Min Wei

Yilin Fan

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

Gilles Flamant

Fluid Flow Distribution Optimization for Minimizing the Peak Temperature of a Tubular Solar Receiver

Keywords: Flow maldistribution, High temperature solar receiver, Peak temperature, Evolutionary algorithm, Optimality criterion

High temperature solar receiver is a core component of solar thermal power plants. However, non-uniform solar irradiation on the receiver walls and flow maldistribution of heat transfer fluid inside the tubes may cause the excessive peak temperature, consequently leading to the reduced lifetime. This paper presents an original CFD-based evolutionary algorithm to determine the optimal fluid distribution in a tubular solar receiver for the minimization of its peak temperature. A pressurized-air solar receiver comprising of 45 parallel tubes subjected to a Gaussian-shape net heat flux absorbed by the receiver is used for study. Two optimality criteria are used for the algorithm: identical outlet fluid temperatures and identical temperatures on the centerline of the heated surface. The influences of different filling materials and thermal contact resistances on the optimal fluid distribution and on the peak temperature reduction are also evaluated and discussed.

Results show that the fluid distribution optimization using the algorithm could minimize the peak temperature of the receiver under the optimality criterion of identical temperatures on the centerline. Different shapes of optimal fluid distribution are determined for various filling materials. Cheap material with low thermal conductivity can also meet the peak temperature threshold through optimizing the fluid distribution.

Introduction

Concentrated Solar Power (CSP) plants are expected to play an important role in the energetic scenarios for more efficient use of renewable energy and for the reduced emission of greenhouse gases [START_REF] Zaversky | Object-oriented modeling for the transient response simulation of multi-pass shell-and-tube heat exchangers as applied in active indirect thermal energy storage systems for concentrated solar power[END_REF][START_REF] Cocco | Performance comparison of two-tank direct and thermocline thermal energy storage systems for 1 MWe class concentrating solar power plants[END_REF]. It is estimated that the CSP would contribute up to 11% of the global electricity production in year 2050 [START_REF] Pitz-Paal | Concentrating solar power in Europe, the Middle East and North Africa: achieving its potential[END_REF][START_REF]Technology Roadmap[END_REF].

In a CSP tower plant, the high temperature solar receiver installed at the top of the tower absorbs and transforms the concentrated solar irradiation delivered from heliostats into heat and then transfers it to a heat transfer fluid. The amount of heat carried by the heat transfer fluid will be used in a downstream thermodynamic cycle to allow electricity generation. The solar receiver, being a key component of CSP systems accounting for about 15% of the total investment [START_REF] Pitz-Paal | ECOSTAR-European Concentrated Solar Thermal Road-Mapping[END_REF], has a decisive influence on the overall efficiency of the plant.

Depending on the geometrical configuration, the solar receivers can be classified into different types, such as volumetric receiver, cavity receiver or particle receiver [START_REF] Behar | A review of studies on central receiver solar thermal power plants[END_REF]. In particular, tubular receivers are widely used in present commercial CSP projects because they belong to a proven technology on concepts from heat exchanger which is relatively inexpensive and durable [START_REF]Concentrating Solar Power Projects[END_REF][START_REF] Li | Compact heat exchangers: A review and future applications for a new generation of high temperature solar receivers[END_REF]. Some numerical or experimental studies on tubular solar receivers in the recent literature are summarized in Table 1. Pressurized air or molten salt are used as heat transfer fluid whereas the receiver is usually made of stainless steel or Inconel. Table 1. Some recent studies on tubular solar receiver (N: numerical study; E: experimental study).

The lifetime of tubular solar receivers depends strongly on the thermal-mechanical stress on the material which is in close relation with the peak temperature of the receiver wall [START_REF] Salomé | Control of the flux distribution on a solar tower receiver using an optimized aiming point strategy: Application to THEMIS solar tower[END_REF].

Based on the study of [START_REF] Fork | Life estimation of pressurized-air solar-thermal receiver tubes[END_REF] on a pressurized-air solar receiver tube made of Alloy 617, the lifetime is estimated about 30 years while operating 10 hours per day under peak temperature of 1100 K. This temperature is also considered as the threshold temperature under which the corresponding mechanical stress reaches the allowable design value that the material can afford for normal operation. When peak wall temperature reaches 1250 K, the real mechanical stress is 7.2 times the allowable design stress, resulting in a reduction of receiver lifetime by a factor of 10 to 20. Moreover, a peak wall temperature of 1250 K instead of 1100 K increases about 40% the thermal loss due to radiation, leading to more than 3% reduction in electricity production. Therefore, how to decrease the peak temperature on the solar receiver wall is becoming a serious concern.

The temperature distribution on the tubular solar receiver wall of a given geometry mainly depends on the following factors.

• The material of the receiver: temperature and pressure-resisting materials with high thermal conductivity are generally expected, but often require additional investment cost and technology breakthrough on fabrication methods;

• The power and distribution of concentrated solar irradiation: uniform heat flux on the receiver aperture is often assumed in theoretical analysis, but in reality very difficult to achieve due to the large size of the receiver even with advanced aiming point strategy [START_REF] Salomé | Control of the flux distribution on a solar tower receiver using an optimized aiming point strategy: Application to THEMIS solar tower[END_REF][START_REF] Grange | Thermal performances of a high temperature air solar absorber based on compact heat exchanger technology[END_REF]. The actual heat flux distribution is generally non-uniform, either similar to a Gaussian shape (e.g. Fig. 3) or to an alternative shape such as "table-top" distribution [START_REF] Boerema | High temperature solar thermal centralreceiver billboard design[END_REF];

• The physical properties and the working conditions of heat transfer fluid: this is usually determined by operational specifications of CSP plants;

• And finally the flow distribution of heat transfer fluid among the multiple parallel tubes: this factor is relatively easier to be managed, but may play an important role on reducing the peak temperature of the receiver wall.

Most of the studies in the literature aimed at uniformly distributing the heat transfer fluid among parallel tubes under an assumed uniform solar irradiation. But when the solar irradiation distribution on the receiver wall is in reality non-uniform, there must exist an optimal fluid flow distribution so that the peak temperature of the receiver can be minimized.

Then the challenge is how to determine this optimal fluid flow distribution. Researches in this area are relatively fewer and less recognized, with the exception of [START_REF] Boerema | High temperature solar thermal centralreceiver billboard design[END_REF].

In their work [START_REF] Boerema | High temperature solar thermal centralreceiver billboard design[END_REF], an optimality criterion was proposed, suggesting that the peak temperature can be largely reduced when the outlet fluid temperatures from parallel tubes are identical.

The theoretical fluid distribution for a single-layer tubular solar receiver could then be determined based on a 1-D heat transfer model, using classic correlations for heat transfer and fluid flow. Various improved receiver designs were also proposed, including multi-diameter or multi-pass configurations. Nevertheless, the analytical solution obtained was more or less approximate because of various simplifications made for the model, i.e. negligible axial heat conduction, established turbulent flow, negligible entrance effect, etc. When dealing with more complicated geometries (e.g. multiple layers of tubes rather than one layer) or transfer mechanisms (e.g. significant non-uniformity in axial/longitudinal heat flux distribution), the formulation of physical models and the solution could become cumbersome, sometimes even impossible.

So emerges the idea that classic analytical approaches could benefit from modern Computational Fluid Dynamics (CFD) simulation. The main objective of the present study is to develop an original CFD-based evolutionary algorithm for the determination of the optimal fluid flow distribution in a tubular solar receiver subjected to a non-uniform net heat flux, so as to minimize the peak temperature on the receiver wall.

For this purpose, a real 3D pressurized-air solar receiver comprising of 45 parallel straight tubes arranged in three layers will be used as a model for study. By comparison with simple tube bundle design, this tube-in-matrix concept allows a better heat dissipation by conduction inside the receiver [START_REF] Garcia | Codes for solar flux calculation dedicated to central receiver system applications: a comparative review[END_REF][START_REF] Grange | Modélisation et dimensionnement d'un récepteur solaire à air pressurisé pour le projet PEGASE[END_REF].

We will firstly use "identical outlet fluid temperatures" proposed by [START_REF] Boerema | High temperature solar thermal centralreceiver billboard design[END_REF] as the optimality criterion (OC-I), appropriate for introducing the receiver geometry, for describing basic principles of the evolutionary algorithm, and for illustrating the effectiveness of the algorithm through an actual numerical example. The reduction of the peak temperature through the optimization of fluid flow distribution will be presented and discussed.

Furthermore, a detailed analysis may reveal that such optimality criterion (identical outlet fluid temperatures) does not necessarily mean minimum peak temperature on the receiver wall. So a large section of this article is devoted to proposing a new evolutionary optimality criterion for further reducing the peak temperature, taking the receiver geometry and the physical properties of the material into account. The shapes of optimal fluid distribution, the corresponding minimum peak temperatures for different filling materials and the effect of thermal contact resistances will also be compared and discussed.

It should be emphasized that the present article is focused on the determination of the optimal fluid flow distribution for a tubular solar receiver. The realization of such optimal distribution by a simple and practical method is certainly an interesting subject with practical significance, but is beyond the present paper. Some discussions may be found in the perspectives section of the article, in connection to our earlier work on the design and optimization of fluid distributors and collectors [START_REF] Luo | Heuristic shape optimization of baffled fluid distributor for uniform flow distribution[END_REF][START_REF] Wei | CFD-based evolutionary algorithm for the realization of target fluid flow distribution among parallel channels[END_REF].

CFD-based Evolutionary Algorithm

In this section, a 3D multi-layer solar air receiver used as a model for study will be briefly described. The basic principles of the evolutionary algorithm and the CFD simulation parameters will be introduced as well.

