Pavel A Dub

Aurélien Béthegnies

Rinaldo Poli
email: poli@lcc-toulouse.fr

Catalyzed Olefin Hydroamination: Effect of Halogen, Amine Basicity, and Olefin on Activity, Regioselectivity, and Catalyst Deactivation

come

Introduction

Hydroamination, the direct addition of an amine N-H bond across an unsaturated C-C bond, is a challenging transformation for which catalytic approaches are attracting much interest in both academia and industry worldwide. 1 Alkenes are the most difficult substrates to hydroaminate, 2 especially for the intermolecular version and when using nonactivated substrates, namely the simple -olefins. [START_REF] Brunet | Catalytic Heterofunctionalization[END_REF] An additional challenge concerns the N-H addition regiochemistry in favour of the anti-Markovnikov product for monosubstituted olefins, [START_REF] Borman | [END_REF] listed as one of the "Ten Challenges for Catalysis". 2 For the homogeneously catalyzed hydroamination of ethylene and other nonactivated olefins, limited success was achieved by using Li, 5 lanthanides, 6,7 Zr, 8 and late transition metals such as Fe, 9 Ru, [10][11][12] Rh, [13][14][15] Ir, 14,15 Pd, 16,17 Ag, 18 Au [19][20][21][22] and notably Pt. 23- 25 Investigations initiated in our team by J.-J. Brunet have shown that the hydroamination of ethylene or 1-hexene by anilines is catalyzed by PtX2 in the presence of n Bu4PX (X = Cl, Br, I) as activator, this being one of the most efficient catalytic systems reported for this challenging transformation. [26][27][28][29] The hydroamination catalytic activity is slightly lower for 1-hexene than of ethylene under the same conditions. The highest productivity (expressed by the turnover number, TON) in the hydroamination of ethylene was obtained with the catalyst PtBr2/nBu4PBr, whereas hydroamination of 1-hexene is most productive with PtBr2/nBu4PI, 29 an observation that has not yet been rationalized. Halide ion effects have also been observed for other types of catalyzed transformations, but poorly understood in most cases. [30][31][32][33] Another interesting observation, made for the hydroamination of ethylene, is that the TON is inversely proportional to the ArNH2 basicity. 26 A greater productivity for less basic amines (anilines, carboxamides and sulfonamides) has also been reported for the intermolecular hydroamination of ethylene catalyzed by other Pt systems, notably PtCl2(C2H4)2 23 and Pt(COD)(OTf)2. 24 Finally, a high Markovnikov selectivity (95%) is observed for the 1-hexene hydroamination. 27 The hydroamination reaction with late transition metal complexes as catalysts can in principle follow an olefin activation or an amine activation mechanism, and the former appears as the most likely option on the basis of experimental evidence for Group 10 metals. 29 The mechanism has been analyzed by a few computational investigations. Restricting the citations to the intermolecular version and to late transition metal catalysts, a first report by Senn et al. addressed the model NH3 addition to ethylene catalyzed by the [MCl(PH3)(C2H4)] z+ complexes of Group 9 (z = 0) and 10 (z = 1) metals. 34 The study addressed only the olefin activation pathway for the group 10 metals and showed that the TOF-determining step is the H-CH2CH2NH2 bond formation by reductive elimination from a Pt IV aminoalkyl hydride intermediate, whereas the NH3 nucleophilic addition to the activated ethylene is thermodynamically and kinetically favourable. More recently, Tsipis and Kefalidis, using the "Pt 0 " model complex Pt(C2H4)(PH3), explored only the amine activation pathway, finding the reaction to be limited by the product reductive elimination step from the Pt II amido alkyl intermediate. 35 Other computational studies (e.g. on gold-catalyzed diene 36 or ethylene 37 hydroamination or palladium-catalyzed vinylarenes hydroamination 38 ) have also explored solely the olefin activation mechanism.

In a recent article, we have reported a detailed DFT analysis of the catalytic cycle for the C2H4/PhNH2 reaction catalyzed by the PtBr2/Br -system. 39 This study indicated that the preferred pathway is the olefin activation, rather than the amine activation. The best pathway, shown schematically in Scheme 1, involves bromide and olefin coordination to give the resting state complex [PtBr3(C2H4)] -(1 Br ). Other off-loop species, namely trans-PtBr2(C2H4)(PhNH2) (2 Br ), cis-PtBr2(C2H4)(PhNH2), trans and cis PtBr2(C2H4)2, and trans-PtBr2(PhNH2)2 (3 Br ) were also found at relatively low energy (in agreement with experimental equilibrium studies), 40 the lowest ones being 2 Br and 3 Br . Scheme 1. Mechanism of the PtBr2/Br -catalyzed hydroamination of C2H2 by aniline. 39 The essential features of the catalytic cycle are also shown as an energetic profile in Figure 1. The aniline nucleophilic addition to 1 Br yields the zwitterionic complex [PtBr3(CH2CH2NH2Ph)] -(4 Br ) through the transition state TS1 Br . Although the nucleophilic addition to other olefin complexes, notably 2 Br The aim of this article is to test the proposed catalytic cycle, by means of additional DFT investigations, to check whether it fits with the other observations highlighted above and to possibly provide their rationalization. Additional catalytic experiments have also been carried out, showing how the observed TON is not only a reflection of the catalytic activity (TOF, related to the ΔG ‡ cycle) but also of catalyst decomposition, which has been traced to the decomposition of the zwitterionic intermediate (4), by reaction with an external bases.

Results and Discussion

As mentioned in the Introduction, the prototypical hydroamination process investigated in our laboratory has been the addition of aniline to ethylene, for which the highest TON were found for the PtX2/X -catalytic system with X = Br. Therefore, the previously reported comprehensive computational study of the catalytic cycle was restricted to this halide and to ethylene as olefin and aniline as amine. 39 The ΔG ‡ cycle value resulting from the study (35.6 kcal mol -1 ) is in relatively good agreement, given the computational approximations, with the experimentally measured catalytic turnover frequency (38 h -1 at 150°C, 26 which yields the upper estimate of 28.9 kcal/mol). The discrepancy between the calculation and the experiment (ca. 7 kcal mol -1 ) is not unreasonable given the usual precision of the DFT method, considering the additional approximation involved in modeling solvation effects for ionic species and the estimation of thermodynamic parameters in a condensed phase. 43 For comparison purposes, the additional calculations presented in this contribution were carried out at the same level of theory, with inclusion of the gas phase free energy and solvent effects at the CPCM level in aniline, but without consideration of ion pairing effects. The reasons for these choices have been discussed

previously. 39,40,43 (a) Halide effect in the hydroamination of ethylene.

All energies are given with respect to the reference complex [PtX3(C2H4)] -, 1 X , taking into account the values of all the species that are added to or subtracted from the system. The results are shown in Figure 2. As mentioned above, the previously reported investigation had shown that complex 1 Br is the catalyst resting state for the X = Br system. 39 The next lowest ethylenecontaining system was found to be trans-PtBr2(C2H4)(PhNH2), 2 Br , only 4.0 kcal mol -1 higher than the resting state, while trans-PtBr2(PhNH2)2 (3 Br ) is even closer at +3.6 kcal mol -1 . Since the effect of X on the relative energy of compounds 1 X , 2 X and 3 X could not be predicted a priori, we have also run calculations of all systems 2 X and 3 X to insure that we always consider the correct form of the resting state. As shown in Figure 2, the lowest energy species is in each case complex 1 X and this remains true at 150°C (catalytic conditions) according to the calculations.

Other potential off-loop intermediates, obtained by ligand exchange processes from 1 X with the molecules present under catalytic conditions (X -, aniline and ethylene) were at even greater energy for X = Br 43 and their optimization for the other halogen systems was not pursued. The energy gap between 1 X and complexes 2 X and 3 X is much greater for X = F (12.4 kcal/mol for 2 F and 15.9 kcal/mol for 3 F ), lower and relatively halogen-independent for the three heavier halogens (ca. 4 kcal/mol for both series). Although this has no relevance to the catalyst activity, since the resting state remains in all case the 1 X system, it is interesting to understand this "fluoride effect". With this objective, we have carried out more detailed analyses of the electronic structure and charge distribution for the 1 X , 2 X and 3 X complexes and the most relevant results are shown in Table 1. 1.580 a Average value for the cis-X atoms in systems 1 X , 2 X and 3 X . b Average value for the two N atoms in system 3 X .

As the halogen becomes less electronegative, the halogen atom natural charge predictably changes to less negative values for all series of complexes. The metal charge changes even more dramatically (for instance, from 1.022 for 1 F to -0.251 for 1 I , a change of more than 1 unit).

Counter intuitively, the ethylene C atoms in the 1 X and 2 X series are more negatively charged in the systems with the more electronegative halogen atoms, where the metal is more positively charged. The change is particularly pronounced on going from F to Cl and greater for the 1 X system (+0.041) than for the 2 X system (+0.021). This suggests that there is greater metalethylene back-bonding, yielding greater contribution of the platinacyclopropane limiting form, for the fluoride system. This result, contrary to expectation based the halide electronegativity and the computed charge on the metal atom, is nevertheless in agreement with the computed longer C=C distance in 1 F and 2 F , decreasing on going down the halogen series. Hence, the fluoride coligands, even though pulling more electron density away from the metal atom via the σ-bonding framework, provide more electron density in return via the π mechanism. In other words, the F - ligands are worse σ-donors but better π-donors than the other halides and allow the metal to provide more back-bonding to the ethylene ligand. Given the orientation of the ethylene ligand perpendicularly to the coordination plane, this interaction occurs via the metal dxz orbital, where x is the axis along the C2H4(center)-Pt-Xtrans vector. A qualitative orbital interaction diagram of the π interaction for the 1 X system is shown in Figure 3. To a first approximation, only the trans X atom participates in the interaction with its pz orbital. The [PtX3] -fragment on the left of the interaction diagram participates with two filled orbitals, resulting from the in-phase and out-of-phase combination of the trans-X pz orbitals and the Pt dxz orbital, which results in no net X-Pt π bonding. The greater electronegativity in the order F > Cl > Br > I is reflected in an energy decrease along the same sequence, with the exception of the F system, for which the out-of-phase combination is raised above those of the other halogen systems, whereas the in-phase combination is unusually stabilized relative to those of the other three systems. This is already clear indication of the stronger π interaction occurring for the F system. Now, both orbitals have the correct symmetry to interact and be stabilized by the ethylene CC π* combination (on the right of the interaction diagram), a phenomenon widely known as push-pull interaction. 44 The net result is transmission of electron density from the halogen pz orbital to the ethylene π* orbital with the consequent increase of C-C distance and natural charge of the ethylene C atoms. As shown in Figure 3, the establishment of these interactions clearly leads to a stronger stabilization for the out-of-phase combination in the case of the F system. The donor orbital (the out-of-phase Ftrrans-Pt xz combination at -0.093 eV) is much more Pt(dxz) in character and higher in energy relative to the other halogen systems, thus resulting in a stronger transfer of electron density from the F atom pz orbital to the CC π* combination. This effect is partly counterbalanced by that of the in-phase combination, which grows in importance on going from F to I because the in-phase combination becomes more and more Pt(dxz)-like as X becomes heavier (hence stronger back-bonding to ethylene), whereas the out-of-phase combination becomes more and more X(pz)-like (hence weaker back-bonding to ethylene). The overall balance is much in favour of stronger (XtransPt)-(C2H4) backbonding as the halogen gets lighter, with the biggest gap between Cl and F.

