domain: spi phys

text:
Dr Benjamin Godard
email: benjamin.godard@onera.fr

Edouard De Jaeghere
email: edouard.de-jaeghere@safrangroup.com

Pr Nicolas Gourdain

EFFICIENT DESIGN INVESTIGATION OF A TURBOFAN IN DISTORTED INLET CONDITIONS

Designing a turbofan to operate in distorted inlet conditions is an issue of growing interest. In such conditions however, fan design can be computationally challenging. Indeed, subject to neither axi-symmetrical nor periodic inlet conditions, computations must be carried out throughout the whole circumferential domain i.e. 360 • . Besides, the classical CFD approach implies the use of URANS computations so as to capture the distortion transfer across the fan stage. Eventually, computations are too onerous to be used in design loops. In this context, this paper presents a methodology to effectively assess a fan blade design domain in distorted conditions. This methodology is based on a body-force source term approach formulated in order to accurately recreate deviations, loads and losses across the fan stage. It notably enables to gain two orders in terms of restitution time and thus the possibility to use optimization tools. The design domain of this study is based on variations of profile chord, blade leading and trailing edges angles applied at two different relative heights. A Latin Hypercube Sampling (LHS) associated with a meta-model based on Radial Basis Functions (RBF) enables to assess the impact of geometric variations on performance and operability. Although this study emphasizes that some modeling improvements are still necessary, it also demonstrates the potential of the body-force methodology to conduct fan design when subject to inlet distortion.

NOMENCLATURE

BLI Boundary Layer Ingestion LHS Latin Hypercube Sampling RBF Radial Basis Functions NSGA2 Non-dominated Sorting GA 2 UHBR Ultra High Bypass Ratio a s Sound velocity [m/s] b Blockage factor [-] c Blade chord length [m] c x Blade axial chord length [m] c x Normalized Blade axial chord length [%] e x Blade axial thickness [m] e t Total energy [J/kg] H Blade to blade staggered spacing [m] h Relative height [-] h t Stagnation specific enthalpy [J/kg] f Body-forces [N/kg] f n Body-force component normal to w [N/kg] f p Body-force component parallel to w [N/kg] f θ Circumferentially averaged and normalized tangential force [%] K p0 Friction drag coefficient at maximum efficiency [-] M Absolute Mach number [-] M rel Relative Mach number [-] M x Axial Mach number [-] ṁ Massflow rate [kg/s] ṁcorr Corrected massflow rate [kg/s] 1 ṁnorm Normalization value for massflow rates [kg/s] ṁre f Reference Massflow rate [kg/s] n Iteration number [-] n p Flow plane normal [-] n s Blade profile normal [-] p Static pressure [Pa] p i Stagnation pressure [Pa] p i,0 Upstream stagnation pressure [Pa] p i,norm Normalization value for stagnation pressure ratio [-] p piv Pivot static pressure [Pa] q Heat flux vector Re xc Blade chordwise Reynolds number [-] S b Blockage source term SM Stability margin [%] SM re f Stability margin of the initial geometry [%] s Grid motion vector [m/s] s Normalized entropy [-] T Static temperature [K] T i Stagnation temperature [K] T i , 0 Upstream stagnation temperature [K] v Absolute flow velocity [m/s] w Relative flow velocity [m/s] (x, y, z) Cartesian coordinates (x, r, θ ) Cylindrical coordinates α Absolute flow angle [ • ] β Relative flow angle [ • ] λ 4 Valve factor [Pa] η is Isentropic efficiency [-] η is,norm Normalization value for isentropic efficiency [-] ρ Density [kg/m 3 ] τ Stress tensor [J/m 3 ] Ω Rotating speed [rad/s]

