Jacques C Vedrine
email: jacques.vedrine@upmc.fr

Heterogeneous catalytic partial oxidation of lower alkanes (C 1 -C 6 ) on mixed metal oxides

Keywords:

This review paper aims at analysing the state of the art for partial oxidation and oxidative dehydrogenation (ODH) reactions of lower alkanes C 1 -C 6 into olefins and oxygenated products (aldehydes, anhydrides, carboxylic acids) on metal oxide catalysts with cations of variable oxidation state, such as Mo and V in particular. Key parameters to be met by the catalysts, such as their redox properties, their structural aspects, active sites composed of ensembles of atoms isolated one from the others, mechanisms of reactions, are discussed. Main features of the different reactions of C 1 -C 6 alkanes and catalysts are analysed and their generalisation for determining more active and more selective catalysts is attempted. Prospective views for the future of the domain are proposed.

Introduction

Considerable effort s have been carried out to the oxidative transformation of alkanes using different classes of catalytic materials or catalytic-system designs [START_REF] Derouane | Catalytic Activation and Functionalisation of Light Alkanes: Advances and Challenges[END_REF][START_REF] Asinger | Paraffins: Chemistry and Technology[END_REF][START_REF] Cavani | Basic Principles in Applied Catalysis[END_REF][START_REF] Arpentinier | The Technology of Catalytic Oxidations: Chemical, Catalytic and Engineering Aspects[END_REF][START_REF] Védrine | [END_REF] . These efforts have been driven by the possibility of developing new environmentally friendly and low cost processes to synthesise olefins and partially oxidised organic compounds. Furthermore, with increase in the world production of medium-to-long chain alkanes, there is a need to convert them into useful products. Selective oxidation of alkanes is difficult, since they are quite inactive compounds, which necessitate to using high temperatures and/or pressures and, hence, lead to dominantly radical reactions. Moreover, the product molecules are more reactive than alkanes, and may easily be sur-oxidised into CO 2 . The gas-phase selective oxidative transformation of light alkanes to alkenes (e.g., C 2 n H 2 n + 2 → C 2 H 2 n ) with n = 2-6 or oxygenates (aldehydes, acids) with n = 1-6 is an important challenge as it aims at reducing the number of process steps, at decreasing both the energy required and CO 2 emissions, and at improving the atom economy.

Up to now, transformation of light alkanes (C 2 -C 6 ) to olefins by direct dehydrogenation constitutes the major industrial processes available, unless strong drawbacks (endothermicy and strong deactivation by coking). Subsequently, both oxidative dehydrogenation (ODH) and O-insertion of C 1 -C 6 alkanes constitute important reactions to deal with, in particular any method for a better selective oxidative transformation of methane (C 1 ) and best catalytic systems constitute also an important challenge.

The importance of olefins in further applications to chemicals is summarised in Fig. 1.

In Table 1 , important processes industrially developed or under development are listed.

In Fig. 2 are reported the numbers of patents vs. open literature scientific publications for C 2 -C 4 alkanes. It is clear that at present simple dehydrogenation is more developed in the industry than ODH at variance with academic researches.

General features of selective catalytic oxidation reactions

The most important catalysts, which have been used or studied

They are listed below, some of their structures being shown in Fig. 3 and their catalytic properties reminded below in the following different sections. Such solid materials are composed, most of the time, of arrangements of MO 6 octahedra (M = Mo,V,Nb,Te,Sb,) creating some empty holes filled by additional cations.

• Basic solids such as VMgO (e.g., for propane to propene).

• Bulk vanadates or molybdates. • V 2 O 5 deposited on oxidic supports (SiO 2 , TiO 2 , ZrO 2 , Al 2 O 3 , HDT, MCM-41 or SBA-15). • Heteropoly oxo metallates (POMs) Mo based (Keggin-type structure mainly). • VPO-type catalysts (e.g., butane to maleic anhydride or pentane to maleic anhydride). • Mixed oxides MoNbTe(Sb)V-O for propane to acrylic acid/acrylonitrile. • NiMoTeP-O, NiNb-O, for ethane to ethylene….

In the case of metal oxides deposited on an oxide support, it is well accepted that the synthesis procedure obeys some specific laws related to pH of the solution vs. the zero charge point (ZPC) of the support surface. In other words, in the case of impregnating a given salt on an oxide surface, when the pH value of the solution is less than ZPC, the surface is positively charged and anionic species can be adsorbed. At variance if pH > ZPC, the surface is negatively charged and cationic species are adsorbed. This is the case, for instance, of Pt cationic complex (Pt(NH 3 ) 4 n + ) vs. anionic complex (PtCl 4 -) adsorbed on alumina (amphoteric) vs. zeolite (acidic) surfaces, to adsorb Pt either on alumina or on zeolite, as H-FAU.

Main aspects of selective oxidation on metal oxides catalysts

An important aspect to consider in oxidation reaction is the number of electrons necessary for the reaction to occur, for 2 assembles such electron numbers involved in oxidative transformation of lower alkanes.

The three main features of metal oxides, which are essential for their application in catalysis, are (i) the coordination environment of surface atoms, (ii) the redox and, subsequently, acid-base properties, number of electrons involved and (iii) the oxidation states of According to this mechanism the substrate is oxidised by the catalysts and not directly by the gaseous oxygen and the role of oxygen is to regenerate or maintain the oxidised state of the catalyst. Lattice oxygen is introduced into the substrate or into H 2 O for ODH reaction. This involves two active sites: an active cationic site and a site active for oxygen reduction. The process starts with the abstraction a proton, accompanied by a two electron transfer which reduces the transition metal cations. This step is followed by a nucleophilic addition of an oxide ion from the catalyst to the oxidised hydrocarbon with formation of oxygen vacancies, which are filled by oxygen with re-oxidation of the metal cation, in a kind of reversible breathing motion. This is schematised in Fig. 4.

2[CatO

] + R-CH 3 → 2[Cat] + R-CHO + H 2 O 2[Cat] + O 2 (gas) → 2[CatO]
In majority of cases, selective oxidation reactions involve Mars and van Krevelen or redox mechanism or minorly other mechanisms as reminded below in Table 2.

• The reaction proceeds at a single site (atoms or clusters/ensembles of atoms) [START_REF] Taylor | [END_REF] . • Principle of site isolation, and phases cooperation introduced by Grasselli et al. [10] . • The reaction proceeds as a sequence of consecutive steps, the first one being usually an H atom abstraction, which is the rate determining step (rds). • Other steps, such as product desorption, O insertion, other H atom abstraction, are usually faster and do not influence the rds. • An acidic surface favours desorption of acidic product whereas a basic surface favours desorption of basic products such as olefins.

• In a landmark contribution to catalytic theory, Haber [START_REF] Haber | Proceedings of the Eighth International Congress on Catalysis[END_REF] has suggested that a catalysed chemical reaction is not catalysed over the entire solid surface of the catalyst but only at certain "active sites" or centres. He has also suggested that chemisorption may be an activated process and may occur slowly. Moreover, he conceived the idea that chemically active sites might be composed of an atom or an ensemble of atoms and could be sparse on the surface of a catalyst and, hence, could be inhibited/poisoned with relatively few molecules. Since that time, many descriptions of active sites have been proposed, in particular for metal oxides used in selective oxidation reactions, as revisited recently [11] .

As a general rule the concept of "7 pillars", as proposed by Grasselli [12] , has to be satisfied:

(1) Nature of lattice oxygen anions: nucleophilic (selective) rather than electrophilic (total oxidation). ( 2) Redox properties of the metal oxide (removal of lattice oxygen and its rapid reinsertion). (3) Host structure (permits redox mechanism to take place without structure collapse). An example is given in Fig. 5 . (4) Phase cooperation in multicomponent catalyst or supported catalyst (epitaxial growth and synergetic effects).

(5) Multifunctionality (e.g., α-H abstraction and O-/NH-insertion). ( 6) Active site isolation (to avoid a too high lattice O surface mobility and thus over-oxidation). ( 7) M -O bond strength (to be not too weak (total oxidation) nor too strong (inactivity) (Sabatier principle).

Catalytic partial oxidation reactions are more efficient on mixed metal oxides based mainly on V and Mo cations attached to other redox cations such as Sb, Nb, Te, Ta, as MO 6 octahedra assembled as shown above in Fig. 3 and which are composed of several phases acting in synergy one with respect to the other(s) and leading to very complex chemical compositions as we will give examples in Section 3.3 below, for instance in the case of the mixture of M1 and M2 phases in MoVTeNb-O catalyst used for propane amm/oxidation to acrylonitrile/acrylic acid.

