Min Wei

Guillaume Boutin

Yilin Fan

Lingai

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

Numerical and experimental investigation on the realization of target flow distribution among parallel mini-channels

Keywords: Parallel mini-channels, Flow distribution, PIV, CFD, Perforated baffle

Numerical and experimental investigation on the realization of target flow distribution among parallel mini-channels

Introduction

Fluid flow maldistribution is considered as one of the main causes of performance deterioration in various energy and process systems, such as heat exchangers (e.g., [START_REF] Lalot | Flow maldistribution in heat exchangers[END_REF]Fan et al., 2008;[START_REF] Tarlet | Entropy generation analysis of a mini heat exchanger for heat transfer intensification[END_REF], fluidized beds (e.g., [START_REF] Ouyang | Circulating fluidized bed as a catalytic reactor: experimental study[END_REF][START_REF] Ong | Computed tomographic investigation of the influence of gas sparger design on gas holdup distribution in a bubble column[END_REF], chemical reactors (e.g., [START_REF] Saber | Microreactor numbering-up in multi-scale networks for industrial-scale applications: Impact of flow maldistribution on the reactor performance[END_REF][START_REF] Guo | Mixing performance assessment of a multi-channel mini heat exchanger reactor with arborescent distributor and collector[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], solar receivers (e.g., [START_REF] Fan | Buoyancy effects on thermal behavior of a flat-plate solar collector[END_REF][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]Wei et al., 2015a) and heat sinks for cooling (e.g., [START_REF] Sehgal | Experimental investigation of the effect of flow arrangements on the performance of a micro-channel heat sink[END_REF][START_REF] Siva | Effect of flow maldistribution on the thermal performance of parallel microchannel cooling systems[END_REF]. Nowadays, increasing attention has been devoted to miniaturized fluidic networks in which the flow distribution among parallel micro-or mini-channels should be properly controlled for enhanced heat and mass transfer and safety reasons [START_REF] Luo | Heat and Mass Transfer Intensification and Shape Optimization: A Multi-scale Approach[END_REF].

In the majority of the existing literature, flow maldistribution is synonymous with "non-uniform flow distribution", implying that uniform shape is assumed, or considered to be the optimal distribution among parallel channels. Various methods have been proposed to improve the fluid flow uniformity, as summarized in [START_REF] Rebrov | Single-phase fluid flow distribution and heat transfer in microstructured reactors[END_REF] and [START_REF] Luo | Heuristic shape optimization of baffled fluid distributor for uniform flow distribution[END_REF]. Through continuous efforts of academic or industrial researches, uniform fluid flow distribution has become a more or less realistic goal. However, recent studies (e.g., [START_REF] Milman | Steam condensation in parallel channels with nonuniform heat removal in different zones of heat-exchange surface[END_REF][START_REF] Boerema | High temperature solar thermal central-receiver billboard design[END_REF]Wei et al., 2015a) clearly show that the optimal flow distribution may not necessarily be uniform, but usually non-uniform corresponding to a defined objective function and constraints for optimization. The extension of the term "flow maldistribution" from "non-uniform distribution" to "non-optimal distribution" (Wet et al., 2015b) raises higher requirements for the design and optimization of flow distribution devices which are beyond the capabilities of conventional empirical propositions or trial-and-error attempts.

In our previous study (Wei et al., 2015b), a Computational Fluid Dynamics (CFD)-based evolutionary algorithm has been developed with the aim of realizing the target flow distribution among parallel channels. The basic idea is to install a thin perforated baffle at the distributing manifold and to optimize the size and distribution of orifices, in order to reach the target flow-rate in every downstream channel. The effectiveness of the proposed evolutionary algorithm has been tested by several 2D examples with different fluidic network geometries (axisymmetric or nonaxisymmetric) and different target curves (uniform, ascending, descending, or step-like). Numerical results show that the optimized flow distributions reached are in good agreement with the target curves. As the first attempt on the realization of target flow distribution, the optimized baffle insertion method has been numerically confirmed as a practical solution to this issue. However, the proposed method depends strongly on the accurate simulations of fluid flow so that experimental validation of CFD models as well as the proposed algorithm is indispensable for bridging the gap between theory and practice. This involves the simulation and optimization of 3D objects, their fabrication and experimental characterizations which are still lacking.

The current study aims at presenting an experimental validation of the optimized baffle insertion method for the realization of target flow distribution among parallel mini-channels. For this purpose, different perforated baffles are designed, optimized, fabricated and installed in a 15-channel fluidic network to test their capability of realizing different target distributions (uniform or non-uniform). In parallel, 3D CFD simulations are also performed, providing a comparison with the experimental results.

The experimental technique used in this study is based on flow visualization using Particle Image Velocimetry (PIV) technique. Through recording the movement of seeding particles illuminated by the pulsed sheet laser, the fluid flow velocity profiles can be obtained. Compared to conventional flow-rate measurement techniques, such as Hot Wire Anemometry (HWA) or Pitot tube, PIV as a non-intrusive method will not disturb the fluid flow streamlines. Another advantage of PIV is the whole-field (or multiple points) measurement which is unique compared with other non-intrusive single point measurement techniques, such as Laser-Doppler Velocimetry (LDV) and Doppler Ultrasonic Velocimetry (DUV). A lot of successful applications of PIV for fluid flow measurement have been reported in the literature, including single phase (air or water) (e.g., [START_REF] Meinhart | PIV measurements of a microchannel flow[END_REF][START_REF] Wen | An experimental and numerical investigation of flow patterns in the entrance of plate-fin heat exchanger[END_REF] or multi-phase flow (e.g., [START_REF] Kiger | PIV technique for the simultaneous measurement of dilute two-phase flows[END_REF][START_REF] Lindken | A novel PIV technique for measurements in multiphase flows and its application to two-phase bubbly flows[END_REF], laminar or turbulent flow (e.g., [START_REF] Sheng | A large eddy PIV method for turbulence dissipation rate estimation[END_REF][START_REF] Westerweel | Particle image velocimetry for complex and turbulent flows[END_REF] and coupled thermo-fluid characteristics (e.g., [START_REF] Carlomagno | Thermo-fluid-dynamics of submerged jets impinging at short nozzle-to-plate distance: A review[END_REF]. Therefore, it is considered as a reliable and accurate technique for flow-rate measurement.

In the rest of this paper, we shall first introduce the 15-channel fluidic network device fabricated for study and optimized perforated baffles used for different operation conditions. Then the experimental set-up and measuring procedure will be presented, as well as the numerical parameters for CFD simulations. After, the numerical and experimental results of several cases for different target distribution curves will be presented, compared and discussed. The effective range of the optimized baffle when the working condition varies from the design value will also be evaluated and discussed to provide some guidelines for future applications. Finally, main conclusions and perspectives will be summarized.

Fluidic network device and optimized baffles

In this section, the 15-channel fluidic network with the distributor and the collector will be briefly introduced. The target distribution curves and optimized baffles will be presented as well.

15-channel device

A mini-channel fluidic network consisting of 15 parallel channels, a distributor and a collector is used for study, as shown in Fig. 1. The overall dimension of fluidic network is 242 mm in length and 90 mm in width. For the convenience of fabrication, the entire fluidic network has the identical depth (e=2 mm). The inlet and outlet tubes located in diagonal position have the same width of 5 mm, but different lengths (50 mm for inlet tube and 100 mm for outlet tube). The length of the distributing manifold is 13 mm and the width is 90 mm. In the distributing manifold, a groove with the thickness of 1 mm and the width of 96 mm is reserved for the installation of a perforated baffle. Identical dimensions are used for the collecting manifold (length of 13 mm and width of 90 mm) but without baffle groove. There are fifteen parallel straight channels having identical width of 2 mm, length of 60 mm and depth of 2 mm. They are evenly spaced at 4 mm between the axis of one channel and another. For the convenience of description, they are indexed by k from 1 to M from the inlet side to the outlet side. The mass flow-rate in kth channel is notated as m k and the flow-rate ratio  is introduced for each channel as follows:

k k m m   ( 1
)
where m is the average mass flow-rate among 15 channels given by:

1 M k k m m M    (2)
Note that it is a real 3D object with all parallel channels located in the same plane, suitable for experimental tests of fluid flow distribution properties using visualization technique.

The device was fabricated in the laboratory LTN, France by digitally-assisted carving the network on the surface of a PMMA rectangular parallelepiped (A), as shown in Fig. 1(a). Six edge fillets (noted by R1 in Fig. 1(a)) having the same radius of 1 mm can be observed at different corners of the fluidic network due to fabrication process. Another PMMA rectangular parallelepiped (B) was also carved to form the enclosed fluidic network, as shown in Fig. 1(b). Note that the depths of the groove reserved for the baffle are 6 mm in rectangular parallelepiped A, and 4 mm in rectangular parallelepiped B to form the effective area of the baffle (the area that may be touched or passed through by the fluid flow). A number of grooves were reserved around the network or between neighboring parallel channels, and filled with rubble strips to prevent the water leakage. Moreover, 29 bolts were used for further sealing. The photo view of the 15-channel device after assembling is shown in Fig. 1(c) and the location of baffle is indicated in Fig. 1(d).

Target distribution curves and optimized baffles

To validate the effectiveness of the optimized baffle insertion method, different target distribution curves are proposed to be realized, including a uniform curve, an ascending curve and a descending curve. The geometric dimensions of the optimized baffles are obtained accordingly using the CFD-based evolutionary algorithm developed in Wei et al. (2015b).