Geometry of the tubular solar receiver: the tube-in-matrix concept

Figure 1 shows the geometry of the studied tubular solar receiver, which is a real prototype designed for the PEGASE project [START_REF] Grange | Modélisation et dimensionnement d'un récepteur solaire à air pressurisé pour le projet PEGASE[END_REF]. It appears to be a cuboid solid monoblock, the overall dimension being 405 mm in length (z direction), 202 mm in width (x direction) and 49 mm in height (y direction). The inlet fluid distributor and the outlet fluid collector are not included in the present study. The tubes are also made of Inconel 600, with identical geometric dimensions (circular crosssection; inner diameter d=6 mm tube length L=405 mm).

The interspaces between the external shell and the tubes are filled with filling materials. Note that various filling materials such as Inconel or copper may be used, aiming at different operating conditions. Inconel 600 is used as the filling material for the first part of the study when "identical outlet fluid temperatures" is selected as the optimality criterion. The influences of different materials on the optimal fluid flow and temperature distributions will also be evaluated and discussed in the latter section.

In actual operation, the upper surface of the shell is irradiated by the concentrated solar power from heliostats. The heat flux received on the heated surface will be transferred through the filling material and the tube walls by conduction, and to the pressurized air (at 15 bar) flowing inside the tubes by convection. Assuming that the solar receiver is well insulated so that there is no heat loss to the environment, i.e. all heat received is transferred to the pressurized air.

The following equations can be obtained based on energy and mass conservation.

C C tot out out in in ij ij ij ij ij ij i A j i A j Q Q m Cp T Cp T = = = = = = - ∑∑ ∑∑ (2) 1 (
) T and out T repeats iteratively until the optimality criterion Eq. ( 5) is achieved.

tot tot out out in in Q m Cp T Cp T = - (3)
Ideally, the increment or decrement of ij m from one optimization step to another should be as small as possible. 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). From t to t+1 step, the variation of ij m is proportional to the difference between out ij T and out T :

( )

, , 1 , , , ,
out out ij t ij t ij t ij t ij t ij t m m m m T m T γ + ∆ = - = - (6)
where γ is a relaxation factor always larger than zero to control the variation amplitude.

Following this rule, the total inlet mass flow-rate is approximately constant during the optimization procedure, with the assumption that the variation of Cp is negligible (actually <2.6%) within the range of tested temperature conditions

( ) ( ) ( ) 15 15 15 , , , , , 1 1 1 ,
0 tot C C C ij t out out ij t ij t ij t ij t out out i A j i A j i A j ij t Q Q m m T m T Cp T Cp T γ γ = = = = = =     ∆ = - = - ≈     ∑∑ ∑∑ ∑∑ (7)
The initial value of γ is determined as the following: ,

...,15

ij t ij t out out ij t ij t ij t ij t m m i A B C j m T m T γ = - = = - = - ==   -   = = =   -   (8)
The initial fluid flow distribution as the beginning case (step t=0) could be arbitrary. Here a uniform inlet fluid flow distribution is used. A conventional shape of flow distribution using conventional rectangular or conic inlet/outlet headers is also introduced for comparison (step t=-1), presenting a certain degree of non-uniformity (as shown in Fig. 5).

The degree of closeness between the optimized results and the optimality criterion is quantified by the non-uniformity of outlet fluid temperatures (MFT) defined as follows:

2 15 T 1 1 MF 1 out out C ij out i A j T T N T = =   - =     -   ∑∑ (9)
The optimality criterion (OC-I) is considered to be achieved when values of MFT approach 0.

Meanwhile, the non-uniformity of fluid flow (MFf) frequently used [START_REF] Fan | Flow distribution property of the constructal distributor and heat transfer intensification in a mini heat exchanger[END_REF][START_REF] Guo | Multi-channel heat exchanger-reactor using arborescent distributors: A characterization study of fluid distribution, heat exchange performance and exothermic reaction[END_REF] can be written as:

2 15 f 1 1 MF 1 C ij i A j m m N m = =   - =     -   ∑∑ ( 10
)
where m is the mean mass flow-rate of all tubes:

N m m C A i j ij ∑∑ = = = 15 1 ( 11
)
We also define Δpmax as the maximum pressure drop among parallel tubes.

Implementation of optimization procedure

The major steps of the CFD-based evolutionary algorithm are schematized in the following flow chart (Fig. 2) and described in detail below.

(1) Input the initial data such as the size and initial geometry of the solar receiver, the general operational conditions (the ambient temperature, operational pressure, net heat flux distribution, etc.) and the physical properties of working fluid (fluid nature, total inlet mass flow-rate, density, viscosity, etc.).

(2) Mesh generation of the whole simulation domain for both fluid and solid zones. (5) Calculation of the quantities mentioned in the optimality criterion (the outlet fluid temperature of each tube); modification of mass flow-rate distribution according to the variation rule of Eq. ( 5).

(6) Recalculation of the exact flow and temperature fields to obtain the updated data.

(7) Check the stable tolerance of the evolutionary algorithm. If the tolerance is satisfied, then the evolutionary procedure is considered to be converged.

Practically, the optimization process will be terminated when the value of MFT becomes less than 0.001. If not so, the procedure goes back to Step 5 for recurrence.

(8) Exportation of results, including the fluid flow distribution, the temperature field, the peak temperature on the heated surface and the maximum pressure drop, etc.

Calculation of flow and temperature fields by CFD method

An essential step of the evolutionary algorithm is the calculation of exact flow and temperature fields. This generally involves solving Navier-Stokes equations and heat transfer equations by CFD methods. For the fluid zone, the equation for conservation of mass or continuity equation is:

( ) 0 u t ρ ρ ∂ + ∇ ⋅ = ∂  (12)
The momentum conservation equation is described by:

( ) ( ) ( ) u uu p g F t ρ ρ ρ ∂ + ∇ ⋅ = -∇ + ∇ ⋅ ∏ + + ∂     (13)
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 given by: ( )

T 2 3 u u uI µ   ∏ = ∇ + ∇ -∇ ⋅        ( 14
)
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 ∂ + ∇ ⋅ + = ∇ ⋅ ∇ + ∏ ∇ + ∂   ( 15
)
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.

For the solid zone, the energy transport equation is:

( ) h T C T S t ρ λ ∂ = ∇ ⋅ ∇ + ∂ ( 16
)
where C is the specific heat and h S is the heat sources within the solid.

Numerical parameters for CFD simulation

Meshes were generated using software ICEM (version 12.1) to build up the geometry model of the studied solar receiver. Note that half of the real object was adopted for the purpose of lessening the computational burden because of the symmetric feature for both the simulation domain and the net heat flux distribution. The grid used in this study had 3 680 940 elements in total, with 1 421 550 structured hexahedral elements for fluid zone and 2 259 390 unstructured hexahedral elements for solid zone. Relatively more mesh elements were used for the fluid zone (38.6% grid for 12.9% volume) since the physical properties of pressurized air are strongly temperature dependent. A grid independence study was conducted to guarantee that the current mesh density used was appropriate and sufficient regarding both accuracy and calculation time.

The working fluid used was pressurized air at 15 bar while the solar receiver manufacturing material was Inconel 600. The physical properties of fluid and solid were considered as temperature dependent, using the polynomial fitting correlations listed in Table 2. It should be noted that the expansion properties of solid material were ignored in our study.

Table 2. Physical properties of fluid and solid used for simulations [START_REF] Li | The optimization of fluid flow and heat transfer in high-temperature pressurized air solar receivers[END_REF].

In this study, 3D fluid flow simulations were performed under steady state with heat transfer, using a commercial code FLUENT (version 12.1.4). The gravity effect was also considered.

Standard k-ε model was used to simulate the potential turbulent flow. For the pressurevelocity coupling, COUPLED method was used. For discretization, standard method was chosen for pressure and first-order upwind differentiation for momentum.

For the fluid zone, the inlet boundary condition was set as mass flow-rate of air at 300 K normal to the boundary of the inlets. The total inlet mass flow-rate (m tot ) of 45 channels was equal to 135 g•s -1 , each tube having an initial mass flow-rate of 3 g•s -1 . Pressure outlet boundary was set for the outlets with the gauge pressure value being zero. All the walls for tubes were defined as non-slip conditions. For the solid zone, a Gaussian heat flux distribution was defined for the upper surface of the solar receiver, as shown in Fig. 3. It should be noted that the heat flux considered here is net heat flux absorbed by the heated surface of the tubular solar receiver. In real operation the heat losses of a solar receiver are dominated by radiation and convection that results in about 15-20% of the incident solar radiation [START_REF] Flesch | Numerical analysis of the influence of inclination angle and wind on the heat losses of cavity receivers for solar thermal power towers[END_REF]. The highest heat flux is located at the center of the heated surface, with the heat flux distribution given by the following equation: The solution was considered to be converged when (i) the outlet fluid temperature from each tube and the inlet static pressure were constant from one iteration to the next (less than 0.5% variation), and (ii) the normalized residuals were lower than 10 -5 for all control equations. At each optimization step, MATLAB was used to take the computed flow and temperature fields data from FLUENT, to perform calculations of the flow-rate change for each tube according to Eq. ( 6) and to pass the data required to FLUENT to recalculate the updated flow and temperature fields. It can also be observed from Fig. 4b that the key parameter of the objective function ̶ the peak temperature Tpeak is reduced from 1335.2 K (step -1) to 1144.7 K (step 6), i.e. by 190.5 K. Generally, the value of Tpeak decreases along with the decreasing MFT. This observation supports the OC-I proposed by [START_REF] Boerema | High temperature solar thermal centralreceiver billboard design[END_REF] in that the peak temperature of the receiver can be largely reduced when outlet fluid temperatures tend to be identical. One exception is that the Tpeak for the uniform fluid distribution case (1346.1 K) is slightly higher than that of the conventional fluid distribution case (1335.2 K). This confirms that uniform fluid flow distribution is apparently not the optimal option regarding peak temperature reduction, under the condition of non-uniform net heat flux distribution.

