant indicative of the intensity of the adsorption. mT[-] is the Toth model exponent. θ [-] = qe / qm, is the surface coverage. ΔQ [kJ.mol -1

	Isotherm				Equation			Linear form
	Langmuir		𝑞 𝑒 =	𝑞 𝑚 𝑏 𝐶 𝑒 1 + 𝑏 𝐶 𝑒	1 𝐶 𝑒	= 𝑏 ( 𝑞 𝑚 𝑞 𝑒	-1)
	Freundlich		𝑞 𝑒 = 𝐾 𝐹 𝐶 𝑒 1 𝑛 ⁄	ln 𝑞 𝑒 = ln 𝐾 𝐹 +	1 𝑛	ln 𝐶 𝑒
	Temkin	𝜃 =	𝑅 𝑇 𝛥𝑄	ln 𝐾 0 𝐶 𝑒	𝜃 =	𝑅 𝑇 𝛥𝑄	(ln 𝐾 0 + ln 𝐶 𝑒 )
	Elovich	𝑞 𝑒 𝑞 𝑚	= 𝐾 𝐸 𝐶 𝑒 𝑒	(-	𝑞 𝑒 𝑞 𝑚	)	ln	𝑞 𝑒 𝐶 𝑒	= ln 𝐾 𝐸 𝑞 𝑚 -	𝑞 𝑒 𝑞 𝑚
	Kiselev	𝑘 1 𝐶 𝑒 =	𝜃 (1 -𝜃) (1 + 𝑘 𝑛 𝜃)	1 𝐶 𝑒 (1 -𝜃)	= 𝑘 1 (𝑘 𝑛 +	1 𝜃	)
	Fowler-Guggenheim	𝐾 𝐹𝐺 𝐶 𝑒 =	𝜃 1 -𝜃	𝑒	( 2 𝜃 𝑊 𝑅 𝑇	)	ln [ 𝐶 𝑒 (1 -𝜃) 𝜃 ] =	2 𝜃 𝑊 𝑅 𝑇	-ln 𝐾 𝐹𝐺
	Hill-de Boer	𝐾 1 𝐶 𝑒 =	𝜃 1 -𝜃	𝑒	( 1-𝜃 𝜃	-	𝐾 2 𝜃 𝑅 𝑇	)	ln [ 𝐶 𝑒 (1 -𝜃) 𝜃 ] -	𝜃 1 -𝜃	= -(ln 𝐾 1 +	𝐾 2 𝜃 𝑅 𝑇	)
	Toth	𝑞 𝑒 𝑞 𝑚𝑇	=	( 𝐾 𝑇 1	𝐶 𝑒 + 𝐶 𝑒 𝑚𝑇 ) 𝑚𝑇 1	ln (	𝑚𝑇 𝑞 𝑒 𝑞 𝑚𝑇 𝑚𝑇 -𝑞 𝑒 𝑚𝑇 ) = 𝑚𝑇(ln 𝐶 𝑒 + ln 𝐾 𝑇 )
	Where :
	qe [g.kg -1 ] is the amount of solute adsorbed per unit weight of adsorbent at equilibrium.
	qm [g.kg -1 ] is the maximum adsorption capacity.
	qmT [g.kg -1 ] is the Toth maximal adsorption capacity.

b [m 3 .g -1 ] is the Langmuir constant related to the free energy of adsorption. Ce [g.m -3 ] is the equilibrium concentration of the solute in the bulk solution. C0 [g.m -3 ] is the initial concentration of the solute in the bulk solution. KF

Table 17 .

 17 The membrane materials used by manufacturer[START_REF] Szymczyk | Les procédés de filtration membranaire appliqués au traitement des eaux dans : Traitement et épuration des eaux industrielles polluées -Procédés membranaires, bioadsorption et oxydation chimique[END_REF]

	The membrane materials	manufacturers
	Cellulose acetate	Grace
	Hydrin C	Zeon
	Pebax	Atochem
	Polyacrylate	Röhm
	Polydimethylsiloxane	Wacker, GKSS
	Polyhydantoin	Bayer
	Polyetherimide	General Electric
	polyethersulfone	Bayer, BASF, Monsanto

Table 18 .

 18 Classification of membrane separation processes[START_REF] Bowen | Properties of microfiltration membranes: Adsorption of bovine serum albumin at polyvinylidene fluoride membranes[END_REF]

	Application	Driving force	Separation size range
	Microfiltration	Pressure gradient	10 -0.1 µm
	Ultrafiltration	Pressure gradient	< 0.1 µm -5 nm
	Reverse osmosis	Pressure gradient	< 5 nm
	Electrodialysis	Electric field gradient	< 5 nm
	Dialysis	Concentration gradient	< 5 nm

Table 19 .

 19 Distribution of membranes according to pore size[IUPAC]

	Membrane type	Pores size (Å)	Physical mechanism	Application
	Dense		Diffusion	Reaction, gas separation
	Microporous	≤ 20	Micropore diffusion	Gas separation
				Ultrafiltration,
	Mesoporous	20 -500	Knudsen diffusion	Nanofiltration, Gas
				separation
	Macroporous	≥ 500	Molecular sieve	Ultrafiltration

Table 20 .

 20 Characteristics of the different geometries of membranes[START_REF] Boucif | Modélisation et simulation de contacteurs membranaires pour les procédés d'absorption de gaz acides par solvant chimique[END_REF]

	Property	Flat sheet membrane	Tubular membrane	Spiral wound membrane	Hollow fiber membrane
	Interfacial area [m 2 /m 3 ]	≈ 100	≈ 1000	≈ 500	5 -10000
	Filling density	Low	High	Moderate	Very high
	Resistance to soiling	Good	Low	Moderate	Low
	Use at high pressures	Difficult	Easy	Easy	Easy
	manufacturing cost	High	Moderate	High	Moderate
	Application limited to	No	Yes	No	Yes
	membranes

Table 21 .