As the selectivity depends on the nature of surface oxygen species (O 2-) associated to cation(s) Me n + , to their bond energy Me -O (of the lattice or with adsorbed O), and to their distribution on the surface, it is reasonable to conclude that oxidation reactions are demanding, i.e., are structure sensitive. They are more demanding when the number of electrons involved in the reaction is high. For instance, if many catalysts are able to transform propane to propylene (reaction with 2e -) only three are able to (Cu -O, Bi -Mo -O, Sn -Sb -O) are able to transform propylene to acrolein (4e -reaction) and only one ((VO) 2 P 2 O 7 ) for butane to maleic anhydride (a 14e -reaction) or (VTi -O) for o-xylene oxidation to phthalic anhydride (a 12e -) reaction. Partial oxidation catalysts contain usually one transition metal ion from first column of the periodical table (Ti, V, Cr, Mo). As this cation is very small, highly charged and very polarising it can form oxo cations, such as oxo vanadium (VO 3 + ), vanadyl (VO 2 + ), molybdenyl. In first approximation, one can estimate that during reduction of V 5 + to V 4 + (or Mo 6 + to Mo 5 + ) and reversely in oxidation, the transfer of one e -is easy, resulting in good catalytic properties. As a matter of fact, a redox catalyst is more active when the principle of small changes in structure and oxidation state is satisfied. By contrast, the situation is less favourable for other redox couples such as Fe 3 + /Fe 2 + et Cu 2 + /Cu + , for which either the size of the ions, (e.g., 0.017 nm difference for Cu 2 + /Cu + with Cu at the same coordination of six), or coordination, are very different. Vanadium is a typical example of structure sensitivity. Ionic radius r i of V 5 + ( r i = 0.054 nm) is very small compared to that of oxygen anion O 2-( r i = 0.140 nm). In an octahedral environment of six O 2-, vanadium get closer to O 2-forming a short bond to be more stable. The V…O bond in trans position being longer, its bond energy is weaker and O can be easily been withdrawn. Then V coordination becomes 5. For the same reason, crystal cuts easily perpendicularly equatorial plane. Considering octahedral environments [VO 6 ] in V 2 O 5 and in V 2 O 4 the bonds O… V 4 + are respectively shorter (0.158-0.164 nm) (0.278-0.270 nm) than their homologs with V 5 + , and the average height of the octahedron changes very little. The same holds true for Mo 6 + /Mo 5 + couple in the same coordination [MoO 6 ] ( r i = 0.059 nm et 0.061 nm pour

Mechanism

Kinetic equation

Power law model r = k 0 P m A P n B ( -E a RT ) Eley-Rideal model (ER) r = k K p A p B 1+ K p A Langmuir-Hinshelwood model (LH) r = k K A K B ( 1+ K A p A + K B p B + K C p C ) 2

Main reaction mechanisms involved in heterogeneous catalytic reactions

The main reaction mechanisms involved in oxidative transformation of alkanes are schematised in Table 3.

A reaction mechanism involving cations such as V or Mo at different oxidation states and lattice oxygen (O 2 -) extraction and reinsertion is proposed below following the scheme detailed above in Fig. 4 , whereas the number of cationic sites involved in a reaction is shown in Fig. 5 below:

C 3 H 8 + 2 {V 5 + -ם -O L 2 -} → C 3 H 6 + 2 OH -+ 2{V 4 + -}ם 2 OH -→ H 2 O + O 2 -+ ם

Cases studies

Natural gas

Depending on its origin, natural gas contains mainly methane but also and in different amounts C 2 -C 4 alkanes, which necessitate some up grading or uses. The following equations and Fig. 5 describe some processes in use at present or in perspective, such as: CH 4 + air → CH 3 OH, HCHO

Case of methane

Ideally many transformations of methane into valuable chemicals could be possible as schematised in Figs. 6 and7 and in the bullet list below, but at present only methane partial oxidation to CO + H 2 mixture (syngas) and its further transformation is industrially viable. [21] , and also to oxidise ethane to ethylene when using O 2 as an oxidant [22] and oxygenated products by O 2 in the presence of H 2 and N 2 O [23] .

Many other examples of methane activation can be found in the literature but none had a chance to be commercialised in a near future. One can cite an example of methane and ethylene selective oxidation in water [24] , for which it has been shown that stable μnitrido di-iron phthalocyanine activates H 2 O 2 to form a high-valent di-iron oxo species. This species is a very powerful oxidant which oxidises methane in pure water at 25-60 °C to methanol, formaldehyde and formic acid [24] . The catalytic activity can significantly be increased in the presence of a diluted acid solution. Thus, a high turnover number of 209 was attained in 0.075 M H 2 SO 4 . Oxidation of ethylene resulted in the formation of formic acid as a major product and formaldehyde with high turnover numbers. The practical and green features of this approach as well as the relevance to biological oxidation (binuclear structure of bio-inspired complex) are of great importance both from practical and fundamental points of view.

Case of lower alkanes C 2 -C 6 oxidatively transformed into olefins (ODH) [25] or oxygenates with O-insertion

This topic has already been largely studied in the literature and many review articles [6 , 26] and books [27] can be found on the subject with vanadium ion being a key element in the solid catalyst.

More recently a new family of catalysts based on NiO was published [START_REF] Heracleous | [END_REF] for ethane selective oxidation. Such catalysts present high catalytic activity for ethane conversion and ethylene selectivity. The preparation procedure of the catalysts (sol-gel or precipitation methods) has been observed to govern its catalytic properties, whatever activity and selectivity [29 , 30] . When Nb was deposited on NiO, the catalytic activity for ethane conversion and ethylene selectivity were improved. This phenomenon was assigned to electron transfer enhancement played by Nb, leading to easier activation of the C -H bond of ethane.

Another aspect of ethane partial oxidation is not only the formation of ethylene but also that of acetic acid. The main pro- or V 5 + to form an ethoxide species which transforms to ethylene by β-elimination, but also possibly oxidises and then acetaldehyde by α-elimination, transformed to surface acetate and then to acetic acid by its hydrolysis. MoVNb-O catalyst family thus appears as the most promising possibility at present for ethane to ethylene and acetic acid, the problem being that both products are formed and limiting acetic acid formation, if desired, is difficult. However, when adding Pd the selectivity to acetic acid may reach 80% [32 , 33] , whereas adding water in the feed increases acetic acid formation [34] . The same effect holds true for propane oxidative dehydrogenation by Pt-H 3 PMo 12 O 40 [35] with O 2 and H 2 , for propane ammoxidation to acrylonitrile. High transformation to acetic acid were claimed by Saudi Basic on MoVNbP-O; MoVLaPd-O; MoVNbPd-O under pressure 1-2 MPa. The technology developed by Saudi Basic is now commercial [START_REF] Mamedov | Handbook of Advanced Methods and Processes in Oxidation Catalysis[END_REF] , as the catalyst presents multifunctionality, ethylene as well as acetic acid are rather stable compounds and do not transform to CO 2 in the catalytic conditions chosen, unconverted ethane and ethylene being recycled.

The basic structure of Mo 5 O 14 material is shown in Fig. 8 . It is known that some amounts of other cations such as V, Ta, Te, Sb or Nb can be incorporated in the structure as determined by XRD analysis (see Fig. 9 ).

At this stage, it is important to note that such a structure initiated in 1978 has been found later on after inserting cations such as V, Nb, Sb and Te [38] to also catalyse propane direct oxidation to acrylic acid and ammoxidation to acrylonitrile and to lead to a new industrial process for acrylonitrile. Two main components of such material have been identified to be determining the phases designated M1 and M2 (shown in Fig. 10 ), who were found to act synergetically in propane oxidation reaction.

The propane to acrylic acid reaction proceeds via the following reaction steps on MoVTeNb-O, whereas the active sites were suggested to be either 4 monomeric or two dimeric units of redox storage to deal with one O 2 molecule (see Fig. 11 right).

• 4V + 4 -OH + 4OH -→ 4V + 5 = O + 4e -+ 4H 2 O. • O 2 + 2H 2 O + 4e -→ 4OH -. • 4 V + 4 -OH + O 2 → 4V + 5 = O + 2H 2 O. • O = V-O-V = O + C 3 H 6 → O = V-O-V-OC 3 H 5 + H + + e -. • O = V-O-V = OC 3 H 5 + H + + e -→ HO-V-O-V-OC 3 H 5 . • HO-V-O-V-OC 3 H 5 + 3OH -→ HO-V-O-V-OC 3 H 4 O + 2H 2 O + 3e -.
A large number of catalysts, in majority V based, have been investigated for propane ODH but propene yield varies in the 8-20% range. The upper yield was attained with VMgO, Mg 2 V 2 O 7 or MgV x Sb y O z , or with orthovanadate on hydrotalcite [42] , while V 2 O 5 supported on Al 2 O 3 , Bi 2 O 3 , La 2 O 3 , Nb 2 O 5 , SiO 2 , Sm 2 O 3 or TiO 2 gives inferior performances. Co and Ni molybdates were also investigated with different dopants such as Fe, Mg, Ni or P and gave yield near 15% with high productivity and no production of undesirable oxidised products. The major problem is that in addition to alkane activation such metal oxide catalysts may possess also O-or NH-insertion properties and thus may give undesired oxygenates or nitriles in addition to the olefin. The use of Pt based catalyst and of short contact times may be a solution to get mainly the olefin, which is known to be a reaction intermediate. At variance, one may add a catalyst known to transform the olefin to unsaturated carboxylic acids or nitriles for complete transformation of alkanes to carboxylic acids or nitriles, but both operating temperature and chemical compatibility of both catalysts need to be optimised.

Another aspect in ODH reaction is the nature and surface density of surface oxygen species. For instance, in the case of propane ODH both propylene and ethylene can be formed depending on the reaction mechanism by dehydrogenation or cracking. Comparing non reducible (rare earth/alkaline earth oxides) and reducible (V based oxides) catalysts, it was shown [43] that the ratio propylene over ethylene depends on the strength and mainly the density of oxidising active sites (DOS), high DOS favouring propylene. The oxidising sites are those active for H-atom abstraction, usually V 5 + = O and or bridged V 5 + -O -M (M = metal cation in the next cationic position, i.e., V or another metal atom from the structure or support).