For systems 2 X , the ligand trans to ethylene (aniline) is common to all complexes and the cis halogen atoms give rise to a weaker π effect. The resulting qC and CC distance variations are smaller, although still in the direction expected for a more important contribution of the π bonding mechanism. Counterintuitively, the calculated natural charge on the aniline N atoms in systems 2 X and 3 X is also more negative for the complexes containing more electronegative halogen atoms, the biggest gap being once again between the F and Cl systems, although this variation is less pronounced relative to that on the ethylene C atoms in system 1 X , which is paralleled by shorter Pt-N distances. In this case, it does not seem possible to attribute the observed change to a π bonding interaction, since aniline is not a π acid. A σ effect through the hardness/softness concept [45][46][47] could be held responsible for this trend, the harder F atom rendering the Pt-N bond more ionic/less covalent. The Pt-N natural charge difference gets smaller on going to the heavier halide systems, which is accompanied by a lengthening of the Pt-N bond. Now the trend observed for the energy change on going from 1 X to 2 X can be rationalized. Indeed, the replacement of a π-donating X -ligand with the π-neutral PhNH2 decreases the effect of the special π-stabilizing effect of the F -ion and flattens the relative stability between the species, probably because of a more balanced compensation between σ and π effects. On going further from 2 X to 3 X , the ethylene ligand is replaced by a second aniline ligand and the residual small π synergism that was present in 2 X is no longer possible. Hence, the fluoride system is destabilized by a greater amount than the other three systems, for which the exchange is essentially thermoneutral.

The transition states TS2 X were also optimized, see Figure 2. The free energy relative to the resting state, G(TS2 X )-G(1 X ), is a direct measure of G ‡ cycle, related to the turnover frequency under standard conditions. 41,42 This value varies in a non monotonous way, increasing from F to Cl, then decreasing from Cl to Br, and finally increasing again from Br to I. Note that for the three systems that were investigated experimentally (Cl, Br, I), the computational result is in qualitative agreement with the experimental results shown in Table 2. 28 However, we must be cautious to make a direct comparison, because only TON values were reported for the different halide systems. The G ‡ cycle parameter can be related by the Eyring expression to the turnover frequency (TOF), which was not measured, and not to the TON. Given that the TON values were obtained under the same conditions and after the same period (10 h), they should be proportional to the TOF assuming no catalyst degradation. However, we know that the catalyst is irreversibly degraded during the catalytic runs to metallic Pt, which is inactive for the hydroamination process. 26,48 Hence, the greater TON observed for the Br system might be simply the result of slower catalyst degradation. Note also that the maximum activity for each halide system (in terms of TON, Table 2) was observed for a different excess amount of the activating salt (65 equiv for the chloride, 10 equiv for the bromide and iodide). The computed values relate, on the other hand, to the standard conditions (1 M concentrations for all species). 28 . Conditions: PtBr2, 0.13 mmol; PhNH2, 45 mmol; p(C2H4) = 25 bar at 298K (ca. 100 mmol), 150°C, 10 h.

One relevant and interesting result of the calculations is that the lowest G ‡ cycle is in fact calculated for the F system, which has not yet been experimentally investigated ( n Bu4PF is less readily available), suggesting that the PtX2/X -catalyst may have a greater catalytic efficiency when X = F. Note that a positive effect of fluoride was experimentally demonstrated for the iridium-catalyzed hydroamination of norbornene by aniline. 49 For this reason, as well as for removing the doubt about catalyst degradation, new experimental measurements were carried out for this catalytic process with monitoring of the kinetic profile, in the presence of an equivalent amount of all halides, I -, Br -, Cl -and F -(see next section).

For the sake of completion, we have also carried out a natural charge analysis on the TOFdetermining transition state TS2 X , see Table 1. The same trends observed above for the halide and metal charge in 1 X are also observed in TS2 X . The negative charge on the Pt-bonded C atom decreases systematically on moving from F to I, indicating that the alkyl group is still sufficiently bonded to the metal to feel the competitive inductive effect of the halide ligands. The H atom moves from Pt to C as a proton (positive charge, approximately halogen-independent) and the transition state has an "earlier" character (the C-H distance becomes longer, while the Pt-C and Pt-H remain approximately unchanged) as the X atoms become heavier.

(b) New catalytic studies.

A comparison of the four halide salts is legitimate if all conditions are perfectly identical except for the nature of the halide ion. Namely, the cation should be the same. The effect of chloride, bromide and iodide was studies using the n-butylphosphonium salts, all being commercially available (Cl, Br) or easily synthesized (I). In the case of the fluoride salt, we did not find a convenient access to nBu4PF and so we wondered whether the nature of the cation is really important. A readily available fluoride salt is nBu4F•3H2O. In order to see whether using this salt for comparison could be justified, we have run a few additional catalytic experiments with the following bromide salts as catalyst additives: nBu4PBr (classical conditions), 26 n-Bu4NBr, and n-Bu4NBr•3H2O. These three salts give essentially equivalent results (see Table 3, runs 1-3), showing that the process is not significantly affected by the cation nature and by the presence of water. Incidentally, we had already established that similar catalytic activities and an identical selectivity could be achieved by activation with aqueous NaBr. 48 Hence, we could proceed to catalytic tests using n-Bu4NF•3H2O. The experiment, carried out under the classical conditions, gave disappointing results, which did not significantly change after drying the salt (Table 3, runs 4 and 5). This result could remain consistent with the high activity suggested by the calculated ΔG ‡ cycle only if the fluoride system is affected by a more severe catalyst decomposition process or if fluoride exerts another function in addition to coordinating the Pt center. The next series of experiments aimed at obtaining the relative initial activities though a determination of the kinetic profile. These experiments were carried out under nearly identical conditions as the previous runs (e.g. those shown in Table 2 and Table 3), with double amounts of all reagents in order to have enough material in the autoclave for multiple sampling. The only exception was C2H4, which was maintained at the same pressure.

However, its available amount relative to aniline is still sufficient to satisfy the reaction stoichiometry. The results are reported in Figure 4. The analysis of these data leads to several considerations. The first one is the limited catalyst lifetime in each case. At the end of the catalytic run, massive amounts of both aniline and ethylene are still present in the autoclave, but the catalytic activity is reduced essentially to zero after just 6 hours, at least for the most active Br and I systems. Metallic platinum, as previously reported, 26,48 was found in the product mixture. The second consideration is that, limited to the heavier halogens Cl, Br and I, the initial activity is in qualitative agreement with the original report by Brunet et al., which was based on the TON after 10 h (activity in the order Br > I >> Cl). Thus, even though the system is heavily affected by catalyst decomposition, a qualitative direct relationship between initial TOF and TON exists. 26 The third observation is that the fluoride system, contrary to the computational prediction, is the least active system. Its initial slope is not any greater than that of the chloride system and furthermore it seems to deactivate faster, the % yield no longer increasing significantly after 45 min, whereas the other halogen systems continue to produce the N-ethylaniline product. The data, therefore, suggest that fluoride not only leads to a faster catalyst decomposition, but also exerts another function in addition to coordinating the Pt center.

In order to rationalize this result, we consider that one important difference between F -and the heavier halides is its basicity. We have recently shown that the use of a basic amine, notably NHEt2, leads to a more favorable nucleophilic addition to the coordinated ethylene, with generation of equilibrium amounts of the zwitterionic intermediate which could be fully characterized (contrary to aniline for which this intermediate is unobservable), but the excess amine leads to a deprotonation equilibrium with the aminoalkyl species, incapable to proceed along the hydroamination cycle, and leads to the system decomposition with generation of metallic platinum, as also pointed out by other previous related studies, [50][51][52] by a pathway that is not yet understood. 53 Metallic platinum was proven catalytically inactive for the hydroamination process. Hence, it may be proposed that the F -ion leads to a faster catalyst deactivation through its stronger basic character. The NHEt2-induced zwitterion deprotonation was also analyzed by DFT calculations. 53 We did not consider it useful to carry out additional calculations in the presence of an extra halide ion. On the other hand, we checked the hypothesis of a base-related deactivation through additional experimental studies.

As we have recently reported and already mentioned above, the [PtBr3(C2H4)] -system is also catalytically active when generated through the use of aqueous NaBr (the K2PtCl4 precatalyst was more conveniently used in this case). 48 The same conditions have now been used in the presence of added bases, namely NHEt2, NEt3 and 1,8-Diazabicyclo[5.4.0]undec-7-ene (DBU). As shown by the results in Table 4, the addition of the base completely suppresses the catalytic activity. This is also and especially true for bases such as NEt3 and DBU that combine high basicity with poor nucleophilicity. In the absence of any external base (fluoride or a stronger amine), the catalyst is still deactivated, though more slowly, by the action of aniline itself. Fluoride is a weaker base (pKa 3.18) than aniline (pKa 4.62) in water, but its basicity is enhanced by up to three orders of magnitude in lower polarity solvents. 54 On the basis of these results, it is also possible to rationalize the previously highlighted positive effect of a catalytic amount of a strong acid as an additive to the PtBr2/Br -system. 26 The strong acid (CF3COOH in the reported study) in the PhNH2-rich medium generates an equivalent amount of PhNH3 + , which buffers the action of free aniline in the deprotonation of the zwitterionic intermediate and therefore retards the deactivation process.

(c) Additional experimental mechanistic studies

In order to learn more about how the basicity affects the catalyst stability, a few additional experiments have been carried out. First, two control experiments probed the catalyst stability in a mixture where one of the two reaction components (aniline and ethylene) is left out. The experiments were run under the same conditions of time (10 h) and temperature (150°C) as the catalytic runs. A first experiment involved warming up PtBr2 in pure aniline in the presence of nBu4PBr without ethylene. This run could be carried out in regular glassware since aniline boils at 184°C. The result is dissolution of PtBr2 to yield a Bordeaux-red solution, from which a powdery brown precipitate deposited upon cooling. This product may correspond to (nBu4P)[PtBr3(PhNH2)], previously obtained by anion exchange from K2PtCl4 in the presence of aniline. 40 For the purpose of this test, however, the exact nature of this product is irrelevant and characterization was not pursued. The important point is that the solution was filtered and the solid washed out completely with a variety of solvents (dichloromethane, acetone, ethanol, …) leaving on the filter only traces of insoluble residue that could possibly correspond to metallic platinum. A second experiment was carried out in the presence of ethylene and absence of aniline. Since aniline is also the catalysis solvent in the classical procedure, 26 the test was carried out by replacing aniline with toluene. The result of this experiment is the complete dissolution of PtBr2, yielding a clear red solution and notably containing no visible amounts of black Pt 0 .