 21 Condensation or solidification temperatures, at atmospheric pressure, for the different compounds present in biogas

	Compound	Condensation temperature [°C]
	H2S	-60.3
	CO2	-78.5
	CH4	-161.5
	N2	-195.8

Table 23 .

 23 Comparison of the different biogas purification and upgrading technologies

	Separation technology	Methane obtained [%] concentration	Methane losses [%]	Process needs
	Chemical absorption	95 -98 [83] > 99.5 [84]	0.1 -0.2	Amines or chemical solvent recharges
	Water wash at high pressure	96 -98 [83] > 98 [84]	10 -20 (The high pressure increases the methane solubility in	Large water requirement
			water)
	Adsorption	95 -98 [83] 98 [84]	2
	Membrane separation	76 -95 [83] 90 -93.5 [85] 98 [86]	6.5 -10 [85] 2 [86]	Change membranes
	Cryogenic technology	> 97 [83]		Refrigerants

Table 24 .

 24 Performances, costs, advantages and disadvantages of separation processes[START_REF] Cloirec | Les Composés Organiques Volatils (COVs) dans l'Environnement[END_REF]

	Separation technology	Efficiency [%]	Investment [€/m 3 .h -1 ] cost	Operating [€/1000 m 3 ] cost	Advantages	Disadvantages
					-Simple operation
					for a wide range of
					flow rates,
	Absorption	95 -98	7 -32	1.7 -8.2	concentrations and compounds.

Table 27 .

 27 Sensitivities of measurement tools

	Properties to measure	Measurement tools	Accuracy
	Pressure drop in the packing column	Differential Pressure Transmitter	± 0.065 % Full Scale (FS)
	Pressure	Pressure sensor	< 0.5 % FS
	H2S concentration	Biogas analyser	± 3 % FS
	Flow rates	Flowmeter	< 1 % FS

Table 28 .

 28 Constants for the Billet and Schultes model[START_REF] Billet | Prediction of mass transfer columns with dumped and arranged packings: Updated summary of the calculation method of Billet and Schultes[END_REF]

	Manufacture	Material	Description a [m 2 .m -3 ]	ε	Clp	CFl	Ch	Cp	CL	CV
	Flexipac	Metal	350Y	350	0.985 3.157
	Mellapak	Metal	250Y	250	0.970 3.157 2.464 0.554 0.292	-	-
	Ralu pak	Metal	YC-250	250	0.945 3.178 2.558	-	0.191 1.334 0.385
	Gempack	Metal	A2T-304	202	0.977 2.986 2.099 0.678 0.344	-	-
	Euroform	Plastic	PN-110	110	0.936 3.075 1.975 0.511 0.250 0.973 0.167
	Impulse	Metal	250	250	0.975 2.610 1.996 0.431 0.262 0.983 0.270
	packing	Ceramic	100	91.4	0.838 2.664 1.655 1.900 0.417 1.317 0.327
	Montz	Metal	B1-200 B2-300	200 300	0.979 3.116 2.339 0.547 0.355 0.971 0.390 0.930 3.098 2.464 0.482 0.295 1.165 0.422
	packing	Plastic	C1-200 C2-200	200 200	0.954 0.900 2.653 1.973 --	--	0.453 1.006 0.412 0.481 0.739 -
	Correlations (19), (

Table 30 .

 30 Liquid holdup and velocities at loading and flooding point[START_REF] Lamprecht | Establishing a facility to measure packed column hydrodynamics[END_REF]

	Parameter								Correlation
	Gas velocity at loading point	𝑢 𝑉,𝑙𝑝 = √ 𝜓 𝑙𝑝 𝑔	( 𝑎 𝜀 1 6	-𝑎 0.5 (	12 𝑔	𝜇 𝐿 𝜌 𝐿	𝑢 𝐿,𝑙𝑝 ) 1 3 ) ( 12 𝑔	𝜇 𝐿 𝜌 𝐿	𝑢 𝐿,𝑙𝑝 ) 1 6 √ 𝜌 𝑉 𝜌 𝐿	(22)
	Liquid velocity at loading point						𝑢 𝐿,𝑙𝑝 =	𝜌 𝑉 𝜌 𝐿	𝐿 𝑉	𝑢 𝑉,𝑙𝑝	(23)
											3
	Gas velocity at flooding point			𝑢 𝑉,𝐹𝑙 = √ 𝜓 𝐹𝑙 2𝑔	(𝜀 -ℎ 𝐿,𝐹𝑙 ) 2 𝜀 0.5	-√ ℎ 𝐿,𝐹𝑙 𝑎	√ 𝜌 𝑉 𝜌 𝐿	(24)
	Resistance coefficient at loading point Resistance coefficient at flooding point				𝜓 𝑙𝑝 = 𝜓 𝐹𝑙 =	𝐶 𝑙𝑝 2 ( 𝑉 √ 𝐿 𝜌 𝑉 𝜌 𝐿 𝐶 𝐹𝑙 2 ( 𝐿 𝑉 √ 𝜌 𝑉 𝜌 𝐿	𝑔 ( 𝜇 𝑉 𝜇 𝐿 𝑔 ( 𝜇 𝐿 𝜇 𝑉	) 0.4 ) 0.2	2𝑛 𝑙𝑝 ) ) 2𝑛 𝐹𝑙	(25) (26)
	Packing specific constant at loading point	𝐿 𝑉	√ 𝜌 𝑉 𝜌 𝐿	𝐿 𝑉 > 0.4 → 𝑛 𝑙𝑝 = -0.723 → 𝐶 𝑙𝑝 = 0.695 𝐶 𝑙𝑝 ( √ 𝜌 𝑉 ≤ 0.4 → 𝑛 𝑙𝑝 = -0.326 → 𝐶 𝑙𝑝 = 𝐶 𝑙𝑝 𝜌 𝐿 𝜇 𝑉 𝜇 𝐿	) 0.1588	(27)
	Packing specific constant at flooding point	𝐿 𝑉	√ 𝜌 𝑉 𝜌 𝐿	𝐿 𝑉 > 0.4 → 𝑛 𝐹𝑙 = -0.708 → 𝐶 𝐹𝑙 = 0.6244 𝐶 𝐹𝑙 ( √ 𝜌 𝑉 ≤ 0.4 → 𝑛 𝐹𝑙 = -0.194 → 𝐶 𝐹𝑙 = 𝐶 𝐹𝑙 𝜌 𝐿	𝜇 𝑉 𝜇 𝐿	) 0.1028	(28)
	Liquid holdup at the loading point				ℎ 𝐿,𝑙𝑝 = (	12 𝑎 2 µ 𝐿 𝑢 𝐿 𝜌 𝐿 𝑔	) 1 3	( 𝑎 ℎ 𝑎	2 ) 3	(29)
	Liquid holdup at the flooding point			ℎ 𝐿,𝐹𝑙 3	(3ℎ 𝐿,𝐹𝑙 -𝜀) =	6 𝑔	𝑎 2 𝜀	𝜇 𝐿 𝜌 𝐿	𝐿 𝑉	𝜌 𝑉 𝜌 𝐿	𝑢 𝑉,𝐹𝑙