Results and discussion

Effectiveness of the evolutionary algorithm

Comparison of three cases

More details on the fluid flow and thermal characteristics under three different fluid flow distribution cases (conventional distribution, uniform distribution and optimal distribution) are shown on Fig. 5. For conventional and uniform fluid flow distribution cases, the outlet fluid temperatures from the tubes on layer A are significantly higher than those on layers B and C. The reason lies in that layer A is geometrically the closest to the heated surface so that it absorbs higher amount of heat than other two layers. In order to achieve the optimality criterion of identical outlet fluid temperatures with a fixed total mass flow-rate of air, more fluid flow should be concentrated in tubes on layer A while the fraction for layers B and C should be reduced. The shape of optimal fluid distribution that leads to identical outlet fluid temperatures is presented in Fig. 5a. Obviously it is far from uniform but presents some specific features.

Firstly, the flow-rates of tubes in layer A range from 5.9 g•s -1 to 8.7 g•s -1 , much higher than those in layer B (from 1.04 g•s -1 to 1.21 g•s -1 ) and layer C (from 0.57 g•s -1 to 0.62 g•s -1 ).

Secondly, the fluid flow distribution among the tubes of layer A shows a parabolic shape with the highest flow-rate in the center (tube A8) and the lowest at the corners (tubes A1 and A15). This trend is in close connection to the Gaussian-shape net heat flux distribution, with its focusing point at the center of the heated surface (as shown in Fig. 2). On the contrary, the fluid flow distribution among the tubes of the layer B or layer C is relatively much more uniform.

The corresponding temperature fields on the heated surface of solar receiver for three fluid distribution cases are shown in Fig. 5c. A Gaussian-like temperature field can be observed for uniform fluid distribution case, with the hot spot (Tpeak=1346.1 K) located at the center and a temperature difference of 961.9 K between the maximum and the minimum. For the conventional fluid distribution case having the highest mass flow-rates in central tubes (A8, B8, C8, etc.), the hot spots (Tpeak=1335.2 K) are no longer located at the center of the heated surface but appear on both sides of the centerline, the max-min temperature difference being 950.0 K. Meanwhile, the optimal fluid distribution case presents a more uniform temperature distribution on the heated surface, with a reduced peak temperature (Tpeak) of 1144.7 K and a much smaller max-min temperature difference of 773.7 K.

Short summary

The numerical results presented above indicate that under our studied conditions (multi-layer tubular geometry, Inconel 600 material, Gaussian net heat flux and fixed total mass flow-rate of air), the property of fluid flow distribution has a significant influence on the temperature distribution on the heated surface of the solar receiver. By simply arranging the fluid flow distribution in a proper way among the parallel tubes, the peak temperature could be reduced by about 200 K. This optimal fluid flow distribution can be easily determined by our CFDbased evolutionary algorithm, using identical outlet fluid temperatures as the optimality criterion.

One may be satisfied with these encouraging results but another question may arise: is the optimality criterion of identical outlet fluid temperatures equivalent to the minimum peak temperature of the heated surface? Note that it is the peak temperature rather than the uniformity of outlet fluid temperature that is the main issue of concern in relation to the lifetime of solar receivers. The answer is perhaps not: the peak temperature is largely reduced but there is no evidence that it achieves the minimum. Then one may ask: can the peak temperature be further reduced by acting on the fluid flow distribution? If yes, could this optimal fluid distribution be determined by the proposed evolutionary algorithm? Furthermore, what will be the effects of fluid flow distribution on the peak temperature reduction when different filling materials are used for the solar receiver? The following section aims at answering these questions by proposing a new evolutionary optimality criterion (OC-II) for the CFD-based evolutionary algorithm.

A new optimality criterion for peak temperature minimization

By carefully examining the temperature field of the heated surface for the optimal fluid distribution case (Fig. 5c), one may observe that it still shows a Gaussian-like shape with the hot-spot located at the center of the heated surface. The temperature distribution is not uniform along with the x direction of the receiver, i.e. the isothermal lines are not perpendicular to the fluid flow direction. It seems that the peak temperature can be further reduced by concentrating more fluid flow into the central tubes (e.g. A7, A8, A9) while decreasing the flow-rates in tubes on the edge (e.g. A1, A15). In brief, the optimality criterion of identical outlet fluid temperatures (OC-I) does not necessarily lead to the minimum peak temperature. If one aims at minimizing the peak temperature, the optimality criterion used should be modified and the corresponding optimal fluid flow distribution will be differed from that shown in Fig. 5a.

The main motivation of this section is thus to propose a new evolutionary optimality criterion (OC-II) that leads to minimum peak temperature of the heated surface. The corresponding optimal fluid flow distribution can still be determined using the developed CFD-based evolutionary algorithm.

Description of the new optimality criterion

The basic idea is to adjust the fluid flow distribution among the parallel tubes so that isothermal lines on the heated surface are perpendicular to the fluid flow direction. Due to the Gaussian-shape net heat flux distribution, the peak temperature is always located on the centerline of the heated surface. As a result, identical temperatures on the centerline perpendicular to the fluid flow direction may be proposed as the new optimality criterion (OC-II).

Once the new optimality criterion is proposed, the next step is to establish the variation rule for the mass flow-rate in each tube to realize this OC-II. This calls for a detailed analysis on coupled heat transfer and fluid flow to reveal the relationship between the flow-rate change in each channel and the temperature variation on the centerline. To do that, we reduce the 3D multi-layer tubular solar receiver problem into 2D, by selecting the middle cross section (A-A plane shown in Fig. 1) as the study plane. Note that at steady state, the mass flow-rate of fluid at different cross-sections (i.e. tube inlet, middle cross-section, tube outlet) of each tube is constant.

The schematic view of the middle cross-section (A-A plane) is shown in Fig. 1. Recall that the tubes are indexed as ij (i represents layer from A to C and j represents the index of column from 1 to 15). The middle cross-section is divided into 15 columns. In each column, there are three tubes of different layers (e.g. tubes A1, B1, C1 in column 1). Figure 6 shows a simplified heat conduction model from the upper line of each column to the inner wall of one tube. We note the mean temperature on the upper line of column j as j T and the average temperature of inner wall of tube ij as ij T . By using an equivalent distance between the upper line and the inner tube wall (as shown in Fig. 6a), the conduction heat flux for the single solid material condition (Inconel 600) can be written as follows:

j ij ij ij ij T T l λ - Φ = ( 24
)
where ij λ is the average thermal conductivity of solid material:

( ) When different filling materials other than Inconel 600 are used (Fig. 6b), we assume that the solar receiver is perfectly fabricated so that thermal contact resistances at material interfaces are negligible. Then Eq. ( 24) could be written as: , , , , , ,

2 j ij T T ij T T λ λ + = = (25)
j ij ij ij shell ij filler ij wall ij shell ij filler ij wall T T l l l λ λ λ - Φ = + + (26)
In order to achieve identical temperatures on the centerline (upper line of the middle crosssection), the mass flow-rate in each tube ij m will be modulated by comparing ij Φ and the average Φ of tubes. More precisely, if ij Φ is higher than Φ , the mass flow-rate in tube ij should be increased to further enhance the heat transfer. Vice versa if ij Φ is smaller than Φ , the flow-rate in the tube ij should be reduced because of the poor heat transfer potential.

To realize the new optimality criterion step-by-step by the CFD-based evolutionary algorithm, a variation rule is introduced as follows:

( )

, , 1 , ij t ij t ij t ij m m m γ + ∆ = -= Φ -Φ (27)
where γ is the aforementioned relaxation factor in Eq. [START_REF]Concentrating Solar Power Projects[END_REF]. It should be noted that during the optimization steps, the mass flow-rate in some tubes may approach zero. Therefore the average Φ is calculated as:

( )

15 1 0 C ij i A j ij for m N = = Φ Φ = > ′ ∑∑ (28)
where N' is the number of tubes whose flow-rates are not equal to zero.

The shape of optimal fluid distribution obtained for identical outlet fluid temperatures (Fig. 5a) is used as the beginning case (step 0) this time. The implementation of the CFD-based evolutionary algorithm is similar as that shown in Fig. 2 and explained in section 2.3. The main distinction is the use of new optimality criterion of identical temperatures on the centerline so that the termination condition of the algorithm is defined as:

( ) 1, 2,...,15 j T T j = = (29)
where T is the mean temperature of all 15 j T .

Optimization results

This sub-section presents the optimization results for OC-II when Inconel 600 is used as filling material. The effects of different filling materials will be discussed in the next subsection.

Figure 7 shows the evolution of the non-uniformity of j T ( j T MF ) and the peak temperature (Tpeak) of the heated surface as a function of optimization step. It can be observed that the value of j T MF decreases from 0.049 (step 0) to less than 0.001 (step 6) during 6 optimization steps, indicating that the temperature distribution on the centerline becomes more and more uniform. The corresponding peak temperature is also reduced from 1144.7 K (step 0) to 1094.0 K (step 6). Compared to the optimality criterion of identical outlet fluid temperatures (OC-I) proposed by [START_REF] Boerema | High temperature solar thermal centralreceiver billboard design[END_REF], the newly proposed OC-II is more effective regarding the peak temperature reduction. The optimal fluid flow distribution corresponding to the OC-II is shown in the first row of Fig.