Precipitation by addition of diethyl ether yielded an orange crystalline product, the 1 H NMR of which confirmed that this is the known nBu4P[PtBr3(C2H4)] by comparison with an authentic sample. 40 These two tests clearly indicate the thermal stability of Pt II and resistance to reduction in the coordination environment provided by bromide anions in combination with either ethylene or aniline. The reduction must therefore be a consequence of the combined action of both reagents.

A working hypothesis is that the base acts on intermediate 4 Br (Table 1), obtained after nucleophilic addition of aniline to the coordinated ethylene. A logical possibility is deprotonation of the ammonium function, but other possibilities may also be considered, such as reversible β-H elimination followed by deprotonation of the resulting Pt(II) hydride species, whereas the deprotonation of the Pt(IV) hydride complex subsequently formed along the hydroamination cycle (5 Br ) would not lead to the generation of metallic platinum. These hypotheses for the catalyst decomposition will be further probed from the experimental and computational viewpoints in forthcoming investigations and published in due course.

(d) Effect of the ArNH2 basicity.

As mentioned in the introduction, the catalytic activity for the Brunet system, as measured by the TON obtained under the same conditions for a variety of substituted anilines ArNH2, is inversely proportional to the amine basicity: 22 for 4-MeOC6H4 (pKa = 5.34), 55 for C6H5 (pKa = 5.08), 90 for 4-ClC6H4 (pKa = 4.15), and 110 for 2-ClC6H4 (pKa = 2.65). 26 In light of the discussion offered in the preceding sections, this trend would simply result from a different rate of catalyst deactivation, faster for more basic anilines. We were interested, however, to see whether the electronic structure of the aniline would also affect the intrinsic activity in terms of the ΔG ‡ cycle. Therefore, we have run additional calculations with two modified anilines, 4-MeOC6H4NH2 and 4-ClC6H4NH2, restricted to the bromide system. These calculations included the transition states TS2 BrOMe and TS2 BrCl , and the corresponding complexes trans-[PtBr2(C2H4)(ArNH2)] (2 BrOMe with Ar = 4-MeOC6H4 and 2 BrCl with Ar = 4-ClC6H4). We did not calculate the corresponding trans-[PtBr2(ArNH2)2] derivatives (3). The results are compared with those of the previously calculated 2 Br and TS2 Br systems in Figure 5. Once again, species 1 Br is in every case the most stable one, both at room temperature and at the temperature used for catalysts (resting state). The calculations yield G ‡ cycle values growing with the aniline basicity, which agrees with the experimental observed greater activity for the addition of less basic anilines. Hence, the observed activity trend may result from an intrinsic aniline basicity effect on G ‡ cycle, or from a lower propensity of the less basic aniline to deactivate the catalyst, or from a combination of these two effects. The natural charge analysis of transition states TS2 BrCl and TS2 BrOMe (Table 5) do not reveal dramatic changes relative to the corresponding parameters of TS2 Br analyzed above in Table 1. In particular, the charge on the Pt center remains approximately unaffected. The most notable change is the less negatively charged Br -for the p-chloroaniline derivative relative to the other two derivatives. As the aniline becomes more basic, the Pt-bonded C atom engaged in the reductive elimination becomes less negatively charged, while the geometry of the PtHC moiety experiences very minor variations.

(e) Hydroamination of 1-hexene

As highlighted in a recent review, 29 many active catalysts for ethylene hydroamination fail when applied to higher olefins. Previous work on the PtX2/X -catalytic system, however, has shown a relatively high activity also for the hydroamination of 1-hexene, even though slightly lower than for the ethylene hydroamination (under the same conditions, namely 10 h at 150°C with 0.3% catalyst and 65 equiv of nBu4PBr per Pt, 1-hexene yields 57 turnovers 27 vs. 131 for ethylene 26 ). This activity difference is not likely due to a difference in catalyst deactivation rate, since the deactivation process is now known to be related to the action of an external base (which ethylene and 1-hexene are not) on the zwitterionic intermediate and the above comparative productivities were obtained under strictly identical conditions except for the olefin. Therefore, the different activity is probably related to a greater activation barrier (ΔG ‡ cycle) when going from ethylene to the higher olefin. It was also shown that the Markovnikov product is highly favored (95:5) 27 and the best results (highest TON) were obtained in this case in the presence of the iodide salt. 28 Our aim is to use computational chemistry to compare the aniline addition to 1-hexene and ethylene with the PtX2/X -catalytic system. The calculation of the thermodynamic gain of the reaction (see Table 6) shows that the hydroamination of 1-hexene is slightly less favourable than that of ethylene, and that the Markovnikov product is slightly favoured on the H and G scales in the gas phase, although the preference reverses in favour of the anti-Markovnikov product when considering the solvent effect of aniline by the C-PCM. 2) -6.0 (-1.4) -8.0 (-6.3) C4H9CH=CH2 + PhNH2  C4H9CH(NHPh)CH3 -41.9 (-41.6) -13.4 (-13.3) -0.9 (4.4) -1.1 (1.0) C4H9CH=CH2 + PhNH2  C4H9CH2CH2NHPh -41.8 (-41.9) -12.1 (-12.1) 0.4 (5.6) -1.5 (0.4) a The data refer to 298.15 K, with those at 423.15 K in parentheses. b Data from ref. 39 .

(e1) Markovnikov vs anti-Markovnikov regioselectivity.

Calculations to address this issue were carried out only for the bromide system, since the 95:5 regioselectivity is essentially independent on the nature of the halide. 28 hexene coordination provides a smaller stabilization than ethylene by 4.7 kcal mol -1 for 1 Br hex (-9.9 kcal mol -1 relative to -14.6 kcal mol -1 for 1 Br ) and by 3.3 kcal mol -1 for 2 Br hex (-7.4 kcal mol -1 relative to -10.7 kcal mol -1 for 2 Br ). Calculations on the 1-hexene analogues of the other, higher-energy systems were not carried out. We recall, however, that compound trans-[PtBr2(PhNH2)2], 3 Br , is placed at -11.0 kcal mol -1 relative to [PtBr4] 2-on the G CPCM (aniline) scale at 298K. 39 Hence, while 1 Br is the resting state for the ethylene system, the analogous complex 1 Br hex is slightly less stabilized than 3 Br . Calculations including the thermochemical corrections at 423.15 K, however, place compound 1 Br hex 0.25 kcal/mol lower in relative G CPCM (aniline) than 3 Br . Therefore, the resting state of the catalytic cycle for the 1-hexene hydroamination is either 1 Br hex or the off-loop species 3 Br depending on the temperature. It should also be mentioned that, under conditions similar but not totally identical to those of the hydroamination catalysis, [PtBr4] 2-was recently shown to afford [PtBr5] 3-in the presence of a large excess of a bromide salt, and that the latter species does not react with 1-hexene. 55 Since it was shown that the only productive zwitterionic intermediate for the ethylene system is that derived from the anionic tribromo complex (Figure 1), we have only examined the aniline addition to 1 Br hex, considering however both the CH2 and CHBu groups to yield zwitterionic complexes of Markovnikov (4 Br hexM) and anti-Markovnikov (4 Br hexA) types.

Several local minima were found for each addition product, the most stable ones being outlined in Figure 6. Both types of PhNH2 additions are slightly less favourable thermodynamically relative to the addition to ethylene (18.2 kcal mol -1 , see Figure 1), with 4 Br hexM being less destabilized than 4 Br hexA. The geometries of the two products are closely related to that of the ethylene analogue 4 Br , with a short N-H•••Pt hydrogen bond, predisposed for proton transfer to the platinum center, and a less tight hydrogen bond connecting a second N-H bond with one of the bromine atoms. We have also analyzed the activation barriers for the aniline nucleophilic addition (TS1 Br hex). The shape of the LUMO of complex 1 Br hex (Figure 7) clearly suggests that the C2 atom should be more easily attacked by the aniline nucleophile. The NBO analysis yields the charges of -0.48 for C1 and -0.24 for C2. Therefore, charge control also favours attack at the C2 atom. The figure also indicates the four possible pathways for nucleophilic addition: two anti attacks (Manti and Aanti) and two syn attacks (Msyn and Asyn). For each type of attack, there are actually two different pathways, depending on the direction in which the attached C atom will twist. Only the lower energy transition state for each type of attack is shown in Figure 8. The calculations indicate that for each C atom, the anti attack is favoured (the same result was previously found 39 for the PhNH2 to ethylene in 1 Br ), and that the Markovnikov attack is kinetically preferred by ca. 1 kcal mol -1 . Hence, the lowest transition state is TS1 Br hexManti, leading to the Markovnikov zwitterionic complex 4 Br hexM. However, the difference between the two barriers is irrelevant for the regioselectivity of the catalysis reaction, because the turnover rate for both Markovnikov and anti-Markovnikov products is determined by the final reductive elimination step (vide infra). The two competitive pathways are compared in the energy diagram of Figure 11. Like for the hydroamination of ethylene, the calculation shows that the TOF-determining transition state is that of the reductive elimination process forming the C-H bond between the aminoalkyl and hydride ligands. The nucleophilic addition step occurs with a transition state TS1 Br hex at lower energy. Therefore, the different energy barriers at the nucleophilic addition step level have no kinetic consequence. The only interesting difference is the relative thermodynamic stability of the zwitterionic products (4 Br hexM is present at greater relative concentration than its isomer 4 Br hexA under catalytic conditions). The overall rate is determined by the free energy difference between TS2 Br hex and the resting state, which is calculated as trans-PtBr2(PhNH2)2 (although complex 1 Br hex is calculated as slightly more stable at 150°C). Since the interesting parameter here is the G ‡ cycle CPCM for the two regioisomers and the resting state is shared by the two cycles, whether the resting state is trans-PtBr2(PhNH2)2 or [PtBr3(1-hexene)] -(or even [PtBr5] 3-) 55 , the result does not change. From this G ‡ cycle CPCM (2.6 and 3.7 kcal mol -1 at 298.15 and 423.15 K, respectively) an expected M/A ratio of 98.8:1.2 can be predicted at both temperatures (the greater energy difference at the higher temperature is balanced off exactly by the different RT term), which can be considered in qualitatively good agreement with the observed 95:5 ratio. In conclusion for this section, the results of the calculations based on the previously proposed mechanism 39 are in agreement with a number of experimental observations. (i) The catalytic activity for the hydroamination of 1-hexene (TON = 37) 27 is lower than for ethylene (TON = 92) 26 under the same conditions, consistent with a greater G ‡ cycle in the former case.

The difference between the two barriers at 150°C (37.7 kcal mol -1 for the Markovnikov addition to 1-hexene and 35.6 kcal mol -1 for addition to ethylene) is actually too large to justify such a small activity difference, but it is in the right direction. Besides computational errors, a possible reason leading to an artificially smaller activity difference is a faster catalyst degradation in the presence of ethylene relative to 1-hexene, but this does not seem likely on the basis of the considerations already made above. (ii) The catalytic activity for addition to 1-hexene is greater for the Markovnikov addition mode, with the calculated G CPCM qualitatively agreeing with the observed 95:5 ratio.