Table 31 .

 31 Liquid holdup in preloading and loading regions[START_REF] Billet | Prediction of mass transfer columns with dumped and arranged packings: Updated summary of the calculation method of Billet and Schultes[END_REF]

	Parameter		Correlation
	Liquid holdup in preloading region		ℎ 𝐿,𝑝𝑙 = (12	𝜇 𝐿 𝑎 2 𝑢 𝐿 𝜌 𝐿 𝑔	)	1 3	( 𝑎 ℎ 𝑎	2 ) 3	(31)
	Hydraulic area of the packing	𝑎 ℎ 𝑎 𝑎 ℎ 𝑎	= 𝐶 ℎ 𝑅𝑒 𝐿 0.15 𝐹𝑟 𝐿 0.1 = 0.85 𝐶 ℎ 𝑅𝑒 𝐿 0.25 𝐹𝑟 𝐿 0.1			𝐹𝑜𝑟 𝑅𝑒 𝐿 < 5 𝐹𝑜𝑟 𝑅𝑒 𝐿 ≥ 5	(32)
	Liquid Reynolds number		𝑅𝑒 𝐿 =	𝑢 𝐿 𝜌 𝐿 𝑎 𝜇 𝐿				(33)
	Liquid Froude number		𝐹𝑟 𝐿 =	𝑢 𝐿 2 𝑎 𝑔				(34)
	Liquid holdup at flooding point		ℎ 𝐿,𝐹𝑙 = 2.2 ℎ 𝐿,𝑝𝑙		(35)
	Liquid holdup in loading region		ℎ 𝐿 = ℎ 𝐿,𝑝𝑙 + (ℎ 𝐿,𝐹𝑙 -ℎ 𝐿,𝑝𝑙 ) (	𝑢 𝑉,𝐹𝑙 𝑢 𝑉	) 13

Table 32 .

 32 Pressure drop in packing columns using Billet and Schultes model[START_REF] Billet | Prediction of mass transfer columns with dumped and arranged packings: Updated summary of the calculation method of Billet and Schultes[END_REF]

	Parameter			Correlation
	Dry pressure drop	( 𝑑𝑃 𝑑𝑧	) 𝑑	= 𝜓 0	𝑎 𝜀 3	𝐹 𝑐 2 2	1 𝐾	(37)
	Resistance coefficient	𝜓 0 = 𝐶 𝑝 (	64 𝑅𝑒 𝑉	+	1.8 𝑅𝑒 𝑉 0.08 )	(38)
	Gas capacity factor			𝐹 𝑐 = 𝑢 𝑉 √𝜌 𝑉	(39)
	Wall factor	1 𝐾	= 1 +	2 3	1 (1 -𝜀)	𝑑 𝑝 𝑑	(40)
	Particle diameter			𝑑 𝑝 = 6	1 -𝜀 𝑎	(41)
	Gas Reynolds number	𝑅𝑒 𝑉 =		𝑢 𝑉 𝑑 𝑝 𝜌 𝑉 (1 -𝜀) 𝜇 𝑉	𝐾	(42)
	Wet pressure drop	𝑑𝑃 𝑑𝑧	= 𝜓 𝐿		𝑓 𝑤 𝑎 (𝜀 -ℎ 𝐿 ) 3	𝐹 𝑐 2 2	1 𝐾	(43)
	Resistance factor	𝜓 𝐿 ′ = 𝜓 𝐿 𝑓 𝑤 = 𝐶 𝑝 𝑓 𝑠 (	64 𝑅𝑒 𝑉	+	1,8 𝑅𝑒 𝑉 0,08 ) ( 𝜀 -ℎ 𝐿 𝜀	) (1.5)

Table 34 .