The thickness of the perforated baffle is 1 mm and the distance between the middle of the baffle and the inlets of parallel channels is 2.5 mm. For the convenience of installation, the baffle has a width of 96 mm and a height of 10 mm (as well as the groove for baffle), which is a bit larger than the cross-section of inlet distributing manifold. Likewise, the height of orifices is 3 mm, which is also larger than the thickness of the fluidic network (e=2 mm). However, the effective area of baffle is kept at 90 mm × 2 mm. 30 orifices are distributed equally on the effective area of baffle, with an identical spacing of 3 mm. For the convenience of description, they are indexed from 1 to 30 from the inlet side to the outlet side. For the initial uniform baffle subject to optimization (shown in Fig. 2), the orifices have identical width of 0.6 mm, the global porosity thus being 20%. Since the thickness of the entire network is fixed at 2 mm, the only variable for optimization is the width of orifice. Note that some of the parameters used here are determined or selected based on the parametric study presented in [START_REF] Luo | Heuristic shape optimization of baffled fluid distributor for uniform flow distribution[END_REF] and Wei et al. (2015b). In this study, four perforated baffles are optimized under two different inlet total fluid flow-rate conditions, i.e. high inlet flow-rate (Q in =1.8 L•min -1 ) and low inlet flow-rate (Q in =0.18 L•min -1 ), for representing both the turbulent and laminar flow patterns. The corresponding Re numbers in the inlet tube are 8500 and 850, respectively. The mean Re numbers in the parallel channels (Re ch ) are 1000 for Q in =1.8 L•min -1 and 100 for Q in =0.18 L•min -1 . The corresponding inlet flow-rate under which the perforated baffle is optimized is named as the baffle's design flow-rate. Baffles 1 and 2 are used for realizing uniform flow distribution among 15 parallel channels respectively under low or high inlet flow-rate condition (Q in =1.8 L•min -1 ; mean Re ch =1000). Baffles 3 and 4 are optimized for respectively achieving ascending or descending flow distribution under low inlet flow-rate (Q in =0.18 L•min -1 ; mean Re ch =100). Baffle 5 is the uniform initial baffle with identical width of orifices whereas Baffle 6 is named as "empty baffle" for representing the configuration of distributing manifold without baffle insertion. The baffle number, the corresponding target curve, the design inlet flow-rate and detailed dimensions of the orifices are summarized in Table 1 and Fig. 3. Note that k m is the target value of mass flow-rate in kth channel.

No. Baffle Target curve

Design flow-rate The perforated baffles, as shown in Fig. 4, were also fabricated in the laboratory LTN by carving the orifices on thin PMMA slices, using a digital drilling machine. Theoretically, more precise optimization results lead to better approaching of target curves, but also set higher requirements on computational abilities and fabrication precisions. For this study, the in-house fabrication tolerance is 0.01 mm and the minimum width of orifice is 0.16 mm, implying a possible deviation between the numerically optimized distribution and the target curve. To minimize this intrinsic maldistribution, more precise fabrication technologies should be introduced, with smaller fabrication tolerances. The degree of closeness between the target curve and the optimized flow distribution curve obtained by CFD-based evolutionary algorithm or by PIV measurement is quantified globally by the maldistribution factor (MF) and locally by the maximum deviation max  defined as follows:

1 uniform   1, 2,...,15 k m m k    (3) Qin=0.18 L•min -1 ; mean Rech=100 2 uniform   1, 2,...,15 k m m k    (3) Qin=1.8 L•min -1 ; mean Rech=1000 3 ascending     0.025 0.8 1, 2,...,15 k m k m k     (4)
2 1 1 MF 1 M kk k mm Mm          (6) max max 1, 2,...,15 100% kk k mm k m            (7)
3 Experimental setup and CFD simulation parameters

Experimental setup

Figure 5 shows a schematic diagram of the experimental set-up built in the laboratory LTN. It is composed of fluid circuit and PIV measurement facility. Pure water was used in this study as the working fluid. In the fluid circuit, water was pumped from a 300 L water tank, then passed through the test section (15-channel fluidic network), and finally returned back to the water tank to form a closed loop. A precision pump (VWR, REGLO-Z, instrumental error less than 0.1%) was installed in the circuit with a measurement range of 32-3200 mL•min -1 and the output mass flow-rate was controlled directly based on its own readings. Fresh water was introduced to clean the 15-channel device after each series of tests and a filter was used to avoid the discharge of sewage. Standard 2D-2C PIV facility used in this study consists of illumination unit, image record unit, synchronous controlling unit and data processing unit. The light source used was Nd-Yag laser (Litron Incorporation, DualPower 65-15 DANTEC laser). The pulse energy was 65 mJ with the duration of 4 ns. The light wavelength was 532 nm and the diameter of light beam was 6.5 mm. The pulse frequency is 5 Hz, indicating that the fluid flow was recorded five times per second. The light scattered from the particles was collected by a CCD camera (DANTEC, FlowSense) with 2048×2048 pixels. An additional macro LAVISION lens was used to yield a magnification of about 230 pixel•mm -1 , corresponding to the view field of about 8.9 mm×8.9 mm. A synchronizer (DANTEC, 80N77) was used to guarantee that the laser and the CCD camera work synchronously. A commercial software FlowManager (v3.41) was used to control the PIV system and to process the data by 2-frame cross-correlation and Fast Fourier Transforms (FFTs). In the current study, the size of the PIV interrogation window was fixed at 16×16 pixels considering a compromise between the measurement accuracy and the computational time. The spatial resolution of PIV for this study is about 0.19 mm, determined by 2.8 times of the size of interrogation window [START_REF] Foucaut | PIV optimization for the study of turbulent flow using spectral analysis[END_REF]. Polyamid Seeding Particles (PSP-5) supplied by DANTEC Company were used as seeding in our study with good chemical stability and environmental friendliness. The particle density is 1030 kg•m -3 and the diameter of particles ranges from 1 µm to 10 µm with the average of 5 µm.

Figure 6 shows the workbench specially designed and built which permits the subtle movements of the test-section. The camera can be moved in x direction whereas the platform to fix the test-section is moveable in y and z directions. The movements in x and y directions were realized using steel orbits while the movement of the platform along z direction was realized using a precise lift and a digital gauge with an accuracy of 0.01 mm. The focus of the lens must be adjusted for each vertical plane due to different scattered light optical paths, leading to different magnification ratios which must be taken into account in the post-processing. The platform can also be rotated around z-axis to ensure a fine overlap between the walls of channels and the boundaries of the PIV interrogation windows.

Measurement procedure and uncertainty

To obtain the flow distribution properties among parallel channels, the flow-rate in every individual channel should be measured when the total inlet flow-rate is stable. To do that, the cross-section of each channel was divided into 9 planes along z direction for measuring the velocity profiles using 2D-2C PIV technique, with a stepping distance of 0.25 mm. It should be noted that the laser source was fixed so that the device should be moved vertically along z direction for different measuring planes, using the adjustable workbench described above. Due to the limitation of the test view of PIV, at most two channels can be measured together each time. For a total of 15 channels, the fluidic network should be moved along y direction and then eight successive measurements should be repetitively performed.

In our pre-test, two numbers of instantaneous frames, 200 and 375, were used for each plane to test the image independence. The maximum deviation is less than 0.1% [START_REF] Boutin | Experimental measurement of flow distribution in a parallel mini-channel fluidic network using PIV technique[END_REF]. Therefore, 200 pairs of frames for each plane were enough to save the experiment and post-processing time. In order to analyze the uncertainty of the 2D-2C PIV technique, the results of total mass flow-rate obtained by PIV in this study are compared to the display of high precise pump. The maximum possible deviation is 5.1% which will be indicated as the error bar for PIV results in the following figures, implying that the PIV technique is relatively accurate for flow-rate distribution measurements among parallel channels. More details on the PIV measurement procedure may be found in our earlier work [START_REF] Boutin | Experimental measurement of flow distribution in a parallel mini-channel fluidic network using PIV technique[END_REF].

CFD simulation parameters

To compare with the PIV measurements, 3D CFD simulations were performed to calculate the flow-rate distribution among 15 parallel channels, using the same geometrical characteristics as the fabricated device and the same working fluid of pure water with density of 998.2 kg•m -3 and viscosity of 1.003×10 -3 kg•m -1 •s -1 . The inlet mass flow-rate was set to be constant according to different operational conditions. The operational pressure was fixed at 101325 Pa. Simulations were performed under steady state, incompressible and isothermal condition without heat transfer. The gravity effect at -z direction was also taken into account. In fact, the optimized perforated baffles in Section 2.2 were obtained with the same simulation parameters.

Navier-Stokes equations were solved by a commercial code ANSYS FLUENT (version 12.1.4), using COUPLED method for pressure-velocity coupling, and second-order upwind differential scheme for discretization of momentum and standard method for pressure. Laminar or standard k-ε model was respectively used to simulate the laminar or turbulent flow. Constant mass flow-rate at inlet surface was given and the boundary condition of outlet was set as pressure-outlet with zero static pressure. Adiabatic wall condition was applied and no slip occurred at the wall. The solution was considered to be converged when 1) sums of the normalized residuals for control equations were all less than 1×10 -6 ; and 2) the mass flow-rate at each channel was constant from one iteration to the next (less than 0.5% variation).

In the study, structured mesh was generated using software ICEM (version 12.1) to build up the geometry model, including about 1.42 million elements in total. A grid independence study was performed with a refined mesh with about 5.49 million elements in total. The maximum difference of the mass flow-rate in each channel is less than 4.9% so that the normal mesh was used as a compromise between the calculation time and the precision while the value of 4.9% was indicated as error bars for CFD results in the following figures. ). With the insertion of empty baffle, a similar distribution curve may be observed compared to that under low-rate condition, the σ k values ranging from 0.04 to 4.00 (PIV results). CFD results are also in good agreement with the PIV results for the general tendency, especially noticeable discrepancies for channels 5-9, implying the potential difficulties in simulation caused by turbulence and vortex flows. Even so, the insertion of the optimized baffle 2 serves as an optimized flow homogenizer to improve the distribution uniformity, indicated by the decreased MF value from 1.100 to 0.041 (PIV results). For each channel, the maximum deviation θ max also decreases from 299.9% to 8.0% (PIV result). For the case of optimized baffle, CFD results show better agreement with PIV results, indicated by almost flat distribution curves at σ k =1.0.

Results and discussion

To better validate the CFD simulations by PIV measurement, an examination on the fluid flow streamlines in the distributing manifold was carried out. Due to the limitation of view field, the inlet distributing manifold was divided into seven independent parts which were successively recorded by the CCD camera and then processed and connected so as to reconstruct the fluid flow streamlines in the whole distributing manifold, except for a narrow bar area which is not transparent due to the existence of the baffle. PIV and CFD results under low inlet flow-rate conditions (Q in =0.18 L•min -1 ; mean Re ch =100) with optimized baffle 1 are presented in Fig. 8. Corresponding flow streamlines obtained with empty baffle can be found in Fig. 7 of [START_REF] Boutin | Experimental measurement of flow distribution in a parallel mini-channel fluidic network using PIV technique[END_REF].