INTRODUCTION

Many new concepts of civil aircraft involve the issue of propulsive system integration. It notably concerns Ultra High Bypass Ratio (UHBR) turbofan set with: (i) short inlet [START_REF] Peters | Ultra-short nacelles for low fan pressure ratio propulsors[END_REF][START_REF] Thollet | Body-force modeling of fanairframe interactions[END_REF]. In this case, it is important to assess whether the profit in terms of load and drag enabled by the slimmer and shorter inlet is not offset by a loss in terms of fan performance and/or operability. Indeed, when subject to specific flight conditions (ex. high angle of attack [START_REF] Cao | Fan-intake interactions under high incidence[END_REF] or cross-wind [START_REF] Colin | Simulation numérique de la distorsion générée par une entrée d'air de moteur civil par vent de travers[END_REF][START_REF] Peters | Ultrashort nacelles for low fan pressure ratio propulsors[END_REF]), the inflow is more inclined to separate, which can result with an incoming heterogeneous flow pattern i.e. an inlet distortion. It is therefore important to assess the fan robustness regarding these kinds of inlet distortion. Nevertheless, most of the flight envelop remains (almost 1 ) absent of distortion. The problem of airframe-engine interaction can however concern the whole flight envelop when considering innovating engine integration such as (ii) Boundary Layer Ingesting (BLI) propulsive systems [START_REF] Plas | Performance of a boundary layer ingesting (bli) propulsion system[END_REF][START_REF] Hardin | Aircraft system study of boundary layer ingesting propulsion[END_REF][START_REF] Liou | Minimizing inlet distortion for hybrid wing body aircraft[END_REF][START_REF] Florea | Parametric analysis and design for embedded engine inlets[END_REF][START_REF] Liou | Aerodynamic design of integrated propulsion-airframe configuration of the hybrid wing-body aircraft[END_REF]. In this kind of configuration, the fan operates constantly within the boundary layer of the fuselage.

Whatever the case, inlet-fan interactions cannot be anymore omitted from the intake or fan design process. It is therefore important to take them effectively and efficiently into account, even during the first steps of the design. The impact of inlet distortion has been assessed in many studies [START_REF] Cousins | Inlet distortion testing and analysis of a high-bypass ratio turbofan engine[END_REF][START_REF] Ferrar | Measurement and uncertainty anal-ysis of transonic fan response to total pressure inlet distortion[END_REF][START_REF] Perovic | Stall inception in a boundary layer ingesting fan[END_REF]. They have in particular emphasized the importance of not only stagnation pressure [START_REF] Fidalgo | A study of fan-distortion interaction within the nasa rotor 67 transonic stage[END_REF] but also swirl distortion to account for performance and stability margin losses and have proposed relevant metrics to assess these distortions [START_REF] Sheoran | A centrifugal compressor operability correlation with combined total pressure and swirl distortion[END_REF][START_REF] Frohnapfel | Experimental quantification of fan rotor effects on inlet swirl using swirl distortion descriptors[END_REF].

In order to mitigate the drop of efficiency and stability margin, several authors have addressed the issue of fan stage design in distorted conditions. In his PhD thesis, Hall [START_REF] Hall | Analysis of civil aircraft propulsors with boundary layer ingestion[END_REF] has explored the impact of loading distribution variations in the propagation of distortion indicators and diffusion factors. This large parametric study has been possible by means of a low fidelity CFD methodology involving body-force source terms in order to account for the fan and OGV load and deviation effects. Among many conclusions, the author indicates that non-axisymmetric stator design is a promising solution in order to improve the fan stage efficiency. Gunn et Hall [START_REF] Gunn | Aerodynamics of boundary layer ingesting fans[END_REF][START_REF] Gunn | Non-axisymmetric stator design for boundary layer ingesting fans[END_REF] confirmed this indication with experimental validations. Incidentally, they demonstrated the ability of URANS computations to effectively reproduce the fan stage behavior in their applied distorted conditions. The NASA BLI subscale high-bypass ratio fan stage tested at the Glenn Research Center [START_REF] Arend | Experimental evaluation of bli propulsion: The boundary layer ingesting inlet/distortion-tolerant fan solution[END_REF] is perhaps the most advanced research attempt to assess a robust design in BLI conditions, not only in terms of aerodynamic performances [START_REF] Hardin | Data analysis techniques for fan performance in highly-distorted flows from boundary layer ingesting inlets[END_REF] but also in terms of aeromechanical response [START_REF] Provenza | Aeromechanical response of a distortion-tolerant boundary layer ingesting fan[END_REF].