In the case of n-and i-butane ODH reaction to butene and isobutene on catalysts of the same type as for ethane or propane such as VMgO or metal molybdates or γ -A1 2 O 3 supported vanadium pentoxide. MgO supported vanadium was reported as a very selective catalyst in propane and n-butane ODH, and to exhibit a good selectivity to ethylene for ethane ODH but a poor selectivity in the ODH of n-butane. The acid-base character of the support was suggested to lead to this different behaviour. On MgO, a support with basic properties, the interaction between vanadium species and the support leads to the formation of vanadate compounds. In the case of more acid supports, such as SiO 2 or A1 2 O 3 a weak interaction is expected leading to less dispersed vanadium species on the surface favouring the formation of V 2 O 5 crystallites.

However, the most important process concerns butane (pentane) partial oxidation on (VO) 2 P 2 O 7 catalyst designated as VPO and which has substituted the old benzene oxidation process in the 1980s. Active sites have been described as ensembles of four dimers of octahedral VO 6 entities isolated by P 2 O 7 groups [44] . In such a reaction, maleic anhydride is relatively stable to further oxidation which permits high yields. This is also why in a moving bed technology, developed in Asturias, Spain by Du Pont Co, the catalyst encapsulated by porous silica to limit attrition of catalyst grains, maleic anhydride was further transformed to less stable tetrahydrofurane (THF) by reduction on Pd based catalyst.

For a more general point of view the role of vanadium atom oxygen arrangement in catalytic properties is schematised in Fig. 12.

The process of propane ammoxidation is an eightelectron oxidation reaction and requires a catalyst or combination of catalysts which are even more complex than those for the ammoxidation of olefins (e.g., propylene):

CH 3 CH 2 CH 3 + NH 3 + 2O 2 (air) → CH 2 = CHCN + 4H 2 O.
There has been great studies to develop an ammoxidation process based on propane rather than propylene, as the price and availability of propane are more favourable than for propylene. The task is difficult from a commercial standpoint, since the wellestablished and well-functioning propylene based process is constantly being further improved, and the catalysts therefore constantly made better, more efficient and environmentally friendlier. It is simply a moving target, difficult to hit, yet not out of reach and will be achieved in due time. Many catalysts have been tested for the ammoxidation of propane to acrylonitrile, and the most effective of them fall, as do propylene ammoxidation catalysts, into two main classes: they are either antimonates or molybdates. Among the antimonates, the most studied belong to the family VSb xMy , where M can be many different elements, with the most frequently used dopants being W, Te, Nb, Sn, Bi, Al, and Ti. Virtually all of these antimonates possess the rutile or trirutile structure. Some of the highest acrylonitrile yielding catalysts are summarised in Table 4.

The molybdate family is represented by VMo xMy O z , where M is most often Bi or Te. Some catalysts are of the scheelite structure, others are multiphase and of more complex structure. Among these molybdates, the highest acrylonitrile yields are claimed for a catalyst composition having the empirical formula V 0.3 Te 0.23 Nb 0.12 MoO x supported on SiO 2 .

Active sites in metal oxides for partial oxidation of light alkanes

From the analysis of many publications dealing with subject and the main features of the solid catalysts one may make further suggestions. It is clear that ensembles of atoms constitute the active site but the size and arrangement of such atoms in the ensemble looks important. If one considers the data in Figs. 12 -14 , it appears that many metal oxides can catalyse many reactions but the activities and selectivities depend on the organic reactant molecule, in particular on its size vs. active site size and the number of electrons involved in the redox process.

Conclusions

Catalytic selective oxidation is a scientific area in which recent advancements have led to remarkable steps forward towards a more sustainable chemical industry. This may require rethinking the chemical productions through an innovative vision. However, the key-issue still remains selectivity. In catalytic oxidation, the recent developments clearly show that better selectivity, better sustainability and improved economics often go hand in hand. One may summarise the general aspects in heterogeneous selective oxidation reactions. For the future, more efficient integration of oxidation reactions with up-or down-stream processes, and more efficient oxidation technologies should be developed. The main rules and advices for the development of more sustainable oxidation processes can be summarised in some bullet points:

(1) Make reactions where the amount of co-product is minimal -first, use reactants with maximum atomic incorporation into final products. (2) Search for solvents, when needed, that are non-toxic, and can be easily separated and re-utilised. (3) Search for methods to immobilise catalysts (anchoring, grafting,…) over supports that can be filtered. (4) Consider that the reaction may be a part of a more complex process. Integration has to be thought from the beginning of the research activity. (5) From a scale-economy approach to a modular design based on parallel small units (intensification, microreactors technology). ( 6) Search for reaction conditions at which the selectivity is the highest (analysis of the reaction scheme), rather than productivity or yield.

All these aspects show that large effort s in researches, process development, reactors design [START_REF] Bordes-Richard | Handbook of Advanced Methods and Processes in Oxidation Catalysis[END_REF] , new catalysts, new activation procedure, are necessary to improve selective oxidation reaction, looking mainly on providing a better selectivity.

Fig. 1 .

 1 Fig. 1. Schematic importance of olefins worldwide applications (Personal communication by J.M. Lopez Nieto, Valencia).

Fig. 2 .

 2 Fig. 2. Number of publications in the last two decades (1989-2009) dealing with short chain alkanes ODH and partial oxidation products including acetic acid, acrylic acid, methacrolein, methacrylic acid, maleic anhydride (From [6] Fig. 24.1). instance CH 3 -CH 2 -CH 3 + O 2 -→ CH 3 -CH = CH 2 + H 2 O + 2e -while the dissociation of molecular oxygen into 2O 2 -requires 4e -. Table2assembles such electron numbers involved in oxidative transformation of lower alkanes.The three main features of metal oxides, which are essential for their application in catalysis, are (i) the coordination environment of surface atoms, (ii) the redox and, subsequently, acid-base properties, number of electrons involved and (iii) the oxidation states of

Fig. 3 .

 3 Fig. 3. Structures (composed of MO 6 octahedra) of the most studied catalysts in alkane partial oxidation: (a) vanadyl pyrophosphate (VPO), (b) VSbO rutile phase, (c) M1 phase, MoVTe(Sb)NbO, and (d) Keggin molybdophophoric acid.

Fig. 4 .

 4 Fig. 4. Schematic representation of the mechanism of O insertion in a hydrocarbon molecule and its consequence in O vacancy reversible creation in a metal oxide (right part) P 2 O 7 4 -+ O 2-↔ 2 PO 4 3 . From J. Haber [8] .

Fig. 5 .

 5 Fig. 5. Over-simplified reaction schemes (a) propene adsorption O -V -O site and formation of an allene, (b) reaction of an alkane according to a redox mechanism, and (c) reaction of an alkane with oxygen.

Fig. 6 .

 6 Fig. 6. Schemes of methane up-grading to valuable chemicals.

  Mars van Krevelen model (MvK) r = k ox k red pO n 2 p P k ox pO n 2 + k red p p Mo 6 + et Mo 5 + , respectively). V 5 + and Mo 6 + ions can easily accept coordination VI, V et IV, but coordination IV is less frequent for V 4 + and Mo 5 + .

Fig. 7 .

 7 Fig. 7. Transformation of methane to propylene: a two-step reaction route catalyzed by modified CeO 2 nanocrystals and F-MFI zeolite from Ref. [17] .

Fig. 8 .

 8 Fig. 8. Crystal structure of Mo 5 O 14 crystal composed of MoO 6 octahedra.

Fig. 9 .

 9 Fig. 9. Lattice parameters vs. x in (Mo 1 -x V x O) 2.8 upon insertion of V (left) and Nb (right) in Mo 5 O 14 from Ekström et al. [37] .

Fig. 10 .

 10 Fig. 10. Representation of the structure of the phases M1 (top) M2 (bottom) along the c -axis MO 6 octahedra, from Ref. [39] .

Fig. 11 .

 11 Fig. 11. Bulk ab planes of M1 phase (right). Refined orthogonal structure had the space group Pba2, formula unit Mo 7.8 V 1.2 NbTe 0.937 O 28.9 , and lattice parameters, a = 2.1134 nm, b = 2.6658 nm, and c = 0.40146 nm with z = 4, from Ref. [40] . Active site in Mo 7.8 V 1.2 NbTe 0.94 O 28.9 (M1): 2 × 2 unit cell structure model of M1 in [001] projection showing four isolated catalytically active centres (left). The active sites are isolated from each other by four Nb-bipyramids that are surrounded by five MO n -octahedral. From Fig. 10 in Ref. [41] .

Fig. 12 .

 12 Fig. 12. Importance of the V-structure in V-catalysts for H-abstraction and Oinsertion. From Fig. 24.3 in Ref. [6] .

Fig. 13 .

 13 Fig. 13. Selectivity to the main partial oxidation products achieved during the partial oxidation of C 2 -C 4 alkanes over MoVTeNbO (a) and VPO (b) catalysts. AA = Acrylic acid; MA = maleic anhydride; MTA = methacrolein. From Fig. 24.5 in Ref. [6] .

Fig. 14 .

 14 Fig. 14. Selectivity to the main partial oxidation product achieved at low conversion (XT = 2%) and high conversion (XT = 20%) during propane oxidation at 400 °C on VO x /Al 2 O 3 ; Mo -V -Nb mixed oxides calcined in air at 450 °C (amorphous); Mo -V -Te-Nb mixed oxides heat-treated at 600 °C in N 2 presenting M1 phase (MoVTeNbO). The main adsorbed species observed during the adsorption of propylene on each catalyst is shown. From Fig. 24.4 in Ref. [6] .