It can be observed that the vast majority of mass flow (97%) is concentrated in tubes of

layer A, showing a parabolic shape with the highest value (11.9 g•m -1 ) in the middle (tube A8)

and the lowest (4.81 g•m -1 ) at the corners (tubes A1 and A15). Several tubes in layer B (B5-B11) have a small proportion of mass flow (3%) ranging from 0.2 g•m -1 to 0.88 g•m -1 , while tubes in layer C are totally empty. By comparing Fig. 8a and Fig. 5a, one may find that increasing the mass flow-rate proportion in tubes closer to the heat source could further reduce the peak temperature of the heated surface.

Figure 8a also shows the corresponding temperature field on the heated surface. It can be observed that the isothermal lines on the heated surface are by and large perpendicular to the direction of fluid flow so that the peak temperature of the heated surface may be considered to be minimized.

Effect of different filling materials

From the viewpoint of heat transfer, Inconel 600 may not be that favorable as filling material due to its relatively low thermal conductivity (about 20 W•m -1 •K -1 ). In order to evaluate the effectiveness of the OC-II and the influence of optimal fluid distribution on peak temperature reduction under different filling material conditions, three filling materials (iron, brass and copper) with higher thermal conductivities are introduced for comparison. The physical proprieties of the studied materials within the tested temperature range used in the CFD simulations are listed in Table 3.

Table 3. Physical proprieties of different filling materials within the tested temperature (K) range [START_REF] Li | The optimization of fluid flow and heat transfer in high-temperature pressurized air solar receivers[END_REF][START_REF] Yang | Heat transfer[END_REF].

Figure 8 shows the shape of optimal fluid distribution, the corresponding temperature field of the heated surface and the minimum peak temperature for different filling materials. For every filling material, the isothermal lines are all perpendicular to the fluid flow direction, without obvious "hot spots" on the heated surface. This confirms the fact that the new optimality criterion is applicable to various filling materials with different thermal conductivities.

It can also be observed from Fig. 8a that the shape of optimal fluid distribution is greatly differed from one filling material to another. The higher the thermal conductivity of filling material, the higher proportion of mass flow will be shared by the layers far from the heated surface. When copper is used as filling material, the proportion of mass flow-rate for layer A, B and C is 35.5%, 36.9% and 27.6%, respectively. Let us notice that for Inconel 600, these values are 97.0% (layer A), 3.0% (layer B) and 0.0% (layer C). This trend is in line with the smaller thermal resistance for heat conduction when high thermal conductivity filling materials are used. Higher mass flow-rates are thus necessary for tubes in layer B or C to evacuate more heat flux transferred from the heated surface.

The obtained results also indicate that when Inconel 600 is used as filling material, multiplelayer design may not be necessary since tubes in layers B and C are almost not used. By optimizing the fluid flow distribution under the new optimality criterion, one layer of parallel tubes is sufficient for saving the materials. On the contrary when brass or copper is used as filling material, all three layers are useful. One or more additional layers could also be arranged for the receiver design with the purpose of reducing the total pressure drop with the same load.

Figure 9 presents the calculated results of peak temperature (Tpeak) for four filling materials, under different fluid flow distribution conditions, i.e. conventional distribution, uniform distribution, optimal distribution for OC-I and optimal distribution for OC-II. For every filling material, the peak temperature is the lowest when fluid flow distribution is optimized for identical temperatures on the centerline, indicating the effectiveness of the new optimality criterion and the evolutionary algorithm. It can also be observed from Fig. 9 that the use of filling material with high thermal conductivity could significant reduce the peak temperature than that with low thermal conductivity. The minimum peak temperature decreases from 1094 K for Inconel 600, to 946.4 K for iron, to 830.4 K for brass and just to 762.8 K for copper. This implies that filling material with high thermal conductivity (e.g. copper) is much more favorable in practical use regarding peak temperature reduction, but usually requires higher cost. In contrast for simpler assembly using one single refractory alloy with low thermal conductivity (e.g. Inconel 600), the peak temperature of the receiver could be minimized from 1349 K to 1094 K (below the threshold value of 1100 K) through simply optimizing the fluid flow distribution. In this sense, the efforts of fluid flow distribution optimization seem to be worthwhile regarding lower material cost and easier fabrication.

Effect of thermal contact resistance

In the above analysis, we assumed perfect contacts between different materials in the tubesin-matrix concept so that the thermal contact resistances at materials' interfaces were neglected. However, due to the fabrication process, thermal contact resistances may not be negligible especially for materials with relatively high thermal conductivity. In this section, possible effects of thermal contact resistances on the peak temperature of the receiver and on the optimal shape of flow distribution will be tested and discussed by some additional cases, using copper as the filling material.

In fact, the thermal contact resistances could be easily taken into account in our CFD-based evolutionary algorithm. To do that, additional values of thermal resistance are assigned as boundary condition for interfaces (between the shell and the filling material, and between the filling material and the Inconel tubes) in CFD simulations. For the evolutionary algorithm, an extra term (1/hc) should be added in Eq. ( 26) for representing the global thermal contact resistance: ,

ij ij ij shell ij filler ij wall ij shell ij filler ij wall c T T l l l h λ λ λ - Φ = + + + , , , , , 1 j
where c h is the global thermal contact conductance and its reciprocal is the global thermal contact resistance.

The thermal contact resistance depends on many factors such as the roughness of contact surfaces, the properties of the contacting materials (i.e. hardness, thermal conductivity, etc.), and the operational conditions (i.e. pressure, temperature, etc.). Therefore, the exact values vary widely in different cases and are difficult to be determined. In this study we used the correlation proposed by [START_REF] Antonetti | An approximate thermal contact conductance correlation[END_REF] to make a rough estimation of the order of magnitude:

0.95 0.257 4200 c s a P h R H λ -  =     ( 36
)
where P is the contact pressure, H is the surface micro hardness, Ra is the combined average roughness and s λ is the harmonic mean thermal conductivity given by [30]:

1 2 1 2 2 s λ λ λ λ λ = + (37)
where 1

λ and 2 λ are the thermal conductivities of two contacted materials (Inconel and copper). Note that under ideal condition when contacted surfaces are perfectly fabricated and very smooth, the value of Ra reaches zero, implying zero thermal contact resistances.

Using proper values for different parameters, the reasonable value of thermal contact resistance between Inconel and copper is estimated in the order of magnitude of 10 -5 m 2 •K•W - 1 . Of course this value augments when surface roughness increases or fabrication defects occur. As a result, two values, 1/hc=1×10 -5 m 2 •K•W -1 and 1/hc=1×10 -4 m 2 •K•W -1 , have been tested to represent low and high thermal contact resistance.

The values of receiver's peak temperature under different thermal contact resistances conditions are shown in Table 4. When uniform inlet flow distribution is provided, the Tpeak increases slightly from 767.6 K (1/hc=0) to 771.3 K (1/hc=1×10 -5 m 2 •K•W -1 ) and jumps towards 829.4 K (1/hc=1 × 10 -4 m 2 •K•W -1 ). This is logic since extra thermal contact resistances augment the global thermal resistance of the receiver. By performing the proposed evolutionary algorithm under the OC-II, the peak temperature of the receiver can always be minimized. As shown in Table 4, the value of Tpeak under optimized flow distribution condition is always lower than that under uniform flow distribution, for the same value of thermal contact resistance. Nevertheless, the reduction of Tpeak by optimizing the flow distribution (from 829.4 K to 821.4 K) is less significant compared to the negative effect (from 767.6 K to 829.4 K) of thermal contact resistances for the highest value of 1/hc (1×10 -4 m 2 •K•W -1 ). The optimal shapes of flow distribution by considering different thermal contact resistances are shown in Fig. 10. Compared to Fig. 8a for zero thermal contact resistance, it can be observed that the fluid flow slightly shifts to the tubes in layer A and also in the middle of solar receiver. However, the global shape of optimized flow distribution (under OC-II) seems similar.

From the previous results, we can conclude that the proposed evolutionary algorithm can determine the optimal fluid flow distribution for a minimum peak temperature, taking the possible thermal contact resistances into account. For a more precise analysis however, the exact values of thermal contact resistance should be experimentally measured rather than rough estimation.

Discussions

Compared to the optimality criterion of identical outlet fluid temperatures (OC-I) proposed by [START_REF] Boerema | High temperature solar thermal centralreceiver billboard design[END_REF], the new optimality criterion (OC-II) proposed in this paper is more effective for peak temperature reduction. It is developed for a multiple-layer tubular solar receiver, surely applicable for single-layer designs. In fact a simplified variation rule could be used for a single-layer receiver design by comparing directly the temperatures on the upper line. Eq. ( 25)

could then be modified as:

( ) , , 1 , j t j t j t j m m m T T γ + ∆ = - = - (38)
It should be noted that a Gaussian shape heat flux distribution is assumed in the current study, which is perfectly symmetrical with its focusing point at the center of the heated surface. In reality, the shape of heat flux distribution could be less regular with deviated focusing point [START_REF] Salomé | Control of the flux distribution on a solar tower receiver using an optimized aiming point strategy: Application to THEMIS solar tower[END_REF]. In this case, the studied A-A plane should be adjusted according to the location of the heat flux peak. The new optimality criterion (OC-II) is still applicable in its more general form: identical temperatures on the upper line of the cross-section where the heat flux peak is located.

For the upper heated surface, we used the net heat flux absorbed by the heated surface as the boundary condition for the simulations and optimization to avoid potential influences. But in real operation conditions, the heat losses due to thermal emissions of the heated surface and convection may play an important role on the temperature distribution thus should be carefully considered but may be studied only for a specific solar receiver geometry.

Results also imply that for a multiple-layer design, uniform fluid flow distribution is not the optimal option even when net heat flux is uniform. The proportions of mass flow in tubes from one layer to another will be varied, depending on their distance to the heat source. In other words, uniform fluid distribution is only a special case of optimal solution for uniform net heat flux and single-layer design.