(e2) Halide effect.

Like for the aniline addition to ethylene, we applied the computational approach to analyze the G ‡ cycle for the addition of aniline to 1-hexene catalyzed by different PtX2/X -systems.

Calculation of the various [PtX3(1-hexene)] -(1 X hex) and comparison with the trans-PtX2(PhNH2)2 (3 X ) systems already discussed above reveals that the lowest energy system (resting state) is the former for X = F and the latter for the other three halogens (see Figure 12).

The much greater stabilisation of 1 X hex relative to 3 X for the fluoride system can be rationalized in the same way as that of 1 X , since the Pt-C2H4 and Pt-hexene bonds have similar constitution. The G CPCM values of the TOF-determining transition state for the two regiochemistries, TS2 X hexM and TS2 X hexA, are also reported in Figure 12 relative to the resting state. The G CPCM (TS2 X hexM) -G CPCM (TS2 X hexA) value is approximately constant (between 2.6 kcal mol -1 for the Br system and 3.8 kcal mol -1 for the I system), in agreement with a small effect of the halide nature on the M/A ratio. When considering only the heavier halides, the G ‡ cycle for the main (Markovnikov) pathway is greater for the Cl system, smaller and essentially identical for the Br and I systems, whereas the iodide system was experimentally found to yield a greater TON. As already concluded above in sections (a)-(c), the experimental result is largely affected by the rate at which the catalyst decomposes, whereas the computational limitations and approximations make the G ‡ cycle differences between the different halide systems little reliable.

Hence, a perfect match between the trends in G ‡ cycle and observed TON is not to be expected.

The results of the calculations qualitatively fit with the observation of a relatively halideindependent M/A regioselectivity, but a finer analysis of such small effects is probably meaningless.

Conclusions

Application of the catalytic mechanism recently established by a DFT study for the PtBt2/Br --catalyzed N-H addition of aniline to ethylene 39 has provided the following results: (i) the activation barrier (G ‡ cycle) for the C2H4/PhNH2 reaction catalyzed by PtX2/X -is similar for Br, Cl and I, though marginally smaller for the Br system which affords the highest activity; (ii) the G ‡ cycle slightly increases as the aniline basicity increases (p-chloroaniline < aniline < pmethoxyaniline); (iii) the G ‡ cycle for the 1-hexene/PhNH2 reaction is slightly greater than for the C2H4/PhNH2 reaction; (iv) G ‡ cycle for the Markovnikov addition of aniline to 1-hexene is lower than for the anti-Markovnikov addition; (v) the G ‡ cycle (M/A) for the 1-hexene/PhNH2 reaction is relatively halide independent. Some of these results are in reasonable agreement, at least qualitatively, with the experimental evidence (notably the marked preference for the Markovnikov addition), whereas others are less reliable or in fortuitous agreement because of the small computed differences where the computational errors may not perfectly cancel out, 56,57 and because the observed trends in catalytic activity for the previously published studies (expressed by the TON) are affected by catalyst decomposition which may not systematically be in direct relation with the turnover frequency.

Overall, it seems that all the available experimental results for the Pt-catalyzed olefin hydroamination are reasonably well rationalized by the proposed mechanism (Figure 1), 39 after allowance is made for the base-induced catalyst decomposition. For instance, decomposition seems to negatively affect the catalytic performance in the presence of fluoride ions, contrary to the computational prediction. The mechanism of Figure 1 may be considered a good basis for further mechanistic elaborations and catalyst design, which also requires however a fuller understanding of the catalyst degradation mechanism.

Additional studies into the catalyst degradation, learning how to prevent it or retard it, are warranted. It is also interesting to conceive other coordination geometries allowing a ΔG ‡ cycle reduction, namely a reduction of the free energy difference between the resting state and the transition state of the TOF-determining C-H reductive elimination step. Work in these directions is currently ongoing.

Experimental section

Computational details. All geometry optimizations were carried out by the DFT approach with the Gaussian03 suite of programs [START_REF] Frisch | Gaussian 03, Revision C.02[END_REF] using the B3LYP functional. [START_REF] Becke | [END_REF][60][61] All atoms except Pt and I were described by the standard 6-31+G* basis set, which includes both polarization and diffuse functions that are necessary to allow angular and radial flexibility to the highly anionic systems.

The I atom was described by the LANL08d basis and the Pt atom by the LANL2TZ(f) basis, which are uncontracted versions of LANL2DZ and include an ECP. 62 The I basis set contains a d polarization and a diffuse function and the Pt basis set contains an f polarization function. The geometry optimizations were carried out on isolated molecules in the gas phase. Transition states were optimized using Synchronous Transit-Guided Quasi-Newton (STQN) method in the gas phase. Frequency calculations were carried out for all optimized geometries in order to verify their nature as local minima, for the calculation of thermodynamic parameters at 298.15 K and at 423.15 K under the ideal gas and harmonic approximations, and for the identification of all transition states (one imaginary frequency). Solvent effects were introduced by means of C-PCM 63,64 single point calculations on the gas-phase optimized geometries in aniline ( = 6.89), which was considered to best approximate the reaction solvent. The solvent cavity was created by a series of overlapping spheres by the default UA0 model (or SPHEREONH for the delocalised proton atoms) and all standard settings as implemented in Gaussian03 were used for the C-PCM calculations. The reaction free-energy changes in solution were calculated as ΔGsolut = ΔGgas+ Δ(ΔGsolv) and corrected for the change of standard state from the gas phase (1 atm) to solution (1 M). 65 The approach of optimizing the geometries in the gas phase and then keeping the geometry frozen for the C-PCM calculation was preferred because of convergence problems of the geometry optimization in the presence of the C-PCM, especially for molecules containing weak interaction (i.e. hydrogen bonds). were added to the autoclave by syringe through a valve equipped with septum and the ethylene pressure was adjusted to 25 bar (ca. 100 mmol). The temperature was then raised to 150°C. After 10 h, the autoclave was allowed to cool to room temperature and slowly vented. The entire reaction mixture was treated with the external standard (N,N-dibutylaniline) and extracted with dichloromethane (ca. 60 ml). The organic layer was analyzed by GC.

  and trans-PtBr2(C2H4)2, has a lower barrier, only 4 Br is the productive intermediate. The latter easily transfers a proton to the Pt atom to yield [PtHBr3(CH2CH2NHPh)] -(5 Br ), which then proceeds to C-H reductive elimination through the TOF-determining transition state TS2 Br with formation of the transient -complex [PtBr3(k 2 :C,H-HCH2CH2NHPh)] -(6 Br ). Liberation of the product from 6 Br (or from a rearranged isomeric form with an N-bonded hydroamination product, [PtBr3(NHEtPh)] -, 7 Br ) via ligand substitution by a new C2H4 molecule regenerates 1 Br . The overall activation barrier is determined by the free energy difference (ΔG ‡ cycle) between the resting state and the TOF-determining transition state, 41, 42 which are respectively 1 Br and TS2 Br .

Figure 1 .

 1 Figure 1. Free energy profile of the aniline addition to ethylene catalyzed by PtBr2/Br -. 39 Values

Figure 2 .

 2 Figure 2. Relative Gibbs free energies of [PtX3(C2H4)] -(1 X ), trans-PtX2(C2H4)(PhNH2) (2 X ),

Figure 3 .

 3 Figure 3. Orbital interaction responsible for the Pt-C2H4 back bonding in complexes 1 X (see

Figure 4 .

 4 Figure 4. Kinetic profiles of the hydroamination of C2H4 by aniline, catalyzed by PtBr2/nBu4PX

Figure 5 .

 5 Figure 5. Relative Gibbs free energy of the trans-PtBr2(C2H4)(ArNH2) (Ar = 4-C6H4OMe,

  Following our previous work on the catalytic cycle for ethylene (see Introduction), at least five 1-hexene complexes are potentially susceptible to aniline nucleophilic addition, namely [PtBr3(1hexene)] -, cis-and trans-[PtBr2(1-hexene)(PhNH2)], and cis-and trans-[PtBr2(1-hexene)2]. The calculations started with complexes [PtBr3(1-hexene)] -(1 Br hex) and trans-[PtBr2(1hexene)(PhNH2)] (2 Br hex), which represent the 1-hexene analogues of the resting state and the next most stable olefin-containing complex for the ethylene system. Relative to [PtBr4] 2-, 1-

Figure 6 .

 6 Figure 6. Geometries and relative Gibbs free energies of the most stable products of PhNH2

Figure 7 .Figure 8 .Figure 9 .Figure 10 .

 78910 Figure 7. Three mutually orthogonal views of the LUMO (E = 0.0424 Hartree) of complex

28

 28

Figure 11 .

 11 Figure 11. Relative Gibbs free energy pathways for the PtBr2/Br -catalyzed hydroamination of 1-

Figure 12 .

 12 Figure 12. Relative Gibbs free energies for systems [PtX3(1-hexene)] -(1 X hex), trans-

  Yield vs. time monitoring. Catalytic experiments were conducted in a 100 ml stainless steel autoclave without glass liner under magnetic stirring. In a typical procedure, the autoclave was charged with PtBr2 (0.26 mmol) and the appropriate salt additive (2.6 mmol), closed, and submitted to vacuum/argon cycles. Aniline (8.3 ml, 90 mmol) was added to the autoclave by syringe through a valve equipped with septum and the ethylene pressure adjusted to 25 bar (ca. 100 mmol). The temperature was then raised to 150°C. Samples were withdrawn at regular time intervals, treated with the external standard (N,N-dibutylaniline) and extracted with dichloromethane. The organic layer was analyzed by GC.(b) In the presence of added base. In a typical procedure, K2PtCl4 (0.13 mmol) and NaBr (19.5 mmol) were introduced in the autoclave, which was then closed and submitted to vacuum/argon cycles. Subsequently, water (15 mL), aniline (4.15 ml, 45 mmol) and additional base(1.3 mmol)

Table 1 .

 1 Selected natural charge and bond distances in systems 1

X , 2 X , 3 X and TS2 X .