Under low inlet flow-rate condition, both the flow streamlines in the distributing manifold obtained by PIV and CFD methods seem smooth and regular. A vortex next to the inlet tube may be easily observed both on PIV and CFD images, with similar location and size. This also explains the good agreement between PIV and CFD results on flow-rate distribution among parallel channels. Due to the optimized dimensions of orifices on the baffle, the fluid flow downstream the baffle is changed significantly so that a relatively uniform fluid flow distribution is achieved. Figures 9 shows the streamlines in the distributing manifold with optimized baffle 2 under high inlet flow-rate condition (Q in =1.8 L•min -1 ; mean Re ch =1000). Corresponding flow streamlines obtained with empty baffle can be found in Fig. 8 of [START_REF] Boutin | Experimental measurement of flow distribution in a parallel mini-channel fluidic network using PIV technique[END_REF]. The flow profiles become irregular, indicated by the existence of more and larger vortices and recirculation flows. The difference of flow profiles under low and high inlet flow-rate conditions can be explained qualitatively based on the pressure distribution in the distributing manifold, as shown in Fig. 10. Under low inlet flow-rate, the pressure decreases gradually from inlet side to outlet side. On the contrary, a relative low-pressure area can be observed in the middle of the distributing manifold under high inlet flow-rate, implying a backflow from outlet side to inlet side which that generates obvious vortices. The inserted perforated baffle mainly influences the downstream pressure distribution to regulate the fluid flow-rates in parallel channels.

With the optimized baffle 2, the sizes of the vortices in y direction are relatively smaller because of the reduced available space in the distributing manifold upstream the baffle. Similar to the cases for low inlet flow-rate condition, the locations of the vortices upstream the baffle are not significantly changed with or without the insertion of baffle 2. Nevertheless, because of optimized sizes of orifices, the fluid flow downstream the baffle is also controlled to achieve uniform flow distribution, illustrating and confirming the effectiveness of the geometrically optimized baffle insertion method. The values of MF and the maximum deviation for empty baffle 6 and for optimized baffles 1 and 2 under different inlet flow-rate conditions are summarized in Table 2. It can be observed that the flow distribution finally achieved is relatively more uniform under low flow-rate condition than under high flow-rate condition. The precise control of fluid flow becomes much more difficult due to the turbulent flow and the existence of various vortices, leading to higher uncertainties in CFD simulations, in evolutionary algorithm and in PIV measurement. Nevertheless, satisfactory results can still be obtained (MF<0.05, θ max about 10%) under high flow-rate condition and we are pretty sure that more satisfying results may be achieved by employing advanced computational facilities and precise fabrication techniques. Meanwhile, the total pressure drops of the fluidic network increase respectively by 4.95% from 391.9 Pa

Effective range of optimized baffle -a robustness study

In the above sub-section, it has been demonstrated that under the design flow-rate conditions, the optimized baffles are really effective on the improvement of flow distribution uniformity. Generally, it is recommended that the optimized baffle should work under its design working condition and should be replaced by a new one when the operation condition changes. In real-world engineering however, the workload (inlet flow-rate) may be fluctuant so that the frequent replacement of baffle would become laborious, sometimes not cost-effective. Actually small deviations from target flow distribution may be acceptable when the tolerance on the distribution quality is relatively high. Therefore, it will be interesting and of practical significance to test the robustness of the optimized baffle when the inlet flow-rate deviates from its design value.

In this study, the optimized baffle 2 designed for uniform distribution (Q in =1.8 L•min -1 ; mean Re ch =1000) was tested both numerically and experimentally under other two inlet flow-rate conditions, i.e. a lower flow-rate of 0.18 L•min -1 (mean Re ch =100) and a higher flow-rate of 3.6 L•min -1 (mean Re ch =2000). Note that for this higher flow-rate, a gear pump (MICROPUMP, O/CGC-M35.PVSF) with a measurement range of 1-5 L•min -1 and a flange flowmeter (KOBOLD, DPL-1P15GL443) are used. PIV and CFD results on the flow distribution among parallel channels obtained under different inlet flow-rate conditions are shown in Fig. 11. It can be obviously observed that the flow distribution becomes less uniform when the inlet flow-rate deviates from the design value. More precisely, when the inlet flow-rate is reduced by a factor of 10 with respect to its design value (Q in =0.18 L•min -1 instead of 1.8 L•min -1 ; mean Re ch =100 instead of 1000), the flow distribution curve appears to be a parabolic shape, with the maximum in the middle channels (channels 6-7) and the minimum on both sides (channels 1 and 15). Contrarily when the inlet flow-rate is doubled (Q in =3.6 L•min -1 instead of 1.8 L•min -1 ; mean Re ch =2000 instead of 1000), the actual distribution curve appears to be a multi-segment line fluctuating around σ k =1.0. The MF and θ max values for the optimized baffle 2 under different inlet flow-rate conditions are listed in Table 3. It may be observed that the optimized baffle works less effectively when the working condition deviates from its design value, indicated by larger values of MF and θ max . When the inlet flow-rate is reduced by a factor of 10 (Q in =0.18 L•min -1 ), the MF value is smaller than 0.095 while the θ max value is lower than 15%. When the inlet flow-rate is doubled (Q in =3.6 L•min -1 ), the MF value is smaller than 0.085 while the θ max value is lower than 20%. Within a large variation range of inlet flow-rate (a factor of 0.1-2.0 with respect to the design value), the maximum deviation will rise to ±20%, which is the double of that under design value (<±10%). However, compared to the empty baffle case (with θ max higher than 170% at least), the distribution non-uniformity is still limited at a relatively low level under our tested conditions by the insertion of a sub-optimal baffle. Results shown in Fig. 11 also confirm the reliability of CFD simulations by PIV measurement. Generally speaking, the results obtained by PIV and CFD are in good agreement with each other, forming similar curves under the tested working conditions. Due to the potential simulation errors, or the uncertainty in PIV measurement, the flow-rates in some individual channels may be slightly different (e.g., channel 14 in Fig. 11(a), and channel 3 in Fig. 11(b)).

Realization of non-uniform target flow distribution

In our previous work (Wei et al., 2015b), we have numerically demonstrated the realization of non-uniform target flow distribution by the insertion of an optimized perforated baffle, which has never been achieved by other conventional methods to the best of our knowledge. In this sub-section, we try to validate this distinguished feature of our method by testing two representative non-uniform target curves, i.e. a linear ascending and a linear descending curve. These target curves are mathematically described in Table 1 and indicated by hollow triangular symbols (∆) in the following figures. Corresponding perforated baffles 3 and 4 were optimized and fabricated, as shown in Table 1 and Fig. 4. These baffles were then installed in the distributor section of the 15-channel fluidic network (Fig. 1) and PIV measurements were performed under the inlet flow-rate Q in =0.18 L•min -1 . The results of empty baffle 6 under the same working condition were also used as the reference case for comparison.

Figure 12 and Table 4 present the CFD and PIV results for realizing the ascending or descending curve, using the optimized baffle 3 or 4, respectively. It can be seen that the CFD results are in good agreement with the PIV measurements, approaching quite well the target curves. Compared to the empty baffle 6 case, the MF value is reduced from about 0.7 to smaller than 0.02 for the ascending target curve, and from about 0.6 to about 0.03 for the descending target curve, by using the optimized baffles. The θ max value also decreases from about 240% to lower than 4% for the ascending target curve, and from about 140% to lower than 7% for the descending target curve. Moreover, the extra pressure drops caused by the inserted perforated baffles are about 5% for both cases. Based on the results obtained above, it is experimentally validated that the CFD-based evolutionary algorithm developed in Wei et al. (2015b) is capable of realizing some non-uniform target flow distributions among parallel channels.

It should be noted that these target curves proposed and tested here are just simple examples. By a logical extension of this point, we are quite confident that other shapes of target distribution curves can also be approached by the insertion of optimized perforated baffles. For some extreme operation situations (e.g., step-like target curve) however, complementary methods (e.g., partition walls, second baffle, etc.) may be necessary and this issue will be further investigated in our future work.

Discussions

A close investigation on the flow streamlines in the distributing manifold indicates that the locations and sizes of vortices are influenced significantly by the total inlet flow-rate. The discrepancies between CFD and PIV results under high flow-rate condition can be observed due to the well-known difficulty of correctly modelling turbulent flow. Consequently, the flow distribution optimization under high flow-rate conditions is relatively more difficult because the evolutionary algorithm used to optimize the perforated baffles relies on CFD simulation results. Therefore, high performance computational facilities are generally required for more accurate simulation of fluid flow. Meanwhile, CFD results must be validated by experimental measurements, especially under turbulent flow condition.

In our evolutionary algorithm, it is basically recommended that each optimized perforated baffle works under its design flow-rate to reach the best performance on flow distribution. However, a deviation of flow-rate with a factor of 0.1-2.0 from the design value will not significantly influence the flow distribution uniformity under our tested conditions. Therefore, the replacement of the perforated baffle can be performed according to the tolerance on flow distribution quality and the costs. If the larger deviation from the target curve cannot be accepted, a new perforated baffle has to be optimized and reinstalled. The proper design of new perforated baffle calls for the running of whole optimization process which might be time-consuming. Alternatively, some rapid designs may also be proposed following the similar pattern or tendency of baffle configurations already obtained.

In actual applications, some practical issues must be carefully considered, such as the orifice blocking or abrasion which may potentially modify the effective sizes of orifices on the perforated baffles. These problems can be solved by adding a filter on the inlet tube or using wear resistant materials for the fabrication of baffles. The flow distribution optimization is a systematic task, not simply depends on local design or structural optimization of the distributor.

Conclusion

In this paper, CFD simulations and experimental (PIV) measurements have been performed in a 15-channel fluidic network to validate the realization of target flow distribution by the method of geometrically optimized baffle insertion. CFD and PIV results obtained indicate that various target flow distributions (uniform, ascending and descending curves) can be approached under different inlet mass flow-rate conditions. Main conclusions are summarized as follows:

 Different CFD models used for simulations are validated by the PIV results. For laminar flow, CFD simulation results (both flow distribution and streamlines) are in excellent agreement with PIV measurements. For turbulent flow, CFD and PIV results are also in good agreement with each other, but with a small deviation.

 The realization of different target fluid flow distributions (uniform, ascending and descending curves) by the insertion of geometrically optimized baffles is validated by PIV measurement. The MF values can be reduced from more than 0.5 to less than 0.05 for all cases by using the optimized baffles. Likewise, the values of θ max are reduced from more than 100% to less than 10%.

 The robustness study implies that for uniform fluid flow distribution, the optimal baffle can be used within a relative wide range of inlet flow-rate condition (by a factor of 0.1 to 2.0 with respect to the design value), without affecting much the flow distribution uniformity (MF<0.1). For non-uniform target distribution, the robustness depends clearly on the shape of target curve so that the influences will be more complex and should be tested case by case.