 34 Liquid holdup in packing columns using SRP model[101]

	Parameter					Correlation
	Dry pressure drop	( 𝛥𝑃 𝛥𝑧	)	𝑑	=	𝐴 𝜌 𝑉 𝑠 𝜀 2 (𝑠𝑖𝑛𝛳) 2 𝑢 𝑉 2 +	𝐵 𝜇 𝑉 𝑠 2 𝜀 𝑠𝑖𝑛𝛳	𝑢 𝑉

Table 37 .

 37 Physical properties of the systems tested[100]

	Component	Density [kg/m 3 ]	Viscosity [kg/m.s]	Surface tension [N/m]
	Air	0.81	18.10 -6	-
	Water	1000	0.001	71.2 x 10 -3
	Kerosol 200	763	2.31 x 10 -3	23.9 x 10 -3

Table 38 .

 38 Dimensions of Flexipac® 350Y [100]

	Property	Value
	Void fraction	0.985
	Corrugation angle	45 °
	Corrugation base	15.5 mm
	Corrugation side	11.5 mm
	Crimp height	8.4 mm
	Height of element	265 mm

Table 40 .

 40 Changes made to calculate liquid holdup

	Equations to modify	New equations
		For liquid density > 900 kg/m 3
		ℎ 𝐿,𝑝𝑙 = [𝟔𝟐𝟖. 𝟒 * (𝒖 𝑳 * 𝟑𝟔𝟎𝟎) -𝟎.𝟗𝟐𝟗 ]	𝟏 𝟑 (	𝜇 𝐿 𝑎 2 𝑢 𝐿 𝜌 𝐿 𝑔	) 1 3	(	𝑎 ℎ 𝑎	2 ) 3
	(79)	For liquid density ≤ 900 kg/m 3
		ℎ 𝐿,𝑝𝑙 = 𝑬𝒙𝒑(-𝟎. 𝟎𝟎𝟓𝟕 * 𝒖 𝑳 * 𝟑𝟔𝟎𝟎 + 𝟏. 𝟐𝟎𝟕𝟒) (	𝜇 𝐿 𝑎 2 𝑢 𝐿 𝜌 𝐿 𝑔	)	1 3	( 𝑎 ℎ 𝑎	2 ) 3
		For liquid density > 900 kg/m 3
		ℎ 𝐿 = ℎ 𝐿,𝑝𝑙 + (𝟏, 𝟑 ℎ 𝐿,𝐹𝑙 -ℎ 𝐿,𝑝𝑙 ) (	𝑢 𝑉 𝑢 𝑉,𝐹𝑙	)
	(80)

𝟏𝟎

For liquid density ≤

900 kg/m 3

  ℎ 𝐿 = ℎ 𝐿,𝑝𝑙 + (𝑬𝒙𝒑 [-𝟎. 𝟎𝟎𝟎𝟔 * (𝒖 𝑳 * 𝟑𝟔𝟎𝟎) 𝟐 -𝟎. 𝟎𝟑𝟏 * 𝒖 𝑳 * 𝟑𝟔𝟎𝟎 + 𝟎. 𝟑𝟕𝟑𝟔] * ℎ 𝐿,𝐹𝑙 -ℎ 𝐿,𝑝𝑙 ) (

	𝑢 𝑉 𝑢 𝑉,𝐹𝑙	)

[-𝟎.𝟎𝟐𝟏𝟓 * (𝒖 𝑳 * 𝟑𝟔𝟎𝟎) 𝟐 + 𝟎.𝟖𝟗𝟐𝟕 * 𝒖 𝑳 * 𝟑𝟔𝟎𝟎 + 𝟓.𝟑𝟑𝟓𝟖]

Table 41 .

 41 Changes made to calculate Pressure drop for liquid density less than 900 kg.m-3

	Equations to modify			New equations
		Preloading region
	(81)	𝜓 𝐿 ′ = 𝜓 𝐿 𝑓 𝑤 = 𝐶 𝑝 ( ℎ 𝐿,𝑙𝑝 ℎ 𝐿	) 𝟎.𝟑	exp ( 200 ) ( 𝑅𝑒 𝐿 𝑅𝑒 𝑉 64	+	1,8 𝑅𝑒 𝑉 0,08 ) ( 𝜀-ℎ 𝐿 𝜀 ) 𝟏.𝟓
		Loading & Flooding regions
		𝜓 𝐿 ′ = 𝜓 𝐿 𝑓 𝑤 = 𝐶 𝑝 ( ℎ 𝐿,𝑙𝑝 ℎ 𝐿	) 𝟎.𝟔	exp ( 200 ) ( 𝑅𝑒 𝐿 𝑅𝑒 𝑉 64	+	1,8 𝑅𝑒 𝑉 0,08 ) ( 𝜀-ℎ 𝐿 𝜀 ) 𝟎.𝟓

Table 43 .

 43 Statistical deviation between the modified model and experimental data for pressure drop and liquid holdup predictions

			Air / Water System
	Liquid load [m/h]	Pressure drop AAD [%] MAD [%]	Liquid holdup AAD [%] MAD [%]
	35.6	10	19	7	11
	28.8	5	22	2	6
	20.5	5	18	6	16
	12.9	8	22	4	12
	6	9	21	7	12
			Air / Kerosol 200 System
	Liquid load [m/h]	Pressure drop AAD [%] MAD [%]	Liquid holdup AAD [%] MAD [%]
	35.6	9	26	3	11
	28.8	7	20	3	12
	20.6	6	13	4	8
	12.7	5	12	6	9
	6.1	12	23	4	11

Table 44 .