	2 I	0.002	-0.822	-0.230	-0.444	2.173	1.398	2.151
		qPt	qN b	qXcis a				Pt-N
		(NBO)	(NBO)	(NBO)				(d/Å)
	3 F	0.754	-0.815	-0.658				2.063
	3 Cl	0.312	-0.794	-0.468				2.075
	3 Br	0.191	-0.795	-0.406				2.081
	3 I	-0.006	-0.796	-0.305				2.092
		qPt	qXtrans	qH	qC	Pt-C	C=C	Pt-H	C-H
		(NBO)	(NBO)	(NBO)	(NBO)	(d/Å)	(d /Å)	(d/Å)	(d/Å)
	TS2 F	0.873	-0.629	0.191	-0.661	2.195		1.584	1.435
	TS2 Cl	0.132	-0.424	0.212	-0.587	2.203		1.571	1.548
	TS2 Br	-0.065	-0.364	0.209	-0.577	2.212		1.571	1.567
	TS2 I	-0.378	-0.261	0.204	-0.569	2.226		1.573
		qPt	qXtrans	qXcis a	qC	Pt-C	C=C
		(NBO)	(NBO)	(NBO)	(NBO)	(d/Å)	(d /Å)
	1 F	1.022	-0.650	-0.640	-0.523	2.089	1.415
	1 Cl	0.269	-0.445	-0.430	-0.482	2.131	1.410
	1 Br	0.070	-0.382	-0.367	-0.480	2.145	1.408
	1 I	-0.251	-0.274	-0.266	-0.478	2.166	1.405
		qPt	qN	qXcis a	qC	Pt-C	C=C	Pt-N
		(NBO)	(NBO)	(NBO)	(NBO)	(d/Å)	(d /Å)	(d/Å)
	2 F	0.875	-0.842	-0.624	-0.467	2.136	1.400	2.123
	2 Cl	0.350	-0.821	-0.406	-0.446	2.158	1.398	2.133
	2 Br	0.216	-0.821	-0.338	-0.446	2.164	1.398	2.137

Table 2 .

 2 Hydroamination of ethylene by aniline catalyzed by [PtBr2]/nBu4PX. a

	nBu4PX	Total TON	Total TON	Total TON
		10eq nBu4PX	65eq nBu4PX	150eq nBu4PX

Table 3 .

 3 Hydroamination of ethylene by aniline catalyzed by [PtBr2]/nBu4PX. a

	Run Additive	TON PhNHEt TON PhNEt2 TON quinaldine
	1	nBu4PBr	97	1	10
	2	nBu4NBr	119	2	8
	3	nBu4NBr•3H2O	104	2	7
	4	nBu4NF	6	0	2
	5	nBu4NF•3H2O	2	0	2
	Conditions: PtBr2, 0.13 mmol; salt additive, 1.3 mmol; PhNH2, 45 mmol; p(C2H4) = 25 bar at
	298K (ca. 100 mmol), 150°C, 10 h.

Table 4 .

 4 [K2PtCl4]/NaBr-catalyzed hydroamination of ethylene by aniline in the presence of bases. a

	Run Base	TON PhNHEt TON PhNEt2 TON quinaldine
	1	-	85	1	8
	2	NHEt2his	1	0	~0.5
	3	NEt3	0.5	0	0
	4	DBU	0.4	0	0
	a Conditions: K2PtCl4, 0.13 mmol; NaBr, 19.5 mmol; PhNH2, 45 mmol; p(C2H4) = 25 bar at 298K
	(ca. 100 mmol); H2O, 15 mL; base, 1.3 mmol; 150°C; 10 h.

4.0 (4.8) 33.7 (35.6) 1.8 (2.6) 34.1 (36.2) 33.4 (35.4) 4.7 (5.6) 2 BrCl 2 Br 2 BrOMe 1 Br TS2 BrCl TS2 Br TS2 BrOMe

Table 5 .

 5 Selected natural charge and bond distances in systems TS2 BrCl and TS BrOMe .

		TS2 BrCl TS2 BrOMe
	qPt(NBO)	-0.067	-0.066
	qtrans-Br(NBO) -0.359	-0.365
	qH(NBO)	0.209	0.208
	qC(NBO)	-0.579	-0.574
	C-H, d (Å)	1.566	1.570
	Pt-H, d (Å)	1.572	1.570
	Pt-C, d (Å)	2.211	2.213

Table 6 .

 6 Thermodynamic data for the hydroamination of ethylene and 1-hexene by aniline in the gas phase and in aniline as solvent. a

	Reaction

b -37.4 (-37.4) -17.2 (-17.

Acknowledgements. We are grateful to the ANR (Agence Nationale de la Recherche, Grant No. NT09_442499), to the CNRS (Centre National de la Recherche Scientifique) and to the IUF (Institut Universitaire de France) for support of this work. This work was granted access to the HPC resources of CINES under the allocation 2010-086343 made by GENCI (Grand Equipement National de Calcul Intensif) and to the resources of the CICT (Centre Interuniversitaire de Calcul de Toulouse, project CALMIP). PD thanks the MENESR (Ministère de l'Éducation Nationale, de l'Enseignement Supérieur et de la Recherche, France) for a Ph.D. fellowship.


Lingai Luo
email: lingai.luo@univ-nantes.fr

Min Wei

Yilin Fan

Gilles Flamant

Heuristic Shape Optimization of Baffled Fluid Distributor for Uniform Flow Distribution

Keywords: Flow maldistribution, Perforated baffle, Distributor, Shape optimization, CFD

This paper presents a CFD-based, heuristic evolutionary algorithm for shape design and optimization of baffled fluid distributor. In this algorithm, the baffle surface is firstly divided with numerous identical control areas (volumes), each control area having an orifice in the middle. Under the constraint of constant global porosity of the baffle, the algorithm adjusts the size distribution of orifices so as to approach identical mass flowrate passing through every control area. An automatic program is processed iteratively so that the baffle configuration evolves toward the optimized shape, providing a uniform flow distribution among parallel outlet channels.

Introduction

Delivering and distributing precise and controlled flows of one or more fluids into a bundle of parallel channels of multi-tubular equipment is a great challenge for many unit operations of process engineering. Single phase fluid flow maldistribution is an important cause of globally poor performance of heat exchangers, fluidized beds and catalytic monoliths. When mixing of different fluids is involved, such as in chemical reactors, in combustors and in absorption columns, the impact of flow maldistribution might be more disastrous because it may change dramatically the proportion between reactants.

Flow maldistribution can be classified into two categories: gross maldistribution and passage-to-passage maldistribution [START_REF] Mueller | Review of various types of flow maldistribution in heat exchangers[END_REF]. The latter usually occurs in highly compact devices mainly caused by manufacturing tolerances among passages, fouling, blockage due to condensable impurities, etc. If properly operated, such differences can be less than 5% [START_REF] Rebrov | Single-phase fluid flow distribution and heat transfer in microstructured reactors[END_REF]. On the other hand, the gross maldistribution is mainly associated with improper design of fluid distributor and/or collector configurations (also referred to as headers or manifolds). The maximum to minimum flowrate ratio among parallel channels, though depending on various factors (eg. inlet velocity, drag coefficient of the distributor, number of parallel channels, etc.), can vary from several tens of percent just up to factors of tens to hundreds [START_REF] Saber | Rapid design of channel multi-scale networks with minimum flow maldistribution[END_REF]Tondeur et al., 2011a). Therefore, many studies have been accomplished and attempted on design and optimization of fluid distributor with the purpose of improving the flow distribution uniformity.

Many studies have been devoted to improve the flow uniformity, as summarized in the review of [START_REF] Rebrov | Single-phase fluid flow distribution and heat transfer in microstructured reactors[END_REF] and an earlier book of [START_REF] Idelchik | Fluid Dynamics of Industrial Equipment: Flow distribution design methods[END_REF]. One of the effective solutions is to optimize the geometrical parameters of fluid distributor, such as the diffuser angle (eg. [START_REF] Commenge | Optimal design for flow uniformity in microchannel reactors[END_REF]Jiao et al., 2003a;Tondeur et al., 2011b;[START_REF] Kumaran | Experimental and numerical studies of header design and inlet/outlet configurations on flow mal-distribution in parallel micro-channels[END_REF]Midoux andTondeur, 2014a, 2014b), the ratio of distributor area to channel area (Jiao et al., 2003b) and the locations of inlet/outlet tubes [START_REF] Chein | Numerical study of the inlet/outlet arrangement effect on microchannel heat sink performance[END_REF][START_REF] Dang | Comparisons of the heat transfer and pressure drop of the microchannel and minichannel heat exchangers[END_REF]. The general idea is to provide sufficient distance between the inlet tube of the distributor and the parallel channels for the development of a regular flow profile. This usually implies that the fluid distributor occupies a large space which is unfavorable regarding volume compactness. Some researchers also proposed to use nature-inspired tree-like structures as fluid distributing network. Various tree-like (or arborescent) distributors were designed to deliver a certain quantity of fluid from a single inlet to a series of parallel channels evenly spaced on a line (point-to-line problem, [START_REF] Ajmera | Microfabricated differential reactor for heterogeneous gas phase catalyst testing[END_REF][START_REF] Barber | Optimal design of microfluidic networks using biologically inspired principles[END_REF][START_REF] Wang | Distribution optimization for plate-fin catalytic combustion heat exchanger[END_REF][START_REF] Yue | Flow distribution and mass transfer in a parallel microchannel contactor integrated with constructal distributors[END_REF][START_REF] Liu | Maintaining equal operating conditions for all cells in a fuel cell stack using an external flow distributor[END_REF][START_REF] Tarlet | Entropy generation analysis of a mini heat exchanger for heat transfer intensification[END_REF], on a surface (point-to-surface problem, Tondeur andLuo, 2004, Fan et al., 2008b;[START_REF] Chen | Thermal and hydrodynamic characteristics of constructal tree-shaped minichannel heat sink[END_REF][START_REF] Tondeur | Channel interlacing: A geometric concept for intensification and design of the internal structure of fluid contactors[END_REF][START_REF] Tesař | Bifurcating channels supplying "numbered-up" microreactors[END_REF][START_REF] Guo | Mixing performance assessment of a multi-channel mini heat exchanger reactor with arborescent distributor and collector[END_REF], or to a volume to be drained (point-to-volume problem, Gheorghiu et al., 2005;[START_REF] Tondeur | Constructal optimization of arborescent structures with flow singularities[END_REF]. It is known that multi-scale tree-like distributor could provide better flow distribution uniformity preferentially for laminar flow [START_REF] Luo | Integration of constructal distributors to a mini crossflow heat exchanger and their assembly configuration optimization[END_REF][START_REF] Luo | Experimental study of constructal distributor for flow equidistribution in a mini crossflow heat exchanger (MCHE)[END_REF]. When the flowrate increases, tree-like distributors become less efficient mainly due to the influence of inertial force. A slight asymmetry velocity profile at one bifurcation would lead to significant maldistribution after successive bifurcations (Fan et al., 2008a). Worse still is the drastic increase of pressure drop due to singular losses in numerous bifurcations, elbows or downcomers in complex tree-like structures (Fan et al., 2008a;[START_REF] Tondeur | Constructal optimization of arborescent structures with flow singularities[END_REF]. Other researchers focused on the multi-scale network based on ladder-type structures [START_REF] Saber | Rapid design of channel multi-scale networks with minimum flow maldistribution[END_REF][START_REF] Saber | Microreactor numbering-up in multi-scale networks for industrial-scale applications: Impact of flow maldistribution on the reactor performances[END_REF][START_REF] Commenge | Methodology for multi-scale design of isothermal laminar flow networks[END_REF]. An equivalent hydraulic resistance method is presented to calculate the flow characteristics of multi-scale microchannel reactors, under laminar and isothermal conditions.

