Djamel Hadbi
email: djamel.hadbi@g2elab.grenoble-inp.fr.

Nicolas Retière

Frederic Wurtz

Xavier Roboam

Bruno Sareni

Comparison between system design optimization strategies for more electric aircraft network

Keywords: System design, integrated optimal design, multilevel optimization, embedded electrical system

The aircraft electric network is a complex system, consisting of many different elements integrated to form a unique entity, designed to perform a well-defined mission. In the current state, the network conceptual design is based on standards defined by the aircraft manufacturer. As a consequence, electric subsystem suppliers are doing local optimizations to fulfill these standards in a separated way through a "mechanistic approach". This results in a set of optimized subsystems which is not necessarily "optimal" with respect to the network level. To overcome this problem, we present a design approach called EPFM (Extended Pareto Front Method) based on separated subsystem optimizations which aims at finding an optimal configuration of the electrical network at the system level. The EPFM is discussed with regard to the computational cost and the collaboration requirements in the aeronautical industrial context and compared with the classical mechanistic approach.

Introduction

Modern aircraft power systems use the following main forms of energy: mechanical, hydraulic, pneumatic, thermal and electrical. The main design drivers of an aircraft are its range (weight and fuel consumption), its production and maintenance costs, and its environmental impacts [START_REF] Liscouet-Hanke | A Methodology for Systems Integration in Aircraft Conceptual Design -Estimation of Required Space[END_REF]. Electrification of aircraft systems is a technological key to reach aircraft industry targets. Electrification is a process of penetration and propagation of electrical energy by reducing or substituting the use of other forms of energy, implementation of new and modernization of existing electrical devices [START_REF] Fraj | Modelling approachs for the simulation-based preliminary design and optimization of electromechanical and hydraulic actuations systems[END_REF][START_REF] Chiesa | Methodology for an Integrated Definition of a System and Its Subsystems: The Case-Study of an Airplane and Its Subsystems[END_REF][START_REF] Avery | Electrical generation and distribution for the more electric aircraft[END_REF].

Besides the aircraft network evolution, new methodologies must be developed in order to optimize in the same time each subsystem and the network operations. In a classical mechanistic design approach, the integrator defines several standards at the conceptual design phase (Fig. 1) in order to ensure a proper operating of the network when subsystems are connected. Even if subsystems are optimized by the electrical suppliers, the network is usually oversized [START_REF] Roboam | More Electricity in the Air: Toward Optimized Electrical Networks Embedded in More Electrical Aircraft[END_REF]6].

Global approaches that integrate all components in a same optimization loop have shown good result on actuator systems [6,[START_REF] Hadbi | Comparison between system design optimization strategies for more electric aircraft[END_REF][START_REF] Roboam | Integrated design by optimization of electrical energy systems[END_REF]. The generalization of such approaches on a network containing a high number of subsystems provided by different suppliers faces the following questions: -Is it possible to gather in one model all system components while each component could be itself represented by a complex model? Is it relevant to use such models in a global optimization loop while ensuring convergence? -On the other side, are subsystems suppliers willing to easily provide their models and their knowledge? Do they accept to transfer the design decision of their own equipment to the integrator? In this paper, these issues are illustrated in a design problem of an aircraft network containing two electric subsystems: Subsystems are simplified and only the sizing of their filters is considered. The mechanistic approach based on the system decomposition is used firstly to show the current state of filter sizing. The global approach (industrially unachievable) is secondly applied to measure the possible mass gain.

Then, the Extended Pareto Front Method (EPFM) is presented. This method was initially introduced in [START_REF] Nguyen-Huu | Optimization of an electrical system using Pareto borders of each component, Application to an automotive drive chain[END_REF] for the optimal design of an actuator devoted to an electrical vehicle. It is a collaborative approach based on a system decomposition that ensures the privacy of subsystem design.

Study case and specifications

A simplified case of 540 V HVDC (High Voltage Direct Current) network containing two subsystems is chosen to illustrate design strategies (Fig. 2). A 40 kW generating channel supplies a 40 kW environmental conditioning system used for air compression, the DC (direct current) current is 74 A. For methodological issues, a simplified representation with couplings is needed. Thus, we will restrict the electrical network description to the filters which are coupled through current and voltage harmonics. The issue here consists in designing the filter elements with minimum mass while fulfilling power quality constraints.

Model of the generating channel: Source modeling

In conventional aircraft networks, electrical energy is supplied by IDGs (Integrated Driven Generator). It is composed of 400 Hz synchronous machine that converts the hydromechanical power from the Engine Bleed Air system to HVDC network throw an ATRU (Auto TRansformer Unit) [START_REF] Nguyen-Huu | Optimization of an electrical system using Pareto borders of each component, Application to an automotive drive chain[END_REF]. The ATRU generates high voltage DC (HVDC) source power using unregulated AC to DC power conversion with natural commutation operation and a voltage step down transformer. A capacitive output filter is used to reduce voltage harmonics before the connection to the 540 V HVDC bus. Current harmonics in the rectifier downstream have been identified for one operating point of the flight mission. Therefore, the part of the generating channel before the capacitive filter has been substituted by corresponding current sources (Fig. 3). Two main harmonics from (6 k × f Generator ) spectrum are considered: 2400 Hz and 108 kHz (Table 1). This subsystem is referred as "source".

The "source" model is only represented by the bus capacity C s . Source and bus currents are the input variables whereas the bus voltage is the output variable (Eq. ( 1)).

---→

V bus ω = ----→ I source ω j • C s • ω - --→ I bus ω j • C s • ω (1)
As the "source" is defined with characteristic frequencies, a matrix representation is used. Voltage and current variables are considered as vectors. Since source and bus current phases are not known, the worst case of phase shift is considered to calculate voltage harmonics (Eq. ( 2)).

⎧ ⎪ ⎨ ⎪ ⎩ ---→ V bus ω = SourceFt Vbus/Isource × ----→ I source ω + SourceFt Vbus/Ibus × --→ I bus ω SourceFt Vbus/Isource = SourceFt Vbus/Ibus = 1 C s • ω (2)
A mass model is required to calculate the mass of the filter. Using datasheet from passive component manufacturer [START_REF]Manufacturer and supplier of electronic components[END_REF], a linear relation between the capacitance value and its mass can be obtained (Eq. ( 3)).

M Source (kg) = 2666.67 × C s (μF )

(3)

Model of the environmental conditioning system: Load modeling

The environmental conditioning system is based on a High Speed Permanent Magnet Synchronous Machine (HSPMSM). This machine is controlled by an inverter. An input filter is used to extenuate the rejected harmonics on the DC bus.

Harmonic current components before filtering have been identified for the same operating point of the flight mission. Based on a PWM (Pulse-Width Modulation) control strategy, the inverter and the HSPMSM have been replaced with a current source delivering those harmonics (Fig. 4).

The characteristic frequencies depend on the switching frequency (here a f swi = 35 kHz switching frequency is used to obtain the relevant current harmonics) and on the HSPMSM frequency (f HSPMSM = 1 kHz) (Fig. 5) [START_REF] Zhang | Switching Harmonics in a Three-Phase PWM Inverter[END_REF]. Two frequencies of high harmonic amplitudes are also included in the spectrum: 70 kHz (2 × f swi ) and 140 kHz. (4 × f swi ). An additional harmonic frequency common to the source at 108 kHz (3 × f swi + 3 × f HSPMSM ) is finally considered to illustrate the case where the bus is polluted at the same time from both subsystems (Table 2). This subsystem is referred as "load". The "load" model is only represented by its filter. The bus voltage and load current are the input variables and the bus current is the output variable of this subsystem (Eq. ( 4)).