The previously mentioned papers involve expensive experimental campaign or high fidelity URANS computations. However, design iterations require more efficient methodologies while still having to account for the performance metric sensitivities to geometric variations. To this end, body-force methods represent an interesting way of reducing computational cost. Several models have already shown promising results not only to reproduce potential fan stage effects on inlet aerodynamics [START_REF] Peters | Ultra-short nacelles for low fan pressure ratio propulsors[END_REF][START_REF] Thollet | Assessment of body force methodologies for the analysis of intake-fan aerodynamic interactions[END_REF][START_REF] Thollet | Body-force modeling of fanairframe interactions[END_REF][START_REF] Godard | Methodologies for turbofan inlet aerodynamics prediction[END_REF] but also to describe fan working conditions [START_REF] Gong | A computational model for rotating stall and inlet distortions in multistage compressors[END_REF][START_REF] Guo | A three-dimensional analysis code of compressor performance and stability for steady inlet distortion[END_REF][START_REF] Thollet | Body force modeling for aerodynamic analysis of intake-fan interactions[END_REF], notably in distorted conditions [START_REF] Hall | Analysis of civil aircraft propulsors with boundary layer ingestion[END_REF][START_REF] Godard | Etude et méthodologies de simulation de doublet entrée d'air -soufflante pour la conception de turbofan de nouvelle génération[END_REF].

In order to address the problem of effectively and efficiently conducting a fan design in distorted conditions, the objectives of this study are to :

(i) assess the extend to which the body-force method can reproduce both fan working conditions and design parameter sensibilities when subject to inlet distortion and to (ii) test its ability to be used in an automated optimization process.

After a brief description of the test-case, the body-force formulation and the numerical set-up, an assessment of the body-force methodology is presented. This validation is based on comparisons with RANS and URANS computations. Then, the optimization process based on an arbitrary large design domain is presented. Results are finally discussed and conclusions are drawn so as to stress what must be improved for the methodology to be used in an industrial environment.

TEST-CASE AND NUMERICAL SET-UP General information

The selected test-case is a Safran Aircraft Engines fan stage scale model representative of UHBR developments. In particular, the fan stage targets very high bypass and low stagnation pressure ratios as presented in TABLE 1.

TABLE 1. Fan Stage Characteristics

Bypass ratio [-]

≈ 16

Nominal pressure ratio [-] < 1.4

Body-force modeling

The body-force approach consists in replacing the bladed region by an axisymmetric volume where blockage, momentum and energy source terms are applied in each cell. Source terms are a combination of external data and flow proprieties so as to reproduce the effects of the blades. The equation system (1) presents these source terms applied in the right-hand side of the simplified Navier-Stokes equations.

       ∂ ρ ∂t + div [ρ ( v -s)] = S b = ρ v b • grad (b) ∂ ρ v ∂t + div ρ v ⊗ ( v -s) + p I -τ = S b v + ρ f ∂ ρe t ∂t + div ρe t ( v -s) + p v -τ • v + q = S b h t + ρrΩ f θ (1) In particular, S b = ρ v b • grad (b)
serves to model blockage effects and depends on the blockage factor b = e x /H which is the ratio between the blade axial thickness e x and the blade to blade staggered spacing H [START_REF] Kottapalli | Development of a body force model for centrifugal compressors[END_REF]. It is then all about finding suitable values for the body-forces f . In this study, the method follows a formulation based on a decomposition of the body-forces f within the flow plane2 as illustrated by FIGURE 1. In this particular FIGURE 1. Body-force principle and decomposition within the flow plane [START_REF] Hall | Analysis of fan stage design attributes for boundary layer ingestion[END_REF] plane, the body-forces can be decomposed in only two components f = f n + f p . The first component f n is perpendicular to the flow relative velocity w and serves to reproduce main deviation and loading effects. The second component f p is parallel to w and serves to add losses as directly proportional to entropy gradients [START_REF] Marble | Three-dimensional flow in turbomachines[END_REF]. The selected formulation is described by equations (2) and (3) [START_REF] Hall | Analysis of civil aircraft propulsors with boundary layer ingestion[END_REF][START_REF] Thollet | Body-force modeling of fanairframe interactions[END_REF]:

f n = w 2 2bH 2πδ ×        1 √ 1-M rel 2 if M rel < 1 4 2π √ M rel 2 -1 if M rel > 1 Compressible effects (2)
f p = w 2 2bH K p0 + 2π (δ -δ 0 ) 2 (3)
in which:

(i) M rel (x, r, θ ) is the relative Mach Number, (ii) K p0 = 2 × 0.0592 × Re -0.2 xc
is the friction drag coefficient and depends on the local Reynolds number Re xc (which is based on the chordwise position x c of the blade) and (iii) δ = arcsin w• n s || w|| is the local deviation in the plane defined by the blade profile normal n s (x, r).

(iv) Also, δ 0 (x, r) is the deviation extracted from a body-force computation at maximum efficiency where only friction losses are applied i.e. f p = w 2 K p0 / (2bH).

This model is particularly easy to implement as it exclusively depends on the blade geometry as well as a single calibration parameter δ 0 , the establishing of the latter "only" requiring the massflow rate value at maximum efficiency. Moreover, this model has already shown promising results [START_REF] Hall | Analysis of civil aircraft propulsors with boundary layer ingestion[END_REF][START_REF] Thollet | Body-force modeling of fanairframe interactions[END_REF]. However, one of the major noted limitation is the inability of the model to accurately reproduce stall behavior at low massflow rate.

Numerical set-up CFD simulations are performed with the elsA solver developed by Safran, Airbus and ONERA on structured meshes [START_REF] Cambier | The onera elsa cfd software: input from research and feedback from industry[END_REF]. Equations are discretized in space using a cell-centered finite volume method with the second order upwind scheme of Roe [START_REF] Roe | Approximate riemann solvers, parameter vectors, and difference schemes[END_REF]. The kl model of Smith [START_REF] Smith | The k-kl turbulence model and wall layer model for compressible flows[END_REF] ensures turbulence closure. Steady simulations are run with an implicit pseudo-time marching method with backward Euler scheme. Unsteady simulations use a Gear scheme for integration in time with 15 subiterations process. Two computational domains are considered. The first one refers to meshed blades (U)RANS computations and is presented in FIGURE 2. The grid was generated using NUMECA FIGURE 2. URANS Computational Domain AUTOGRID T M software [START_REF] Numeca | User manualautogrid5 release 8.4 : A full automatic hexahedral grid generator for all types of rotating macmanual[END_REF] and is relatively coarse with a domain mesh size of approximately 60 × 10 6 elements (2.5 × 10 6 elements for periodic sector RANS computations). The second domain presented in FIGURE 3 is used for body-force RANS computations. The grid was generated using the Cassiopee software [START_REF] Benoit | Cassiopee: a cfd pre-and post-processing tool[END_REF] with a domain mesh size of approximately 4.0×10 6 elements. Sectors of stagnation pressure losses are applied as inlet conditions. The inlet cartography applied is a constant stagnation pressure drop of ∆p i /p i,0 = 10% applied in the last ∆h = 35% of 3). The outflow boundary conditions are set with a radial equilibrium defined by a pivot static pressure p piv located at the hub. Iso-rotating speed characteristics are obtained by adjusting the pivot static pressure through a valve factor λ 4 :

p piv,n+1 = p re f + λ 4 ṁn ṁre f 2 ( 4
)
where ṁn is the massflow rate at iteration n and ṁre f and p re f are massflow rate and static pressure values of reference.