Another common solution to the flow maldistribution problem is adding "packings" as additional hydraulic resistance inside the distributor. Packings could be either thick wall screens (Rebrov et al., 2007a(Rebrov et al., , 2007b)), or thin wall screens like meshes or gauzes. In particular, a thin perforated baffle is widely introduced in the inlet distributor to improve the flow distribution uniformity. Table 1 summarizes various studies in the literature on fluid distributors with insertion of perforated baffle. It is generally reported that the baffled fluid distributor could effectively improve the flow distribution uniformity with an acceptable pressure drop increase. Moreover, they can be easily fabricated and installed with less cost compared to tree-like distributors.

Nevertheless, by exploring in more detail the perforated baffle, we may find out that the configurations proposed could be very different, even arbitrary, as illustrated in Fig. 1. The orifices drilled in the plate could have identical or different diameters (usually small orifices at the centre of baffle and big orifices on the edge), and be arranged in single line or in multiple lines. In any case, the topology of the orifices' distribution, more specifically, the arrangement of the orifices on the plate and their size distribution are either empirical or intuitive. To the best of our knowledge, neither theoretical guidelines nor basic principles were proposed or established for the design and optimization of the baffle configuration. This will result in two consequences since the equidistribution effect of perforated baffle is generally restricted to a certain range of flow circuit geometry (eg. number of parallel channels) and working condition (eg. inlet velocity). Firstly, the best topology configuration of the perforated baffle can not be determined for one specific circuit geometry and under certain working condition. Secondly, once the circuit geometry or working condition changes, the equidistribution effect would probably be no longer in force. Then trial-and-error tests are inevitable until finding a non-optimal but acceptable new configuration. As a result, the development of a general method for the design and optimization of baffled fluid distributor is of great interests in fields wherever the flow maldistribution problem exists.

In this study, a heuristic evolutionary algorithm [START_REF] Luo | Procédé de détermination de caractéristiques d'orifices à ménager à travers une plaque et programme correspondant[END_REF] is proposed for the design and shape optimization of baffled fluid distributor for uniform flow distribution. The algorithm is CFD-based, and can be considered as a small extension in the general framework of Cellular Automata (CA) [START_REF] Wolfram | A new kind of science[END_REF]. With this algorithm, we try to automatically evolve the size and distribution of orifices on the perforated baffle in order to determine the optimal topology configuration subject to each particular case. Compared to traditional arbitrary or empirical propositions, this algorithm is more general, precise and flexible, and can be easily applied to different distributor geometries and different working conditions.

We shall first present the heuristic algorithm with a general 2D case, appropriate for introducing basic principles and for describing in detail each step of the algorithm. A numerical benchmark will then be developed and optimization results will be presented. The influences of various design and operational parameters will be analyzed and discussed. The theoretical basis for the extension of the current algorithm to tackle 3D problem are also established. Finally, remarks and main conclusions will be summarized. [START_REF] Lalot | Flow maldistribution in heat exchangers[END_REF]. (b) Adapted from Jiao et al. (2003b). (c) Adapted from Wen et al. (2006). (d) Adapted from [START_REF] Sheik Ismail | Numerical study of flow patterns of compact plate-fin heat exchangers and generation of design data for offset and wavy fins[END_REF]. (e) Adapted from [START_REF] Wang | Characteristics of flow distribution in compact parallel flow heat exchangers, part II: Modified inlet header[END_REF]. (f) Adapted from [START_REF] Krichnavaruk | Analysis of gas-liquid mass transfer in an airlift contactor with perforated plates[END_REF]. (g) Adapted from [START_REF] Maharaj | The effect of distributor configuration on the hydrodynamics of the teetered bed separator[END_REF].

Table 1 Selected studies on fluid distributor with insertion of perforated baffle.

Heuristic evolutionary algorithm

The basic idea of this study is to homogenize the velocity profile in the distributor by the insertion of a perforated baffle instead of natural development, as shown in Fig. 2. The size and distribution of orifices should be arranged in such a way that a regular velocity profile on the whole surface could be achieved at the downstream of the baffle (before the inlets of parallel channels). To do that, we first divide the baffle into numerous small sections, each section having an orifice in the middle. The objective of the algorithm is to obtain identical fluid flux in every section by modulating the size of every orifice. Locally uniform velocity profiles will then merge into a global regular one. We will describe in detail in this section the basic principles and steps of this heuristic algorithm with a 2D example. To simplify the numerical algorithm, we assume steady Newtonian fluid flow in this study.

Introduction of notations

l d i i = ε (2) N L d N i i N i i ∑ ∑ = = = = 1 1 ε φ (3)
where εi is the local porosity in the ith control area, Φ the global porosity of the baffle. The mean flowrate among orifices q can be calculated as:

N q N Q q N i i ∑ = = = 1 (4)
Note that the departure of qi from q depends on the location of the ith control area on the baffle and the size of the orifice di.

Basic principles

The objective of the evolutionary algorithm is to determine the optimal size distribution of orifices so that the flowrate in every control area is identical. l q l q l q j i = =

(5) Practically, the diameter of each orifice di will be modulated by comparing qi and q , as explained in Fig. 4. More precisely, if the flowrate in the ith control area qi is larger than the mean value, then the size of the orifice di should be reduced (Fig. 4a). Vice versa if qi is smaller than q , di should be enlarged so that it could receive higher amount of fluid flux (Fig. 4b). The variation of different di according to the difference between qi and q repeats literally until the optimality criterion Eq. ( 5) is achieved. To realize this evolutionary procedure, a cellular automata based iterative program is developed and will be described in next subsection. Ideally, the increment or decrement of di from one optimization step to another should be as small as possible (usually elemental cell length). However, that will by far lengthen the calculation time thus it is not highly efficient. Therefore, we introduce a variation rule as presented in Eq. ( 6). Precisely from t to t+1 step, the variation of di is proportional to the departure of qi,t from the mean value:

        - = - = ∆ + q q d d d t i t i t i t i , , 1 , , 1 γ (6)
γ is a relaxation factor to control the variation amplitude. Following this rule, the global porosity of the baffle is kept constant during the optimization procedure:

0 1 1 , 1 , =         - =         - = ∆ ∑ ∑ = = q Q N q q d N i t i N i t i γ γ (7) t N i N i t i t i N i t i t L d d L d φ φ = ∆ + = = ∑ ∑ ∑ = = = + + 1 1 , , 1 1 , 1 (8)
We also restrict that the boundary of the control area does not move during optimization (l is constant and uniform) and that the boundary of orifice can not overstep the boundary of control area:

1 , , ≤ = l d t i t i ε (9)

Implementation of optimization procedure

The optimization procedure is described in detail using the flow chart shown in Fig. 5.

1.

Definition of the geometry of fluid distributor (width of inlet tube, number of channels, etc.) as well as the general operational conditions such as the ambient temperature, operational pressure, amount of heat exchange, etc.

2.

Input the initial data relative to working fluid, such as the nature of fluid, the viscosity, compressibility and density of the fluid (temperature and/or pressure dependent or independent), gravity effect and viscous heating effect. The total inlet mass flowrate Q should also be specified.

3.

Definition of the parameters relative to the perforated baffle, such as the location and thickness of baffle, number of control area N and the initial distribution of di.

4.

Mesh generation of the whole simulation domain, thus the baffled fluid distributor.

Structured mesh is preferable for easy modification of fluid-solid interfaces.

5.

Calculation of the exact flow field at steady state, by solving the Navier-Stokes equations.

6.

Calculation of the mass flowrate qi passing through each control area and the mean value of flowrate q ; modification of the baffle configuration by comparing qi to q , according to the rule presented in Eq. ( 6).

7.

Regeneration of the mesh for updated simulation domain; recalculation of the exact flow field.

8.

Check the stable tolerance of the algorithm. If the tolerance is satisfied, then the heuristic procedure is terminated. If not, the procedure goes back to Step 4 for recurrence. The result is considered to be stable when the size variation at t step Δdi,t is smaller than elemental cell size of the mesh. This also implies that the values of qi,t and q are very close.

9.

Exportation of results, including the final configuration of perforated baffle, the flow distribution uniformity among parallel outlet channels and the pressure drop of the baffled distributor.

The flow maldistribution is quantified by two parameters: the relative flow rate deviation Di and the maldistribution factor MF.

f f f D i i - = (10) ∑ =         - - = M i i f f f M 1 2 1 1 MF (11)
where M stands for the number of parallel outlet channels. fi is the local mass flow rate in ith outlet channel and f is the mean flowrate between parallel outlet channels. Uniform flow distribution is achieved when values of Di and MF approach 0.

M f M Q f M i i ∑ = = = 1 (12)

Calculation of flow field by CFD method

One important step in the algorithm is the calculation of exact flow field in the baffled fluid distributor.

This generally involves solving Navier-Stokes equations by finite volume methods. The equation for conservation of mass or continuity is:

( ) 0 = ⋅ ∇ + ∂ ∂ u t  ρ ρ (13)
The momentum conservation equation is described by:

( ) ( ) ( ) u uu p g F t ρ ρ ρ ∂ + ∇ ⋅ = -∇ + ∇ ⋅ ∏ + + ∂     (14)
where u  is the velocity, p is the static pressure, g ρ  and F  are the gravitational body force and external body forces, ∏ is the stress tensor which is given by: ( )

T 2 3 u u uI µ   ∏ = ∇ + ∇ -∇ ⋅        ( 15
)
where μ is the molecular viscosity, I is the unit tensor.

The energy equation is:

( ) ( ) ( ) ( ) : H E u E p T u Q t ρ ρ λ ρ ∂ + ∇ ⋅ + = ∇ ⋅ ∇ + ∏ ∇ + ∂   (16)
where E is the internal energy, λ is the thermal conductivity, and QH includes the heat of chemical reaction, radiation and any other volumetric heat sources.

To predict turbulent flow pattern, additional turbulence models should be employed. It is worth noting that besides the traditional CFD methods, new simulation techniques such as the Lattice Boltzmann Method (LBM) can also be used in this regard for more complex fluid domains [START_REF] Chen | Lattice Boltzmann method for fluid flows[END_REF][START_REF] Wang | Heuristic optimality criterion algorithm for shape design of fluid flow[END_REF]2014).

Having described the heuristic algorithm developed for uniform flow distribution, its effectiveness and sensibility will be tested with an actual example in the next section.

Simulation and results

Numerical benchmark

Here a simple example of baffled fluid distributor restricted in a two-dimensional domain is considered. Fig. 6 shows the schematic view of the case with main geometrical characteristics indicated. The inlet tube is 10 mm (w) in width and 40 mm in length while the distributor chamber is 100 mm in width (L) and 25 mm in length (H). Ten parallel outlet channels (M=10) with identical dimension of 2 mm in width and 40 mm in length are connected to the distributor chamber, spaced 10 mm from each other. Note that the inlet tube is located in the middle so that the whole geometry is symmetrical.

In this study, the "sudden expansion configuration" of fluid distributor is selected as example because it is the most compact but most difficult to realize the uniform flow distribution. However, the proposed algorithm is not limited to this configuration but suitable for various geometries such as distributors with 30° or 45° diffuser angle. The geometry of distributor may vary from case to case, but the optimization procedure remains the same.