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ --→ I bus ω --→ V C1 ω --→ V C2 ω ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎣ j • L • ω 1 0 1 -C 1 • ω -C2 • ω 0 1 -(j • R • C 2 • ω + 1) ⎤ ⎥ ⎥ ⎦ -1 ⎡ ⎢ ⎢ ⎢ ⎣ ---→ V bus ω ---→ I oload ω 0 ⎤ ⎥ ⎥ ⎥ ⎦ (4)
Since load current and bus voltage phases are not known, we consider the worst case of phase shift to calculate the bus current harmonics (Eq. ( 5)).

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ --→ I bus ω = LoadFt IL/Vbus × ---→ V bus ω + LoadFt IL/Iinv × --→ I load ω LoadFt Ibus/Vbus = -R • C 1 • C 2 • ω 2 + j(C 1 + C 2 )ω 1 -(L • C 1 + L • C 2 ) • ω 2 + j • (R • C 2 • ω -L • R • C 1 • C 2 • ω 3 ) LoadFt Ibus/Iload = 1 + j • R • C 2 • ω 1 -(L • C 1 + L • C 2 ) • ω 2 + j • (R • C 2 • ω -L • R • C 1 • C 2 • ω 3 ) (5)
The capacity sizing is similar to that presented for the source. It is used to design C 1 and C 2 too. The resistor mass is estimated by calculating its losses (Eqs (6) and ( 7)).

P R = R • I 2 R (6)
M R = k R • P R k R = 1 2800 kg/W (7)
The inductance mass is calculated using the geometric model presented in [START_REF] Leplus | Bobine à noyau de fer en régime variable[END_REF]. The load filter mass is the sum of passive elements masses (Eq. ( 8)).

M Load = M R + M C1 + M C2 + M L (8)

Design specifications

Both source and load must cooperate in safe and light weight operation. Light weight operation means that one objective which resides in the system weight minimization is considered (i.e. minimization of the source + load filter masses). Safe operation means that source and load must fulfill voltage and current quality standards. Table 3 recalls source and load harmonics that must be filtered to obtain a clean HVDC bus.

In our example, harmonic standards on bus voltage and current are considered as constraints. Figure 6 shows an illustration of Airbus Directives standards [START_REF]ABD0100-Equipment-Design[END_REF]. The requirements for the HVDC voltage source are described with regard to a particular bus current: in steady-state conditions, the maximum differential-mode voltage ripple at the power terminals of the HVDC source shall be within the limits of Fig. 6-bottom when considering a current with a ripple within the envelope defined in Fig. 6-top. The spectral component amplitude is respectively given in dB from 1.0 VRMS and in dB from 1.0 ARMS.

Similarly, requirements for HVDC load currents are described with a particular bus voltage: In steadystate conditions, the differential-mode current ripple at HVDC load input shall be within the limits of Fig. 6-top when supplied with a steady-state voltage within the envelope defined in Fig. 6-bottom. Let us notice that current requirements for the source are higher than for the loads because the standards are planned for more than one load HVDC network. These standards are used in aircraft networks to ensure safe operations to all electric devices connected to the HVDC bus. Four main frequencies were chosen in the spectrum for the design problem. We summarize in Table 4 the values of standards for these particular frequencies.

Classical sizing of the filters

In a classical design approach called also "mechanistic approach", subsystems are separately designed. The system integrator (i.e. the aircraft manufacturer) only assembles the network components designed from quality standards by other suppliers.

Design of the source

Optimization problem formulation

The optimal design of the source is obtained by finding the value of its decision variable (here the bus capacity C s ) that minimizes the source weight while fulfilling quality standards (Fig. 6). In the source model, the bus current is an input variable. In the classical design approach, bus current harmonic values considered in the model are those of the worst case of quality standards (Fig. 7 and Eq. ( 9)) Voltage harmonics must remain under the maximal value permitted by the standards (Fig. 6bottom).

The formulation of the optimization problem of the source in the classical approach is as follows:

⎧ ⎨ ⎩ min Cs W source -----→ V bus ω -----→ V bus STANDARD ω (9)
This optimization problem is solved using the SQP (Sequential Quadratic Programming) algorithm and the CADES package [START_REF]Vesta-System[END_REF]. Mass and voltage are convex functions (Eqs (2), (3) and Fig. 8). Therefore, a deterministic algorithm can easily find the optimum.

Optimization results

The mass capacity obtained is 2.17 kg. The optimal design of the source is limited by the constraint of the first voltage harmonic at 2400 Hz. All other harmonic values are far under standards (Fig. 9).

Design of the load

Optimization problem formulation

The optimal design of the load is obtained by finding the values of its decision parameters (here the passive elements values C 1 , C 2 , L and R) that minimize the load weight while fulfilling quality standards (Fig. 10 and Eq. ( 10)).

In the load model, bus voltage is an input. In the classical design approach, bus voltage harmonic values considered in the model are those of the worst case of standards. Current harmonics must remain under the maximal value permitted by the standards.

The formulation of the optimization problem of the load in the classical approach is given in Eq. ( 10):

⎧ ⎨ ⎩ min C1, C2, L, R W load --→ I bus ω --→ I bus STANDARD ω ( 10
)
The mass function (Eq. ( 8)) and the transfer function of the load (Eq. ( 5)) are convex functions in the range [START_REF] Liscouet-Hanke | A Methodology for Systems Integration in Aircraft Conceptual Design -Estimation of Required Space[END_REF]150] kHz (Fig. 11). Therefore, a deterministic optimization algorithm can be used.

The mass obtained is 2.24 kg and the design of the load is limited by the constraint of the last current harmonic at 140 kHz. The other harmonic values are under standards (Fig. 12).

Simultaneous design approach

Optimization problem formulation

In the simultaneous design approach, the system is considered as a whole. The source and the load are described in the same system model (Eq. ( 11)).

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ --→ I bus ω ---→ V bus ω --→ V C1 ω --→ V C2 ω ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ j • L • ω -1 1 0 1 j • C s • ω 0 0 1 0 -j • C 1 • ω -j • C 2 • ω 0 0 1 -(1 + j • R • C 2 • ω) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ -1 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 ----→ I source ω --→ I load ω 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (11)
Both bus voltage and current are now calculated in the model. Input variables are the source current and the load current harmonics only. There is no worst case assumption on the bus voltage and current values (Fig. 13).

The optimal design of the system is obtained by finding the values of all decision variables that minimize the sum of source and load weights while fulfilling both voltage and current harmonic constraints (Fig. 14).

Equation [START_REF] Zhang | Switching Harmonics in a Three-Phase PWM Inverter[END_REF] gives the optimization problem formulation with the simultaneous design approach:

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ min Cs,C1, C2, L, R W System ---→ V bus ω ---→ V bus STANDARD ω --→ I bus ω --→ I bus STANDARD ω ( 12
)
The optimization problem in the simultaneous design approach may be non-convex because the transfer functions (Eqs ( 13) and ( 14)) are multimodal (Fig. 15). As a consequence, deterministic optimization algorithm could be trapped in a local optimum. To avoid this problem, 10 random initializations of the SQP algorithm are used to find the global optimum.

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ V busf = SystemFt Vbus/Isource (C s , L, C 1 , C 2 , R) × I sourcef +SystemFt Vbus/Iload (C s , L, C 1 , C 2 , R) × I load SystemFt Vbus/Isource = 1 -L • (C 1 + C 2 ) ω 2 -j L • C 1 • C 2 • R • ω 3 -C 2 • R • ω D SystemFt Vbus/Iload = -1 -j • C 2 • R • ω D D = C s • L • C 1 • C 2 • R • ω 4 -C 2 • R (C s + C 1 ) ω 2 -j C s • L (C 1 + C 2 ) ω 3 -(C s + C 1 + C 2 ) ω (13) ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ I busf = SystemFt Ibus/Isource (C s , L, C 1 , C 2 , R) × I sourcef +SystemFt Ibus/Iload (C s , L, C 1 , C 2 , R) × I load SystemFt Ibus/Isource = -C 1 • C 2 • R • ω 2 + j (C 1 + C 2 ) ω D SystemFt Ibus/Iload = -C s • C 2 • R • ω 2 + j • C s • ω D (14)

Optimization results

The total mass obtained is 2.56 kg. The system design is limited by the constraint of the last current harmonic at 140 kHz and the constraint of the first voltage harmonic at 2400 Hz (Fig. 16).