 44 Comparison between modified correlations and experimental data for the prediction of pressure drop in a structured packing column

	No. of point	L [kg/h]	V [kg/h]	(𝚫𝐏) 𝐞𝐱𝐩 [𝐏𝐚]	(𝚫𝐏) 𝐦𝐨𝐝𝐢𝐟𝐢𝐞𝐝 𝐦𝐨𝐝𝐞𝐥 [𝐏𝐚]	(𝚫𝐏) 𝐨𝐫𝐢𝐠𝐢𝐧𝐚𝐥 𝐦𝐨𝐝𝐞𝐥 [𝐏𝐚]	Absolute value of relative deviation between experimental and modified model [%]	Absolute value of relative deviation between experiment al and original model [%]
	1	818	89.7	304.4	289.5	200.0	4.89	34.29
	2	809	90.2	310.4	292.2	201.6	5.88	35.07
	3	809	90.9	312.3	296.5	204.3	5.06	34.58
	4	809	91.2	312.4	298.8	205.8	4.36	34.13
	5	850	91.8	321.2	307.8	211.1	4.18	34.28
	6	854	92.6	325.5	313.3	214.5	3.74	34.09
	7	870	93.0	323.9	318.6	217.6	1.63	32.81
	8	821	93.8	325.6	318.0	217.7	2.33	33.13
	9	797	94.9	326.2	322.6	220.8	1.10	32.31
	10	855	95.4	345.0	334.0	227.1	3.16	34.15

Table 45 .

 45 Coefficients used by Aspen Plus® to calculate Henry's constantHenry's constants obtained with Aspen Plus® were compared to experimental data from the research report RR-48 of the Gas processors Association[105]. Results of Henry's constants were also compared to the semi-empirical Equation (84) proposed by Harvey[START_REF] Harvey | Semiempirical correlation for Henry's constants over large temperature ranges[END_REF].

	Component i	CH4	CO2	H2S
	Component j	H2O	H2O	H2O
	Low temperature [°C]	1.85	-0.15	-0.15
	High temperature [°C]	79.85	226.85	149.85
	aij	183.7811	159.1997	346.625
	bij	-9111.67	-8477.711	-13236.8
	cij	-25.0379	-21.957	-55.0551
	dij	0.0001434	0.00578	0.05957
	eij	0	0	0

  and (86) for water.

	𝑃 𝑠,𝑤𝑎𝑡𝑒𝑟 = 𝐸𝑥𝑝 (73.649 -	72582 𝑇	-7.3037 ln 𝑇 + 4.1653. 10 -6 𝑇 2 )	(85)
		𝑇 * =	𝑇 𝑇 𝑐,𝑤𝑎𝑡𝑒𝑟

Table 46 .

 46 Coefficients used by Harvey to calculate Henry's constants

	Component i	CH4	CO2	H2S
	Component j	H2O	H2O	H2O
	aij	11.01	9.4234	5.7131
	bij	4.836	4	5.3727
	cij	12.52	10.32	5.4227

Table 47 .

 47 Values of the constants used by Equation (87) to calculate the heat capacity of carbon dioxide

	Constants	Values
	C1i	29.37
	C2i	34.54
	C3i	1428
	C4i	26.4
	C5i	588

( ____ ) Aspen Plus® ; (♦) Experimental values

  5 𝑉 𝑒 𝑐 𝑐𝑎 𝑎 + 10.05 𝑉 𝑒 (𝑐 𝑐𝑎 𝑎 ) 2

Table 48 .

 48 Coefficients used in the calculation of the equilibrium constantThese coefficients are unavailable in literature for Reactions (R.18) and (R.19). Therefore, equilibrium constants of these two reactions were calculated by Aspen Plus® using Gibbs free energies. Results for Reaction (R.18) were verified using equilibrium constants of Reactions (R.14) and (R.16) by using Equation[START_REF] Lamprecht | Establishing a facility to measure packed column hydrodynamics[END_REF]. Fig.55shows the results obtained.

	Reaction	A	B	C
	R.13	231.456	-12092.1	-36.7816
	R.14	132.899	-13445.9	-22.4773
	R.15	216.05	-12431.7	-35.4819
	R.16	214.582	-12995.4	-33.55471
	R.17	-9.74	-8585.47	0
		𝐾 𝑒𝑞,𝑅.6 =	𝐾 𝑒𝑞,𝑅.4 𝐾 𝑒𝑞,𝑅.2

Table 49 .

 49 Coefficients used in the calculation of the equilibrium constant of Reaction (R.[START_REF] Walsh | Utilization of Biogas[END_REF]

	Reaction	A	B	C
	R.18	147	-1930	-21.15

Table 50 .

 50 Parameters k and E for kinetic-controlled reactions[START_REF] Pinsent | The kinetics of combination of carbon dioxide with hydroxide ions[END_REF]

	Reaction	k	E [cal/mol]
	R.24	4.32E+13	13249
	R.25	2.83E+17	29451

Table 51 .

 51 Details of the simulated process

	Packed column
	Diameter [m]	0.15
	Type of the packing / Size / Material / Vendor	Flexipac / 500Y / Metal / KOCH
	Packing height [m]	2.354
	Gas inlet
	Temperature [°C]	9
	Pressure [atm]	1
	Mass flow rate [kg/h]	90
	Volume flow rate [m 3 /h]	77
		CH4 (60 %)
	Molar composition	CO2 (39.997 %)
		H2S (30 ppm)
	Liquid inlet
	Temperature [°C]	4
	Pressure [atm]	1
	Mass flow rate [kg/h]	420
	Composition	Water with NaOH (0.5 g/l)

Table 52 .