There are of course other advanced numerical optimization methods available for the same objective. One obvious option is brute-force search or exhaustive search, consisting of systematically enumerating all possible fluid flow distributions for the objective function and choosing the best one. Another option is the Genetic Algorithm (GA) that mimicries the natural selection of species to generate useful solution (fluid distribution) to minimize the objective function (peak temperature). Both options are characterized by a poor sensitivity to local minima so that the global best solution can theoretically be proposed. Nevertheless, thousands or even more simulations should be performed for one case study, which make them prohibitive regarding the requirements on calculation time and facilities. In contrast, the CFD-based evolutionary algorithm coupled with the new optimality criterion could propose the optimal solution after a limited number of optimization steps. In this sense, this method is effective, general and flexible.

Conclusion and perspectives

In this article, an original CFD-based evolutionary algorithm is developed to determine the optimal fluid flow distribution for minimizing the peak temperature of a multiple-layer tubular solar receiver (tube-in-matrix concept) subjected to non-uniform net heat flux. Two optimality criteria are used and tested, i.e. identical outlet fluid temperatures (OC-I) and identical temperatures on the centerline (OC-II). Different shapes of optimal fluid flow distribution under different filling material conditions are determined and discussed. Based on the numerical results obtained, main conclusions can now be summarized as follows.

• Identical outlet fluid temperatures can be conveniently achieved by performing the evolutionary algorithm. Using this optimality criterion, the peak temperature of the heated surface may be reduced but not be minimized.

• Identical temperatures on the centerline of the heated surface can also be achieved with the evolutionary algorithm. The isothermal lines on the heated surface are byand-large perpendicular to the fluid flow direction, suggesting the minimization of peak temperature.

• The shape of the optimal fluid flow distribution depends strongly on the thermal conductivity of the filling material. The higher the thermal conductivity of filling material, the higher fraction of mass flow should be concentrated in the layer closest to the heated surface, and vice versa. As a result, the number of layers is also a parameter for optimization in solar receiver design.

• Filling material with high thermal conductivity is much more favorable in practical use regarding peak temperature reduction, but often requires higher cost. Moreover, using different filling materials between tubes would raise the question of galvanic action, especially at high temperature. For the purpose of cost saving, manufacturing simplification and ageing resistance, single material with low thermal conductivity can also be used through optimizing the fluid flow distribution, so that the peak temperature could be minimized to be below the threshold value.

• The thermal contact resistances at different interfaces may have a great impact on the peak temperature of the receiver. Thus, the thermal contact resistance should be carefully measured and taken into account in the analysis.

The application fields of the developed evolutionary algorithm are rather vast and promising, and are not limited in CSP solar receivers. It can determine the optimal fluid flow distribution for various engineering equipment or processes, such as for chemical reactors, energy storage systems or heat sinks for cooling of electronic devices. Nevertheless, the objective function should be defined case by case and the corresponding optimality criterion should be well selected.

Once the shape of optimal fluid flow distribution is determined, the next question is how to realize it. This issue involves the novel design of proper fluid distributors/collectors to realize the non-uniform but optimized fluid flow distribution. One practical method is to install a geometrically optimized perforated baffle at the distributing manifold to reach the target flow distribution among downstream parallel channels. Some preliminary work on this subject may be found in the references [START_REF] Luo | Heuristic shape optimization of baffled fluid distributor for uniform flow distribution[END_REF][START_REF] Wei | CFD-based evolutionary algorithm for the realization of target fluid flow distribution among parallel channels[END_REF]. A detailed design and optimization of baffled fluid distributor/collector for the multiple-layer solar receiver is a direction of our future work.

It should be noted that the efficiency of the CFD-based evolutionary algorithm depends largely on simulated flow and temperature fields while there is always a departure from the reality. Experimental validation of the evolutionary algorithm using the same tubular solar receiver is our ongoing work.

Notations

C specific heat of solid J•kg -1 •K -1
Cp specific heat of fluid

•s -1 •m -3 t time step - T temperature K u velocity m•s -1
Greek symbols γ relaxation factor - The surface peak temperature can be reduced by 100 °C and the thermal efficiency increases by 2% due to the bayonet receiver.

λ thermal conductivity W•m -1 •K -1 s λ harmonic mean thermal conductivity W•m -1 •K -1 µ viscosity kg•m -1 •s -1 ∏ stress tensor - ρ density kg•m -3 Φ conduction heat flux W•m -2 Subscripts 1,2 material index f fluid i, j tube index max maximum Superscripts in inlet out outlet tot total

List of figures

List of tables

The corrosion rate and salt decomposition ratio have decreased. T in =300 °C; Heat flux=400-1000 kW•m -2 ; mass 800 °C/-The pressurized air can be heated up to 800 °C and further heated with natural gas to reach the working temperature of Table 2. Physical properties of fluid and solid used for simulations [START_REF] Li | The optimization of fluid flow and heat transfer in high-temperature pressurized air solar receivers[END_REF].

Property Fitting correlation

Pressurized air (300 K<T<1000 K) Density (kg•m -3 )

Table 3. Physical proprieties of different filling materials within the tested temperature (K) range [START_REF] Yang | Heat transfer[END_REF][START_REF] Li | The optimization of fluid flow and heat transfer in high-temperature pressurized air solar receivers[END_REF].

c h - = × m 2 •K•W -1 m 2 •K•W -1 m 2 •K•W -

Fig. 1 .

 1 Fig.1. Geometry and dimension of the solar receiver based on tube-in-matrix concept (unit: mm).

Figure 1

 1 Figure 1 also shows the sectional view of the A-A plane along the centerline of the upper surface. It may be observed that the solar receiver based on tube-in-matrix concept consists of 3 parts from the outside to the inside: the external shell, the filling material and a bunch of parallel straight tubes. The external shell of the receiver is made of Inconel 600. Inside the external shell are 45 parallel straight tubes (N=45) arranged symmetrically in three aligned layers, each layer having 15 tubes. For the convenience of description, these tubes are indexed by i as layer A to C from top to bottom, and by j as 1 to 15 from left to right for each layer.

2 where m 2 .

 2m2 is the fluid mass flow-rate, Q the heat transfer rate, Cp the specific heat of the fluid 3 and T the fluid temperature. The superscripts tot, in and out stand for total, inlet and outlet, 4 respectively. in T is the identical inlet fluid temperature ( Basic principle for identical outlet fluid temperature distribution 9 The evolutionary algorithm is developed to determine the fluid flow distribution among parallel tubes corresponding to the selected optimality criterion (OC-I: identical outlet fluid temperatures): fluid temperature is directly linked to the mass flow-rate of fluid in the tube, the optimality criterion could be reached by augmenting or reducing the flow-rate in the tubes by comparing out ij T and out T , while keeping the total mass flow-rate (m tot ) constant. More precisely, if the outlet fluid temperature of the tube ij is lower than the mean value, then the flow-rate of this tube ij m should be reduced. Vice versa if out ij T is higher than out T , ij m should be increased to reach a lower outlet fluid temperature. The variation of different ij m according to the difference between out ij

( 3 )

 3 Input an initial inlet fluid flow distribution among parallel tubes, e.g. a uniform flow distribution as the beginning of evolutionary algorithm.

( 4 )

 4 Calculation of the exact flow and temperature fields by solving the Navier-Stokes and heat transfer equations.

Fig. 2 .

 2 Fig. 2. Flow chart of the evolutionary algorithm.

  r was the distance to the center point, e the base of the natural logarithm (2.71828) and h the local heat flux. The maximum local heat flux was 650 kW•m -2 and the average heat flux of the whole surface was 284.6 kW•m -2 . Note that this Gaussian-shape distribution is just an example used for the current study. It could be replaced by any other types of distribution without influencing the evolutionary algorithm. Other external walls of the receiver were considered as adiabatic without heat exchange with the environment.

Fig. 3 .

 3 Fig. 3. Gaussian net heat flux (kW•m -2 ) absorbed by the heated surface of the solar receiver.

Figure 4

 4 Figure 4 shows the evolution of various parameters as a function of the optimization step of the evolutionary algorithm, including the non-uniformity of the outlet fluid temperatures (MFT), the peak temperature of the receiver (Tpeak), the non-uniformity for fluid flow distribution (MFf) and the maximum pressure drop (Δpmax) of the receiver. Note that step -1 stands for the aforementioned conventional fluid flow distribution shown in Fig. 5a as reference case introduced for comparison and step 0 represents the uniform fluid flow distribution (3 g•s -1 in each tube) used as the initial distribution for optimization. As shown on Fig. 4a, the values of MFT are 0.159 for the conventional fluid distribution case (step -1) and 0.140 for the uniform fluid distribution case (step 0). The distribution of outlet fluid temperatures is obviously non-uniform for both cases. When the evolutionary algorithm proceeds from one step to the next, the value of MFT decreases rapidly, implying more uniform distribution of outlet fluid temperatures. Finally identical outlet fluid temperatures (MFT<0.001) can be reached after six steps of optimization, showing that the CFD-based evolutionary algorithm is effective and rapid to achieve the selected optimality criterion. Also shown on Fig. 4a is the evolution of MFf as a function of optimization step. The value of MFf increases from 0 for the uniform fluid distribution case (step 0) to 1.053 for the optimal case (step 6), indicating a less uniform flow distribution with respect to the conventional fluid distribution case (step -1, MFf=0.328). The shape of the optimal fluid flow distribution will be shown and discussed later. It can be observed from Fig. 4b that the evolution of Δpmax shows a similar trend as that of the MFf curve. This is obviously in line with the actual situation since the maximum pressure drop is directly linked to the highest flow-rate of all tubes. As a result, the lowest value of Δpmax (1343.3 Pa) is reached when fluid flow distribution is uniform whereas Δpmax increases with the increasing non-uniformity of fluid flow distribution with its highest value of about 7332 Pa.

Fig. 4 .