Why the optimal solution of the simultaneous design approach is better?

The total mass of the system in the mechanistic design approach is 4.41 kg (2.17 + 2.24) while it equals 2.56 kg in the simultaneous design approach (Table 5).

A difference of 42% is due to the local view in the mechanistic design approach. Indeed, in this approach, either bus current (in the design of the source) or voltage bus (in the design of the load) are considered as standards limit values (Figs 9 and12). This leads to the oversizing of both filters. Conversely, in the simultaneous design approach, the real values of voltage and bus current are calculated depending on all network variables.

Extended Pareto Front Method (EPFM)

In the previous section, it has been shown that the simultaneous design approach succeeds in reaching the best design because it gets the optimal values of the harmonic components. Let us remember that the simultaneous design approach is industrially no achievable because it requires sharing the models and the design variables. EPFM is a decomposed based approach. There are several approaches based on hierarchical decomposition, some of them have been developed in MDO (Multidisciplinary Design Optimization) context [START_REF] Sobieski | Structural optimization by multilevel decomposition[END_REF][START_REF] Tosserams | A specification language for problem partitioning in decomposition-based design optimization[END_REF][START_REF] Berbecea | Multi-level approaches for optimal system design in railway applications[END_REF]. EPFM is different because there is no system optimizer at the integrator level. EPFM aims at reaching the optimal system design without sharing the models. In the considered case of filter design, it consists in identifying the same bus current and voltage values as in the simultaneous design approach (Fig. 16). A collaborative process exploiting those variables to meet the system optimal design is needed. Such variables are qualified as coupling variables [START_REF] Nguyen-Huu | Optimization of an electrical system using Pareto borders of each component, Application to an automotive drive chain[END_REF] (Fig. 17).

In the HVDC studied network, bus voltage and current are the coupling variables represented by their frequency harmonics. Their values depend on all decision variables (C s , C 1 , C 2 , L and R). Instead of looking for the optimal values of the decision variables, we try to find the corresponding values of the coupling variables (Eq. ( 15)).

⎧ ⎪ ⎨ ⎪ ⎩ ---→ V bus source ω = ---→ V bus load ω = ---→ V bus system ω --→ I bus source ω = --→ I bus load ω = --→ I bus system ω (15)

Assumptions related to this approach

The following condition should be met for designing a system with N components: The system objective function should be computed from the corresponding objective functions of its subsystems (Eq. ( 16)). In our case, N = 2 which means:

M system = M source + M load (16)

Analysis of component couplings in a system

There are many ways to describe the couplings in a system. The matrix of dependencies is a one of analysis methods that describes the couplings between functions and variables [START_REF] Barthelemy | Engineering design application of multilevel optimization methods[END_REF].

The direct dependencies between decision variables and other variables depend on the physic modeling of the system. Source and load masses are depending on their decision variables only. On the other hand, voltage and current bus depend on all decision variables. Thus, the coupling between source and load masses is indirect through electric variables of the bus (Table 6).

Coupling variables can be inputs or outputs according to the subsystem design. In EPFM, only coupling input variables are considered because the output coupling variables are resulting from the sizing optimization. Considering 4 characteristic frequencies (Table 3) N f = 4 harmonic variables are used to represent the coupling variables in order to ensure the consistency at the integrator (system) level. Those coupling variables are discretized with a uniform sampling of N s points in the range defined by the standards.

Source space solution

A mechanistic optimization design of the source calculates the optimal value of the source mass from overestimated harmonic currents. In the EPFM, the current bus values are not fixed by the standards but sampled in a range with N s points. It should be noted that this number of sampling points per coupling variables could be different. The resulting source solution space is defined as the total number of discrete combinations of the coupling variables (Γ source ) according to Eq. [START_REF] Tosserams | A specification language for problem partitioning in decomposition-based design optimization[END_REF].

Γ source = N Nf s ( 17
)
For each combination in the source solution space, an optimization consisting in minimizing the source filter weight is performed with the corresponding values of the bus current harmonics.

Load space solution

The load solution space is built using the same method as the source solution space. Given a number of sampling points N s , all bus voltage configurations are used to perform Γ load independent optimizations (Eq. ( 18)).

Γ load = N Nf s [START_REF] Berbecea | Multi-level approaches for optimal system design in railway applications[END_REF] In our case, the load and source solution spaces have the same size.

Consistency test and calculation of the system solution

Given both source and load solution spaces, each solution from the source space is compared to all solutions of the load space. The number of comparisons N comp is defined by Eq. [START_REF] Barthelemy | Engineering design application of multilevel optimization methods[END_REF].

Γ comp = Γ source × Γ load ( 19
)
The aim of this comparison step is to identify the consistent solutions that fulfill Eq. ( 15) with 1% of tolerance, ensuring the consistency of the coupling variable constraints. For those solutions, the total mass is calculated by adding the source and load filter masses. The number of consistent solutions identified is denoted as N syst . Among those consistent solutions, the optimal system solution is that with the smallest mass M system .

Optimization results

Table 7 gives the EPFM results for number of sampling points N s . When N s increases, the solution space becomes larger. In the same time more comparisons are required to find the consistent solutions. The smaller the discretization, the better the accuracy achieved on the optimal system solution. However it should be noted that EPFM with the less accurate sampling (N s = 3) provides a better solution than the mechanistic approach (23% of mass gain). Finally, the rate between consistent solutions and all comparisons performed between the source and load solutions decreases with the increase of N s .

Comparison between all approaches

Computational cost

The system mass obtained using the three approaches has already been given in the previous sections. Let us focus on now on time computation. All CPU time presented in the following have been obtained under the same conditions with the same computer. 1 In the mechanistic approach, subsystem optimizations could be done in parallel. Therefore, the CPU time required for obtaining the mechanistic solution is imposed by the highest CPU time between the source and load optimizations. The load filter is more complex than the source, so its CPU time is higher (T mecha.load = 183 ms versus T mecha.source = 43 ms). Similarly, due to a higher level of coupling in the global system model, the CPU time of the simultaneous design approach (T simultaneous = 399 ms) is higher than the cumulative CPU time of both designs in the mechanistic approach. The EPFM CPU time is the sum of the time (T opti ) required for generating the solution space from the independent optimizations and the CPU time (T comp ) dedicated to the solution comparisons for identifying the consistent configurations and the optimal solution (Eq. ( 20)).

T EPFM = T opti + T comp ( 20
)
For N s different values, T opti and T comp are given in Fig. 18(a). Figure 18(b) displays the total CPU time of EPFM (T EPFM ). When using N s = 8 the best EPFM system mass is obtained (2.69 kg) with a computational cost of 6282 s (almost two hours).

What about a more complex system with more loads and more harmonic frequencies for better accuracy?

If a case study with more than one load is considered, the EPFM CPU time will not necessarily increase since all load optimizations can be performed in parallel: T opti will typically depend on the slowest load optimization. The consistency condition will change as the bus current will be defined as the sum of all load currents (Eq. ( 21)). However, the number of comparisons, only depending on two equality constraints on the bus current and voltage, will be the same as for one single load.