Fig. 6. Geometry and dimension of the numerical benchmark (unit: mm).

The 5 mm-thickness (e) perforated baffle is inserted at the middle of the distributor chamber (h/H=0.5). The baffle is divided into 10 control areas (N=10), the length (l) of each being 10 mm. The initial diameters of orifices are identical and equal to 2 mm (di,0=dj,0=2 mm), the global porosity (Φ) being 20%. Note that the initial diameter distribution can be non-uniform, which will be discussed in later subsections. The distribution of di evolves along with optimization steps while the global porosity of the baffle is kept constant.

Liquid water is used as working fluid. The operation pressure is set at 101325 Pa. Simulations are performed under steady state, isothermal condition without heat transfer. Gravity effect and viscous heating are neglected for simplification. The physical properties of water used in the computation are constant (density ρ=998.

2 kg•m -3 ; viscosity μ=1.003×10 -3 kg•m -1 •s -1 ).
Structured mesh is generated using software ICEM TM (version 12.1) to build up the geometry model.

Note that half of the real object is adopted for the purpose of lessening the computational burden because of the symmetric feature of the simulation domain. Generally, one millimeter is divided into ten segments to create square elemental cells. The influence of mesh density on the optimization results is also tested and will be discussed later.

Navier-Stokes equations are resolved by a commercial code ANSYS FLUENT TM (version 12.1.4) to calculate the flow field. Standard k-ε model is used to simulate the turbulent flow. For the pressure-velocity coupling, COUPLED method is used. For discretization, standard method is chosen for pressure and second-order upwind differentiation for momentum. Boundary conditions are set as velocity inlet (1.0 m•s -1 ) normal to the inlet boundary. Outlets are set to be static pressure boundary, with pressure value being the same as ambient pressure. Adiabatic wall condition is applied and no slip occurs at the wall. The solution is considered to be converged when (i) the mass flow-rate at each outlet channel and the inlet static pressure are constant from one iteration to the next (less than 0.5% variation) and (ii) sums of the normalized residuals for control equations are all less than 1×10 -6 .

The optimization algorithm is written in MATLAB TM . At each optimization step, it takes the computed flow field results from FLUENT TM , performs calculations of the displacement according to the variation rule and updates the coordinates of simulation domain. Then, it passes the data required to ICEM TM to regenerate the mesh for the updated geometry. Examining the evolution of diameter distribution of orifices along with optimization time steps (Fig. 8),

Optimization results

Fig

we observe that the sizes of orifices at the centre (No. 5; 6) and at the edge (No. 1, 10) of the baffle decrease while those at intermediate zone (No. 2-4; 7-9) increase. This observation differs from the conventional proposition of a monotonously increasing size distribution from centre to the edge of the baffle (eg. Jiao et al., 2003a;[START_REF] Wen | Study of flow distribution and its improvement on the header of plate-fin heat exchanger[END_REF]. In fact, the size reduction of central orifices is logical since fluid should be guided toward the edge. As to the size reduction of peripheral orifices, it is a bit counterintuitive owing to the backflow from the distributor walls (boundary effect). Anyway, the optimal configuration of baffle is closely related to the geometry of fluid distributor. The improved flow distribution uniformity is along with the increased pressure drop, as shown also in increase is caused by installing the baffle (step -1 to step 0). The pressure drop increase caused by the optimization process is relatively small (20.5%). The heuristic algorithm realizes the optimal allocation of pressure drop, which is the driving force to achieve flow equidistribution.

Sensibility analysis

Having presented the efficiency of proposed heuristic algorithm for uniform flow distribution, the robustness of the algorithm regarding different influencing factors is now examined.

The choice of initial shape is usually important, especially for numerical optimization problems. In order to investigate the sensibility of the algorithm to the choice of initial configuration, a comparative test is carried out. This time a non-uniform size distribution of orifices is provided as initial configuration of perforated baffle. All the boundary conditions and physical properties for fluid flow are kept the same. It can be observed from Fig. 10 that by executing the optimization algorithm, different initial configurations lead to a definite and unique optimized configuration. This justifies the robustness of the proposed algorithm to the choice of the initial shape. Since in our algorithm the minimum size variation is defined as one cell of mesh, it is necessary to assess the sensibility of the optimized results relative to mesh density. A comparative test regarding the "resolution" of the simulated domain has been carried out. We have refined the mesh (20 segments instead of 10 for 1 mm) for the above distributor case, while keeping other numerical parameters unchanged. Results on the improvement of flow distribution uniformity are presented in Fig. 11. It may be observed that the flow distribution uniformity may be further improved when the mesh density increases. This may be attributed to two reasons. On one hand, the calculation of flow field is usually more accurate when more mesh nodes are available. On the other hand, the size variation of orifices could be more precise with a smaller size of elemental cell. However, this further improvement is at the cost of much longer computational time, since more computational time will be consumed to calculate the flow field at each optimization step. For practical applications of the algorithm in real engineering, the mesh density could be determined considering the fabrication precisions and tolerance of maldistribution.

A "progressive refinement technique" may be considered to further improve the efficiency of our algorithm. That is to use firstly coarse mesh and then progressively refine the flow domain in order to reach the necessary density. This will be the subject of our future work.

Effect of design parameters

In this section, the effects of design and operational parameters on the efficiency of the proposed heuristic algorithm will be systematically evaluated. These parameters include the total inlet flowrate (Q), the baffle location (h/H), the number of outlet tubes (M) and the global porosity (Φ).

Total inlet flowrate

Total inlet flowrate of a fluid distributor is one of the key operational parameters which is associated with the throughput or capacity of processes. Meanwhile, it is also an influencing factor for flow maldistribution since it is closed related to the flow patterns (laminar, transitional or turbulent) and velocity profiles in the distributor., A parametric study is carried out by varying the inlet velocity from 0.1 m•s -1 to 2.0 m•s -1 in order to test the efficiency of our optimization algorithm within a wide range of inlet flow rate. The corresponding Re number at inlet tube varies from 1000 to 20 000 while the mean Re number at outlet channel changes from 100 to 2000.

Fig. 12a shows the evolution of flow distribution uniformity as a function of optimization time steps for different values of inlet velocity. It can be easily observed that flow maldistribution augments with increasing inlet velocity when no baffle is inserted in the distributor (step -1). This is mainly due to the stronger inertial force at higher velocity so that the fluid enters the central outlet channels more preferentially. The insertion of a uniformly perforated baffle (step 0) can reduce the maldistribution to some extent, indicated by the reduction of maldistribution factor MF from 0.128 to 0.044 for 0.1 m•s -1 inlet velocity and from 0.324 to 0.189 for 2.0 m•s -1 inlet velocity. However, it is clear that the initial configuration of baffle is not efficient enough to achieve uniform flow distribution under high inlet flowrate conditions.

By applying the heuristic algorithm, the flow maldistribution can be further reduced. For the optimized configurations, the MF value is 0.012 for 0.1 m•s -1 inlet velocity and 0.096 for 2.0 m•s -1 inlet velocity. The improvement of distribution uniformity is always significant with respect to the initial configuration. It should also be noted that the absolute distribution uniformity of the optimized configuration decreases when total flowrate increases. This is mainly due to the higher inertial forces and locally irregular and chaotic turbulences. Higher mesh density and more subtle size variation rule may be useful to achieve a more uniform flow distribution under high flowrate conditions.

Fig. 12b shows the velocity of actual flow fields under 0.5 m•s -1 inlet velocity at three sections of the distributor. The flow fields are changed when the flow passes the perforated baffle. Five uniform peaks correspond to five channels which make identical flow flux in each channel.

Fig. 13 shows that the total pressure drop in the distributor increases along with the increase of the inlet velocity and higher pressure drop is inevitable to improve the distribution uniformity. As mentioned above, the main pressure drop increase is caused by installing the baffle (Step -1 to Step 0), and the proportion caused by the optimization procedure is relatively small.

Location of the baffle

It is reported that location of the perforated baffle in the distributor chamber may influence the flow distribution uniformity and the best location suggested is in the midway [START_REF] Lalot | Flow maldistribution in heat exchangers[END_REF]. To verify this influence, we have tested two additional cases with the baffle located at the upper-middle (h/H=0.25) or at the bottom-middle (h/H=0.75) of the distributor chamber, to compare with the midway case. Other numerical parameters are kept the same as the benchmark.

Number of outlet channels M

The number of parallel outlet channels has certainly a great impact on the flow maldsitribution, since it is by far more difficult to evenly divide one stream into dozens of small streams than into two or three sub-streams. In order to test the efficiency of the heuristic algorithm within a wide range of number of outlet channels (M), we studies two extra cases with M being 5 or 20. Other numerical parameters are kept the same as the benchmark except that the distance between two neighboring outlet channels is adjusted accordingly.

Fig. 16 shows the evolution of flow distribution uniformity as a function of optimization time steps for different numbers of outlet channels. It can be observed that the larger the number of outlet channels, the poorer the flow distribution uniformity will be. The value of maldistribution factor (MF) for the empty distributor augments from 0.153 to 0.736 when M increases from 5 to 20. It is worth noting that the optimized configuration obtained by the heuristic algorithm can always maintain the flow maldistribution at a relatively low level, especially when the number of channels is large. It can also be observed that the optimized distribution becomes relatively less uniform when M increases. A factor defined as the ratio of number of control areas to the number of outlet channels (N/M) is introduced to explain these observations. Theoretically speaking, the insertion of perforated baffle changes the "topological connectedness" from 1-M to 1-N-M. The heuristic optimization algorithm could guarantee an equidistribution for the stage 1-N. For the stage N-M, it can be roughly analogized as N empty distributors with topological connectedness of 1-M/N. The distribution uniformity in this stage thus depends on the value of M/N. Specifically for the three cases tested, the distribution uniformity is significantly improved compared to the reference case (1-M). Examining in more detail the topological connectedness for each case, as schematically illustrated in Fig. 17, we notice that it corresponds to 1-2 (bifurcation, M=20), 1-1 (straight, M=10) and 2-1 (merging, M=5) with internal flow remixing, respectively. It will be easier to achieve equidistribution when fewer sub-streams should be formed. Further discussion could be found in next subsection.

Global porosity Φ

A series of comparative tests were performed to study the effect of global porosity Φ on the performance of the optimization algorithm. The control parameters for different cases are listed in Table 2. It can be noticed that the tested global porosity (Φ) ranges from 10%, 20% to 40%. Higher values of porosity may not be accepted since it may violate the constraint of Eq. ( 9 This is coherent to the issue of "topological connectedness" discussed earlier. It seems that when the number of orifices is large, the optimal distribution uniformity that can be achieved for different porosities is very close. However, it should be noted that the absolute values of MF for these cases are so small (less than 0.01) that they might be easily influenced by calculation errors or by the mesh flaw.