 4 Fig. 4. Evolution of different parameters as a function of optimization step. (a) non-uniformity of outlet fluid temperatures (MFT) and non-uniformity of fluid flow distribution (MFf); (b) surface peak temperature (Tpeak) and maximum pressure drop (Δpmax).

Fig. 5 .

 5 Fig. 5. Fluid flow and thermal characteristics under three different fluid flow distribution cases. (a) fluid flow distribution; (b) outlet fluid temperature distribution; (c) temperature field on the heated surface of the receiver.

Fig. 6 .

 6 Fig. 6. Simplified heat transfer model for the A-A plane: (a) single material; (b) shell and filler.

  consider that the new optimality criterion is achieved when

Fig. 7 .

 7 Fig. 7. Evolution of surface peak temperature (Tpeak) and non-uniformity of j T (

Fig. 8 .

 8 Fig. 8. Effects of different filling materials on the optimal fluid distribution (a) and the corresponding temperature field of the heated surface (b)

Fig. 9 .

 9 Fig. 9. Peak temperature reduction for different filling materials.

Fig. 10 .

 10 Fig. 10. Optimal fluid flow distribution considering thermal contact resistances (copper as filling material). (a) 1/hc=1×10 -5 m 2 •K•W -1 ; (b) 1/hc=1×10 -4 m 2 •K•W -1

Fig. 1 .

 1 Fig.1. Geometry and dimension of the solar receiver based on tube-in-matrix concept (unit: mm).

Fig. 2 .

 2 Fig. 2. Flow chart of the evolutionary algorithm.

Fig. 3 .

 3 Fig. 3. Gaussian net heat flux (kW•m -2 ) absorbed by the heated surface of the solar receiver.

Fig. 4 .

 4 Fig. 4. Evolution of different parameters as a function of optimization step. (a) non-uniformity of outlet fluid temperatures (MFT) and non-uniformity of fluid flow distribution (MFf); (b) surface peak temperature (Tpeak) and maximum pressure drop (Δpmax).

Fig. 5 .

 5 Fig. 5. Fluid flow and thermal characteristics under three different fluid flow distribution cases. (a) fluid flow distribution; (b) outlet fluid temperature distribution; (c) temperature field on the heated surface of solar receiver.

Fig. 6 .Fig. 7 .Fig. 8 .

 678 Fig. 6. Simplified heat transfer model for the A-A plane: (a) single material; (b) shell and filler.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Peak temperature reduction for different filling materials.

  °C; Centered heat flux distribution, offcentered flux distribution and alternative flux distribution, average irradiance=737 kW•m -2 570 °C/~90% The ideal flow receiver has the lowest surface peak temperature. The multi-pass receiver out-performs the other designs by reducing the risk from irradiance changes. diameter=8.4 m; total number of panels=18; number of tubes per panel=22; external diameter of outer tube=60.3 mm; external diameter of inner tube=52 mm Molten salt T in =290 °C; Average heat flux=800 kW•m -2 , maximum heat flux=1200 kW•m -2 ; Total mass flow=290 kg•s -1 565 °C/~75%

Table 4 .

 4 Tpeak under different thermal contact resistances Shape of fluid flow distribution 1

Table 4 .

 4 Tpeak under different thermal contact resistances

1 Table 1 .

 11 Some recent studies on tubular solar receiver (N: numerical study; E: experimental study).This low temperature receiver module design results in an average temperature increase of about 200 K for air and an associated pressure drop of 100 mbar. The maximum tube surface temperature is 950 °C at a module air outlet temperature of 550 °C.The design outlet fluid temperature is reached in both open aperture and quartz window configuration. The low efficiency in the experiments is due to heat loss through the cavity walls and lower-than-expected mass flow.

	2
	Study	Study type	Geometry & material	Heat transfer fluid	Inlet working and test conditions	Outlet efficiency temperature /thermal	Remarks
	Heller et al. (2006)	E	SOLGATE low temperature receiver module: number of tubes=16, multi-tube coil attached to a hexagonal secondary concentrator	Pressurized air	T in =300 °C; average irradiance=770 kW•m -2 kg•s -1 45 heliostats; air mass flow=1.4-1.6	960 °C/~70% (overall SOLGATE)
					T in =600 °C; input power=230-290 kW;
	Amsbeck et al. (2008; 2010)	N, E	SOLHYCO-receiver: length=2.5 m, number of tubes=40, nickel based super alloy Inconel 600	Air	open aperture and with quartz window; 38-46 heliostats; mass flow=0.8 kg•s -1 (N), 0.51 kg•s -1 (E)	800 °C/~70% (N), ~40% (E)

Acknowledgement

One of the authors M. Min WEI would like to thank the French CNRS and "Région Pays de la Loire" for their financial support to his PhD study.


Nathalie Debono

Jean-Claude Daran

Rinaldo Poli

Agnès Labande
email: agnes.labande@lcc-toulouse.fr

A rhodium

Keywords:

  L'archive ouverte pluridisciplinaire

Introduction

Wrighton et al. first demonstrated in the early 1990s that fine tuning of the electronic properties of a metal complex could be achieved by the introduction of redox-active ligands, as they were able to enhance the reactivity of a carbonyl ligand coordinated to rhenium towards nucleophiles by changing the oxidation state of cobaltocene [1]. In the area of catalysis, Mirkin also described the use of hemilabile redox-active ligands that presented charge-dependent behaviour [2]. Later, Gibson and Long showed that they could reversibly modulate the activity of a titanium polymerisation catalyst with the help of a substitutionally inert, redox-active ligand [3]. More recent examples include the work of Bielawski [4][5][6], Diaconescu [7,8] and Plenio [9,10].

Many redox-active groups have been incorporated, such as the ubiquitous ferrocene [2][3][4][5][9][10][11][12][13], or other metallocenes [1,14], quinone [6], terthienyl [15], amidophenolates [16], etc. However, it was only recently that the redox properties of N-heterocyclic carbenes (NHC) bearing electroactive groups were exploited in catalysis. The NHC can either be in direct electronic communication with the electroactive moiety (ferrocene [5], quinone [6]), or simply act as an ancillary ligand or "anchor" to guarantee the stability of the complex whatever the oxidation state of the electroactive component [9].

Our group has a strong interest in the chemistry and catalytic activity of functionalised Nheterocyclic carbene complexes [17][18][19][20][21]. We have recently investigated the behaviour of a 1,5-cyclooctadiene (COD) rhodium(I) complex bearing a ferrocenyl phosphine-NHC ligand, 1, towards oxidation (figure 1) [21]. We have been able to demonstrate that iron oxidation has dramatic consequences on the reactivity of the rhodium complex, and we then wished to extend this work to a similar rhodium(I) complex possessing carbonyl ligands in place of COD. The results of these studies are described here.

Experimental

General considerations

All manipulations were carried out under an inert atmosphere of dry argon by using vacuum line and Schlenk tube techniques. Solvents for all syntheses were dried and degassed by standard methods before use, unless otherwise stated.

The 1D-and 2D-NMR spectra were recorded on Bruker AV500 and AV400 spectrometers. 1 H and 13 C chemicals shifts (δ) are given in ppm (the residual peak of deuterated solvents was used as reference). 31 P chemicals shifts are reported in ppm. Peaks are labelled as singlet (s), doublet (d), triplet (t), multiplet (m) and broad (br). The proton and carbon assignment were performed by COSY, HSQC, 1 H- 13 C HMBC, 1 H-31 P HMBC experiments. The IR spectra were acquired on a Bruker Alpha-P spectrometer equipped with an ATR module (compound 2) or Perkin-Elmer Spectrum 100 FT-IR with an ATR module (compound 3). The MS spectra were obtained from the mass spectrometry service of the Paul Sabatier University. The cyclic voltammetry (CV) experiment was carried out with a PGSTAT-302N potentiostat (Metrohm) at 25°C using a Pt disk working electrode, a Pt wire counter electrode and a saturated calomel reference electrode (SCE); a 0.1 M n-Bu4BF4 solution was used as the supporting electrolyte. The solution used during the electrochemical study was typically 10 -3 M in rhodium compound. All electrochemical data are referenced versus ferrocene (in our hands, E°[FcH/FcH + ] = 0.54 V/SCE in CH2Cl2/n-Bu4BF4, 0.1 V.s -1 , 25°C).

In a glovebox, AgBF4 (10 mg, 0.049 mmol, 1.5 equiv.) was added, as a solid, to a solution of complex 2 (23 mg, 0.033 mmol) in CH2Cl2 (3 mL). The solution turned from yellow to deep green in a few seconds. Stirring was stopped to allow the silver precipitate to settle and an aliquot of the solution was deposited on the IR cell. After evaporation of the solvent, the green solid was analysed: IR (ATR) 2097 (s), 2045 (s) cm -1 (C=O).

Oxidation of [Rh(CO)2(L)] + BF4 -(2) in MeCN: generation of [Rh(CH3CN)3(L)](BF4)2 (3)

A solution of thianthrenium tetrafluoroborate (27 mg, 8.9•10 -5 mole, 1.5 equiv.) in CH3CN (2.5 mL) was added dropwise to a solution of 2 (42 mg, 5.9•10 -5 mole, 1 equiv.) in CH3CN (5 mL). The mixture was stirred for 10 min at room temperature and then the solvent was evaporated in vacuo. The complex was precipitated with a CH3CN/Et2O (1/9) mixture several times until the solution was colourless, then taken up into CH3CN and left standing for 48h.