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ---→ V bus source ω = ---→ V bus load ω = ---→ V bus system ω --→ I bus source ω = Nloads -→ I L load ω = --→ I bus system ω (21)
Conversely, considering more harmonic frequency will increase the CPU time due to the increase of the number of comparisons required for identifying all consistent solutions.

Collaboration level

Regarding the information shared and needed in each approach three collaboration levels can be identified (Table 8). The highest collaboration level is for the simultaneous design approach where it is required to share all subsystems modeling to build a single optimization loop at the system level. EPFM presents an intermediate level of collaboration providing more than one solution through a set of optimizations guided by the standards and the coupling variables at the system level. Finally, the classical mechanistic approach presents the smaller level of collaboration only sharing the HVDC standards. As presented in Table 8, using the EPFM allows finding a system mass close to the best mass obtained with the simultaneous design approach. Compared with the mechanistic design, there is 39% of mass gain with N s = 8. The CPU time calculation is quite high for this simple study case. Regarding collaboration level, it would be acceptable for electric equipment suppliers to give more than one design. Once their design problems are built, they could easily change input values to obtain more than one solution. Even if their time calculations are high, solutions exist to accelerate optimization process by using parallel computing.

Conclusion

In this work, three different design approaches have been applied for the sizing of electric filters of a typical aircraft network. Through this simple study case, advantages and drawbacks of each approach have been discussed with regard to multiple criteria: system optimality (here the filter mass reduction), CPU time and level of collaboration. It has been shown that the EPFM can constitute a good compromise between those criteria in comparison with conventional design approaches such as mechanistic or simultaneous designs. The main drawback of the EPFM resides in the increase of the CPU time with the increase of the number of coupling variables and their associated sampling. However, some improvements of the EPFM could probably be done by exploiting combinatorial optimization for a quicker identification of the consistent solutions.

Fig. 1 .Fig. 2 .

 12 Fig.1. The system design process[START_REF] Giraud | Load allocation problem for optimal design of aircraft electrical power system[END_REF].

Fig. 4 .

 4 Fig. 4. Load filter model.

Fig. 5 .

 5 Fig. 5. Generic pattern of switching harmonics of DC link current idc [12].

Fig. 8 .

 8 Fig. 8. Source transfer function shape.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Voltage and current harmonics in the mechanistic optimization of the source.

Fig. 11 .

 11 Fig. 11. Load transfer function shape.

Fig. 12 .

 12 Fig. 12. Voltage and current harmonics in the mechanistic optimization of the load.

Fig. 13 . 2 Fig. 14 .

 13214 Fig. 13. Global system model.

Fig. 15 .

 15 Fig. 15. System transfer functions.

Fig. 17 .

 17 Fig. 17. Couplings building in EPFM.

Fig. 18 .

 18 Fig. 18. EPFM time calculation.

Table 1

 1

	Rectifier current harmonics considered for the			[I bus ]				[I bus ]
	source Characteristic frequencies (Hz) 2400	[Isource]ω (A) 3.40	[I source ]	C s	C s	[V bus ]	[I source ]	F t (C s )	[V bus ]
	108 000	2.59 10 -3
					Fig. 3. Source filter model.

Table 2 Inverter

 2

	current harmonics considered for
	the load
	Characteristic	[Iload]ω (A)
	frequencies (Hz)
	70 000	51.78
	108 000	14.53
	140 000	17.47

Table 3 Current

 3

		Harmonics to be filtered
	System characteristic frequencies (Hz)	[Isource]ω (A)	[Iload]ω (A)
	2400	3.40	0
	70 000	0	51.78
	108 000	2.59 10 -3	14.53
	140 000	0	17.47

Fig. 6. Requirements on bus voltage and current ripple.

Table 4

 4 Harmonic limit of the 540 V DC bus

	f	[Vbus] STANDARD (dBV) [Ibus] STANDARD for "source"(dBA) [Ibus] STANDARD for "load"(dBA)
	2400 kHz	10.00	31.00	25.00
	70 kHz	-11.13	-12.98	-17.05
	108 kHz	-38.37	-43.68	-45.47
	140 kHz	-54.67	-62.10	-62.48
		[I bus ] STANDARD [I source ]	"Source" Model	M source [V bus ]
		C
			CADES
			Optimizer

s Fig. 7. Mechanistic design of the source filter.

Table 5

 5 Optimal solution comparisons

		Mechanistic design approach Simultaneous design approach
	Msource (kg)	2.17	0.35
	Mload (kg)	2.24	2.21
	Msystem (kg)	4.41	2.56

Fig. 16. Simultaneous design approach results.

Table 6

 6 Matrix of dependencies[START_REF] Barthelemy | Engineering design application of multilevel optimization methods[END_REF] related to the studied system

						"Source"
						Model
			Decision variables
	Objectives	Msource Cs	0	0	0
		Mload	0	C1 C2 L
	Coupling variables [Vbus]	Cs C1 C2 L
		[Ibus]	Cs C1 C2 L	"Load"
						Model

Table 7

 7 EPFM results with different sampling of the coupling variables

	Ns	3	4	5	6	7	8
	Γsource or Γload	81	256	626	1296	2401	4096
	Γcomp	6 561 65 536 390 625 1 679 616 5 764 801 16 777 216
	Nsys	92	967	5 023	19 054	47 239	114 822
	Nsys Γcomp (%)	1.40	1.48	1.29	1.13	0.82	0.68
	Msystem	3.41	3.07	2.91	2.80	2.74	2.69

Table 8

 8 Comparison of the different design approaches

	Approach	System mass (kg) Time calculation (s) Collaboration level
	Mechanistic approach	4.41	0.183	Standards
	EPFM	2.69	6282	Standards + solution spaces
	Simultaneous design approach	2.56	0.399	Standards + modeling

Intel R Core(TM) i5-3317U CPU @ 1.70 GHz, avec 6 GB de RAM, sur Windows 8.1 64-bit.


Yingying Gu

Patrice Bacchin

Jean-Franc ¸ois Lahitte

Jean-Christophe Remigy

Isabelle Favier

Montserrat G Omez

Douglas L Gin

Richard D Noble

Catalytic Membrane Reactor for Suzuki-Miyaura C2C Cross-Coupling: Explanation for Its High Efficiency via Modeling

Keywords: polymeric catalytic membrane, palladium nanoparticles, forced flow-through, modeling, mechanism

A polymeric catalytic membrane was previously prepared that showed remarkable efficiency for Suzuki-Miyaura C-C cross-coupling in a flow-through configuration. A mathematic model was developed and fitted to the experimental data to understand the significant apparent reaction rate increase exhibited by the catalytic membrane reactor compared to the catalytic system under batch reaction conditions. It appears that the high palladium nanoparticles concentration inside the membrane is mainly responsible for the high apparent reaction rate achieved. In addition, the best performance of the catalytic membrane could be achieved only in the forced flow-through configuration, that, conditions permitting to the reactants be brought to the catalytic membrane by convection.