Note that the fluidic network used in this paper is just a model example without any practical application background. The method of realizing target flow distribution by the insertion of a geometrically optimized baffle, which has been proposed and developed in Wei et al. (2015b) and validated experimentally in this study, can be applied in various engineering fields dealing with fluid distribution problem. This will be the main direction of our future work.

Acknowledgement

This work is financed by the BPI France and the Region Pays de la Loire, and monitored by the SATT OUEST VALORISATION within the project OPTIFLUX (No. DV2054). One of the authors M. Min WEI would also like to thank the French CNRS for its partial financial support to his PhD study. The authors also thank Mr. Gwenaël BIOTTEAU and Mr. Nicolas LEFEVRE for their great assistances during the device fabrication and the building of the experimental set-up.

List of figures

List of tables

Figure 1 :

 1 Figure 1: Geometry and dimensions of 15-channel fluidic network (unit: mm): (a) Parallelepiped A; (b) Parallelepiped B; (c) Photo view after assembling; (d) Zoom on the inlet distributor and the location of the perforated baffle. Adapted from Boutin et al. (2016).

Figure 2 :

 2 Figure 2: Geometry and dimensions of the uniform baffle with identical width of orifices (unit: mm).

Figure 3 :

 3 Figure 3: Width distribution of orifices on the baffles.

Figure 4 :

 4 Figure 4: Photography of baffles used in this study.

Figure 5 :

 5 Figure 5: Schematic diagram of the experimental setup built for this study.

Figure 6 :

 6 Figure 6: Experimental workbench for subtle movements of the test-section.

4. 1 Figure 7 :

 17 Figure 7: Realization of uniform fluid flow distribution under different inlet flow-rate conditions: (a) Qin=0.18 L•min -1 , mean Rech=100; (b) Qin=1.8 L•min -1 , mean Rech=1000.Under low inlet flow-rate condition (Q in =0.18 L•min -1 ; mean Re ch =100) with empty baffle, the values of flow-rate are relatively high (σ k >1.5) in channel 1 and 2 facing to the inlet tube and then decrease rapidly to the lowest in channel 4, as shown in Fig.7(a). After that, a slow increase of flow-rate appears till the channel 15 which is the farthest channel from the inlet tube. The flow distribution in the parallel channels is obviously non-uniform, with the σ k values ranging from 0.55 to 2.72 (PIV results). Meanwhile, CFD and PIV results are in good agreement, implying that under low flow-rate condition, the laminar model is capable of correctly simulating the fluid flow. With the insertion of the optimized baffle 1, the fluid flow distribution among parallel channels is greatly homogenized and tends toward uniform, indicated by a small variation of σ k values (both CFD and PIV) around 1.0. The maximum deviation θ max among 15 channels is lower than 5.0% and the corresponding MF value is smaller than 0.02.

Figure 7

 7 Figure 7(b) shows the CFD and PIV results on flow distribution under high inlet flow-rate condition (Q in =1.8 L•min -1 ; mean Re ch =1000). With the insertion of empty baffle, a similar distribution curve may be observed compared to that under low-rate condition, the σ

Figure 8 :

 8 Figure 8: Streamlines in the distributing manifold under low inlet flow-rate condition (Qin=0.18 L•min -1 , mean Rech=100) with optimized baffle: (a) PIV measurement; (b) CFD simulation.

Figure 9 :

 9 Figure 9: Streamlines in the distributing manifold under high inlet flow-rate condition (Qin=1.8 L•min -1 , mean Rech=1000) with optimized baffle: (a) PIV measurement; (b) CFD simulation.

Figure 10 :

 10 Figure 10: Pressure distributions in the distributing manifold.

  (empty baffle) to 411.3 Pa (optimized baffle) under low inlet flow-rate, and by 17.2% from 16786.8 Pa (empty baffle) to 19673.0 Pa (optimized baffle) under high inlet flow-rate. Extra energy consumption is inevitable to change the initial fluid flow distribution toward the target curve.

  Figure 11: Effective range tests for optimized baffle (design flow-rate of 1.8 L•min -1 , mean Rech=1000): (a) lower flow-rate (Qin=0.18 L•min -1 , mean Rech=100); (b) higher flow-rate (Qin=3.6 L•min -1 , mean Rech=2000).

Figure 12 :

 12 Figure 12: Realization of non-uniform target flow distribution using optimized perforated baffle (Qin=0.18 L•min -1 , mean Rech=100): (a) ascending target curve; (b) descending target curve.

Figure 1 :

 1 Figure 1: Geometry and dimensions of 15-channel fluidic network (unit: mm): (a) Parallelepiped A; (b) Parallelepiped B; (c) Photo view after assembling; (d) Zoom on the inlet distributor and the location of the perforated baffle. Adapted from Boutin et al. (2016).

Figure 2 :

 2 Figure 2: Geometry and dimensions of the uniform baffle with identical width of orifices (unit: mm).

Figure 3 :

 3 Figure 3: Width distribution of orifices on the baffles.

Figure 4 :

 4 Figure 4: Photography of baffles used in this study.

Figure 5 :

 5 Figure 5: Schematic diagram of the experimental setup built for this study.

Figure 6 :

 6 Figure 6: Experimental workbench for subtle movements of the test-section.

Figure 7 :

 7 Figure 7: Realization of uniform fluid flow distribution under different inlet flow-rate conditions: (a) Q in =0.18 L•min -1 , mean Re ch =100; (b) Q in =1.8 L•min -1 , mean Re ch =1000.

Figure 8 :

 8 Figure 8: Streamlines in the distributing manifold under low inlet flow-rate condition (Q in =0.18 L•min -1 , mean Re ch =100) with optimized baffle: (a) PIV measurement; (b) CFD simulation.

Figure 9 :

 9 Figure 9: Streamlines in the distributing manifold under high inlet flow-rate condition (Q in =1.8 L•min -1 , mean Re ch =1000) with optimized baffle: (a) PIV measurement; (b) CFD simulation.

Figure 10 :

 10 Figure 10: Pressure distributions in the distributing manifold.

Figure 11 :

 11 Figure 11: Effective range tests for optimized baffle (design flow-rate of 1.8 L•min -1 , mean Re ch =1000): (a) lower flow-rate (Q in =0.18 L•min -1 , mean Re ch =100); (b) higher flow-rate (Q in =3.6 L•min -1 , mean Re ch =2000).

Figure 12 :

 12 Figure 12: Realization of non-uniform target flow distribution using optimized perforated baffle (Q in =0.18 L•min -1 , mean Re ch =100): (a) ascending target curve; (b) descending target curve.

Table 1 : Target curves and design flow-rates for different perforated baffles

 1

Table 2 : Maldistribution factor (MF), maximum deviation (θmax) and total pressure drop of the fluidic network (Δp) for empty baffle 6 and for optimized baffles 1 and 2 under different inlet flow-rate conditions

 2

	Operation condition	Parameter for maldistribution	Empty baffle CFD results PIV results	Optimized baffle for uniform distribution CFD results PIV results
		MF	0.637	0.577	0.011	0.019
	Qin=0.18 L•min -1 mean Rech=100		max	183.6%	172.2%	1.8%	4.7%
		Δp (Pa)	391.9	-	411.3	-
		MF	0.973	1.100	0.049	0.041
	Qin=1.8 L•min -1 mean Rech=1000		max	261.9%	299.9%	10.2%	8.0%
		Δp (Pa)	16786.8	-	19673.0	-

Table 3 : Maldistribution factor (MF) and maximum deviation (θmax) values for the same optimized perforated baffle under different flow-rate conditions Parameter for maldistribution Baffle 2 (Re ch =1000) Baffle 2 (Re ch =100) Baffle 2 (Re ch =2000)

 3

			CFD results	PIV results	CFD results	PIV results	CFD results	PIV results
	MF	0.049	0.041	0.093	0.090	0.085	0.065
		max	10.2%	8.0%	14.1%	14.8%	19.6%	13.5%

Table 1 :

 1 Target curves and design flow-rates for different perforated baffles

Table 2 :

 2 Maldistribution factor (MF), maximum deviation (θ max ) and total pressure drop of the fluidic network (Δp) for empty baffle 6 and for optimized baffles 1 and 2 under different inlet flow-rate conditions

Table 3 :

 3 Maldistribution factor (MF) and maximum deviation (θ max ) values for the same optimized perforated baffle under different flow-rate conditions

Table 4 :

 4 Maldistribution factor (MF), maximum deviation (θ max ) and total pressure drop of the fluidic network (Δp) for realizing non-uniform target flow distribution (Q in =0.18 L•min -1 , mean Re ch =100)


Cyril Pistoresi

Yilin Fan

Julien Aubril

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

Fluid flow Characteristics of a Multi-Scale Fluidic Network

Keywords: Flow distribution, Multi-scale, Fluidic network, Parallel channels, Process Intensification

Fluid flow characteristics of a multi-scale fluidic network

INTRODUCTION

Facing the challenges of climate change and energy shortage issues, Process Intensification that leads to smaller, less costly, cleaner, safer, higher productivity and more energy efficient technologies has now become a widely accepted concept in chemical and process engineering. Particularly, the innovative designs of high yield processes or compact equipments through integration and miniaturization are effective ways to realize process intensification [START_REF] Luo | Heat and Mass Transfer Intensification and Shape Optimization: A Multi-Scale Approach[END_REF][START_REF] Commenge | Methodological framework for choice of intensified equipment and development of innovative technologies[END_REF][START_REF] Van Gerven | Structure, Energy, Synergy, Time -The Fundamentals of Process Intensification[END_REF].