 52 Influence of liquid flow rate on the pressure drop

		35
		30
	outlet gas	25
	concentration in	15 20
	H 2 S	10
		5
		0
		40	140	240	340	440	540	640	740	840
					Liquid flow rate [kg/h]
									Absolute
									value of
									relative
	No. of point	L [kg/h]		V [kg/h]	(𝜟𝑷) 𝒆𝒙𝒑 [Pa]	(𝜟𝑷) 𝒎𝒐𝒅𝒊𝒇𝒊𝒆𝒅 𝒎𝒐𝒅𝒆𝒍 [Pa]	deviation between
									experimental
									and modified
									model [%]
		1	1185		74.9	232		234	0.86
		2	1026		77.1	217		229	5.4
		3	852		74.5	204		200	1.9
		4	766		73.0	180		187	3.8
		5	686		75.0	182		191	5.1
		6	423		77.0	204		186	8.7
		7	329		78.9	211		190	9.8

Table 53 .

 53 Activated carbon properties used for this study[53]

	Properties	Unit	Value
	Shape	-	Pellet or cylindrical shaped
	Particle density	kg/m 3	460
	Particle size	mm	4
	BET surface area	m 2 /g	1134
	Micropore volume	cm 3 /g	0.48

Table 56 .

 56 Estimation of the binary diffusion coefficients[START_REF] Bird | Transport phenomena, 2 nd edition[END_REF] 𝑃 𝑐,𝐻 2 𝑆 𝑃 𝑐,𝐶𝐻 4 ) 1 3 (𝑇 𝑐,𝐻 2 𝑆 𝑇 𝑐,𝐶𝐻 4 )

	Binary systems							Correlations
		𝐷 𝐻 2 𝑆-𝐶𝐻 4
	H2S -CH4	=	2.745 × 10 -4 (5 12 ( 𝑀 𝐻 2 𝑆 1 𝑃	+	1 𝑀 𝐶𝐻 4	0.5 )	( √ 𝑇 𝑐,𝐻 2 𝑆 𝑇 𝑐,𝐶𝐻 4 𝑇	) 1.823	(118)
		𝐷 𝐻 2 𝑆-𝐶𝑂 2
	H2S -CO2	=	2.745 × 10 -4 (𝑃 𝑐,𝐻 2 𝑆 𝑃 𝑐,𝐶𝑂 2 )	1 3 (𝑇 𝑐,𝐻 2 𝑆 𝑇 𝑐,𝐶𝑂 2 ) 𝑃	5 12 ( 𝑀 𝐻 2 𝑆 1	+	1 𝑀 𝐶𝑂 2	0.5 )	( √ 𝑇 𝑐,𝐻 2 𝑆 𝑇 𝑐,𝐶𝑂 2 𝑇	) 1.823	(119)
			𝐷 𝐻 2 𝑆-𝑁 2
			=	2.745 × 10 -4 (𝑃 𝑐,𝐻 2 𝑆 𝑃 𝑐,𝑁 2 )	1 3 (𝑇 𝑐,𝐻 2 𝑆 𝑇 𝑐,𝑁 2 ) 𝑃	5 12 ( 𝑀 𝐻 2 𝑆 1	+	1 𝑀 𝑁 2	0.5 )	( √ 𝑇 𝑐,𝐻 2 𝑆 𝑇 𝑐,𝑁 2 𝑇	) 1.823	(120.a)
	H2S -N2
				𝐷 𝐻 2 𝑆-𝑁 2 =	𝑃 (𝑉 𝑚,𝐻 2 𝑆 1 3 ⁄ 10 -7 𝑇 1.75 √ 𝑀 𝐻 2 𝑆 + 𝑉 𝑚,𝑁 2 1 3 ⁄ ) 2 1 + 𝑀 𝑁 2 1	(120.b)
		𝐷 𝐻 2 𝑆-𝐻 2 𝑂
	H2S -H2O	=	3.64 × 10 -4 (𝑃 𝑐,𝐻 2 𝑆 𝑃 𝑐,𝐻 2 𝑂 )	1 3 (𝑇 𝑐,𝐻 2 𝑆 𝑇 𝑐,𝐻 2 𝑂 ) 𝑃	5 12 ( 𝑀 𝐻 2 𝑆 1	+	1 𝑀 𝐻 2 𝑂	0.5 )	( √ 𝑇 𝑐,𝐻 2 𝑆 𝑇 𝑐,𝐻 2 𝑂 𝑇	) 2.334

Table 57 .

 57 Equations used to estimate internal diffusion coefficients

	Type of diffusion mechanism			Equation
	Knudsen diffusion	𝐷 𝐾𝑛𝑢𝑑𝑠𝑒𝑛 = 97 𝑟 𝑝𝑜𝑟𝑒 √ 𝑀 𝐻 2 𝑆 𝑇	(122)
	Surface diffusion	𝐷 𝑠 = 1.1 × 10 -8 𝑒	( -5.32 𝑇 𝐵,𝐻 2 𝑆 𝑇	)	(123)
	Poiseuille diffusion	𝐷 𝑃𝑜𝑖𝑠𝑒𝑢𝑖𝑙𝑙𝑒 =	2 𝑃 𝑟 𝑝𝑜𝑟𝑒 8 𝜇 𝐻 2 𝑆	(124)
	Pore diffusion (Global internal diffusion)	1 𝐷 𝑝	=	1 𝐷 𝑚	+	1 𝐷 𝐾𝑛𝑢𝑑𝑠𝑒𝑛

Table 58 .