Introduction

Catalytic membranes have been extensively studied in the last two decades because they represent a process intensification. More recently, catalytic polymeric membranes, which were less studied than inorganic ones, have attracted growing interest because of their relatively low cost and high efficiency. 1 They were found catalytically active on a variety of reactions (e.g., alcohol and ether syntheses, 2 C-C cross-couplings [START_REF] Gu | High catalytic efficiency of palladium nanoparticles immobilized in a polymer membrane containing poly(ionic liquid) in Suzuki-Miyaura cross-coupling reaction[END_REF][START_REF] Faria | Palladium nanoparticles supported in a polymeric membrane: an efficient phosphine-free "green" catalyst for Suzuki-Miyaura reactions in water[END_REF][START_REF] Seto | Membrane reactor immobilized with palladium-loaded polymer nanogel for continuous-flow Suzuki coupling reaction[END_REF] hydrogenations, [START_REF] Carlin | Ionic liquid-polymer gel catalytic membrane[END_REF] chemical reductions [START_REF] Domenech | Polymer-stabilized palladium nanoparticles for catalytic membranes: ad hoc polymer fabrication[END_REF][START_REF] Emin | Influence of UV grafting conditions and gel formation on the loading and stabilization of palladium nanoparticles in photografted polyethersulfone membrane for catalytic reactions[END_REF] ). Some polymeric catalytic membranes even gave full conversion of substrates within residence time of seconds, [START_REF] Gu | High catalytic efficiency of palladium nanoparticles immobilized in a polymer membrane containing poly(ionic liquid) in Suzuki-Miyaura cross-coupling reaction[END_REF][START_REF] Emin | Influence of UV grafting conditions and gel formation on the loading and stabilization of palladium nanoparticles in photografted polyethersulfone membrane for catalytic reactions[END_REF] showing prospective potential of the catalytic polymeric membranes containing palladium nanoparticles (PdNPs). Whereas there is a lack of investigations into the reasons why those polymeric catalytic membranes are so efficient. This understanding is undoubtedly important to provide guidelines to design catalytic membrane reactors.

One major type of catalytic membrane reactors are membrane contactors. Due to their nonpermselectivity toward reactants and products, membrane contactors can offer higher throughput than extractor and distributor catalytic membranes. It is generally accepted that main function of membrane contactors consists of favoring mass transfer by intensifying the contact between reactants and catalyst. [START_REF] Tan | Inorganic Membrane Reactors: Fundamentals and Applications[END_REF][START_REF] Zaspalis | Reactions of methanol over catalytically active alumina membranes[END_REF] Nagy established several mathematical models to study the mass transfer accompanied by reactions in the catalytic membrane. [START_REF] Nagy | Diffusive plus convective mass transport through catalytic membrane layer with dispersed nanometer-sized catalyst[END_REF] Herein, we deduced using a similar model that the intensified contact between catalyst and reactants was not the only factor responsible for the high catalytic activity observed in a forced-flow membrane contactor using a catalytic membrane with immobilized PdNPs. The results obtained provide further understanding into the principles involved in catalytic membrane contactors. In this work, PdNPs were used as catalyst for their catalytic performance, especially in carbon-carbon bond formation. [START_REF] Deraedt | Homeopathic" palladium nanoparticle catalysis of cross carbon-carbon coupling reactions[END_REF]

Materials and Methods

Catalytic polymeric membrane, PdNPs colloidal solution and the corresponding Pd-catalyzed Suzuki-Miyaura C-C cross-coupling reactions were previously reported [START_REF] Gu | High catalytic efficiency of palladium nanoparticles immobilized in a polymer membrane containing poly(ionic liquid) in Suzuki-Miyaura cross-coupling reaction[END_REF] The catalytic membrane was prepared through the functionalization of a microfiltration membrane whose nominal pore size is 0.2 lm, with a very narrow pore-size distribution.

To run the reaction using the catalytic membrane, the reaction mixture was premixed and then filtered through the membrane. The permeate flow rate was varied by using the peristaltic pump (with the flux density j fell in the range of 27-1300 L h 21 m 22 ). The transmembrane pressure was in the range of 1-150 mbar. The mean flow velocity inside the membrane v m can be calculated as v m 5j=e, with e as membrane porosity, which is considered to be 0.8 in our case. As for the calculation of the flow velocity in the bulk solution, porosity is taken as 1.

The same reaction was carried out under batch conditions using a PdNPs colloidal solution.

Mass-transfer model for the catalytic membrane

The fluid followed a laminar flow pattern in the membrane pores (pore size 0.2 lm), giving rise to a parabolic velocity profile. When the radial diffusion across the pore is much faster than convective mass transfer in the axial direction, the parabolic flow velocity profile can be reconciled with plug flow behavior (flat concentration profile). [START_REF] Aris | On the dispersion of a solute in a fluid flowing through a tube[END_REF][START_REF] Nagy | Mixing and dispersion in small-scale flow systems[END_REF][START_REF] Taylor | Dispersion of soluble matter in solvent flowing slowly through a tube[END_REF] For our catalytic membrane, the residence time was 10 4 to 10 5 longer than the characteristic mixing time (to diffuse halfway across the pore, s mix 5 d 2 4D , with d as pore diameter 0.2 lm and D as diffusion coefficient). Hence, a plug flow pattern was readily achieved inside the membrane, permitting a simplification of modeling of the catalytic membrane to one dimension (the concentration can be considered as homogeneous in a slice of the membrane parallel to the surface). The external masstransfer resistance through the boundary layer around the catalyst was neglected (concentration on the catalyst surface equals to the bulk concentration, see section the Calculations on Mass-Transfer Resistance). Since the PdNPs are dense particles (unlike porous pellets), no internal diffusion needs to be considered. In addition, isothermal conditions (fluid and membrane temperature were constant and in complete agreement) were also achieved under the experimental conditions according to Westernann's model. [START_REF] Westermann | Heat transfer and temperature profiles in flow-through catalytic membrane reactors[END_REF] Therefore, a constant intrinsic reaction rate k mem was imposed. The differential mass balance for the catalytic membrane at steady state taking into consideration the convective flow, diffusion and a first-order reaction can be then expressed by Eq. 1.

D d 2 C dz 2 2v m dC dz 2ak mem C50 (1)
where D is the diffusion coefficient (4.1 3 10 210 m 2 s 21 ); v m is the convective velocity inside the membrane; C is concentration of the limiting reactant (1-iodo-4-nitrobenzene); a is the specific surface area of the catalyst (total catalyst surface area divided by the membrane volume, m 2 m 23 ); k mem is the surface intrinsic reaction rate constant; and z is the space coordinate. The D value of the solute molecules at 608C can be calculated by the Stokes-Einstein equation (Eq. 2)

D5 k B T 6plr (2)
where k B is Boltzmann's constant; T is the absolute temperature; l is the dynamic viscosity of the solvent (5:9310 24 kg m 21 s 21 for ethanol at 608C 17 ; and r is the solute molecule radius (on the order of 1 nm). Equation 1 can be transformed into Eq. 3:

d 2 c dZ 2 2Pe M Á dc dZ 21 2 c50 (3)
where c ð5C=C e Þ and Z ð5z=LÞ are the dimensionless concentration and coordinates, respectively; with C e being the limiting reactant concentration on the feed side surface of the membrane and L being the membrane thickness. Pe M is the P eclet number for mass transfer inside the membrane (Pe M 5v m L=D); and 1 is the Thiele modulus 15 ffiffiffiffiffiffiffiffi ffi ak mem D q L. The boundary conditions for this differential equation are [START_REF] Langmuir | The velocity of reactions in gases moving through heated vessels and the effect of convection and diffusion[END_REF] (Figure 1):

Z50 : c51 (4)
Z51 :

dc dZ 50 (5)
The solution for Eq. 3 is therefore:

c5 e Pe M 2 ffiffi D p 2 Z 12 Pe M 2 ffiffi ffi D p PeM1 ffiffi ffi D p e 2 ffiffi ffi D p 1 e Pe M 1 ffiffi D p 2 Z 12 Pe M 1 ffiffi ffi D p PeM2 ffiffi ffi D p e ffiffi ffi D p (6)
where

D5Pe 2 M 141 2 (7)
When there exists a concentration boundary layer of a thickness d C over the feed side of the membrane, a concentration gradient in the boundary layer is produced, and C e does not equal to the limiting reactant concentration in the bulk solution C b (Figure 1). The conversion X can be then expressed as:

X512 C f C b 512 C e C b Á C f C e 512 C e C b Á cj Z51 (8)
where C f is the limiting reactant concentration at Z51. The differential mass balance in the concentration boundary layer without reaction at steady state can be given by:

D d 2 C dz 2 2v dC dz 50 (9)
with boundary conditions:

C z52dC 5C b (10)
C z50 5C e (11) and v as the convective velocity in the bulk solution.