( 1 )

 1 Flexibility of the oxide surface under catalytic (redox) conditions (living surface) avoiding any structure collapse during the redox process [45] . (2) Collective properties: electron transfer (electrical conductivity), changes in adsorption properties. (3) Influence of the co-adsorbate on the bond between the adsorbate and the surface. (4) Role of catalyst surface adlayer composition on activity and selectivity as very often met [46] . (5) Oxygen species involved in the reaction: lattice oxygen anions, weakly adsorbed oxygen species. (6) Mechanism and anisotropy of oxygen migration. (7) Importance of surface species which do not play a direct role in the reaction mechanism, e.g., H 2 O, carbon deposit (fouling), as spectators.

Table

.

  Industrial processes, processes under study or development for the oxidative transformation of light paraffins (C 1 -C 6 ).

	Raw material	Product	Stage of development
	Methane	Methanol	Pilot plant
	Methane	Syngas	Pilot plant
	Methane	Ethylene	Pilot plant
	Ethane	1,2-Dichloroethane,vinyl chloride	Pilot plant

Ethane tion mechanism is the Mars and van Krevelen one

[START_REF] Mars | [END_REF]

, schematised below, and which involves cations of variable oxidation state such as Ce, Co, Cr, Cu, Fe, Mo, Ni, V, W.

Table 2 .

 2 Main catalytic heterogeneous oxidation reactions industrialised (I), at pilot stage (P) or not industrialised yet (NI) or under research (R). At steady state, r red = r ox or k red p HC (1-θ ) = k ox p O2 θ with p HC , p O2 partial pressure of HC and O 2 and k red , k ox rate constants of reduction of catalyst (1st step) and r ox rate of oxidation by O 2 (2nd step). The relative rate value of r red and r ox is important for the selectivity in the product and involves lattice oxygen anions, which may be incorporated in the reactant and the corresponding vacancy created is then replenished by gaseous oxygen in the re-oxidation step. If k red p HC >> k ox p O2 , re-oxidation of the surface is the rate determining step; if k red p HC << k ox p O2 , reduction of the surface is the rate determining step. Thus, reoxidability and reducibility of the catalyst will leave to different kinetic expressions.

	Reaction	Reactants	Products	Number e-	Catalysts	Y (mol%)	Status
	Oxidative coupling	CH 4	Ethane	2	Li 2 O/MgO	25	NI
	-	-	+ ethylene	4	-	-	-
	ODH	Ethane	Ethylene	2	Pt/oxides	60	NI
	-	P r o p a n e	Propene	2	Pt/oxides	40	NI
	-	n-butane	Butenes	2	Metal molybdates	38	NI
	-	-	Butadiene	4	-	-	-
	Oxychloration	Ethane,Cl 2	Vinyl chloride	4	AgMnCo-O	15	NI
	Hydroxylation	CH 4	CO + H 2	2	Pt,Ni	90	R
	-	-	HCHO	4	MoSnP-O	16	R
	Partial oxidation	Ethane	CH 3 COOH	6	MoVNb-O	10	P
	-	P r o p a n e	Acrylic acid	6	MoV(Nb,Te,Sb)-O	8	NI
	-	n-butane	Maleic anhydride	14	(VO) 2 P 2 O 7	70	I
	-	i-butane	Methacrylic acid	8	Heteropolyacids	11	R
	Ammoxidation	Propane + NH 3	Acrylonitrile	8	VSb-O,MoV-O, MoVTeNb-O	30	I
	where [CatO] represents the oxidised catalyst surface and [Cat] its
	reduced state, r red the rate of catalyst reduction by reactant and
	r ox rate of its re-oxidation by co-fed oxygen, R -CH 3 and R-C -O the
	reactant and the product. The kinetic equation involves the relative
	concentration of reduced ( θ ) and oxidised (1-θ ) sites of the cata-
	lyst surface.

Table 3 .

 3 Main reaction mechanisms in heterogeneous catalysis.

  dimethyl ether by known processes (e.g., Cu/ZnO catalyst) and further MTO or MTG, processes or hydrocarbon mixtures in Fisher-Tropsch processes, etc.• Methane to ethane/ethene by oxidative coupling: studied for many decades since pioneered works on Li/MgO system by Ito et al.[13] and on Ln-type oxides (Ln = Ne, Sm, Gd) without any process developed up to now[14] . • Methane oxidation to oxygenate compounds: methanol, formaldehyde still in research domain although some successes have been gained . • Non oxidative methane aromatization (to benzene, naphthalene, etc.) at high temperature > 700 °C upon Mo/MFI catalyst [15] . • Two-step conversion of methane to methanol via methyl bisulfate using oleum as an oxidant [16] . • Methane to propylene in a two-step route via methane oxy chlorination/bromination to CH 3 Cl or CH 3 Br on CeO 2 and H-MFI catalyst [17] (See Fig. 6 ). • Methane to formaldehyde on VO x /SiO 2 prepared according to a new method [18] . • NiO small clusters/ZrO 2 -CeO 2 acting as Lewis acid sites for methane to methanol [19] . • Silica-supported tantalum hydride, ( ≡SiO) 2 Ta -H for the direct non-oxidative coupling transformation of methane into ethane and hydrogen at 250 °C, with a high selectivity ( > 98%) [20] . • Iron phosphate (FePO 4 ) to give methanol by oxidation of methane with O 2 when co-fed with H 2

• Methane oxidation to syngas:

CH 4 + 1 / 2O 2 → CO + 2H 2

( H °298 = -36 kJ/mol) followed by CO + H 2 transformation to methanol/

Table 4 .

 4 Summary of the best results in the literature for partial oxidation of C 2 -C 3 alkanes from Ref.[START_REF] Ivars | Advanced Methods and Processes in Oxidation Catalysis: From Laboratory to Industry[END_REF] p.772 and references herein.

		T ( °C)	Conversion (%)	Selectivity (%)	Yield (%)
	Ethane → ethylene
	MoVTeNbO	400	85	88	75
	NiNbO	400	51	90	47
	VCoAPO-18	600	57	60	34
	Ethane → acetonitrile
	NbSbO/Al 2 O 3	540	40	50	20
	Co-BEA	476	47	57	26
	Ethane → acetic acid
	MoVSbNbReCaO	277	14	78	11
	MoVNbPdO x	300	5	82	4.1
	Propane → propylene
	VMgO	540	62	38	24
	V-silicalite	550	30	70	21
	Propane → acrylonitrile
	MoVTeNbO	420	86	72	62
	VSbWO x /SiO 2 -Al 2 O 3	500	67	60	40
	Propane → acrylic acid
	MoVTeNbO	432	69	67	46
	MoVTeNbO A relevant	380	80	61	49


Antonio C P Brasil Junior
email: antonioc.p.brasiljuniorbrasiljr@unb.br

Rafael C F Mendes

Théo Wirrig

Ricardo Noguera
email: ricardo.noguera@ensam.eu

Taygoara F Oliveira
email: taygoara@unb.br

On the design of propeller hydrokinetic turbines: the effect of the number of blades

Keywords: Hydrokinetic turbines, Propeller horizontal axis turbines, BEMT methods, Wind tunnel experiments, Glauert theory

A design study of propeller hydrokinetic turbines is explored in the present paper, where the optimized blade geometry is determined by the classical Glauert theory applicable to the design of axial flow turbines (hydrokinetic and wind turbines). The aim of the present study is to evaluate the optimized geometry for propeller hydrokinetic turbines, observing the effect of the number of blades in the runner design. The performance of runners with different number of blades is evaluated in a specific low-rotational-speed operating conditions, using blade element momentum theory (BEMT) simulations, confirmed by measurements in wind tunnel experiments for small-scale turbine models. The optimum design values of the power coefficient, in the operating tip speed ratio, for two-, three-and four-blade runners are pointed out, defining the best configuration for a propeller 10 kW hydrokinetic machine.

Introduction

Hydrokinetic energy conversion devices have become an important choice of renewable energy resource in many worldwide regions. Applying different concepts of properly designed hydro turbines (e.g., Laws and Epps [1]), the kinetic energy of natural water streams (marine or river) can be converted into electricity for the supply of ocean, coastal or riverine societal needs, with low environmental impact and as a sustainable energy resource.

This technology has been reviewed by recent important number of works [1][2][START_REF] Kumar | A review on the technology, performance, design optimization, reliability, techno-economics and environmental impacts of hydrokinetic energy conversion systems[END_REF][START_REF] Yuce | Hydrokinetic energy conversion systems: A technology status review[END_REF] in which the emergence of the technology has been pointed out. One has reported several possibilities of technology applications (tidal and riverine), different machine design concepts and pre-commercial or R & D experiences, in the framework of the development of an important number of innovative hydrokinetic turbine converters.

Yuce and Muratoglu [START_REF] Yuce | Hydrokinetic energy conversion systems: A technology status review[END_REF] had reported in 2015 about 300 relevant international project initiatives in hydrokinetic energy technology. Among them, 60% of the turbine designs are based in horizontal axis conception, mainly in the case of tidal applications. The hydrodynamical design methods for horizontal axis free flow hydrokinetic turbines are similar to those employed in wind turbines or to the propulsion propeller turbomachines. The design approaches can be adapted for the specific situations of hydrokinetic turbines, inheriting the important scientific rationale of the blade design of wind turbines [START_REF] Rehman | Horizontal axis wind turbine blade design methodologies for efficiency enhancement-a review[END_REF][START_REF] Schubel | Wind turbine blade design[END_REF].