The solvent was evaporated in vacuo to give 3 as an orange solid (38.8

General procedure for the hydrosilylation of acetophenone derivatives

The rhodium catalyst (4•10 -6 mole, 1 mol%) was placed in a dry Schlenk tube under argon and THF (0.2 mL) and acetophenone (3.9•10 -4 mole, 1 equiv., 2.0 M in THF) were added. Diphenylsilane (4.3•10 -4 mole, 1.1 equiv.) was then added dropwise at room temperature and the resulting mixture was stirred for 17h. The solvent was concentrated in vacuo, the residue taken up into CH2Cl2 and MeOH (1 mL) and stirred for 1h. HCl (2 M, 1 mL) was then added and the mixture was stirred for an additional hour. The phases were separated, the aqueous phase extracted with CH2Cl2 and the organic phases were dried (MgSO4), filtered and concentrated in vacuo. The residue was analysed by 1 H NMR (CDCl3) and the conversions were calculated by integration and comparison of the characteristic signals of acetophenone ( 2.63, s, 3H) and 1-phenylethanol ( 4.90, q, 1H).

Kinetic follow-up: the rhodium catalyst (8•10 -6 mole, 1 mol%) was placed in a dry NMR tube under argon and d8-THF (0.4 mL) and acetophenone (7.9•10 -4 mole, 1 equiv., 2.0 M in THF)

were added. Diphenylsilane (8.7•10 -4 mole, 1.1 equiv.) was then added dropwise at room temperature and the resulting mixture was shaken. The reaction mixture was analysed by 1 H NMR and the conversions were calculated by integration and comparison of the characteristic signals of acetophenone ( 2.50, s, 3H) and diphenyl(1-phenylethoxy)silane ( 1.53, d, 3H).

X-ray structural analyses

A single crystal of compound 2 was mounted under inert perfluoropolyether at the tip of a glass fiber and cooled in the cryostream of an Oxford-Diffraction Xcalibur CCD diffractometer. The structure was solved by direct methods (SIR97 [23]) and refined by leastsquares procedures on F 2 using SHELXL-97 [24]. All H atoms attached to carbon were introduced in calculation in idealized positions and treated as riding models. The drawing of the molecules was realized with the help of ORTEP32 [25].

Crystal data for compound 2: yellow needles, C29H25FeN2O2PRh

Results and discussion

1. Synthesis and characterisation of the rhodium(I) dicarbonyl complex 2

The synthesis of the precursor rhodium complex 1 was reported in a previous work [17]. It is perfectly air stable and was not altered after several months of storage. Its 31 P NMR spectrum shows a doublet at δ 19.5 (JRh-P= 158 Hz), whereas the 13 C NMR spectrum shows a doublet of doublets situated at δ 176.9 ( 1 JRh-C= 49.7 Hz, 2 JP-C= 15.5 Hz), typical of the C2 carbon in Rh(I) NHC complexes.

Scheme 1. Synthesis of rhodium(I) carbonyl complex 2.

The dicarbonyl complex 2 was prepared in quasi-quantitative yield, as verified by 31 P NMR monitoring, by bubbling CO gas through a solution of 1 in CH2Cl2 (scheme 1). As observed for other cationic Rh(I) complexes [26,27], the resulting pale yellow solid is moderately stable in solution and decomposes in a few days when left in air. It is stable under argon in the solid form, however, and was fully characterised by conventional methods. Its 31 P NMR spectrum shows a doublet at δ 20.5, with a smaller coupling constant to rhodium (JRh-P= 132 Hz) than that of complex 1. The 1 H NMR spectrum revealed the disappearance of the COD proton signals, whereas typical signals for coordinated CO ligands were attributed in the 13 C NMR spectrum at δ 186.7 (dd, 1 JRh-C= 55. 7 Hz, 2 JP-C= 16.0 Hz, cis to P) and 181.8 (dd, 1 JRh-C= 61.2 Hz, 2 JP-C= 102. 3 Hz, trans to P). The characteristic 13 C NMR signal of the C2 carbon (dd, δ 165.9, 1 JRh-C= 42.1 Hz, 2 JP-C= 16.2 Hz) was found upfield compared to that of complex 1, which is consistent with similar Rh(I) NHC complexes [26,28]. Finally infra-red spectroscopy, which is a very useful tool for the characterisation of carbonyl compounds, showed two intense bands for CO stretching frequencies at 2084 and 2029 cm -1 , which are in accord with other cationic NHC-Rh complexes [26,27]. The presence of two bands of equal intensity is consistent with the expected cis arrangement of the two carbonyl groups.

X-ray quality crystals of 2 were obtained by slow diffusion of pentane into a CH2Cl2 solution of the complex under 1 atm of CO, the diffraction analysis of which confirmed the expected structure (fig. 2), with a typical square-planar geometry about the rhodium atom. The structure does not present any unusual features. Selected bond lengths and angles are listed in Table 1. The carbene-rhodium and phosphorus-rhodium bond lengths are consistent with those measured in related Rh(I)-NHC complexes [28,29] and are similar, within errors, to those measured for the previously described Rh(I)-NHC COD complex 1 [17]. The Rh-carbonyl bond lengths are also within the expected range for this type of compounds.

Oxidation studies

The rhodium complex 2 was analysed by electrochemical methods. The cyclic voltammogram of 2 was recorded in CH2Cl2 (fig. 3). Similarly to that of complex 1 [21], the voltammogram shows a reversible redox wave for the ferrocene moiety at E1/2 = 0.43 V/[FcH/FcH + ] followed by an irreversible process at 1.06 V/[FcH/FcH + ], that could be assigned to the oxidation of Rh(I). By analogy with 1, we can assume that the oxidation process at 0.43 V is a one-electron oxidation of ferrocene to ferrocenium. Although it was possible to generate the oxidised complexes by bulk electrolysis, we ruled this procedure out for preparative purposes since it could be difficult to separate the expected product from the ionic supporting electrolyte. We thus turned to chemical oxidation. The choice of oxidant was guided by the E1/2 values of the ferrocene/ferrocenium redox couple, obtained via electrochemical methods, but other parameters had to also be taken into account, such as the "innocent" character of the reagent and the ease of purification of the oxidised Rh complex [30]. NOBF4 is a strong oxidant but reacted with the phosphine group, giving a phosphine oxide. AgBF4 was the most obvious choice, however its oxidation potential is very solvent-dependent and should perform the oxidation of the ferrocenyl moiety in 1 and 2 in CH2Cl2 (E°= 0.65 V/[FcH/FcH + ]), but not in a coordinating solvent like MeCN (E°= 0.04 V/[FcH/FcH + ]). We therefore focused our attention on thianthrenium tetrafluoroborate [Th][BF4] (E°= 0.86 V/[FcH/FcH + ] in MeCN), the oxidation potential of which does not vary much with the solvent [22].

[Th][BF4] gives deep purple solutions in acetonitrile and is insoluble in ether, whereas its reduced form is colourless and soluble in ether. This means that the reactions can be followed visually and that the oxidised rhodium complexes may be purified by simple precipitation and washing with Et2O.

Chemical oxidation of complexes 1 and 2 in dichloromethane

Complex 1 was oxidised in CH2Cl2, either by addition of solid, anhydrous AgBF4 [31], or by a [Th][BF4] solution in the same solvent at room temperature (scheme 2). The orange dichloromethane solution turned rapidly olive-green, at which point it was concentrated to ca. 0.5 mL, and diethyl ether was added. A dark green precipitate separated from the light yellow solution, which was filtered off. The green solid was further rinsed with diethyl ether and analysed by 1 H and 31 P NMR in CD2Cl2 (fig. 4).

Scheme 2. Oxidation of rhodium(I) complex 1 with [Th]

[BF4] in CH2Cl2. It has been demonstrated by Wrighton et al. that infra-red CO values in transition metal carbonyl complexes with ferrocenylphosphino ligands could be increased by up to 20 cm -1 upon oxidation of ferrocene [11]. More recently, Bielawski et al. measured the values for a carbonyl rhodium complex bearing a diaminocarbene-ferrocenophane ligand in both reduced and oxidised states, and found on average a difference of 20 cm -1 between the two oxidation states [33]. Infra-red thus appeared to us as a powerful tool to demonstrate the influence of ferrocene oxidation state on the electron density at rhodium in our complexes. In a typical procedure, solid, anhydrous AgBF4 (1.5 equiv.) was used to generate the oxidised form of carbonyl complex 2 in CH2Cl2 (scheme 3). The colour turned in a few seconds from light yellow to forest green [START_REF]2 ox is very sensitive to moisture and turned rapidly from forest green to brown when exposed to traces of water to give a complex mixture[END_REF].

Scheme 3. Oxidation of rhodium(I) carbonyl complex 2 with AgBF4 in CH2Cl2.

Figure 5. IR spectra (ATR), oxidation of complex 2 with AgBF4 in CH2Cl2. CO(2 red ), cm -1 : 2084, 2029; after 45min, CO(2 ox ), cm -1 : 2097, 2045. The small band at 1997 cm -1 for the initial solution is attributed to a monocarbonyl impurity (see Experimental section).

IR spectra were taken at intervals and showed the shift of the original CO bands to higher stretching frequencies by 13 cm -1 for the higher frequency band and by 16 cm -1 for that at lower frequency (fig. 5) [35]. This is characteristic of a metal centre with a lower electron density, since decreased backbonding from the d-orbital of the metal to the * antibonding orbital strengthens the CO bonds and shifts the CO bands toward higher frequencies. The average difference between the CO values for 2 red and 2 ox is 14 cm -1 , which is significant enough to assert that the reactivity at the rhodium centre may be affected by the oxidation of the ferrocene unit.

Chemical oxidation of complexes 1 and 2 in acetonitrile

As AgBF4 has a redox potential of 0.04 V/[FcH/FcH + ] in MeCN [29], it should not be capable of oxidizing the ferrocene moiety of 2. We therefore turned to [Th][BF4] in MeCN was added dropwise to a solution of complex 2 in the same solvent (scheme 4). As reported earlier in the case of 1, we observed a very different behaviour from the reaction in dichloromethane: no colour change to green was detected. Rather, a brownorange solution was obtained after the addition of ca. 1.5 equivalent of oxidant. At the end of the addition, the solvent was evaporated to a minimum and diethyl ether was added to precipitate the rhodium complex and wash off the products of [Th][BF4] reduction as well as unreacted 2.