The mass-transfer flux J can be obtained by means of Eq. 12, taking into account both the diffusive and the convective flows as follows: The relation between C e and C b can be thus deduced from Eqs. 9 to 12 as:

J52D dC dz 1vC (12)
C e 5 Jj z50 v Á ð12e PeBL Þ1C b Á e PeBL (13)
where Pe BL is the P eclet number of the concentration boundary layer: Pe BL 5vd C =D. The concentration boundary layer thickness is a function of the momentum boundary layer thickness d and Schmidt number [START_REF] Landau | Fluid Mechanics[END_REF][START_REF] Becht | Evaluation and comparison of protein ultrafiltration test results: dead-end stirred cell compared with a cross-flow system[END_REF] (Eqs. 14 and 15): 20

d5 ffiffiffiffiffiffi ffi l qx r 5190 lm ( 14
)
where x is the agitation velocity (22 rad s 21 ), and l and q are the dynamic viscosity and density of the solution, respectively.

d C 5d=Sc 1 3 515:4 lm (15)
We denote Sh as Sh5Jj z50 Á Pe M =ðv m Á C e Þ: Then from Eqs. 6 to 12, the expression of Sh can be deduced as:

Sh5ePe M 2 Pe M 2 ffiffi ffi D p 2 12 PeM2 ffiffi ffi D p PeM1 ffiffi ffi D p e 2 ffiffi ffi D p 2 Pe M 1 ffiffi ffi D p 2 12 PeM1 ffiffi ffi D p PeM2 ffiffi ffi D p e ffiffi ffi D p (16)
By combining Eq. 13 with Eq. 16, the ratio of C e =C b can be given as follows:

C e C b 5 e PeBL 12 Sh PeM Á 1 e ð12e Pe BL Þ 5 e Pe BL 12 12 Pe M 2 ffiffi ffi D p 2eÁAÁPeM 2 Pe M 1 ffiffi ffi D p 2eÁBÁPeM Á ð12e Pe BL Þ (17)
where

A512 Pe M 2 ffiffiffi ffi D p Pe M 1 ffiffiffi ffi D p Á e 2 ffiffi ffi D p (18) B512 Pe M 1 ffiffiffi ffi D p Pe M 2 ffiffiffi ffi D p Á e ffiffi ffi D p (19)
The concentration profile inside the concentration boundary layer can be obtained based on Eqs. 9-11 as follows:

C BL 5 e Pe BL ÁY=2 sinhðPe BL =2Þ sinh Pe BL Á 12Y 2 Á C b 1e 2Pe BL =2 Á sinh Pe BL Á Y 2 Á C e ( 20
)
with Y5ðz1dÞ=d.

Results and Discussion

Reaction regime of the colloidal solution

The Stokes number of the colloidal system was calculated to be 5 3 10 212 , indicating that the PdNPs follow the streamline so closely that the relative velocity between PdNPs and the liquid phase is nearly zero. [START_REF] Gu | High catalytic efficiency of palladium nanoparticles immobilized in a polymer membrane containing poly(ionic liquid) in Suzuki-Miyaura cross-coupling reaction[END_REF] Thus, the convection becomes inefficient and actually negligible. The mass transfer is therefore effectuated only by diffusion. The mass-transport coefficient (k b d in the batch reactor) and the diffusion coefficient (D) can be correlated by the Sherwood number Sh (Eq. 21), which can be calculated using Eq. 22. [START_REF] Friedlander | A note on transport to spheres in stokes flow[END_REF][START_REF] Nagy | Spherical effect on mass transfer between fine solid particle and liquid accompanied by chemical reaction[END_REF]

k b d Á d p D 5Sh (21)
Sh521 d p ffiffiffiffiffiffiffiffi pDt p % 2 (22)
where d p is the diameter of PdNPs. At t > 1 3 10 23 s, then d p = ffiffiffiffiffiffiffi ffi pDt p < 0:02, and hence Sh ffi 2. D 5 1.22 3 10 211 m 2 s 21 for the colloidal system (l % 2:0310 22 kg m 21 s 21 for the IL at 608C [START_REF] Atilhan | Viscous behavior of imidazolium-based ionic liquids[END_REF][START_REF] Jacquemin | Density and viscosity of several pure and water-saturated ionic liquids[END_REF] .

The mass-transfer coefficient was calculated to be:

k b d 55:8310 23 m s 21 (23)
The boundary layer thickness around the particle is hence d p 5D=k b d 52 nm. The specific surface area a (total surface area of the PdNPs divided by the reaction mixture volume, m 2 m 23 ) of the colloidal system was estimated to be in the range of 5.3 3 10 3 m 2 m 23 to 3.6 3 10 5 m 2 m 23 (i.e., the PdNP mean diameter varying from 4 nm to 100 nm, taking into consideration the presence of aggregates). The mass-transfer flux (a Á k b d ; s 21 ) was thus found much higher (five to seven orders of magnitude greater) than the apparent reaction-rate constant. Therefore, the activity of the colloidal system was in reactionlimited regime. The intrinsic kinetic a Á k batch value was approximately the same as the apparent kinetics (Eq. 24). a Á k batch % k app 52:3310 [START_REF] Jacquemin | Density and viscosity of several pure and water-saturated ionic liquids[END_REF]

s 21 (24)
The intrinsic reaction-rate constant on the catalyst surface for the batch reactor k batch was estimated to be in the range of 6.0 3 10 210 m s 21 to 4.1 3 10 28 m s 21 . (The exact value of specific surface area a cannot be determined; only a range could be given, see above.). This means that the concentration on the PdNP surface could be considered to be the same as the bulk concentration (k batch =k b d ( 1).

Proposed mechanism of operation of the catalytic membrane

Calculations on Mass-Transfer Resistance. The masstransfer coefficient around the catalyst supported on the membrane k m d can be estimated by Eq. 25 and found to be 0.41 m s 21 , which is much larger than that in batch reactor k b d (5.8 3 10 23 m s 21 ). Calculations deduced from Sh52 (which is also valid for the catalytic membrane [START_REF] Leybros | Extraction solide-liquide, aspects th eoriques[END_REF] gives the same k m d value. The intrinsic reaction rate constant should be approximately the same for PdNPs in the batch reactor and inside the membrane (maybe lightly higher in the membrane since NPs in the membranes are smaller). The mass-transfer resistance imposed by the boundary layer around the catalyst particles is therefore also negligible in the case of the catalytic membrane.

k m d 5D=e50:41 m s 21 (25)
In Eq. 25, D is the limiting reactant diffusion coefficient in ethanol at 608C, 4.1 3 10 210 m 2 s 21 , since obvious swelling of the poly(IL) occurs in ethanol (i.e., the diffusion coefficient is mainly determined by the solvent viscosity when the molecule size is smaller than polymer correlation length n [START_REF] Cai | Mobility of nonsticky nanoparticles in polymer liquids[END_REF] ; and e the boundary layer thickness, considered to be half interparticle distance (e % 1 nm). Even if D is 100 times smaller, the masstransfer resistance imposed by the boundary layer around the catalyst would still not be a limiting factor. Reaction Kinetics. The Suzuki-Miyaura cross-coupling reaction was tested at different flow rates at 608C with a 1iodo-4-nitrobenzene concentration of 0.016 mol L [START_REF] Friedlander | A note on transport to spheres in stokes flow[END_REF] .