In general, the classical design approach for the optimized free flow runners is based on the Glauert blade element momentum theories [START_REF] Glauert | Airplanes propellers[END_REF]. It is a properly consolidated approach, presented in several wind turbine aerodynamics textbooks (e.g., Burton et. al. [START_REF] Burton | Wind energy handbook[END_REF], Hansen [9] or Schaffarczyk [START_REF] Schaffarczyk | Introduction to wind turbine aerodynamics[END_REF]).

Important improvements in BEMT blade design approaches for wind turbines [START_REF] Liu | Development and application of an improved blade element momentum method model on horizontal axis wind turbines[END_REF][START_REF] Buhl | A new empirical relationship between thrust coefficient and induction factor for the turbulent windmill state[END_REF], propellers [START_REF] Benini | Significance of blade element theory in performance prediction of marine propellers[END_REF] and hydrokinetic devices [START_REF] Chapman | The Buhl correction factor applied to high induction conditions for tidal stream turbines[END_REF][START_REF] Goundar | Design of a horizontal axis tidal current turbine[END_REF][START_REF] Li | Review on the blade design technologies of tidal current turbine[END_REF][START_REF] Batten | Experimentally validated numerical method for the hydrodynamic design of horizontal axis tidal turbines[END_REF][START_REF] Silva | Analysis of cavitation for the optimized design of hydrokinetic turbines using BEM[END_REF] have been explored in the literature, and some relevant ad hoc corrections associated with complex physical flow features (wake hydrodynamics, tip 3D effects, cavitation, etc.) have been proposed and integrated in BEMT approach. One has promoted robustness and accuracy in the methodology that may be associated with other advanced fluid mechanics methodologies such as CFD (computational fluid dynamics) and vortex methods, in a complementary framework for the blade hydrokinetic turbine design and its hydrodynamical assessment.

The design of the runners of horizontal axis hydrokinetic turbines allows an optimized geometry of their blade surfaces (here generated by the airfoil section piling in the blade axis), for a chosen number of blades. For larger machines (wind or tidal turbines), runners with three blades have been consolidated as the design selection, taking into account not only the hydrodynamics, but also others interdisciplinary aspects such as costs, weight and dynamics stability. Some small machines (medium and small size) have been proposed with a number of blades greater than three (see the review of Laws and Epps [1], for instance). The optimum number of blades for the hydrokinetic turbine runners is the basic question of the present paper.

The discussion of the number of blades in hydrokinetic or wind turbine is not too explored. Only a few references discussing the hydrodynamical effects on the choice of the number of blades are reported in the literature. Duquette et. al. [START_REF] Duquette | Solidity and blade number effects on a fixed pitch, 50 W horizontal axis wind turbine[END_REF] discussed the effect of the number of blades and the runner solidity for small wind turbines. This work concludes that the optimum design three-blade rotors produced an increase in the experimental power coefficient-Cp as solidity increased, with reduced tip speed ratio (TSR) at the optimum operating point. As blade number was increased at a constant solidity, the aerodynamic efficiency and power sharply decreased, contrary to the classical suggestion of Glauert [START_REF] Glauert | Airplanes propellers[END_REF] that pointed out the increase in Cp with the number of blades. In fact, the experimental work of Duquette does not explore properly the best optimum design for the given number of blades. Duquette et. al. [START_REF] Duquette | Solidity and blade number effects on a fixed pitch, 50 W horizontal axis wind turbine[END_REF] just change the number of blades in the rotor without making modification in the blade design. In the Glauert theory, the solidity has to be changed for the optimum design geometry, for each different number of blades.

In the work of Badshah et. al. [START_REF] Badshah | Hydrodynamical design of tidal current turbine and the effect of solidity on performance[END_REF], for tidal turbines with two-four blades, the experimental and BEMT results showed that the maximum value for nominal Cp remains the same to the different number of blades. Otherwise, a change in the higher TSR design point for rotors with more blades is observed and their torque is considerably higher for low TSR.

The wind tunnel results of Muhle et. al. [START_REF] Muhle | The effect of the number of blades on wind turbine wake a comparison between 2-and 3-bladed rotors[END_REF] for wind turbines with two and three blades present the same behavior. The performances of both runners are quite the same, with a slight reduction in Cp for two-blade runner. The wake influence region did not change with the number of blades.

Previous results of BEMT and CFD computations of propeller hydrokinetic turbines with two-four blades were reported by Brasil Junior et. al. [START_REF] Junior | Hydrokinetic propeller turbines. How many blades?[END_REF]. A small increase in Cp is observed for runners with the higher number of blades, changing the operation of low-TSR point. The wake influence region is not enlarged, but a considerable difference in the wake vortex pattern was observed, which influences the time-dependent flow behind the machine in different characteristics of frequencies.

The present paper explores the discussion of the effect of the number of blades in hydrokinetic turbines, using new wind tunnel experimental results, complemented to numerical simulations with BEMT approach. The revisiting of this problem, only in a hydrodynamical point of view, can promote the best design configuration of hydrokinetic turbines.

This work is organized as follows: In Sect. 2, the model of propeller hydrokinetic turbine is presented. Section 3 presents the design approach using the Glauert theory and BEMT formulations. In these sections, complete presentation of the classical method is developed in order to promote the completeness of the structure of the paper. In Sect. 4, the experimental methodology is presented, which is based on wind tunnel experiments. At last, Sect. 5 explores the results and their discussion.

Hydrokinetic propeller turbine HK-10

The present study is based on a propeller hydrokinetic turbines called HK-10 presented in Fig. 1. This machine, with 2 m diameter, has been designed to reach a rated power of 10 kW at a water stream velocity of 2.5 m/s. The Energy and Environmental Laboratory, at University of Brasilia, had held the hydrodynamic and mechanical design of the machine in collaboration with a group of R&D partners. The main purpose of the proposed kinetic energy converter system is to obtain a sustainable technology to provide energy for small remote communities or for the recovery of the remaining energy potential at downstream of hydroelectric power plants. The modular and floating aspects of this system permit the adaptation of a group of turbines which work together, taking into account the local requirements of the power demand.

The hydrodynamical design parameters for this machine are presented in Table 1. The hydrodynamical design approach is applied considering those target variables. The analysis of the runner design with a different number of blades will be explored in the context of the optimized blade design approach. Two-, three-and four-blade runners will be considered in the present study (see Fig. 2) 3 Blade element momentum theory

Runner kinematics

The performance of a hydrokinetic turbine is expressed by the power coefficient, which is parameterized by its dependence to the tip speed ratio, denoted, respectively, by Cp and classically defined as

(1)

Cp ≡ P 1∕2 AV 3 0 ,
In those equations, P, A and R denote, respectively, the extract hydrodynamical power, area and radius of the runner and is the water density. The notation TSR (tip speed ratio) is also used alternatively for . The variables V 0 and are, respectively, the water flow velocity and the runner angular speed.

A parameterization of the axial velocity on the rotor cross-flow plane is written as Additionally, on the outflow plane, the tangential velocity in the rotating wake in opposite to the rotor rotation can be expressed by a model of free vortex written as Equations 3 and 4 introduce the axial and tangential induction factors denoted by a and a ′ , respectively. Those vari- ables are dependent on the radial position r.

In the rotor middle plane and at the radial position r, the velocity triangle on the blade section can be determined as illustrated in Fig. 3. The main velocity triangle is

(2) ≡ R V 0 .
( composed by the axial velocity (1a)V 0 and the tangential component. The tangential component is equivalent to the blade velocity r added to the induced free vortex a ′ r , which turns in the opposite direction. Thus, the inflow angle can be computed by with From the second small triangle, as illustrated in Fig. 3b, an additional relation yields

) V = (1 -a)V 0 . (4) V = 2 ra � . 3
From equations 5 and 7, a kinematic relationship between a and a ′ can be obtained as or

Axial and angular momentum balance

The axial and angular momentum conservation principles can be formulated in the inflow and outflow sections through the runner. The integral 1D balance in an annular section of radius dr gives the expressions for the thrust dT 1 and torque dM 1 differentials, which are written as and The hydrodynamical extract power in dr can be computed from Eq. 11 as or in a dimensionless form Therefore, the power coefficient for entire runner can be computed by integrating this equation in all its radius range given

(5) tan = (1 -a)V 0 (1 + a � ) r = (1 -a) (1 + a � ) 1 r , (6) r
≡ r V 0 . (7) tan = a � r V 0 a = a � a r . ( 8
) a � (1 + a � ) 2 r = a(1 -a) (9) a � = 1 2 √ 1 + 4 -2 r a(1 -a) -1 . ( 10
) dT 1 = 4 r V 2 0 a(1 -a)dr, ( 11
) dM 1 = 4 r 3 V 0 (1 -a)a � dr. (12) dP = dM = 4 r 3 V 0 2 (1 -a)a � dr, ( 13
) dCp = 8 2 (1 -a)a � 3 r d r . ( 14
) Cp = 8 2 ∫ 0 (1 -a)a � 3 r d r .
In Eq. 14, a direct dependence of the runner performance to the radial distribution of the pair a, a ′ is figured out. Therefore, the optimization of the performance, which lies in the maximization of the Cp, is directed related to finding the maximum value of the term f (a, a � ) ≡ (1a)a � .

Considering the relationship between a ′ and a, given in Eq. 9, the optimization problem related to the search of the maximum condition for f (a, a � ) can be simplified to a one- dimensional problem, only dependent on the variable a. It is the aim of the Glauert theory.