 58 Estimation of dimensionless numbers and diffusion coefficients

	Properties	Unit	Value
	Reynolds number, Re	-	17
	Schmidt number, Sc	-	0.81
	Sherwood number, Sh	-	7.66
	Peclet number, Pe	-	13.77
	Molecular diffusivity	m 2 /s	1.35 x 10 -5
	Knudsen diffusivity	m 2 /s	1.84 x 10 -7
	Surface diffusivity	m 2 /s	2.34 x 10 -10
	Poiseuille diffusivity	m 2 /s	4.35 x 10 -10
	Pore diffusivity	m 2 /s	1.81 x 10 -7

  DL [m 2 .s -1 ] is the axial dispersion coefficient. C [mol.m -3 ] is the concentration of H2S in the biogas. L [m] is the length of the adsorption column. z [m] is the axial dimension. t [s] is the time. u [m.s -1 ] is the superficial velocity of the biogas.

	𝐷 𝐿 𝐿 2	𝜕 2 𝐶 𝜕𝑧 2 +	𝑢 𝐿	𝜕𝐶 𝜕𝑧	+	𝜕𝐶 𝜕𝑡	+	𝜌 𝜀	𝜕𝑞 𝜕𝑡	= 0	(126)
	Where:

ρ [kg.m

  Similarly, the expression (137) is obtained by combining equations (129) and[START_REF] Gnielinski | Gleichungen zur Berechnung des Wärmeund Stoffaustausches in durchströmten ruhenden Kugelschüttungen bei mittleren und grossen Pecletzahlen[END_REF].

	𝐿	𝜕𝐶 𝜕𝑧	+	𝜕𝐶 𝜕𝑡	+	1 -𝜀 𝜀	𝑘 𝑔 𝑎 𝑝 (𝐶 -𝐶 𝑒 ) = 0	(136)
	𝜕𝐶 𝑒 𝜕𝑡	=	𝑘 𝑔 𝑎 𝑝 (1 -𝜀) 𝜌 𝑞 𝑚 𝑏	(1 + 𝑏 𝐶 𝑒 ) 2 (𝐶 -𝐶 𝑒 )

Table 60 .

 60 Input parameters for the simulation of the breakthrough curve

	Parameters	Unit	Value
	Dimensions of the adsorption column
	Diameter	m	0.8
	Height	m	0.8
	Biogas composition and flow rate
	Methane	mol%	60
	Carbon dioxide	mol%	33 / 30.5 / 28
	Hydrogen sulfide	mol%	5 / 7.5 / 10
	Nitrogen	mol%	0.89
	Water vapor	mol%	1.11
	Biogas flow rate	Nm 3 /h	80
		Pressure and temperature
	Inlet pressure	bar	1.103
	Inlet temperature	°C	21
		Langmuir constants
	b	m 3 /g	0.9
	Activated carbon maximal capacity	g/kg	110
		Mesh
	Number of mesh nodes over the column height	-	20
	Time step	s	0.01
	Number of time steps	-	2 x 10 6

]Vapor capacity factor [(m/s).(kg/m 3 ) 0.5 ]

] Liquid load [m/h]

]Gas capacity factor [(m/s) x (kg/m 3 ) 0.5 ] (c')

]Gas capacity factor [(m/s) x (kg/m 3 ) 0.5 ] (e')

]Gas capacity factor [(m/s) x (kg/m 3 ) 0.5 ] (d')

Acknowledgements

I am grateful to Prof. Denis Clodic, supervisor of this thesis for the suggestions that helped to shape my research skills and for the responsibility he granted to me. Without him, this dissertation would not have been possible.

Chapter 4: Industrial demonstrator description

Résumé :

Ce chapitre présente le procédé industriel dans lequel se situe l'étape de désulfurisation étudiée. Il s'agit d'un démonstrateur industriel appelé « BioGNVAL » développé par la société Cryo Pur® en partenariat avec SUEZ et l'Agence De l'Environnement et de la Maitrise de l'Energie (ADEME). GNVert et IVECO sont également partenaires du projet BioGNVAL.

Ce pilote traite 85 Nm 3 /h de biogaz issu de la station d'épuration du Syndicat Interdépartemental pour l'Assainissement de l'Agglomération Parisienne (SIAAP) à Valenton, la deuxième plus grande en France.

Une des voies de valorisation du biogaz est la production de bio-GNL (GNL : Gaz Naturel liquéfié) qui se présente comme un carburant neutre en émissions de gaz à effet de serre avec plusieurs avantages économiques et environnementaux. La production de ce type de carburant requiert de très basses températures afin de liquéfier le bio-méthane, ce qui peut induire la solidification des impuretés et donc des problèmes de fonctionnement des installations. Ces impuretés doivent donc être séparées du biogaz, en particulier l'élimination de l'hydrogène sulfuré est impérative afin de garantir un fonctionnement optimal et une grande pureté des autres composés à valoriser comme le dioxyde de carbone. De surcroît, la présence de H2S dans le biogaz est une source de corrosion pour les équipements comme les pompes et les échangeurs de chaleur.

Because of the lack of predictive models, and because of the imprecision of existing ones to accurately predict the hydrodynamic parameters for some specific applications such as biogas purification, most distillation and packing columns are still being designed based on experimental data from a pilot plant [START_REF] Schultes | Research on mass transfer columns "Old hat or still relevant ?[END_REF].

The objective of this work is to find a model adapted for the representation of the experimental results obtained on the BioGNVAL pilot plant. To this aim, three literature models for the hydrodynamics in structured packing columns have been compared: Billet and Schultes model [START_REF] Billet | A physical model for the prediction of liquid hold-up in two-phase counter-current columns[END_REF], SRP model [START_REF] Fair | A comprehensive model for the performance of columns containing structured packings[END_REF] and Delft model [START_REF] Olujić | Predicting the efficiency of corrugated sheet structured packings with large specific surface area[END_REF]. These models have been developed on dimensionless analysis and experimental data obtained using a distillation column. The two first models are implemented in the process simulator Aspen Plus®. The three models are described in detail in the following section.