Conversion and productivity (interpreted as the product of P eclet number inside the membrane and conversion) were plotted as a function of P eclet number inside the membrane (Figure 2). When flow rate increased (reflected by the increase of the P eclet number inside the membrane), the conversion decreased while the productivity increased.

According to the model described in the Materials and Methods section (Eqs. 6-8, 17), the conversion is a function of P eclet numbers, Thiele modulus and membrane porosity: X5f ðPe M ; 1; Pe BL ; eÞ. Hence, Thiele modulus is the only unknown parameter. By fitting the model to the experimental data, Thiele modulus, 15 ffiffiffiffiffiffiffiffi ffi ak mem D q L, was deduced to be 3.19. This result indicates that the reaction is still limited by diffusion along the membrane thickness. By taking the diffusion coefficient of ethanol at 608C, the apparent reaction rate constant was deduced to be:

k app 5ak mem 53:4310 21 s 21 (26)
When normalized to the PdNP mass and surface area, the apparent reaction-rate constant is then 431 s 21 g 21 and 0.53 s 21 m 22 . Compared with catalytic batch reaction conditions (where k app;batch 5 0.22 s 21 g 21 ), the catalytic membrane reactor exhibited an acceleration of the apparent reaction rate by three orders of magnitude.

The specific catalyst surface in the membrane was calculated as a54:66310 6 m 2 m 23 ; and hence, the intrinsic reaction rate constant on the catalyst supported on the membrane is k mem 57:4310 28 m s 21 , which is the same order of magnitude as corresponding value k batch in the batch reactor.

Concentration profiles

Once the ratio of C e to C b is determined from Eq. 17, concentration profiles in the boundary layer and in the membrane can be plotted as a function of flow velocity using Eq. 20 and Eqs. 6-8. Figure 2 shows the concentration profiles in the concentration boundary layer at the feed side and inside the membrane, predicted by the model with 153:19 at different residence times (flow rates or P eclet numbers). The concentration boundary layer significantly reduced the diffusion flux from the bulk solution to the membrane surface and became a limiting step.

At high flow rates (Pe BL ! 1:5), the convection was largely dominant over diffusion. The reactant was effectively brought to the membrane surface by convection so that C e % C b . Higher flow rates also lead to the increase of reactant concentration inside the membrane?thus increasing the productivity. However, the short residence time was insufficient to achieve full conversion. For P eclet numbers smaller than 1.5, the reaction was limited by diffusion. The concentration at the membrane feed surface C e differentiated from the bulk solution concentration C b (C e < C b ), less reactants reached the inside of the membrane, exerting negative influence on the productivity.

As shown in Figure 3, the reaction productivity increased with the flux until the convection became largely dominant, where a maximum plateau was thus reached. The transition P eclet value (Pe BL 51:5) where (C b 2C e ) became less noticeable corresponded to the flux (146 L h 21 m 22 ) from which the productivity began to approach the maximum plateau. The reaction on the membrane became kinetically limited at maximum productivity. Comparative study between the catalytic membrane reactor and PdNP colloidal solution in the batch reactor and proposed explanation for the high efficiency of the catalytic membrane reactor

The differences between the colloidal system (with PdNPs dispersed in [MMPIM][NTf 2 ], 1,2-dimethyl-3-propylimidazolium bis(trifluoromethylsulfonyl)imide) in batch reactor and the catalytic membrane reactor are summarized in Table 1. The catalytic membrane reactor works with higher efficiency (entry 6) and selectivity (entry 7) than the colloidal system. The apparent reaction constants of the two systems were given in entry 8. The values normalized to catalyst surface were also given (entry 10). The apparent reaction constant is almost 2000 times larger in the catalytic membrane with the same amount of palladium (entry 9). This result can be attributed to the fact that palladium nanoparticles are well-dispersed and distributed very close to each other inside the membrane (with small interparticle distances, entry 2), leading to an extremely high particle number or a high number of catalyst active sites per unit volume of the reactor (entry 3). One direct result is the large increase in local active sites/substrate ratio when the liquid is forced through the membrane pores (entry 5). The apparent reaction rate constant k app (entry 8) is the product of the specific catalyst surface a (entry 11) and the intrinsic reaction rate on the catalyst surface k (entry 12). The large catalyst number per unit volume value (entry 3) leads to a high specific surface (entry 11), which is mainly responsible for the reaction rate increase in the membrane. The difference on the intrinsic k (entry 12) between the two systems depends on the particle size and distribution (entry 1), or more specifically, on the number of atoms on the PdNP surface. The ratio of number of atoms on the PdNPs surface (catalytic membrane/batch reactor) is estimated to be 1.4, and the ratio of number of vertex and edge atoms is calculated to be 2.9 (see the Supporting Information), which is close to the ratio of intrinsic reaction rate constant k (entry 12). The influence of k on apparent kinetics (a factor of 1-3) should be much less remarkable than that exerted by the specific surface area a.

The conventional understanding of a forced flow-through membrane reactor is based on intensified contact between the catalyst and reactants being responsible for the reaction rate increase as local mass-transfer resistance can be effectively eliminated. [START_REF] Tan | Inorganic Membrane Reactors: Fundamentals and Applications[END_REF][START_REF] Zaspalis | Reactions of methanol over catalytically active alumina membranes[END_REF] In consequence, improvements on mass transfer should not have obvious contributions to the apparent reaction rate when reactions were not diffusion-limited. However, for the Suzuki-Miyaura cross-coupling between 1-iodo-4nitrobenzene and phenylboronic acid (kinetic-limited in the batch reactor), the kinetics was largely enhanced nonetheless in the catalytic membrane. A simple calculation can help to understand how the apparent reaction rate was accelerated in the catalytic membrane. The volume of the membrane used in the experiments was 0.14 cm 3 . Hence taking into account the porosity, the liquid volume retained in the membrane was around 0.1 cm 3 . For a concentration of 0.016 mol L -1 , the limiting reactant amount inside the membrane volume was 1.6 3 10 26 mol. When the reaction solution was filtered through the membrane (where the catalyst got into contact with the reagents), the Pd/substrate molar ratio in the local environment of the membrane is ( 5 4.7) 470 times higher than that in the batch reactor ( 5 0.01). The real Pd/substrate molar ratio in the membrane can be even higher due to the heterogeneous catalyst distribution inside the membrane (there exist reactive zones inside the membrane where the catalyst is highly concentrated). To be more precise in describing the relative amount of reactant and catalyst, the ratio of catalyst surface to substrate amount should be compared (to reflect number of active sites) instead of Pd/substrate molar ratio (Table 1, entry 5).

It is also noteworthy that despite the high Pd/substrate ratio, the total palladium amount in the membrane is fairly low. The catalyst surface area ratio (catalytic membrane reactor/batch The ratio between the reactive zone of the membrane and the batch system taking into consideration the presence of aggregates in the latter should be in the range of 143 -2.35 3 10 6 .

f

Local ratio inside the membrane environment, porosity 5 0.8.

reactor) is only around 2 (entry 4). It is important to note that the reaction time for full conversion in entry 6 refers to the contact time of reagents with the membrane. [START_REF] Gu | High catalytic efficiency of palladium nanoparticles immobilized in a polymer membrane containing poly(ionic liquid) in Suzuki-Miyaura cross-coupling reaction[END_REF] The total filtration time (treating time) of the catalytic membrane reactor for the same amount of substrate as that in the batch reactor is around 30 min. [START_REF] Gu | High catalytic efficiency of palladium nanoparticles immobilized in a polymer membrane containing poly(ionic liquid) in Suzuki-Miyaura cross-coupling reaction[END_REF] Hence the treating time of the catalytic membrane reactor is 12 times shorter than for the batch reactor.