Optimum flow conditions for runner design

Considering the turbine nominal design variables given by , V 0 , R , which defines the values of , and for each radial position r . The maximum value of dCp, related to the maximum condition of f (a, a � ) , is simply obtained by the deriva- tive of the function f to a, i.e., Using this equation with the derivative of Eq. 9, a cubic polynomial expression can be obtained as Therefore, for a blade radial position expressed by r , the unique real root greater than zero of Eq. 16 defines the optimum flow condition for the runner. The equivalent value for the radial induction factor can also be obtained as Using this reasoning, for a given value of r , equations 16 and 17 can be used to obtain the values of a, a ′ . Thus, Eq. 5 can be employed complementary to obtain the value of the inflow angle .

The analytical solution to find the root for the Glauert polynomial in Eq. 16 can be simply implemented using (e.g., [START_REF] Maalawi | A direct method for evaluating performance of horizontal axis wind turbines[END_REF]) with and Considering the Glauert approach to determining optimum flow conditions for the runner, Algorithm A1 can be proposed. ( 15)

df da = (1 -a) da � da -a � = 0. (16) 16a 3 -12a 2 + 3(3 -2 r )a -(1 -2 r ) = 0. ( 17
) a � = (1 -3a) (4a -1) . ( 18
) a = 1 2 1 -r cos + -cos -, (19) r
= √ (1 + r ) (20) ± = 1 3 cos -1 (± -1 r ).
Algorithm A1: Optimized flow conditions 1. Given the design parameters , V 0 , R ; 2. Split the blade in N sec , with radial positions r i ;i = 1 ∶ N sec ; (20 to 50 sections are good) 3. Compute blade sections speed ratios: = R∕V 0 and r,i = r i ∕V 0 ;i = 1 ∶ N sec ; 4. Compute the axial induction factor for each blade section a i ;i = 1 ∶ N sec , using Eq. 18; 5. Compute the tangential induction factor for each blade section a � i ;i = 1 ∶ N sec , using Eq. 17; 6. Compute the inflow angles for each blade section i ;i = 1 ∶ N sec , using Eq. 5; 7. Compute the estimate of C p for ideal flow conditions, using numerical integration of Eq. 14.

Hydrodynamical forces on blade sections

The second component of BEMT is the estimate of the hydrodynamical forces on each blade section. It is obtained computing the lift and drag forces on airfoil sections which composes the blades, using the standard airfoil theory and data.

In the BEMT, the computation of the hydrodynamic forces in a blade element dr is obtained by the properly vectorial projection of the drag and lift on the axial and tangential directions. Thus, for a rotor with N b blades, the thrust and the moment on blade sections can be computed by and In those equations, C denotes the chord length on the blade section and C n and C t are defined by and At this moment, the lift and drag coefficients of the foil sections are introduced, where C L = C L ( ) and C D = C D ( ) are available from the data for standard airfoils. Here, denotes the angle of attack of the flow relative on the blade section (see Fig. 3b). The NACA four-digit foils are used in the present work, and their polar curves are obtained from the open-source code XFOIL [START_REF] Drela | A two-dimensional viscous aerodynamic design and analysis code[END_REF].

In order to compute hydrodynamical losses in the framework of complex fluid flow phenomena in the rotating blades (e.g., tip and hub losses), a F(r, N b ) correction function is

(21) dT 2 = 1 2 N b C V 2 0 (1 -a) 2 sin 2 C n dr (22) dM 2 = 1 2 N b C V 0 (1 -a) r(1 + a � ) sin cos C t rdr. ( 23
) C n = C L cos + C D sin (24) C t = C L sin -C D cos .
introduced. In the present work, Prandtl's tip and hub losses corrections are considered, changing equations 10 and 11 to and with and Using those two corrections functions, only acting, respectively, on the blade sections close to the tip and close to the hub, one can write Remark 1 For the blade design phase, an optimum angle of attack has to be selected in the airfoil data. This angle, denoted by op , is obtained from the airfoil polar data where the ratio C L ∕C D is maximum.

Remark 2 An estimate of the chord Reynolds number has to be considered, in order to use XFOIL. It can be obtained iteratively from all steps of geometry design procedures. In general, the airfoil polars are not too sensitive to the close values of Reynolds number and a first estimate of Reynolds can be considered from the values of the chord dimensions of existing similar machines.

Remark 3

For runners with high solidity, defined here by a correction considering the effects of cascade flow has to be taken into account. It was introduced in the design of propeller hydrokinetic turbines, only with a greater number of blades N b ≥ 5 and 𝜎 -1 r < 3 , when the effect of the proximity of the blades has to be included in the formulation (see [START_REF] Junior | Hydrokinetic propeller turbines. How many blades?[END_REF] or [START_REF] Mesquita | An improved approach for performance prediction of HAWT using the strip theory[END_REF] ).

Optimized blade geometry

From a design point of view, Algorithm A1 provides the values of the induction factors (a and a ′ ) and the inflow angle , which establishes the optimized flow conditions for each radial position. The runner design has to be complemented with the definition of the blade shape, expressed [START_REF] Mesquita | An improved approach for performance prediction of HAWT using the strip theory[END_REF]

dT 1 = 4 r V 2 0 a(1 -a)F(r, N b )dr (26) dM 1 = 4 r 3 V 0 (1 -a)a � F(r, N b )dr, (27) F tip = 2 cos -1 exp - N b 2 (R -r) r sin (28) F hub = 2 cos -1 exp - N b 2 (r -R hub ) r sin . (29) F(r, N b ) = F tip F hub . (30) r ≡ N b C 2 nr ,
by the chord C and twist angle for different sections distributed through its length, i.e., C i , i ;i = 1 ∶ N sec . From equations 25 and 21, with dT 1 = dT 2 , a closed expression for the chord length distribution C(r) can be obtained At last, the blade twist angle can be determined from the optimum foil angle of attack (see Fig. 3b), using Therefore, the algorithm for the blade design can be proposed Algorithm A2: Optimized blade geometry 1. Given the design parameters , V 0 , R, N b ; 2. Given optimum flow conditions from Algorithm A1 a i , a � i , i ;i = 1 ∶ N sec ; 3. Using XFOIL, obtain op from maximum condition of C L ∕C D , and C L ( op ) and C D ( op );

4. Compute chord lengths using Eq. 31

C i ;i = 1 ∶ N sec ;
5. Compute blade twist angle using Eq. 32 i ;i = 1 ∶ N sec ;

Performance analysis using BEMT

The formulation presented in the two previous subsections was implemented by means of a MATLAB© code using Algorithms A1 and A2. It permits the complete shape design of the blade geometry for axial flow hydrokinetic turbines.

The runners designed by this approach will be evaluated with two different methodologies: numerical simulations using BEMT and small-scale wind tunnel experiments.

The BEMT formulation can also be used for performance analysis, permitting numerical simulations of the behavior of the turbine performance, in a large range of TSR. The second approach used in the present paper will explore the experimental measurements in wind tunnel experiments. Hence, a good validation of the estimates of machine performance can be assessed, obtaining the performance curves ( Cp× TSR), torque and thrust characteristics.

In performance simulations, two additional equations can be obtained using dT 1 = dT 2 and dM 1 = dM 2 . Thus, from Eq. 25 equal to Eq. 21 and Eq. 26 equal to Eq. 22 the following relations can be obtained and ( 31)

C(r) R = 8 N b a (1 -a) F(r, N b ) sin 2 C n r R . (32) = -op . (33) a 1 -a = r C n 4F sin 2 (34) a � 1 + a � = r C t 4F sin cos .
Or alternatively and An iterative approach can be employed to compute the values of {a, a � } , using the nonlinear problem formulated by Eq. 35 and Eq. 36, for all blade radial positions and for any machine operating point, given by (for details, see Hansen [9], for instance). Therefore, the power coefficient can be computed by the numerical integration of Eq. 14 , correcting the losses formulating by Eq. 26, i.e.,

In this equation,

h ≡ R h ∕V 0 .
The method has to be complemented by an important condition when the values of the axial induction factors surpass 0.4. In this situation, the formulation of the BEMT is not valid due to the complex and important 3D effects in the runner flow (Liu and Janajreh [START_REF] Liu | Development and application of an improved blade element momentum method model on horizontal axis wind turbines[END_REF], for instance). In this situation, if C T ≥ 0.96 , an empirical condition for the axial induction factor has to be formulated as This correction was properly formulated by Buhl [START_REF] Buhl | A new empirical relationship between thrust coefficient and induction factor for the turbulent windmill state[END_REF], using the basic ideas of Glauert and an alternative curve fit of experimental observations for the thrust coefficient, C T , which is defined here by and can be computed in the BEMT approach, for a specific radial section using In the present paper, for the BEMT computations, the opensource numerical tool QBLADE is employed (Marten et. al. [26]). In this code, the BEMT equations are implemented in a robust and friendly simulation platform, equipped with a XFOIL interface for the polar curve computations. This numerical tool permits the performance analysis in the entire range of TSR, considering the corrections for tip and hub

(35) a = 4F sin 2 r C n + 1 -1 (36) a � = 4F sin cos r C t -1 -1 . ( 37
) Cp = 8 2 ∫ h F(r, N b )(1 -a)a � 3 r d r . (38) a = � 18F -20 -3 √ C T (50 -36F) + 12F(3F -4) � (36F -50) . ( 39
) C T ≡ T 1∕2 AV 2 0 , (40) dC T = 4Fa(1 -a).
losses, as well as the Buhl-Glauert correction for high values of axial induction factor. All designed blade geometries have been considered for the BEMT simulations using properly QBLADE setting parameters.