Researches in the field of miniaturization, thanks to the advances in fabrication and observation tools, target at miniaturized structures (e.g., micro or mini-channels) because of their enhanced heat and mass transfer properties and improved controllability. Miniaturized process and energy equipments include heat exchangers [e.g., [START_REF] Fan | Recent Applications of Advances in Microchannel Heat Exchangers and Multi-Scale Design Optimization[END_REF][START_REF] Khan | A review on microchannel heat exchangers and potential applications[END_REF][START_REF] Tilley | On microchannel shapes in liquid-cooled electronics applications[END_REF][START_REF] Dixit | Review of micro-and mini-channel heat sinks and heat exchangers for single phase fluids[END_REF], chemical mixers or reactors [e.g., [START_REF] Hessel | Micromixers -A review on passive and active mixing principles[END_REF][START_REF] Falk | Performance comparison of micromixers[END_REF][START_REF] Sturm | Microwaves and microreactors: Design challenges and remedies[END_REF][START_REF] Plouffe | Liquid-liquid flow regimes and mass transfer in various microreactors[END_REF], catalytic monoliths or fuel cells [e.g., [START_REF] Meyers | Design considerations for miniaturized PEM fuel cells[END_REF][START_REF] Stankiewicz | Process intensification in in-line monolithic reactor[END_REF][START_REF] Lopez-Montesinos | A three-dimensional numerical model of a micro laminar flow fuel cell with a bridge-shaped microchannel cross-section[END_REF] and integrated multifunctional systems [e.g., [START_REF] Rebrov | Optimization of heat transfer characteristics, flow distribution, and reaction processing for a microstructured reactor/heat-exchanger for optimal performance in platinum catalyzed ammonia oxidation[END_REF][START_REF] Dreimann | Highly integrated reactor separator systems for the recycling of homogeneous catalysts[END_REF][START_REF] Anxionnaz | Heat exchanger/reactors (HEX reactors): Concepts, technologies: State-of-the-art[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][START_REF] Schwolow | Design and application of a millistructured heat exchanger reactor for an energy-efficient process[END_REF] with the purpose of realizing a lab-on-a-chip. Nevertheless, to obtain a high throughput or production rate, a large number of micro/mini-channels should be installed in parallel. This so-called scaling-up by numbering-up process is far from trivial but is a key issue involving how to design and develop globally macro-sized, locally micro-structured process equipment by using multi-scale fluidic networks [START_REF] Matlosz1 | Structured multi-scale process systems design and engineering -the role of microreactor technology in chemical process design[END_REF][START_REF] Hohmann | Modules in process industry-A life cycle definition[END_REF].

One typical geometry commonly used to form multi-scale fluidic networks is the elementary Zladder structure, which consists of an inlet fluid distributor, an outlet fluid collector and a number of perpendicular parallel cross-channels (as shown in left part of Fig. 1). Abundant literature is available on the hydrodynamic, heat transfer or reaction characteristics of such structure. These studies distinguish themselves by the presence of analytical [e.g., [START_REF] Commenge | Optimal design for flow uniformity in microchannel reactors[END_REF][START_REF] Kee | A generalized model of the flow distribution in channel networks of planar fuel cells[END_REF][START_REF] Pan | Trend prediction in velocity distribution among microchannels based on the analysis of frictional resistances[END_REF][START_REF] Solovitz | Manifold design for micro-Channel cooling with uniform flow distribution[END_REF], numerical [e.g., [START_REF] Kim | The effect of header shapes on the flow distribution in a manifold for electronic packaging applications[END_REF][START_REF] Tonomura | CFD-based optimal design of manifold in plate-fin microdevices[END_REF][START_REF] Pan | CFD-based study of velocity distribution among multiple parallel microchannels[END_REF][START_REF] Zeng | Qualitative investigation on effects of manifold shape on methanol steam reforming for hydrogen production[END_REF] or experimental results [e.g., [START_REF] Huang | On flow uniformity in various interconnects and its influence to cell performance of planar SOFC[END_REF][START_REF] Cho | Experimental study on microchannel heat sinks considering mass flow distribution with non-uniform heat flux conditions[END_REF][START_REF] Manikanda Kumaran | Experimental and numerical studies of header design and inlet/outlet configurations on flow mal-distribution in parallel micro-channels[END_REF], by the continuous [e.g., [START_REF] Wang | Pressure drop and flow distribution in parallel-channel configurations of fuel cells: Z-type arrangement[END_REF][START_REF] Wang | Theory of flow distribution in manifolds[END_REF] or discrete [e.g., [START_REF] Tondeur | Flow and pressure distribution in linear discrete "laddertype" fluidic circuits: An analytical approach[END_REF][START_REF] Midoux | The theory of parallel channels manifolds (Ladder networks) revisited part 1: Discrete mesoscopic modelling[END_REF][START_REF] Midoux | The theory of parallel channels manifolds (ladder networks) revisited part 2: Design for uniform cross-flow distribution[END_REF][START_REF] Wang | Discrete approach for flow field designs of parallel channel configurations in fuel cells[END_REF][START_REF] Wang | Discrete method for design of flow distribution in manifolds[END_REF] description of the network, and by different aimed applications. Relatively fewer studies were devoted to the construction of a more complex multi-scale fluidic network through parallel arrangement of elementary Z-ladders by introducing intermediate connections, as shown in the right part of Fig. 1 for example. Noteworthy is the group of Falk and Commenge who first performed hydrodynamic modeling of the multi-scale network under isothermal and laminar flow conditions, so as to provide design charts for rapid and flexible numbering-up of parallel micro-reactors [START_REF] Saber | Rapid design of channel multi-scale networks with minimum flow maldistribution[END_REF][START_REF] Commenge | Methodology for multi-scale design of isothermal laminar flow networks[END_REF]. In their later works, the heat transfer characteristics [START_REF] Saber | Heat-transfer characteristics in multi-scale flow networks with parallel channels[END_REF] and the selectivity of consecutive catalytic reactions [START_REF] Saber | Microreactor numbering-up in multi-scale networks for industrial-scale applications: Impact of flow maldistribution on the reactor performances[END_REF] of such multi-scale networks have also been investigated. A few other studies focused on the global flow resistance minimization [START_REF] Kim | Vascularized materials: Tree-shaped flow architectures matched canopy to canopy[END_REF][START_REF] Cho | Vascular design of constructal structures with low flow resistance and nonuniformity[END_REF][START_REF] Cho | Fluid flow characteristics of vascularized channel networks[END_REF][START_REF] Cho | Transient thermal-fluid flow characteristics of vascular networks[END_REF] or thermal performance [START_REF] Cho | A numerical and experimental study to evaluate performance of vascularized cooling plates[END_REF] of similar multi-scale geometries. Nevertheless, most of these works are pure analytical while detailed numerical and experimental results on the fluid flow characteristics are still lacking.

One of the key issues for parallel multi-channel devices is the fluid flow maldistribution that deteriorates their global performances. Extensive efforts have been made on the uniformization of fluid distribution among parallel channels of an elementary Z-ladder. Some studies focused on determining the optimal ratio between hydraulic resistances of the distributor, the collector and the cross-channel [e.g. , 35]. Others worked on triangular or trapezoidal-shaped distributor and collector instead of the basic rectangular (or cylinder) shape [e.g., [START_REF] Kim | The effect of header shapes on the flow distribution in a manifold for electronic packaging applications[END_REF][START_REF] Pan | Modeling of velocity distribution among microchannels with triangle manifolds[END_REF]. Still others introduced more degrees of freedom and proposed stairway-type [e.g., [START_REF] Commenge | Optimal design for flow uniformity in microchannel reactors[END_REF][START_REF] Solovitz | Manifold design for micro-Channel cooling with uniform flow distribution[END_REF][START_REF] Tondeur | Uniform flows in rectangular lattice networks[END_REF][START_REF] Pistoresi | Numerical study on the improvement of flow distribution uniformity among parallel mini-channels[END_REF][START_REF] Saeed | Header design approaches for mini-channel heatsinks using analytical and numerical methods[END_REF] or curvature-shaped [e.g., [START_REF] Pan | Optimal design of complex manifold geometries for uniform flow distribution between microchannels[END_REF][START_REF] Tong | Geometric strategies for attainment of identical outflows through all of the exit ports of a distribution manifold in a manifold system[END_REF][START_REF] Jackson | Discrete geometry optimization for reducing flow non-uniformity, asymmetry, and parasitic minor loss pressure drops in Z-type configurations of fuel cells[END_REF] obeying some scalingrelations or subjected to geometric optimization. More detailed description of these approaches may be found in recent review papers [START_REF] Rebrov | Single-phase fluid flow distribution and heat transfer in microstructured reactors[END_REF][START_REF] Siddiqui | Efficient energy utilization through proper design of microchannel heat exchanger manifolds: A comprehensive review[END_REF]. With these methods, a relatively uniform flow distribution is achievable at the price of lower compactness. Moreover, it seems very difficult to prevent the flow distribution non-uniformity under higher flow-rate conditions (e.g., mean Rech>500) [START_REF] Saber | Rapid design of channel multi-scale networks with minimum flow maldistribution[END_REF][START_REF] Pistoresi | Numerical study on the improvement of flow distribution uniformity among parallel mini-channels[END_REF]. For multiscale fluidic networks comprising of a number of elementary ladders, only some preliminary thoughts [START_REF] Saber | Rapid design of channel multi-scale networks with minimum flow maldistribution[END_REF][START_REF] Kim | Vascularized materials: Tree-shaped flow architectures matched canopy to canopy[END_REF][START_REF] Cho | Fluid flow characteristics of vascularized channel networks[END_REF] were reported, but without further supplementary numerical or experimental data to the best of our knowledge.

Recently, an innovative method based on the insertion of a geometrically optimized perforated baffle in the distributing manifold is proposed by Luo et al. [START_REF] Luo | Heuristic shape optimization of baffled fluid distributor for uniform flow distribution[END_REF][START_REF] Luo | Method for determining characteristics of holes to be provided through a plate and corresponding programme[END_REF]. This method, associated with an evolutionary algorithm to determine the optimal sizes of orifices, is in similar relation to the study of Tong et al. [START_REF] Tong | Attainment of flowrate uniformity in the channels that link a distribution manifold to a collection manifold[END_REF] and that of Liu and Yu [START_REF] Liu | Numerical study on performances of mini-channel heat sinks with non-uniform inlets[END_REF] which adjust the dimension of the inlet/outlet opening of every mini-channel as a supplemental hydraulic resistance. Numerical and experimental results [START_REF] Wei | CFD-based evolutionary algorithm for the realization of target fluid flow distribution among parallel channels[END_REF][START_REF] Wei | Numerical and experimental investigation on the realization of target flow distribution among parallel mini-channels[END_REF][START_REF] Wei | Design and optimization of baffled fluid distributor for realizing target flow distribution in a tubular solar receiver[END_REF] confirmed that this method is practical, easy to implement, effective and robust in realizing different target (both uniform and non-uniform) flow distributions among parallel channels. For complex multiscale fluidic networks such as shown in Fig. 1, it could be a promising option but hasn't yet been applied nor fully tested.

The main objective of this study is to investigate the fluid flow characteristics of a multi-scale fluidic network consisting of a number of elementary Z-ladders. Both Computational Fluid Dynamics (CFD) simulations and local flow visualization using fast camera and tracers were performed. A special attempt is made to improve the flow distribution uniformity among the parallel mini-channels, by the insertion of geometrically optimized perforated baffles. Note that neither the local fluid flow patterns nor the flow distribution properties in such multi-scale fluidic networks have been reported in the literature before; these are also the main originalities of this paper.