Besides, the catalytic membrane also exhibits the advantage of operating in a continuous manner. With the comparative factor of total catalyst surface area (entry 4) between the two systems much smaller than that of total treating time and reaction time (entry 6), it is clearly proved that the PdNPs size plays only a minor role in the high efficiency of the catalytic membrane reactor, which is in coherence with the small difference of the intrinsic reaction rate constant between the two systems (entry 12). The high local Pd/substrate ratio can only be beneficial in forced flow-through configuration: every single volume of reaction solution is forced into the local high catalyst concentration environment. When the same membrane is submerged in a batch reactor, the catalytic performance will be no more outstanding. [START_REF] Gu | Membrane reactor based on hybrid nanomaterials for process intensification of catalytic hydrogenation reaction: an example of reduction of the environmental footprint of chemical synthesis from a batch to a continuous flow chemistry process[END_REF] In brief, the Suzuki-Miyaura cross-coupling between 1iodo-4-nitrobenzene and phenylboronic acid is not masstransfer limited under batch conditions. The substantial reaction rate increase by the catalytic membrane reactor in forced flow-through configuration can be certainly attributed to the concentrating effect of the membrane (i.e., packing a large number of particles into a tiny volume). This is in agreement with the observation of Seto and coworkers, [START_REF] Seto | Membrane reactor immobilized with palladium-loaded polymer nanogel for continuous-flow Suzuki coupling reaction[END_REF] who attributed the small rate constant of their Pd-loaded continuous-flow membrane reactor to the low concentration of Pd catalyst in the membrane reactor. They suggested that densification of the Pd catalyst in the membrane could lead to catalytic improvements. The convective flow in our case serves to eliminate the concentration gradient in the boundary layer at the membrane feed side surface and inside the membrane, bringing reactants from the bulk solution into the reactive membrane.

Conclusions

The polymeric catalytic membrane reactor that we prepared was similar to, and thus can be considered as, a micro-reactor for its small characteristic dimensions (ca. 0.2 lm). Theoretical calculations showed that a plug-flow behavior was always expected in the catalytic membrane pores. Therefore, there will be virtually no preferential flow pathways. The membrane reactor was also adapted for moderately exothermic reactions, achieving isothermal conditions.

It is indispensable to adopt the flow-through configuration when using the catalytic membrane reactor to attain a high catalytic activity. The convective flow brings the reagents into the catalyst highly concentrated membrane local environment where the reaction takes place. It also helps to eliminate the mass-transfer limit (in the boundary layer at the feed side as well as inside the membrane).

When the catalytic membrane reactor was employed, the Suzuki-Miyaura cross-coupling using 1-iodo-4-nitrobenzene as substrate was complete within 10 s without formation of any by-product. This reaction was in the reaction-limited regime in the batch reactor. Mathematical modeling of the catalytic membrane showed that there is a concentration boundary layer on the feed side of the membrane and that the reactant concentration gradient inside this boundary layer and along the membrane thickness can be eliminated at high flux (by convection). These features explain the productivity increase with the flux increase. The much higher efficiency of the catalytic membrane (compared to the PdNPs dispersed in an IL under batch conditions) is principally attributed to the high local catalyst concentration inside the small membrane reactor volume since the intrinsic reaction rate constants of the membrane reactor and the batch reactor are of the same order of magnitude. PdNPs size (in the range of 1-6 nm) seems to exert a negligible effect on the reactivity.

Figure 1 .

 1 Figure 1. Schematic representation of the modeled catalytic membrane (black dots representing PdNPs; figure components not drawn to scale).

Figure 2 .

 2 Figure 2. Concentration profiles along the coordinate at various flow rates (simulated by the model).

X 5 Figure 3 .

 53 Figure 3. Variation in dimensionless productivity (Pe M Á X) and conversion (X) values as a function of P eclet number for the catalytic membrane reactor at 608C.

Notationa

  = catalyst specific surface area, m 2 m 23 C = limiting reactant concentration in the membrane, mol m 23 C b = limiting reactant concentration in the bulk solution, mol m 23 C BL = limiting reactant concentration in the boundary layer on the feed side of the membrane, mol m 23 C e = limiting reactant concentration on the feed side surface of the membrane, mol m 23 C f = limiting reactant concentration in the permeate, mol m 23 c = dimensionless concentration D = diffusion coefficient, m 2 s 21 d = pore diameter, m d p = diameter of nanoparticles, m e = interparticle distance, m J = molar flux density, mol m 22 s 21 j = filtration flux density, L h 21 m 22 k = intrinsic reaction rate in either the batch reactor (k batch ) or the membrane (k mem ), m s 21 k B = Boltzmann constant (1:38310 223 J Á K 21 ), J K 21 k app = apparent reaction rate, s 21 k d = mass-transfer coefficient in either the batch reactor (k b d ) or the membrane (k m d ), m s 21 L = membrane thickness, m Pe M = P eclet number inside the membrane Pe BL = P eclet number in the boundary layer r = solute molecule radius, m S = total catalyst surface area, m 2 Sc = Schmidt number Sh = Sherwood number T = temperature, K v m = convective velocity in the membrane, m s 21 v = convective velocity in the bulk solution, m s 21 X = conversion Z = dimensionless coordinate z = space coordinate, m 1 = Thiele modulus e = membrane porosity q = liquid density, kg m 23 d = momentum boundary layer thickness on the feed side of the membrane, m d C = concentration boundary layer thickness on the feed side of the membrane, m x = agitation velocity, rad s 21 l = dynamic viscosity, kg m 21 s 21 s mix = characteristic mixing time, sLiterature Cited 1. Ozdemir SS, Buonomenna MG, Drioli E. Catalytic polymeric membranes: preparation and application. Appl Catal A Gen. 2006;307: 167-183. 2. Song IK, Lee WY. Heteropolyacid (HPA)-polymer composite films as heterogeneous catalysts and catalytic membranes. Appl Catal A Gen. 2003;256:77-98.

Table 1 .

 1 Comparison Between the Batch Reactor and the Catalytic Membrane Reactor: PdNPs Size and Distribution and Performance on Suzuki-Miyaura Cross-Coupling Between 1-Iodo-4-Nitrobenzene and Phenyboronic Acid Number of palladium particles/aggregates per unit volume. c Values calculated taken into consideration the presence of aggregates (diameter 5 4-100 nm). d Value for reactive zones where palladium is highly concentrated (4.0 3 10 14 ) and an average value obtained by (amount of Pd)/(membrane volume) (1.4 3 10 14 ). e 143 5 4.2 3 10 14 /2.8 3 10 12 (comparative factor between the reactive zone of the membrane and the Pd concentration in the batch reactor calculated with d 5 4 nm).

	Entry

b

[Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] AIChE Journal February 2017 Vol. 63, No. 2 Published on behalf of the AIChE DOI 10.1002/aic

[Color figure can be viewed in the online issue, which is available at wileyonlinelibrary.com.] AIChE Journal February 2017 Vol. 63, No. 2 Published on behalf of the AIChE DOI 10.1002/aic

g In the membrane reactor, the reaction time is defined as the contact time of reagents with the membrane. 3 h Selectivity toward the cross-coupling product. i Apparent reaction rate constant normalized to palladium mass. j Specific surface: catalyst surface area divided by the reactor volume. k Intrinsic reaction rate constant on the catalyst surface. 702 DOI 10.1002/aic Published on behalf of the AIChE February 2017 Vol. 63, No. 2 AIChE Journal

AIChE Journal February 2017 Vol. 63, No. 2 Published on behalf of the AIChE DOI 10.1002/aic

Acknowledgments

The authors gratefully acknowledge the French Ministry of Education and Research, Paul Sabatier University and the National Center for Scientific Research (CNRS) for providing financial support through the FOAM2 project (Paul Sabatier University).