Examining the pressure drop of the distributor for different cases as shown in Fig. 19, one may observe that the increase of pressure drop due to the insertion of perforated baffle is inversely related to the global porosity of the baffle. The higher the global porosity, the smaller the pressure drop increase will be caused. This analysis indicates that higher global porosity with larger number of orifices (while always keeping local porosity below 1) is beneficial to achieve both uniform flow distribution and minimal pressure drop increase.

Extension to 3D

This problem will be addressed briefly in order to indicate how it can be handled, but without going into details. In fact, the extension of current 2D algorithm to 3D geometry is astonishingly simple and requires practically no rewriting. Let us merely illustrate the general cases of this point-to-surface problem: a bundle of parallel channels evenly distributed on a rectangular or an oval surface to be irrigated. Uniform flow distribution is required by a proper design of baffled fluid distributor, whose inlet tube is located facing the centre of the surface.

The most delicate part of the exercise is actually to divide the surface (rectangular or oval) by control volumes (instead of control areas for 2D cases), as shown in Fig. 20. Equations (1-3) can then be easily rewritten by replacing linear parameters with surface parameters.

N S s = (17) s a i i = ε (18) N S a N i i N i i ∑ ∑ = = = = 1 1 ε φ (19)
where S is the surface area of the baffle, s the surface area of the control volume and ai the size of the orifice in ith control volume. The basic principles of the optimization algorithm and evolution rules for the 3D case are:

s q s q s q j i = = (20)         - = - = ∆ + q q a a a t i t i t i t i , , 1 , , 1 γ (21)
It can be easily verified that the global porosity does not change along with the optimization steps.

Implementing the same optimization procedure as for 2D cases, uniform flow distribution may be expected using our heuristic algorithm. Of course, more computational time is also needed for the calculation of flow field at each optimization step by CFD code.

Note that the shape of control volumes may be rectangular, regular hexagon or those may uniformly pave the entire baffle surface. As to the orifices, the simplest shape is circular or square since the area variation at each optimization step can be realized by modulating single variable (diameter or side length). Of course, other shapes are also realizable regarding different geometries, with the constraint that the boundary of orifice does not overstep the boundary of the control volume during optimization.

Discussion

In the current algorithm, uniform and identical value of pressure is assigned as boundary condition for outlet channels. This approximation is appropriate for the shape optimization of inlet fluid distributor as an individual device. However, in compact systems where inlet distributor, parallel channels for unit process operation and outlet collector are highly integrated and closely connected, the configuration of the downstream collector may have a noticeable impact on the upstream flow distribution uniformity between parallel channels. In this case, the algorithm should be modified so that the simulation domain covers the entire geometry from the single inlet to the single outlet. The optimization rule should be updated accordingly as well.

Basic constraints for the proposed algorithm are that the control areas (volumes) are evenly divided and that the boundary of orifices can not overstep the boundary of control areas during optimization steps (local porosity less than 1). These constraints are useful and necessary to guarantee a uniform flow distribution among parallel channels evenly spaced between one another on a line or on a surface. However, when non-uniform but controlled flow distribution is required or unevenly spaced parallel channels are involved in certain operational equipment, these constraints should be modified accordingly. In that case, an additional degree of freedom may be added, i.e. the lengths of control area li (surface areas ai for 3D application) can be locally varied (li≠lj; li,t≠li,t+1) corresponding to specified optimization criterion, while keeping the global porosity constant during optimization.

The baffle configuration optimization with our algorithm will surely cause a certain increase of pressure drop in the distributor. However, this pressure drop increase which depends on the configuration of baffle could be relatively small (higher porosity; smaller thickness but within the limit of mechanical deformation). Moreover, this increase can be compensated by the reduced pressure drop of downstream equipment owing to more uniform flow distribution.

It should be noted that the efficiency of the heuristic algorithm depends largely on the precision of the transport phenomena simulation. The more realistic the simulated flow, the more satisfactory results may be achieved. When more complex flows such as varied fluid composition or more complicated situations such as fouling or channel blockage are involved, the selection of adaptive and reliable numerical models is very important. However, the heuristic algorithm is somewhat independent to the CFD codes used.

Conclusion and prospects

This paper presents a CFD-based, heuristic evolutionary algorithm for shape design and optimization of baffled fluid distributor. In this algorithm, the baffle surface is firstly divided with numerous identical control areas (volumes), each control area having an orifice in the middle. Under the constraint of constant global porosity of the baffle, the algorithm adjusts the size distribution of orifices so as to approach identical mass flowrate passing through every control area. An automatic program is processed iteratively so that the baffle configuration evolves toward the optimized shape, providing a uniform flow distribution among parallel outlet channels.

Compared to traditional arbitrary or empirical propositions, the advantages of our optimization algorithm lie in two main aspects. On the one hand, it leads to an optimal baffle configuration for a given distributor geometry under certain working condition. On the other hand, it may be easily applied to different flow circuit geometries (eg. different numbers of outlet channels) and different working conditions (eg. different total flowrate). In this sense, this algorithm is robust, effective, general and flexible.

The examples presented in this study deal with single phase flow. The extension of the proposed algorithm for multi-phase flow maldistribution problem is interesting but challenging. This is one direction of our future work and some guidelines may be followed based on the recent works of Schouten's group (Al-Rawashdeh et al., 2012a, 2012b, 2012c, 2014).

It should be noted that higher global performances of devices (in terms of higher thermal efficiency, presence of less hot spots, enhanced mixing, controlled reaction, etc.) for different applications may need different optimal criterion [START_REF] Mies | Experimental validation of the performance of a microreactor for the high-throughput screening of catalytic coatings[END_REF][START_REF] Al-Rawashdeh | Phenylacetylene hydrogenation over [Rh(NBD)(PPh3)2]BF4 Catalyst in a numbered-up microchannels reactor[END_REF]. The development a pertinent performance criterion for special unit operation, and the experimental validation of the optimization algorithm for an actual application are the directions of our ongoing work.

Fig. 1 .

 1 Fig. 1. Different baffle configurations proposed in the literature. (a) Adapted from Lalot et al. (1999). (b) Adapted

Fig. 2 .

 2 Fig. 2. Schematic view of the effect of perforated baffle on the homogenization of velocity profile.

Fig. 3

 3 Fig. 3 shows a representative 2D baffled fluid distributor with single inlet tube (width w) and certain

Fig. 3 .

 3 Fig. 3. Representative example of a fluid distributor with insertion of perforated baffle.

Fig. 4 .

 4 Fig. 4. Basic principles of the optimization algorithm. (a) Reduction of the orifice. (b) Enlargement of the orifice.

Fig. 5 .

 5 Fig. 5. Flow chart of the algorithm.

  Fig.7ashows the evolution of shape and flow field of different optimization steps at three sections of

Fig. 7 .

 7 Fig. 7. Shape and flow field evolution of different optimization steps at three sections (a); Relative flowrate

Fig. 8 .

 8 Fig. 8. Evolution of size distribution of orifices on the baffle along with optimization steps. N=10; M=10; Φ=20%;

Fig. 9 .

 9 Fig. 9. Maldistribution factor (MF) and pressure drop (Δp) as a function of optimization steps. N=10; M=10;

Fig. 9 .

 9 Fig. 9. The value of pressure drop increases from 467 Pa (empty) to 676 Pa (initial configuration), and

Fig. 10 .

 10 Fig. 10. Evolution of size distribution for different initial shapes.

Fig. 11 .

 11 Fig. 11. Influence of mesh density on the performance of optimization algorithm.

Fig. 12 .

 12 Fig. 12. Influence of inlet flowrate on the performance of optimization algorithm. N=10; M=10; Φ=20%; h/H=0.5

Fig. 13 .

 13 Fig. 13. Influence of inlet flowrate on the pressure drop in the baffled fluid distributor. N=10; M=10; Φ=20%;

Fig. 14

 14 Fig.14shows the optimized flow distribution among parallel outlet channels for three different baffle

Fig. 14 .

 14 Fig. 14. Relative flowrate deviation for optimized configurations with different baffle locations. N=10; M=10;

Fig. 15 .

 15 Fig. 15. Pressure drop as a function of optimization step, for different baffle locations. N=10; M=10; Φ=20%;

Fig. 16 .

 16 Fig. 16. Influence of number of outlet channels on the distribution uniformity. N=10; Φ=20%; h/H=0.5; inlet

Fig. 17 .

 17 Fig. 17. Schematic view of the topological connectedness between the number of orifices and the number of

  ) during the evolution procedure (the boundary of the orifice may overstep the boundary of the control area). For each value of global porosity, different numbers of control areas (N=10, 20, 40, 50) with corresponding initial size of orifices are tested. Other parameters are kept identical, i.e. M=10, h/H=0.5, inlet velocity at 1 m•s -1 .

Fig. 18 .Fig. 19 .

 1819 Fig. 18. Influences of global porosity and number of orifices on the optimized flow distribution uniformity. M=10;

Fig. 20 .

 20 Fig. 20. Division of a baffle surface by numerous control volumes for 3D application.

Fig. 1 .

 1 Fig. 1. Different baffle configurations proposed in the literature. (a) Adapted from Lalot et al. (1999). (b)

Fig. 2 .

 2 Fig. 2. Schematic view of the effect of perforated baffle on the homogenization of velocity profile.

Fig. 3 .Fig. 4 .Fig. 5 .Fig. 7 .

 3457 Fig. 3. Representative example of a fluid distributor with insertion of perforated baffle.

Fig. 8 .

 8 Fig. 8. Evolution of size distribution of orifices on the baffle along with optimization steps. N=10; M=10;

Fig. 9 .

 9 Fig. 9. Maldistribution factor (MF) and pressure drop (Δp) as a function of optimization steps. N=10;

Fig. 10 .Fig. 11 .Fig. 12 .

 101112 Fig. 10. Evolution of size distribution for different initial shapes.

Fig. 13 .

 13 Fig. 13. Influence of inlet flowrate on the pressure drop in the baffled fluid distributor. N=10; M=10;

Fig. 14 .Fig. 16 .Fig. 17 .Fig. 18 .

 14161718 Fig. 14. Relative flowrate deviation for optimized configurations with different baffle locations. N=10;

Fig. 19 .

 19 Fig. 19. Influences of global porosity and number of orifices on the pressure drop increase. M=10;

Table 2

 2 Control parameters for different tested cases.

Table 2 . Control parameters for different tested cases.

 2

	1
	Case	Global porosity Φ (-)	Number of orifices N (-)	Initial size of orifices di,0 (mm)	Number of outlets M (-)	Inlet velocity (m•s -1 )	Baffle location h/H
		10%	10	1
		10%	20	0.5
		20%	10	2
		20%	20	1
		20%	40	0.5	10	1.0	0.5
		20%	50	0.4
		40%	10	4
		40%	20	2
		40%	40	1
	10	40%	50	0.8

Acknowledgement

This work is partially financed by the French CNRS within the project AAP-Energie 2012 of INSIS. One of the authors M. Min WEI would like to thank the "Région Pays de la Loire" for its partial financial support to his PhD study.

The perforated plate sparger produces a higher gas holdup 1 2