The rest of the paper is organized as follows. Section 2 introduces the geometry, the detailed dimensions and a transparent prototype of the multi-scale fluidic network tested in this study. Section 3 presents the used methodologies including the CFD simulation parameters, the experimental set-up and the procedure on the sizing and optimization of perforated baffles. Numerical and experimental results on the hydrodynamic characteristics of the basic network (without baffles) and the optimized network (with insertion of baffles) are compared and discussed in section 4. Finally in section 5, main findings of this study as well as the future research directions are summarized.

GEOMETRY AND DEVICE

Geometry of the multi-scale fluidic network

Figure 1 shows the geometry and dimensions of the multi-scale fluidic network. The basic component of this network is a Z-ladder circuit. Ten mini-channels having a square cross-section of 1 mm × 1 mm and a length of 50 mm are arranged in parallel, spaced one from each other by 4 mm. These parallel channels are connected together by an inlet fluid distributor and an outlet fluid collector to form an elementary 2-scale circuit. Both the elementary distributor and the collector have a rectangular-shape manifold, connected to an inlet/outlet port of 5 mm in length and 5 mm in width. The length of the collecting manifold is 46 mm and the width is 7 mm whereas for the distributing manifold, the length is 46 mm and the width is 9 mm. A groove of 1 mm in thickness is reserved for the insertion of an elementary perforated baffle, situated in the distributing manifold with 2 mm distance away from the entrance of mini-channels. Note that regarding the convenience of prototype fabrication, the channel depth is identical (1 mm) for the elementary circuit and for the entire multi-scale network. Three elementary Z-ladders are then arranged in parallel connected by a distributor and a collector to form a 3-scale circuit. The dimensions of the intermediate distributor and collector are listed in Table 1. A groove is located at 2 mm distance away from the inlet port of elementary distributors, permitting the insertion an intermediate perforated baffle of 1 mm in thickness. Likewise, three 3-scale circuits are arranged in parallel to form a 4-scale fluidic network, connected by a principal distributor (with groove reserved for 1 mm thickness principal baffle) and a principal collector. Their detailed dimensions are also listed in Table 1. Following the above-mentioned scaling-up measure, a 4-scale fluidic network with a topology [START_REF] Sturm | Microwaves and microreactors: Design challenges and remedies[END_REF][START_REF] Van Gerven | Structure, Energy, Synergy, Time -The Fundamentals of Process Intensification[END_REF][START_REF] Van Gerven | Structure, Energy, Synergy, Time -The Fundamentals of Process Intensification[END_REF] has been built. Here "10" is the number of parallel mini-channels in an elementary ladder, the middle "3" is the number of 2-scale ladders and the second "3" is the number of 3-scale circuits. The total number of parallel mini-channels equals to 90. For the convenience of description, 9 elementary Z-ladders are indexed from A to I from top-left to bottom-right, as shown in Fig. 1. For each elementary ladder, 10 mini-channels are indexed by j from 1 to 10 from left to right.

Prototype fabrication

The prototype was fabricated in the laboratory LTEN, France by digitally-assisted carving of the multi-scale fluidic network (shown in Fig. 1) on the surface of a PMMA rectangular parallelepiped. Another PMMA rectangular parallelepiped was used as the cover. Numerous grooves were reserved around the whole fluidic network or between neighboring parallel mini-channels, and filled with rubber strips to prevent the water leakage. Moreover, a number of bolts were used for further sealing. The overall dimension of prototype is 285 mm in length, 260 mm in width and 35 mm in thickness. The photo view of the multi-scale fluidic network prototype after assembling is shown in Fig. 2.

Parameter definition

In this study, the flow distribution uniformity is quantified by two parameters: the relative flowrate deviation (σi,j) and the maldistribution factor (MF). The relative flow-rate deviation is defined as:

𝜎𝜎 𝑖𝑖,𝑗𝑗 = 𝑚𝑚 𝑖𝑖,𝑗𝑗 -𝑚𝑚 � 𝑖𝑖,𝑗𝑗 𝑚𝑚 � 𝑖𝑖,𝑗𝑗 (1)
where mi,j is the mass flow-rate in the mini-channel i,j with i varying from A to I and j varying from 1 to 10. 𝑚𝑚 � 𝑖𝑖,𝑗𝑗 represents the average mass flow-rate among all 90 parallel channels:

𝑚𝑚 � 𝑖𝑖,𝑗𝑗 = ∑ ∑ 𝑚𝑚 𝑖𝑖,𝑗𝑗 10 𝑗𝑗=1 𝐼𝐼 𝑖𝑖=𝐴𝐴 90 (2)
Maldistribution factor is also introduced, for each elementary Z-ladder (MFA to MFI) and for the whole fluidic network (MFnetwork):

MF i = � 1 9 ∑ � 𝑚𝑚 𝑖𝑖,𝑗𝑗 -𝑚𝑚 � 𝑖𝑖,𝑗𝑗 𝑚𝑚 � 𝑖𝑖,𝑗𝑗 � 2 10 𝑗𝑗=1 (i = A…I) (3)
MF network = � 1 89 ∑ ∑ � 𝑚𝑚 𝑖𝑖,𝑗𝑗 -𝑚𝑚 � 𝑖𝑖,𝑗𝑗 𝑚𝑚 � 𝑖𝑖,𝑗𝑗 � 2 10 𝑗𝑗=1 𝐼𝐼 𝑖𝑖=𝐴𝐴 (4)

METHODOLOGIES

In this section, simulation parameters and the experimental set-up are described. In particular, the detailed procedure on the sizing and optimization of perforated baffles for the multi-scale network is presented.

CFD Simulation parameters

The geometry of the basic and optimized networks and the corresponding meshes were generated using different modules of ANSYS Workbench 16.1. Note that due to symmetrical character of the network in the direction of the depth, half of the real object was adopted in the simulation, the gravity effect being neglected. Pure water at constant temperature of 293 K was chosen as working fluid (density ρ=998.3 kg•m -3 and viscosity µ=1.003×10 -3 kg•m -1 •s -1 ). The operational pressure was fixed at 101325 Pa. In this study, simulations were performed under steady state, incompressible and isothermal condition without heat transfer. For simplification, viscous heating was also neglected.

Fluent code V.16.1 was used to solve the Navier-Stokes equations. Constant mass flow-rate normal to the inlet surface of the principal distributor was given. Eight mass flow inlet (min) were tested: 4.5×10 -

4 kg•s -1 ; 9×10 -4 kg•s -1 ; 1.8×10 -3 kg•s -1 ; 4.5×10 -3 kg•s -1 ; 9×10 -3 kg•s -1 ; 1.8×10 -2 kg•s -1 ; 4.5×10 -2 kg•s -1 ; 9×10 -2 kg•s -1 ,
corresponding to a value of mean channel Reynolds number (Rech) of 10; 20; 40; 100; 200; 400; 1000; 2000, respectively. The boundary condition of outlet was set as pressure-outlet with zero static pressure. Non-slip wall condition was applied.

Laminar flow model was used under very low velocity conditions while the k-ε RNG model was used for higher velocity conditions because micro turbulences and local vortex may exist. For the pressure-velocity coupling, standard SIMPLE method was used. For discretization, standard method was chosen for pressure and first-order upwind differentiation for momentum. The solution was considered to be converged when the flow-rate at each channel and the inlet static pressure were constant from one iteration to the next (less than 0.5% variation) and the normalized residuals of all monitoring parameters were lower than the order of magnitude of 10 -6 .

A grid independence study was performed using three structured meshes with different densities: coarse mesh (10 segments per millimeter; 0.64 million elements in total); medium mesh (20 segments per millimeter; 2.8 million elements in total) and refined mesh (30 segments per millimeter; 7.21 million elements in total). Simulation results with an inlet velocity of 9×10 -2 kg.s -1 (mean Rech=2000) indicated a difference of 5.4% on the inlet pressure between the coarse mesh and the refined mesh, and of 1.1% between medium mesh and refined mesh. Mass flow-rates in each mini-channel were also compared, showing a maximum difference of 2.2% between the coarse mesh and the refined mesh, and of 1.4% between the medium mesh and the refined mesh. Hence, the medium mesh was selected as a compromise between the calculation time and the precision.

Experimental set-up and measuring procedure

Fluid flow visualization was performed on the transparent prototype using fast camera and tracers so as to compare with the CFD results. Figure 3 shows the experimental set-up built in laboratory LTEN for this purpose. Pure water was pumped from a 200 liters water tank to pass through the prototype from its inlet to its outlet. A precision pump (VWR, REGLO-Z) with a range from 32 to 3200 mL•min - 1 was used and the flow-rate was measured by a flow meter (KOBOLD, DPL-1P15GL443) with a precision of ±2.5% full-scale. A pressure sensor Huba Control 692 with a measuring range from 0 to 1.6 bar (±0.4% full-scale) was installed to measure the global pressure loss between the inlet and the outlet of the prototype. The flow meter and the pressure sensor were powered by a 24 V Calex 52024 and linked to an Agilent card 34901a for data recording.

Black ink was used as a tracer. The injection was done once the fluid flow was stabilized using a syringe. A fast camera AVT Prosilica GE (40 pictures per second) was used to film and record the ink progression through the multi-scale fluidic network. An acquisition program has been developed under the LabVIEW software to pilot the fast camera. The size of the acquisition window was set on 640 × 480 pixels. The contrast, the exposure, the gain and the white balance were also adjusted in order to have a white background and to clearly visualize the ink flow patterns through the multi-scale network. Air bubbles, being observed mainly in stagnant zones of the network but also in parallel minichannels, could unfortunately disturb the flow patterns. Their presence was frequently encountered at the beginning of liquid circulation and might keep blocking the mini-channels if they had not been successfully removed. To avoid the presence of air bubbles, the prototype was assembled in water environment and the inlet/outlet pipes were also connected to the prototype under water. The tightness was checked before each recording to prevent the water leakage.

For convenience of comparison, all the tests were carried out at room temperature (about 25 °C). For each measurement, it took about several minutes for the stabilization of fluid flow. The acquisition time was set as 10 seconds and it launched simultaneously when ink was injected. Same procedure was performed for both the basic network (without baffles) and the optimized network (with baffles). Local ink flow patterns were captured and analyzed image by image in order to make a comparison with the CFD simulation. Serval tests were performed for the same inlet flow-rate condition and good reproductivity was observed.