The 31 P NMR spectrum of the crude reaction mixture showed one major species appearing as a doublet at δ 41.5, with a coupling constant of 138 Hz, along with some unreacted 2, whereas the 1 H NMR spectrum was typical of a diamagnetic compound, and revealed the presence of bound acetonitrile molecules. We demonstrated earlier, for the oxidation of 1 in acetonitrile, that the species showing a doublet at δ 41.5 is a rhodium(III) complex resulting from the oxidation of the ferrocene unit to ferrocenium and subsequent electron transfer from rhodium to ferrocenium [21]. The reorganisation of the coordination sphere around rhodium to adopt the preferred octahedral geometry implied coordination of acetonitrile molecules. An C-H activation at ferrocene followed this reorganisation to give the air-stable rhodium(III) complex 3. Here we demonstrate that the oxidations of complex 1 (bearing a COD ligand) and the dicarbonyl complex 2 lead to the same species (scheme 4). The full characterization of the product 3, which includes the X-ray diffraction analysis of a bipy derivative, is reported in our previous contribution [21]. The reaction of the carbonyl complex 2 was also followed by 31 P NMR (fig. 6). The species giving a doublet at δ 28.7 was previously shown to be the intermediate rhodium(III) species (4) that has not yet undergone the C-H activation at ferrocene (scheme 5): it was the main species observed after 10 min, along with some unreacted 2 and an unidentified by-product at δ ca. 25. Oxidation of rhodium(I) to rhodium(III) is thus very quick and is followed by the slow C-H activation at ferrocene to give complex 3 as the main species after 24 h. In order to check that ferrocene could efficiently act as an electron relay, complex 2 was first oxidized in CH2Cl2 with an excess of AgBF4 (1.5 equiv.), giving rise to the forest-green, paramagnetic ferrocenium-Rh(I) complex 2 ox (scheme 6). After filtration of insoluble silver, the solution was evaporated in vacuo to give a green solid: upon addition of MeCN, the colour instantly turned to orange-brown. After 24h standing at room temperature, the solvent was evaporated and the orange residue analysed by 31 P and 1 H NMR. The 31 P NMR spectrum revealed the presence of two species in ca. 75:25 ratio: the doublet situated at δ 41.6 (JRh-P = 138 Hz), characteristic of complex 3, and a doublet at δ 33.2 (JRh-P= 120 Hz) that might arise from the reaction of 2 red with acetonitrile. The 1 H NMR spectrum confirmed the presence of complex 3 as the main product. The product distribution is consistent with the oxidation stoichiometry (two oxidizing equivalents are necessary to convert 2 red to 3).

(a) Reagents and conditions: acetophenone (1.0 equiv., 2M in THF), diphenylsilane (1.1 equiv.), Rh cat. (1mol%), rt, 17 h. (b) Average of two runs, with % difference between the two runs in parentheses; conversion calculated by integration of characteristic 1 H NMR signals of acetophenone derivative and 1-phenylethanol derivative in the crude reaction mixture after hydrolysis of the silyl ether.

As a general observation, the rhodium(III) complex 3 shows good activity for the hydrosilylation of both electron-rich and electron-deficient acetophenones and is a better catalyst than the rhodium(I) complex 1. The reaction of diphenylsilane with acetophenone was followed by 1 H NMR in d8-THF. The conversion of acetophenone into the expected silyl ether was plotted against time and the reaction profiles turned out surprisingly different for 1 and 3 (fig. 7) [START_REF]silyl enol ether by-product was neglected in the calculations since it represents less than 1% of the total[END_REF]. Comparison of the initial rates shows that the catalytic activity of 3 is 20 times greater than that of 1. For complex 1, the reaction follows a first order rate law, whereas the data for 3 do not appear consistent with a simple rate law.

Conclusions

A new dicarbonyl rhodium(I) complex bearing an electroactive ferrocenyl phosphine-NHC ligand has been synthesized and fully characterized. Its behaviour in the presence of a chemical oxidant has been studied, proving that the electron density at the rhodium centre could be tuned by oxidation of the ferrocenyl group on the bidentate ligand. This difference has been highlighted by infra-red spectroscopic measurement of the CO stretching frequencies. The dicarbonyl rhodium(I) complex has also shown an interesting reactivity in acetonitrile which parallels that of the related COD complex 1, giving a stable rhodium(III) complex after several oxidation-electron transfer processes and complete reorganisation of the coordination sphere.

A

  rhodium(I) dicarbonyl complex with a redox-active ferrocenyl phosphine-NHC ligand: Enhanced reactivity of the metal centre through ferrocene oxidation Nathalie Debono, Jean-Claude Daran, Rinaldo Poli, A. Labande To cite this version: Nathalie Debono, Jean-Claude Daran, Rinaldo Poli, A. Labande. A rhodium(I) dicarbonyl complex with a redox-active ferrocenyl phosphine-NHC ligand: Enhanced reactivity of the metal centre through ferrocene oxidation. Polyhedron, 2015, 86, pp.57-63. 10.1016/j.poly.2014.04.050. hal-01919614

Figure 1 .

 1 Figure 1. Rhodium(I) complex 1 with redox-active ferrocenyl phosphine-NHC ligand.

Figure 2 .

 2 Figure 2. ORTEP representation of the cation in the dicarbonyl complex 2. Ellipsoids are shown at the 30% probability level. All hydrogen atoms are omitted for clarity.

Figure 3 .

 3 Figure 3. Cyclic voltammogram on a Pt electrode of complex 2: 1 mM in CH2Cl2 with nBu4NBF4 (0.1 M) at a scan rate of 0.2 V s -1 .

Figure 4 .

 4 Figure 4. 31 P NMR spectra: oxidation of complex 1 by [Th][BF4] in CD2Cl2.The 1 H NMR spectrum showed very broad signals stretching from δ 25 to -10, whereas no or very weak shifted signals were detected by 31 P NMR. The characteristic colour and the absence of 31 P NMR signal suggested to us that we had generated paramagnetic ferrocenium. The reversible nature of this oxidation was demonstrated when excess water was added to the green CD2Cl2 solution [32]: the solution became orange again and its 31 P NMR spectrum showed a doublet at δ 19.5 with a coupling constant of 158 Hz. Purification of the NMR sample by simple drying and filtration on neutral alumina allowed the complete recovery of the starting Rh(I) complex.

Scheme 4 .

 4 Scheme 4. Oxidation of complexes 1 and 2 with [Th][BF4] in MeCN.

Figure 6 .Scheme 5 .

 65 Figure 6. 31 P NMR spectra: oxidation of complex 2 by [Th][BF4] in MeCN.

Figure 7 .

 7 Figure 7. Reaction of Ph2SiH2 with acetophenone (1 mol% catalyst, d8-THF, rt). Conversion (%) of acetophenone vs. time, circles: complex 1; triangles: complex 3.

Table 1 .

 1 Selected bond distances (Å) and angles (°) for the experimental structure of 2 and comparison with those of 1[17].

	Bond	Complex 1 [17]	Complex 2
	Rh(1)-C(NHC)	2.047(3)	2.062(2)
	Rh(1)-P(1)	2.3345(8)	2.3547(6)
	Rh(1)-C(L)	2.215(3), 2.231(3)	1.901(3)
	(1, L=(COD); 2, L=CO)	2.220(3), 2.217(3)	1.886(3)
	O(31)-C(31)	-	1.124(3)
	O(41)-C(41)	-	1.128(3)
	Angle
	C(NHC)-Rh(1)-P(1)	90.83(8)	89.92(7)
	C(41)-Rh(1)-P(1)	-	91.93(7)
	C(31)-Rh(1)-C(62)	-	86.78(10)
	C(31)-Rh(1)-C(41)	-	91.97(10)

  [Th][BF4], which possesses the additional advantage of being very soluble in this solvent. Typically, a solution of

				Complex 2
				AgBF4 + 5min
				AgBF4 + 20min
				AgBF4 + 45min
	2300	2200	2100	1900	1800

Acknowledgements

This work was supported by the Agence Nationale de la Recherche (ANR-07-JCJC-0041). We are also grateful to the CNRS (Centre National de la Recherche Scientifique) for further financial support.

Scheme 6.

Stepwise oxidation of complex 2 in CH2Cl2 and reaction of 2 ox in MeCN.

The above combined results confirm that the reactivity of the rhodium centre can be enhanced dramatically, simply by changing the oxidation state of a ligand electronically connected to Rh. Thanks to this stepwise process and to the irreversible character of the reaction, it even becomes possible to use AgBF4, a very common and convenient oxidant which is too weak to operate in acetonitrile.

Hydrosilylation catalysis

Rhodium(III) complexes, particularly half-sandwich Rh III complexes, are catalysts of choice for the functionalization of Csp2-H bonds [START_REF] Colby | [END_REF][37][38] and we recently showed the potential of complex 3 for this type of reaction [21]. We envisioned that this system may also be suitable for other types of catalytic reactions, such as the hydrosilylation of carbonyl compounds. We previously described the use of complex 1 and its analogue bearing a norbornadiene ligand in place of COD as catalysts for this reaction but both proved rather sluggish, probably due to their high stability and the difficulty to remove the diene ligand [17].

Thus hydrosilylation of acetophenone and its derivatives were carried out with 1 mol% of complex 1 or 3 in THF at room temperature (scheme 7, table 2). All reactions were stopped after 17 h to allow for direct comparison of conversions.