Hydrodynamical losses

The hydrodynamical process to extract energy from the flow on the ideal condition is expressed by the dimensionless integral in Eq. 14. The maximum amount that it could have occurred is given by the Betz limit ( 59.26% ). Glauert [7] has shown that for low-TSR regimes, this limit is more restrictive, due to the rotational effects in the wake flow.

Considering the amount of energy which can be converted mechanically by the runner, one part of its power is consumed by the drag on the turbine blades. Another part is consumed by other complex flow phenomena associated with the 3D effects in the flow through the runner (mainly in tip and hub regions).

An approximated formulation can be obtained to express mathematically those effects. Firstly, Eq. 24 is simplified neglecting the drag, i.e., Substituting this approximation in Eq. 34, the following relation can be obtained Substituting it in Eq. 22 and integrating on the entire blade, the following expression can be obtained

The different components in this equation quantify the relative losses compared to the ideal flow (given by Eq. 14)

-Drag losses Given by 1 -

C D C L cot , it explicit depends on C D ∕C L ratio.
-Tip and hub losses Estimated by the Prandtl term F(r, N b ) , it has to be integrated in entire blade length.

Some remarks can be pointed out:

Remark [START_REF] Yuce | Hydrokinetic energy conversion systems: A technology status review[END_REF] The design procedure that has selected the best foil standard geometries op , considering a maximum value of C L ∕C D , allows the better choice to minimize the drag losses, as is shown in Eq. 43.

Remark 5

The term F(r, N b ) is the only part in the formula- tion which accounts for the number of blades. This term (41)

C t ≈ C L sin . (42) (1 + a � ) ≈ a � 8 rF N b C cos C L . (43) Cp ≈ 8 2 ∫ h F(r, N b )(1 -a)a � 1 - C D C L cot 3 r d r .
tends to one when N b goes to a higher value (ideal condition related to an infinite number of blades).

The formulation presented in this section allows a complete theoretical modeling approach to design and assess the energy conversion on the hydrokinetic turbines. Simply numerical tools can be developed, in the framework of the evaluation of the number of blades effects in these machines performance, which is closely related to the influence of the correction factor F(r, Nb).

Experimental setup and procedures

In this section, the experimental methodology is presented, which is based on small-scale tests in the wind tunnel facility. The proposed approach was developed as an alternative of experimental performance evaluation of hydrokinetic turbines, generally based on experiments in flumes (e.g., [START_REF] Kolekar | Performance characterization and placement of a marine hydrokinetic turbine in a tidal channel under boundary proximity and blockage effects[END_REF]).

Wind tunnel experiments, complemented by a modelto-prototype transposition approach, are employed in the present study. The results for the small-scale experiments in airflow can be extrapolated to the water flow on the prototype scale in water. It is obtained by a calibrated methodology using also BEMT arguments. It has been validated by means of CFD numerical simulations in both scaling levels [START_REF] Machuca | Wind tunnel experiments of free flow turbineshydrokinetic and wind turbines[END_REF].

Wind tunnel facilities and turbine small-scale models

The experiments were performed in the open-loop wind tunnel of the Laboratory of Energy and Environment of University of Brasilia. With a test section of 1.2 m × 1.2 m with 2 m long and a contraction ratio of 8:1, the 10 kW wind tunnel can reach a controlled air speed up to 20 m/s, with turbulence intensity less than 0.2% . Figure 4 presents a general overview of the test facility. Inside the wind tunnel, a pitot tube is used to measure the free-stream speed. The air temperature and relative humidity are monitored using calibrated electronic sensors. The experimental uncertainty of velocity measurements is ±0.1 m/s.

A small-scale one-tenth turbine model was installed inside the wind tunnel test section as shown in Fig. 5. The different runner models are shown in Fig. 6. Those runners are manufactured in an ABL 3D printer system and finished by epoxy paint. Its dimensions are checked after the manufacturing process ensuring the variations between the 3D CAD design and the final model keep under 0.1 mm.

Control, torque and rotation measurements

A small brushless permanent magnet generator is used as a brake inside the nacelle of the turbine model. It maintains the turbine rotor operation at a constant rotational speed. The control was based on an electronic-resistive load circuit, as illustrated in Fig. 7, where it is possible to set the rotational speed changing the switch time by means of a PWM-PID controller. The closed-loop control system achieves a stable rotation for given wind speed, using different settings for the load levels on the generator. It is completely controlled by an Arduino© code.

The torque, T, was measured using a torsional load cell, as illustrated in Fig. 8. The turbine model was constructed to permit only one inertia referential support for the generator. By this way, all the generator's torque is transmitted directly to the torsional cylinder, where four strain gauges, of 120 ohms each, were fixed, integrating a complete Wheatstone bridge. Any torque variation in generator shaft causes a deformation in the load cell and in the strain gauges, producing an unbalancing in the bridge. The voltage drops are related to torques acting in the shaft by a static calibration procedure. As the rotor is connected to the generator, the assembly allows to measure rotor torque. The torque meter system has an accuracy of ±10 -4 N ⋅ m.

The operation of the turbine model maintains the rotational speed, N, constant in a fixed value within ±10 rpm variations in a range of 300-2500 rpm (measured by an optical infrared rotation sensor, with accuracy of 0.1 rpm).

Therefore, the power converted in the turbine is calculated by (44) P = T , = N 30 .

Experimental procedure

The experimental runs aim to produce curves of power coefficient as a function of the tip speed ratio, Cp× TSR. In this work, TSR variations are generated by keeping the freestream velocity V 0 constant and varying the rotation speed, , in such way that the Reynolds number based on the averaged chord length is kept constant.

For performance evaluation, the experimental runs start by setting the free flow speed at a fixed value (12 m/s is used). The PWM control system was programmed to sweep a pre-settled range of runner rotational speed (300-2500 rpm), by changing automatically the electrical load. The experimental control system varies the rotation for entire range of the turbine TSR (0.25-2.5). The data logger registers the turbine torque for each situation. By this way, an entire Cp× TSR curve is obtained. All experimental proce- dure is completely controlled by a MATLAB© interface and data logging code. All results were obtained for the three different configurations of runners, and the error report is presented in "Appendix A."

Results and discussion

Optimized blade geometry

The design methodology, presented in Sect. 3 and summarized in Algorithms A1 and A2, was implemented in a MATLAB© code. It allows us to compute the blade shape for a given set of input parameters {V 0 , , R, N b } . The com- putation of the blade radius is performed considering the design power-extract target in the machine requirements. An iterative approach can be used, computing successively the power coefficient and the nominal tip speed ratio, up to the convergence to the selected value of the power.

In the present paper, the blade design employs the foil NACA4415, with its polar curves obtained from the software XFOIL, with a typical Reynolds number around 2.0e+06. Only the value of op and the evaluation of C L ( op ) and C D ( op ) are needed as input data in Algorithm A2.

In Fig. 9, the curve of Cp×TSR for an infinite number of blades and for ideal flow conditions is illustrated, highlighting the values of three different design conditions for runner machines in terms of TSR (1.5, 2.0 and 4.0). Those choices represent different specific characteristics for slow and fast rotational speed machines analyzed here. The values of Cp in this plot do not take into account realistically the hydrodynamical losses through the runner. It is only computed from the optimum flow conditions of {a, a � } , issued from Algorithm A1 and by means of the integral in Eq. 14. It can be considered as a limit curve for the behavior of the designed runner, which replaces the Betz limit, in particular for low-TSR regimes (classical Glauert curve).

In Fig. 10, losses are taken into account as formulated in Eq. 43. The flow continues to be ideal in the sense of the Glauert polynomial (i.e., the best conditions for a and a ′ ). Otherwise, the terms of drag losses and tip/hub losses are considered. This last term influences the reduction in Cp values for runners with a low number of blades, corroborating with many theoretical, numerical and experimental observations (e.g., [START_REF] Badshah | Hydrodynamical design of tidal current turbine and the effect of solidity on performance[END_REF][START_REF] Junior | Hydrokinetic propeller turbines. How many blades?[END_REF]).

In Table 2, the values of Cp in different TSR conditions are presented. Those values consider the losses in a optimum flow condition for {a, a � } using Eq. 43.

For low-speed machines, an important difference in the Cp levels is observed for the different number of blades. In the runner of two blades, the lowest level of performance induces the use of design with runners with three or more blades. This difference is not too important for faster (or larger) machines, with greater TSRs. For this condition, runners with two or three blades can be designed, as suggested by Muhle et. al. [START_REF] Muhle | The effect of the number of blades on wind turbine wake a comparison between 2-and 3-bladed rotors[END_REF] for wind turbines, without a relevant reduction in the extract power performance. In Fig. 11, a general overview of the optimized shapes for different design TSRs and number of blades is illustrated. This result shows that thinner shapes are obtained for faster machines.

The optimized runner shapes, for the design conditions of HK-10 machine, as presented in Table 1, are close to TSR=1.5. The 3D runners for this condition are presented in Fig. 2. It is a slow machine, in which the blades are wider and the maximum power coefficient is limited around 0.4. Unfortunately, other design requirements for that machinerelated to the mechanical and electrical components in the powertrain-restrain the rotational speed of the turbine at a lower level. It has a constraint on the power coefficient up to 0.4; otherwise, it gives reliability for a machine operating in lower rotation speeds.