Sizing and optimization of perforated baffles

The basic idea is to install a perforated baffle in the distributing manifold and to optimize the size of orifices by an evolutionary algorithm in order to reach the target flow distribution. This method has been proposed and tested in our earlier studies [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][START_REF] Wei | Numerical and experimental investigation on the realization of target flow distribution among parallel mini-channels[END_REF] for the purpose of achieving target flow distribution among parallel channels of a two-scale elementary ladder, but is applied in the first time to a multi-scale fluidic network.

Regarding the complex multi-scale geometry, the simultaneous optimization of 13 baffles (9 for elementary distributors, 3 for intermediate distributors and 1 for principal distributor) is a highly challenging task, even prohibitive due to the difficulty in generating adapted meshes, complicated fluidstructure interactions and high requirement on computing facilities. In this study, the optimization process is divided into four steps with reasonable simplifications and assumptions.

• Step 1: Simulation on the intermediate collectors and the principal collector part (green color in Fig. 4) in order to evaluate the outlet pressure of each elementary ladder.

• Step 2: Sizing and optimizing the elementary perforated baffle in the distributor (red color in Fig. 4) of the elementary ladder and evaluating the pressure loss. • Step 3: Sizing and optimizing the intermediate perforated baffle in each intermediate distributing manifold (blue color in Fig. 4) and evaluating the inlet pressure of intermediate distributors.

• Step 4: Sizing and optimizing the principal baffle in the principal distributor (yellow color in Fig. 4). Figure 6 shows that the static pressure at each inlet of the simulation domain increases rapidly when mean Rech increases. At a certain mean Rech, the pressure at inlet A is always the highest while that at inlet I is the lowest. This is mainly due to the distance difference between each inlet to the common outlet. The calculated pressure data were used in the following steps.

-Step 2: Elementary baffle

The results obtained in step 1 indicate that the outlet pressure of elementary ladder varies with the position, but it will not influence the flow field and the pressure drop (∆pelementary) when inlet mass flowrate is identical for all ladders. So one elementary ladder (e.g., A) is studied here and the elementary baffle, once optimized, is applicable for all elementary ladders. The studied geometry in this step is then shown in Fig. 7. Constant velocity normal to the boundary of inlet surface is given (inlet mass flow-rate=m) and the boundary condition of outlet is set as pressureoutlet with the static pressure value (Pout, i) calculated in step 1. The initial baffle has 18 orifices of uniform width (0.6 mm) and a global porosity of 0.23. Its geometry optimization follows the evolution algorithm developed in [START_REF] Wei | CFD-based evolutionary algorithm for the realization of target fluid flow distribution among parallel channels[END_REF]. Optimized dimensions of the orifices under different mean Rech conditions are reported in the supplementary material of this paper.

-Step 3: Intermediate baffle

The aim of this step is to optimize the intermediate baffles. The studied geometry as shown in Fig. 8 is the intermediate distributor having one inlet and three outlets that correspond to the inlets of elementary ladders. It was observed that the pressure differences among three outlets are identical for three intermediate distributors (A-C; D-F; G-I). So the intermediate baffle, once optimized, is identical for all three intermediate distributors.

Constant velocity normal to the boundary of inlet surface is given (mass flow-rate=3m) and the boundary condition for three outlets is set as pressure-outlet (Pin, A; Pin, B and Pin, C). The initial intermediate baffle has 6 orifices of uniform width (0.4 mm) and a global porosity of 0.02. Optimized dimensions of the orifices under different mean Rech conditions are given in the supplementary material of this paper.

RESULTS AND DISCUSSION

In this section, the numerical results on the flow field in different parts of the multi-scale network will be presented, together with the experimental visualization of the ink invasion for comparison. Then the flow distribution properties among parallel mini-channels in the basic and optimized fluidic networks will be compared and discussed. Finally the total pressure drop of the prototype obtained by CFD simulation and by experimental measurement will also be presented.

Flow field

CFD simulation results

Figure 10 shows the contours of velocity magnitude on the mid-depth surface (symmetry) of the basic multi-scale fluidic network under different mass flow inlet conditions. Recall that optimized baffles corresponding to a certain inlet mass flow-rate (mean Rech) are inserted at distributing manifolds of different scales and results obtained are shown in Fig. 10.

At low inlet flow-rate (min=4.5×10 -4 kg•s -1 ; mean Rech=10), the velocity profiles in the distributors and in the collectors of different scales are regular and symmetric, implying that the flow patterns are rather laminar. It seems that the converging or diverging bifurcations do not much disturb the flow profiles and the impact of singular losses is not significant. The flow distribution among parallel minichannels is relatively uniform for both basic and optimized networks, which will be further discussed in later sub-section.

When the inlet flow-rate is increased by a factor of 10 (min=4.5×10 -3 kg•s -1 ; mean Rech=100), the velocity profiles in the distributor and in the corresponding collector of different scales are no longer symmetrical without baffles insertion (left side of Fig. 10b). A noticeable difference on the velocity profiles at the distributors or collectors of the same scale (elementary or intermediate) can also be observed. Some stagnant flow zones appear, particularly in the right corner of the elementary distributing manifolds and in the principal collector. This is mainly due to the stronger inertial force at the higher velocity which has been discussed in detail in our previous work on the elementary ladder [START_REF] Pistoresi | Numerical study on the improvement of flow distribution uniformity among parallel mini-channels[END_REF]. On the contrary, regular and symmetric velocity profiles can still be observed for the optimized network. In fact, the contours of velocity magnitude seem rather similar under mean Rech=10 and mean Rech=100 conditions, as shown in the right side of Figs. 10a and10b.

With a further increase of inlet flow-rate by a factor of 10 (min=4.5×10 -2 kg•s -1 ; mean Rech=1000), the inertial force becomes still stronger. Together with the effects of singular losses at numerous converging/diverging bifurcations, stagnant flow zones, local vortex and micro-turbulences are easily observable at the left side of Fig. 10c. In the optimized network, symmetrical velocity profiles can no longer be maintained in the distributor and in the corresponding collector of different scales. Stagnant flow zones and local vortex can also be clearly observed at the right side of Fig. 10c, but the velocity profiles in 9 elementary distributors are similar in general. The resulted difference on flow distribution uniformity among 90 mini-channels due to the insertion of optimized baffles will be discussed in detail in later sub-section.

Ink flow pattern visualization

Figure 11 presents the images recorded by the fast camera, showing the ink invasion process through the multi-scale fluidic network with or without baffles insertion, at inlet mass flow-rate of 4.5×10 -3 kg•s -1 (mean Rech=100). Note that the time t=0 s represents the moment when ink front arrives at the inlet port of the fluidic network.

At t=0.175 s, only two elementary ladders A and D next to the global inlet begin to be irrigated by the tracer. The ink front arrives at the elementary distributing manifold of ladders A and D while their parallel mini-channels have not been irrigated yet. At t=0.375 s, ink progresses towards the global outlet and more elementary ladders are involved. At that moment, ink front has arrived at the distributing manifold of elementary ladder C in optimized network (right side of Fig. 11) while that of basic network is still free of tracers. This tendency becomes clearly observable at figures of t=0.575 s, which show that parallel mini-channels of elementary ladder C in optimized network (right side) have already inkinvaded while those in the basic network are still untouched. The tracer remixing and redistribution at the downstream of elementary baffles next to the inlets of mini-channels can also be observed. Finally at t=0.775 s, ladder C in the optimized network has been entirely irrigated by tracers and the ink front arrives at the principal collecting manifold, which is not the case in the basic network.

In brief, differences on the ink progression and development behaviors between the basic and optimized fluidic networks may be clearly observed. The insertion of baffles tends to modify the flow profiles in distributors of different scales, leading to the redistribution of the fluid flow among different elementary ladders as well as among parallel mini-channels. Moreover, the additional hydraulic resistance by the insertion of baffles tends to guide the fluid flow towards the bottom-left corner of the network, which is initially the most difficult to be irrigated due to the inertial forces of the fluid flow.

The transit time for ink front from the global inlet to the global outlet is measured at 1.67 s for the optimized network and 1.92 s for the basic one. It seems that the residence time distribution is narrower in the optimized network compared to that in the basic one, but difficult to be credibly measured in the current test-rig. The connection between the residence time distribution and the flow distribution uniformity among parallel channels has been discussed in detail in our earlier study [START_REF] Guo | Study of transfer intensification of a miniature multi-channel multifunctional heat exchanger[END_REF].

Comparison between CFD results and visualization

Figure 12 presents a comparison between local flow behaviors simulated by CFD and those recorded by fast camera for the basic network under mean Rech=100. Recall that ink progression experimentally visualized indicates the transient fluid flow behaviors in the multi-scale network whereas the contours of velocity magnitude simulated show the steady-state flow patterns. For a reasonable comparison, representative images showing details of local fluid flow have been captured at different moments of the video recorded by the fast camera. These local snapshots have then been placed next to the corresponding positions of the CFD flow field for comparison. Note that darker color represents higher flow velocity (more ink molecules) while light color means low flow velocity (fewer ink molecules). The gray scale is somewhat proportional to the velocity magnitude but cannot be quantitatively correlated.

Similar flow patterns may be observed on Fig. 12 between CFD results and visualization images at various locations of the multi-scale fluidic network, i.e., at the distributors and collectors of different scales. The micro-turbulences, the segregation of velocity vectors at bifurcations and the stagnant zones can be clearly identified, indicating the complex characteristics of fluid flow due to the complex multiscale structure. The good agreement on the local flow profiles shown in Fig. 12 may serve as a qualitative experimental validation of the CFD results. Nevertheless, detailed information on the flow-rate or velocity magnitude in parallel mini-channels cannot be extracted from the recorded visualization images due to the small size of the mini-channels (1 mm in width) and inadequate resolution of the fast camera (640 × 480 pixels). As a result, only CFD results on the flow distribution uniformity among parallel mini-channels are presented and discussed in the following Figs 13-15.

Flow distribution uniformity

In this sub-section, CFD results on the relative flow-rate deviation σ (Eq. 1) and the maldistribution factors MF (Eqs. 3, 4) will be reported, so as to quantify the fluid flow distribution uniformity among the parallel mini-channels.

Standard deviation

Figure 13 shows the flow-rate deviation of an individual mini-channel (i,j) compared to the mean value of all 90 mini-channels under mean Rech=10, 100 and 1000 conditions. Note that the hollow square curve (□) shows the σ values for the basic network while the solid square curve (■) represents those for the optimized network.