Billet and Schultes model

The Billet and Schultes model [START_REF] Billet | A physical model for the prediction of liquid hold-up in two-phase counter-current columns[END_REF] was at the base founded for random packings. Then, it was extended to cover structured packings. Based on semi-empirical correlations, this model assumes that the packing void fraction is represented by vertical tubes where the liquid is sprayed from the top as a film that meets the gas flow in a counter-current configuration. The angle between the corrugations of the packing is not taken into account by the Billet and Schultes model. The refined model is then compared to an extended range of experimental data retrieved also from the work of Erasmus [100] in order to validate the new model.

Comparisons to validate the modified model were made at various liquid loads and using two different systems: Air -Water and Air -Kerosol 200. Results of liquid holdup and pressure drop of the two systems over Flexipac® 350Y are presented in Fig. 41 and42.

The same conditions used to evaluate the three models (Type of packing: Flexipac® 350Y, system: Air -Water, Liquid load: 20.5 m/h) are used again in order to evaluate the new model and compare it to the other ones and to the experimental data. In Fig. 39, the experimentally determined pressure drop and liquid holdup [100] are compared to the results obtained with all the models including the new one.

Breakthrough curve modeling

Fig. 65: Schematic representation of an adsorption column [START_REF] Sigot | Epuration fine des biogaz en vue d'une valorisation énergétique en pile à combustible de type SOFC : Adsorption de l'octaméthylcyclotétrasloxane et du sulfure d'hydrogène[END_REF] The operating conditions used in the model are the same measured on the BioGNVAL pilot plant, in order to adjust the estimated parameters, such as the overall mass transfer coefficient and the maximum adsorption capacity of activated carbon. These operating conditions include the temperature and the pressure of the process, the adsorption column dimensions, the relative humidity of biogas and its composition. They are presented in Table 54. The adsorption mechanism is governed by different types of mass transfer: external and internal diffusion. That's why, before modeling the breakthrough curve, mass transfer coefficients should be correctly estimated.

Table 54. Biogas composition and operating conditions of the adsorption process

Properties

Estimation of mass transfer coefficients

The estimation of mass transfer coefficients is an essential step in order to simulate the dynamic behavior of adsorption.

 External mass transfer coefficient

The evaluation of the external mass transfer coefficient depends on the H2S molecular diffusion coefficient, the activated carbon particle size and dimensionless numbers as seen in Equation (101).

The particle size is indicated in Table 53. The Sherwood number (Sh) estimation is based on correlations using Reynolds (Re) and Schmidt (Sc) numbers. Some correlations defining the Sherwood number are presented in Table 55. Re < 15000 [START_REF] Dwivedi | Particle-Fluid Mass Transfer in Fixed and Fluidized Beds[END_REF] (105) Re ≥ 500 [START_REF] Onashi | Correlation of Liquid-Side Mass Transfer Coefficient for Single Particle and Fixed-Beds[END_REF]

Table 55. Sherwood number estimation [123]

Correlations

Reynolds and Schmidt numbers are calculated using Equations ( 114) and ( 115) respectively.

In Expression (107) defined by Doytchava et al. [START_REF] Doytchava | Mass Transfer from Solid Particles to Fluid in Granular Bed[END_REF], α is the available surface coefficient, φ is the particle shape factor, τ is the tortuosity and Pep is the Peclet number of the particle calculated using Equation [START_REF] Gel'perin | The relation between the surface tension of aqueous solutions of inorganic substances and concentration and temperature[END_REF].

Where: kg [m.s -1 ] is global mass transfer coefficient. ap [m 2 .m -3 ] is the external surface of the adsorbent particle per unit volume of adsorbent. qe [g.kg -1 ] is the amount of H2S in equilibrium with the gas phase concentration. Ce [g.m -3 ] is the H2S concentration in the gas phase, in equilibrium with the solid phase.

The adsorption equilibrium between the two phases is defined by means of the Langmuir isotherm Equation (type I).

Where qm and b are the Langmuir parameters.

Fig. 66 shows that the equilibrium parameters are sensitive parameters. They have a great influence on the simulation results. These parameters must be adjusted with experimental data. (----) b = 1 Boundary and initial conditions are summarized in Table 59

Abstract

Biogas must be purified for becoming a renewable fuel. At now, the most part of the purification techniques are not satisfactory because they imply hydrogen sulfide (H2S) rejection to the atmosphere. One example of these methods is the treatment with high pressure water.

The first objective of the thesis is modeling the conventional methods for separating H2S from methane. Typical concentrations of H2S in methane vary from 200 to 5000 ppm. Separation methods must decrease the concentration of H2S in methane to less than 1 ppm. At the same time, methods for H2S treatment will be studied.

Once the most appropriated separation methods will be selected, some test will be carried out on a pilot plant capable of treating 85 Nm 3 /h of methane, where quantities of H2S ranging from 1 and 100 ppm will be injected. These tests will allow validating the modeling of the separation process.

The thesis work requires simulating the separation process using the software Aspen Plus® or an equivalent one. The effectiveness of different operative conditions will be tested, varying also the parameter temperature. The energy necessary for the separation will be one of the most important criteria for the comparison, as well as the mass consumption of the different fluids involved in the process.

A system approach is fundamental for evaluating the backward effect of the H2S valorization method on the separation techniques. The process simulator (Aspen Plus® or equivalent) will allow the system approach. The study will involve modeling and experimental parts.

The experimental part will be carried out taking advantage of a semi-industrial size test bench, allowing studying the separation methods down to -90°C.