The volume of the designed runners can be calculated from an integral of chord length distribution, using the standard foil geometry. Its values (expressed in dimensionless form) for a different number of blades up to 5 are shown in Fig. 12. It can be observed a reduction in the volume of the runner, which diminishes with the number of blades. This reduction is more accentuated for low-speed runners and it is very slight for faster runners-with higher values of nominal TSR. For large hydrokinetic turbines (with high TSR), the use of two-or three-blade machines can be eventually justified in terms of the blade material weighing, if the performance can be remained the same (as actually is discussed on the use of two-blade machines for offshore wind turbines [START_REF] Muhle | The effect of the number of blades on wind turbine wake a comparison between 2-and 3-bladed rotors[END_REF]).

Hydrodynamical performance

The experimental and BEMT results were obtained for the runners with two, three and four blades. Figure 13 presents the power coefficient curves for the different runner configurations, a direct extract from the wind tunnel measurements.

The experimental error bars are figured out. The slight difference in the extract power performance for the three-and four-blade runners can be observed. It is not the case for the two-blade runner, where the levels of the C p curve are clearly lower. The observed experimental uncertainties are compatible with the curves behaviors and other equivalent studies (e.g., [START_REF] Kolekar | Performance characterization and placement of a marine hydrokinetic turbine in a tidal channel under boundary proximity and blockage effects[END_REF]). The error bars are greater near the stall points on the blades, where the dynamical effects on the torque can be observed and the variances in the torque measurements are higher.

In Fig. 14, the extrapolation of the experimental results to the prototype scale is considered. This transposition of the results is performed using a Reynolds number correction, as suggested by [START_REF] Machuca | Wind tunnel experiments of free flow turbineshydrokinetic and wind turbines[END_REF] using the parameters presented in Table 3. The reference radial position r ref = R + R h ∕2 is employed to obtain the Reynolds number for the transposition formula, with its respectively chord length. The experimental results are compared to the numerical simulations of QBLADE, for the different number of blades and for the Reynolds number in the prototype scale.

The adherence of the results is quite well, and it is similar to other reported works involving comparisons between BEMT and wind tunnel experiments-in particular with the use of QBLADE (see Monteiro et. al. [START_REF] Monteiro | Wind tunnel testing of a horizontal axis wind turbine rotor and comparison with simulations from two blade element momentum codes[END_REF] for instance) .

Near the design point ( TSR = 1.6 ), the experimental and numerical results are very close, which assures the achievement of the blade design methodology. One can observe that the runner with four blades has the higher power coefficient, reaching the maximum value of 0.39 at the target TSR (11% higher than the three-blade runner), even though the torque coefficient for runners with N b = 3, 4 is very close (consid- ering the experimental results), as shown in Fig. 16. On the other hand, the results for two-blade runner have Cp close to 0.3, poorest than the behavior of the other two runners. A small displacement in the TSR design point to higher values is observed for runners with a smaller number of blades. It is more clearly observed in the BEMT results.

In order to refine the best results for the four-blade runner, new sets of experiments were performed, obtaining more points in the TSR range. The results are presented in Fig. 15 where the CFD simulations performed in the work of Brasil Junior et. al. [START_REF] Junior | Hydrokinetic propeller turbines. How many blades?[END_REF], for the same runner, are included. The good coherence of all methodologies for the runner performance analysis is observed, assuring a reliable validation of the runner design.

Torque coefficient results

The evaluation of the torque the machine is crucial information to determine the mechanical project to match the runner operation to the electrical generator, selecting also a compatible gearbox and all mechanical requirements for the main shaft.

The torque coefficient is defined by

The values of M are directly measured in the turbine model load cell, and in a numerical point of view and in the framework of BEMT, it is computed by the integral of Eq. 26 in all blade sections. In Fig. 16, the torque coefficient results are plotted as a function of TSR. Both results, experimental and numerical, have shown a typical curve of C M , where the high values are (46)

C M ≡ M 1∕2 AV 2 0 R
. encountered in the regions of low TSRs. It is good in terms of the turbine operation, mainly for the starting condition of the rotating electromechanical set. A needed torque level is required for the low rotation regimes, in order to compensate the friction on the shaft components and powertrain. An important remark is that for N b = 3, 4 , a high torque condi- tion is achieved allowing turbines starting even for relatively slow current velocities, which is necessary to the specific requirements of the HK-10 machine.

Conclusions

A design approach for hydrokinetic axial flow turbines is presented. It was formulated in detail from the basic equations of the blade element moment theory, using the Glauert [START_REF] Glauert | Airplanes propellers[END_REF] reasoning for the optimum blade design. In this framework, axial runners for low-rotational-speed propeller hydrokinetic turbine are designed, considering two, three and four blades. The optimum blade shapes have been obtained by the optimized approach described by the cited theories. All runners have firstly evaluated in a qualitative point of view regarding its amount of materials, the feasibility in terms of manufacturing and the limits of the extracted power in the framework of Glauert theory, considering the losses formulations.

The performance of the runners is assessed by means of numerical simulations using BEMT and by means of wind tunnel experiments. In both methodologies, the results have the same behavior and are adherents in terms of tendencies and levels. Those approaches have also been compared to the CFD computations for the four-blade runner, and the agreement of the results allows a good validation or the machine performance, in terms of power coefficient around 0.4 (for the four-blade runner). The results showed a tendency of the reduction in Cp for runners with two and three blades, comparatively to the runners with more blades. The torque coefficient results show the same behavior, and an important level of values of C M is observed in the region of low TSRs, in particular for the runners with three and four blades. Those results agree with the classical observations of Glauert and other authors [START_REF] Liu | Development and application of an improved blade element momentum method model on horizontal axis wind turbines[END_REF][START_REF] Muhle | The effect of the number of blades on wind turbine wake a comparison between 2-and 3-bladed rotors[END_REF], more recently.

The results allow good results for the runner design of the turbine HK-10, taking into account its specific characteristics of the low-TSR design point. Considering this requirement, runners with more than three blades are more adequate for the machine performance and also for the torque characteristics, needed in the starting operating regime.

Fig. 1 Fig. 2 Fig. 3

 123 Fig. 1 HK10 propeller hydrokinetic turbine with four-blade runner (10 kW@2.5 m/s) and floater

Fig. 4 Fig. 5 Fig. 6 PWMFig. 7

 4567 Fig. 4 Experimental setup

Fig. 8 Fig. 9

 89 Fig. 8 Torsional load cell parts. 1: generator shaft, 2: generator support, 3: torsional cylinder, 4: fixation plate, 5: nacelle shell, 6: strain gauge

Fig. 10

 10 Fig. 10 Optimum power coefficient-considering losses

5 Fig. 12

 512 Fig. 11 Optimized blade geometries

Fig. 13

 13 Fig.[START_REF] Benini | Significance of blade element theory in performance prediction of marine propellers[END_REF] Power coefficient for wind tunnel experiments-without model-to-prototype extrapolation

Fig. 15 Fig. 16

 1516 Fig.[START_REF] Goundar | Design of a horizontal axis tidal current turbine[END_REF] Power coefficient for four-blade runner. BEMT simulations (-), experiments of Fig.12(o) and CFD simulations[START_REF] Junior | Hydrokinetic propeller turbines. How many blades?[END_REF] (--)

Table 1

 1 Design parameters

	Water flow velocity	2.5 m/s
	Rotational speed	35 rpm
	Rated power	10 kW
	Powertrain-generator efficiency	0.9
	Power coefficient	0.38-0.4
	TSR op	1.4-2.0
	Number of blades	2-4

Table 2

 2 Cp for ideal flow conditions at selected TSR points (with drag and tip/hub losses)

	Number of Blades TSR=1.5	TSR=2.0	TSR=4.0
	2	0.36	0.41	0.50
	3	0.39	0.45	0.51
	4	0.41	0.46	0.52
	∞	0.44	0.48	0.53

Table 3

 3 Model-to-prototype transposition parameters

	Number of blades	Parameter	Value
	2	Chord length (m)	0.856
		Re prot	2.86e-06
		Re model	2.86e-04
	3	Chord length (m)	0.59
		Re prot	1.97e-06
		Re model	5.88e-04
	4	Chord length (m)	0.41
		Re prot	1.36e-06
		Re model	4.08e-04

Fig.

[START_REF] Chapman | The Buhl correction factor applied to high induction conditions for tidal stream turbines[END_REF]

Power coefficient for BEMT simulations and experimentswith scaling up extrapolation. The error propagation procedure is presented in "Appendix A"

Acknowledgements This work is partially financed by the HYDRO-K project consortium in a context of the ANEEL P&D Grant, with the partnership of AES-Brasil Company. The authors are grateful to the support of Brazilian Ministry of Education, by means of CAPES, for the PhD scholarships. The France-Brazil cooperation program lying ENSAM-ParisTech and UnB has maintained the international mobility of the researchers.

Propagation of uncertainties

The experimental uncertainties of this work are given by the sum of instrumental and random errors. For the digital sensors, the instrumental errors are given by the smallest scale value displayed by the instrument and for the analog instruments by the smallest half scale. Random uncertainties were defined as being the standard deviation a set of realizations under the same conditions. In this work, the derived uncertainties can be expressed according to the following primary measurements errors: wind velocity ( V 0 ), rotational speed ( ) and torque ( T ). The uncertainty in Cp measurements ( Cp ) is computed with the general formula for error propa- gation [START_REF] Taylor | An introduction to error analysis[END_REF], as it is given in Eqs. 47, 47 and 48.