Under low inlet flow-rate condition (min=4.5×10 -4 kg.s -1 ; mean Rech=10), the flow distribution curve for each elementary ladder of the basic fluidic network appears a somewhat different shape, as shown in Fig. 13a. The values of flow-rate are relatively high in channels close to the elementary outlet (e.g., CH8, 9, 10) while relatively low flow-rates may be observed in the first channel (CH1) and the middle channels (e.g., CH4, 5). The values of σi,j range from -31% (E5) to 50% (I10), indicating that the flow distribution among 90 parallel mini-channels is already non-uniform at mean Rech=10 . Regarding the flow distribution among different elementary ladders, C, G, I are relatively overfed whereas B, D, E are underfed. For the optimized fluidic network, small fluctuation of σi,j values (±9%) may be observed, implying that much more uniform flow distribution may be achieved owing to the insertion of optimized baffles.

When the inlet flow-rate is increased by a factor of 10 (min=4.5×10 -3 kg.s -1 ; mean Rech=100), nonuniform flow distribution becomes clearly observable among 10 parallel mini-channels of each elementary ladder as well as among 9 elementary ladders of the basic network. Ladders A, D and G of the first row are relatively under-fed while ladders H, F, I close to the principal outlet are over-supplied. This is mainly due to the inertial effect so that the fluids tend to skip the nearest ladders they pass by. For each elementary ladder, the flow distribution curve generally appears an increasing shape form CH1 to CH10 with some exceptional local salients (e.g., CH3 in ladders B, E, H). This is because of the entrance effect so that fluids tend to develop in the elementary distributing manifold and skip the first two mini-channels next to the inlet, as can be seen in Fig. 10. Regarding the optimized network equipped with baffles, a smaller variation of flow-rate among parallel channels may be achieved, as indicated by the σi,j values ranging from -22% to 29%, (with respect to -54% to 88% for the basic network). The additional hydraulic resistances by the insertion of baffles alleviate the impact of inertial effect and regularize the flow profiles in the distributing manifolds of different scales, as also shown in Fig. 10. Under high inlet flow-rate condition (min=4.5×10 -2 kg.s -1 ; mean Rech=1000), the shapes of the flow distribution curve for different elementary ladders of the basic network are irregular with big rises and falls. For example, the value of σI,j of ladder I varies from -86% to 173%, with a big flow-rate jump from CH2 to CH3. Similar behaviors may also be observed in ladders F and H. Flow distribution nonuniformity is also clearly visible among elementary ladders: those close to the principal collector (F, H, I) are largely overfed while those close to the principal distributor (A, D, G) are underfed, again mainly due to the stronger inertial effect under higher velocity condition. The insertion of optimized baffles in the multi-scale network could smooth the troughs and crests of the flow-rate, leading to much smaller fluctuation of flow distribution curve in each elementary ladder. The values of σi,j range between -26% to +35% for the optimized network compared to -99% to 173% for the basic network.

Maldistribution Factor

The value of relative deviation σi,j indicates the departure on the flow-rate of individual channel to the mean value of all 90 channels whereas the value of maldistribution factor (MF) represents the flow maldistribution of a fluidic network at a global level [START_REF] Anxionnaz | Heat exchanger/reactors (HEX reactors): Concepts, technologies: State-of-the-art[END_REF][START_REF] Fan | Flow distribution property of the constructal distributor and heat transfer intensification in a mini heat exchanger[END_REF]. Figure 14 shows the evolution of MFi for each elementary ladder of the multi-scale network as a function of mean Rech.

For a mean Rech ranging from 10 to 2000, the values of MFi for all elementary ladders of the optimized network are smaller than 0.27, showing relatively uniform flow distribution. For the basic network however, MFi values generally increase rapidly with increasing mean Rech, and their internal gap is also getting bigger. Again, ladder I of the basic network always gets the highest MFi values, mainly due to the impact of inertial forces evoked above. The noticeable augmentation in MFnetwork values for the optimized fluidic network under high mean Rech condition is mainly due to three reasons. Firstly, the CFD results are less accurate due to the influences of local turbulent flows and vortex. Hence the size optimization of baffles' orifices depending strongly on the CFD results is less performant. Secondly, the smallest size of orifices is limited at 0.2 mm considering the fabrication uncertainty. The subtle size variation of orifices is not adequate facing high flow velocity. Finally, constant velocity profile normal to the inlet surface of elementary ladder is assumed during optimization, which is obviously not realistic at high mean Rech.

Pressure losses

Figure 16 shows the evolution of the pressure drop of the multi-scale network (∆pnetwork) as a function of the mean Rech. Both CFD results and experimental results are presented for comparison. It can be observed that the ∆pnetwork values for both basic and optimized networks increase exponentially with the increasing mean Rech. The CFD results are generally in good agreement with the experimental results. The slightly higher data experimentally measured compared to those simulated results are mainly due to the fact that the connection pipes between the measurement points and the inlet or outlet of the network are not taken into account in the simulation.

Noticeable augmentation on the pressure drop is caused by different baffles equipped. At mean Rech=100, ∆pnetwork increases by 58 % from 5.5 kPa (basic network) to 8.7 kPa (optimized network). More significant augmentation is also observed at mean Rech=1000: ∆pnetwork increases by a factor of 2.3 from 333 kPa (basic network) to 777 kPa (optimized network). More pumping power should be consumed to overcome the stronger inertial forces of fluid flow at high flow-rate, in order to homogenize the initial non-uniform distribution into uniform among the 90 parallel mini-channels.

CONCLUSIONS AND PERSPECTIVES

This study investigates the fluid flow characteristics of a multi-scales network having 90 minichannels in parallel. In order to improve the flow distribution uniformity in the multi-scale network, geometrically optimized perforated baffles are inserted in the distributing manifolds of different scales. The main results on the distribution uniformity and on the global pressure drop for both basic and optimized networks are summarized in Table 2. Based on the results presented above, the following conclusions may be reached.

• For the basic fluidic network, fluid flow distribution is non-uniform even at very small mean Rech (e.g., 10). The distribution non-uniformity increases rapidly with increasing flow-rate.

• For the optimized fluidic network equipped with perforated baffles, relatively uniform distribution could be maintained even at high mean Rech. But higher pressure drop of the network will be induced. • Local flow profiles simulated by CFD are in good agreement with the visualization results obtained by fast camera and tracers.

Our current work is focused on the investigation of heat transfer characteristics of such multi-scale fluidic network used as a heat sink to evacuate the heat generated in a surface. The development and implementation of adequate characterization techniques for quantitative analyses of fluid low and heat transfer is one of our on-going work. Moreover, alternately stacking a number of multi-scale structured plates to construct a novel plate heat exchanger as proposed by Luo and Fan [START_REF] Luo | Plate heat exchanger[END_REF] is also interesting. The characterization of its hydrodynamic and thermal performances and the comparison with a commercially available standard plate heat exchanger will also be carried out to illustrate the improvements achieved by multi-scale conception and by optimized fluid distribution.

ACKNOWLEDGEMENT

This work is financed by the "Région Pays de la Loire" within the project "Nouvelle équipe de recherche". Authors are grateful to Mr. Gwenaël BIOTTEAU for the fabrication of the prototypes and to Dr. Min WEI for his help on the CFD-based evolutionary algorithm.

NOMENCLATURE

Figure 1 .

 1 Figure 1. Geometry and dimensions of the multi-scale fluidic network [10, 3, 3] (unit: mm).

Figure 2 .

 2 Figure 2. Photo view of the transparent prototype of the multi-scale fluidic network.

Figure 3 .

 3 Figure 3. Experimental set-up for the visualization of the ink progression through the multi-scale fluidic network.

Figure 4 .

 4 Figure 4. Division of optimization process in 4 steps.

Figure 5 .

 5 Figure 5. Simulation geometry for step 1.

Figure 6 .

 6 Figure 6. Static pressure at the outlet of each elementary ladder as a function of mean Rech.

Figure 7 .

 7 Figure 7. Simulation geometry of step 2.

Figure 8 .

 8 Figure 8. Simulation geometry of step 3.

Figure 9 .

 9 Figure 9. Simulation geometry of step 4.

Figure 10 .

 10 Figure 10. Contours of velocity magnitude in the multi-scale fluidic network under different inlet flow-rate conditions. Left side: basic network without baffles; right side: optimized network with baffles. (a) mean Rech=10; (b) mean Rech=100; (c) mean Rech=1000.

Figure 11 .

 11 Figure 11. Visualization of the ink progression and development through the multi-scale fluidic network for mean Rech =100 (left: Basic network without baffles; right: optimized network with baffles).

Figure 12 .

 12 Figure 12. Comparison of local flow behaviors between CFD results and experimental results in the basic network under mean Rech=100.

Figure 13 .

 13 Figure 13. Fluid flow distribution among parallel mini-channels of the multi-scale fluidic network. (a) mean Rech=10; (b) mean Rech=100; (c) mean Rech=1000.

Figure 14 .

 14 Figure 14. MFi values of each elementary ladder as a function of mean Rech (blue symbols for optimized network; red symbols for basic network).

Figure 15

 15 Figure15shows the evolution of the MFnetwork for a whole multi-scale network as a function of mean Rech. The benefit of the insertion of optimized perforated baffles on the improvement of distribution uniformity can be highlighted. Generally, the MFnetwork value for the basic network increases exponentially from 0.195 to 0.747 when the mean Rech increases from 10 to 2000. For the optimized network, the MFnetwork value is only 0.043 at mean Rech=10, and can be maintained under 0.19 for mean Rech up to 2000.

Figure 15 .

 15 Figure 15. Evolution of MFnetwork values for the multi-scale fluidic network as a function of mean Rech with or without perforated baffles.

Figure 16 .

 16 Figure 16. Pressure drop for the multi-scale fluidic network as a function of mean Rech with or without perforated baffles.

Table 1 .

 1 Detailed dimensions for the multi-scale fluidic network.

Table 1 .

 1 Detailed dimensions for the multi-scale fluidic network.

	Parameter	Value
	Number of mini-channel	90
	Mini-channel cross-section	1 mm × 1 mm

Table 2 .

 2 Summary of distribution uniformity and pressure drop in the basic network and in the optimized network.

	Mean Rech	σi,j			MFnetwork		∆pnetwork
		Basic	Optimized	Basic	Optimized	Basic	Optimized
		network	network	network	network	network	network
	10	-31%~50%	±9%	0.195	0.043	-	-
	100	-54%~88%	-22%~29%	0.351	0.138	5.5 kPa	8.7 kPa
	1000	-86%~173%	-26%~35%	0.737	0.172	333 kPa	777 kPa