BODY-FORCE MODELING ASSESSMENT

In order to validate the body-force modeling formulation used in this study, fan stage simulations have been run in both isolated and distorted inlet conditions. The objectives are to assess the ability of the body-force methodology to account for:

(i) sensitivities of design parameters and (ii) fan working behavior in distorted conditions. Sensitivities of fan design parameters have been assessed in clean inflow conditions. Selected parameters are respectively the leading and trailing edges metal angles β sq,1 and β sq,2 and the chord length c (cf. FIGURE 4 for illustration). Design variations have firstly been applied uniformly throughout the blade span. The relative blade thickness as well as centers of gravity have been conserved through the process. FIGURES 5, 6 and 7 present iso-speed fan characteristic comparisons between the "meshed blades" and the "body-force" RANS computations respectively for variations of ∆β sq,1 = +/ -2 • , ∆β sq,2 = +/ -2 • and ∆c/c = +/ -5%. The stagnation pressure ratio p i /p i,0 and have been normalized (via the values p i,norm and η is,norm ) and are plotted against the normalized corrected massflow rate ṁcorr defined by equation ( 6):

η is = (p i /p i,0 ) γ-1 γ -1 T i /T i,0 -1 (5)
ṁcorr = ṁ √ T i,0 /T i,re f p i,0 /p i,re f ṁnorm ( 6
)
where p i,0 and T i,0 are the stagnation pressure and temperature taken upstream while p i,re f = 101325 Pa and T i,re f = 288.15 K are standard values at ISA-0. Also, ṁ is the massflow rate and ṁnorm is a normalization value. Globally, conclusions are the same for the three studied parameters. Sensitivities are relatively well reproduced. Yet, some notable differences can be highlighted. First, the body-force formulation underestimates the stagnation temperature ratio. As it also underestimates the stagnation pressure ratio, errors balance each other out and the isentropic efficiency is well accounted for. This underestimation of stagnation temperature ratio can be explained by the change of flow behavior along the span i.e. subsonic near the hub and transonic near the shroud, which is not accounted for by the body-force model. Profiles in the transonic part of the blade span denote an absence (or even a change of sign) of camber approximately in the fist 60% of chordwise coordinate, which is typical of transonic fan blade design. Figure 8 presents the evolution of the circumferentially averaged and normalized tangential force f θ of the fan against the normalized axial chord length c x for both the body-force and RANS computations taken at maximum efficiency. It notably shows that at h = 20%, f θ behaves similarly in both cases. Indeed, at this relative height (and especially at maximum efficiency), the flow that crosses the fan remains subsonic and therefore the assumption that the flow deviation follows the blade profile is well respected. Therefore, the formulation (2) which assumes a proportionality between the normal force applied f n and the local deviation δ manages well to reproduce the flow behavior in this region. However, when a shock occurs within the blade-to-blade channel, additional deviation is generated because of a decrease of the flow relative velocity component normal to the shock front. Thus, it locally implies additional loads which are not reproduced by the bodyforce model. This can be observed in Figure 8 by noting that tangential forces f θ at h = 90% are drastically different. While the body-force model induces a tangential force only in the region where the blade camber is not close to zero (which is the case outside of the grey area plotted in Figure 8), deviations occur alongside the entire channel in the RANS computations due to the effect of the shock. Eventually, the integrated tangential forces applied to the flow and therefore the stagnation temperature ratio differ between the two approaches. Besides, it must be noted that maximum efficiency values evolution is not always conserved. It is notably the case for variation of ∆β sq,1 . The maximum isentropic efficiency slightly decreases when ∆β sq,1 increases for the "meshed blades" computations while the maximum efficiency values are lower than the reference for both applied variations for the "body-force" computations (cf. FIGURE 5 right caption). Nevertheless, the body-force methodology manages well to reproduce the massflow rate blockage values predicted by the meshed blades computations. A first assessment in distorted conditions has been performed on the reference geometry. In this kind of configuration (cf. FIGURE 2 and FIGURE 3), assuming that the fan stage working conditions remain axisymetric is not defendable anymore. Therefore, it is necessary to assess variations of working conditions throughout the circumference. In order to properly associate upstream and downstream conditions, a specific postprocessing, firstly presented by Fidalgo et al. (2012) [START_REF] Fidalgo | A study of fan-distortion interaction within the nasa rotor 67 transonic stage[END_REF], has been applied. It consists in dividing a reference section upstream of the fan leading edge into multiple sectors. Then, for each upstream sector, an associated sector located downstream of the fan trailing edge is identified using the intersections between the streamsurfaces generated from the upstream sector boundaries and the downstream section. Upstream and downstream sections have been selected at axial locations close to the fan leading and trailing edges. FIGURE 9 shows an example of the sectors result- ing from this post-processing. Compared to Blade URANS computations, using body-force modeling methods greatly simplifies this kind of post-processing by providing flow field directly in the absolute frame. Therefore, there is no need for complex averaging process to translate results from the rotating to the nonrotating absolute frame. Averaging the flow solution within the identified sectors eventually enables to plot the evolution of the fan working conditions throughout an entire revolution, which is represented by an orbit in the performance map. In the study, the upstream section has been divided between 96 sectors. This quantity results from a convergence study on orbit shapes and stability margins. FIGURE 10 presents the performance map in which both blade and body-force simulations are compared. Stagnation pressure orbits of both Blade URANS and body-force RANS computations are in good agreement. It demonstrates the ability of the body-force model to reproduce fan working conditions in distorted area in terms of stagnation pressure ratio. In particular, the closest local working condition M to the stability limit S3 is well captured. This enables an efficient assessment of the stability margin defined in this study by equation [START_REF] Hardin | Aircraft system study of boundary layer ingesting propulsion[END_REF]:

SM = p i /p i,0 ṁcorr S - p i /p i,0 ṁcorr M p i /p i,0 ṁcorr M (7)
However, larger differences arise with the prediction of the isentropic efficiency. This is mainly attributed to the previously emphasized fact that the body-force formulation used in this study Altogether, the body-force modeling formulation, although not perfect, has been reckoned reliable enough to assess fan working conditions when subject to inlet distortion. In particular, this methodology has been used in order to efficiently explore a design domain so as to assess the impact of geometric parameters on fan efficiency and stability margin in distorted conditions. The description and the results of this study are discussed in the next section.

APPLICATION FOR DESIGN OPTIMIZATION IN DIS-TORTED CONDITIONS

The design domain has been arbitrary chosen as such :

∆β sq,1 ∈ [-2 • , 2 • ], ∆β sq,2 ∈ [-2 • , 2 • ]
and ∆c ∈ [-5%, 5%] at two control relative heights h ∈ {50%, 100%}. Six design parameters are therefore considered and only concern the fan geometry.

One of the challenges of this study is to correctly identify the stability margin for each assessed points. Indeed, it has been emphasized that the body-force formulation does not reproduce the fan behavior outside of the stability limit i.e. the positive slope of the characteristic. Therefore, it is necessary to determine these instability limits for each new design. This is quite inconvenient as it a priori requires to compute a new characteristic with "blades" RANS computations for each evaluated design. In order to reduce the computational cost, it has been decided to indeed compute these characteristics but only at the domain border, which still implies 2 6 = 64 designs. Then, the stability limit conditions have been estimated via an interpolation by means of Radial Basis Functions (RBF) over these border values. FIGURE 13 presents the computed characteristics (left caption) with the associated stability limits (right caption) and those of 320 designs interpolated from the border values. These designs have been defined by a Latin Hypercube Sampling (LHS) which enables to efficiently explore the domain (i.e. with few assessed points) [START_REF] Mckay | Comparison of three methods for selecting values of input variables in the analysis of output from a computer code[END_REF]. Then, each design has been assessed by a bodyforce computation in the same distorted conditions. Design variations can modify the current massflow rate and stagnation pressure ratio. Therefore, comparisons are made at different working conditions, which make the whole exercise easily questionable. However, a same valve factor λ 4 (cf. equation ( 4)) has been applied for all computations. This enables to locate the working conditions at a relatively same (close) distance from their maximum efficiency reference. Therefore, this study must be understood as rather an assessment of how a given design (at its own defined working condition) is impacted by a distortion than an optimization focus on a single set of specifications. FIGURE 14 and FIGURE 15 present the impacts of the different design parameters respectively in terms of stability margin ∆SM and isen- tropic efficiency ∆η is . These objective functions ∆SM and ∆η is are differences between the assessed geometry and the reference design both in distorted conditions. Thus, a positive value means a more robust design than the reference. In FIGURE 14 and FIG-URE 15, global tendency slopes have been added to the graphs. These global tendencies are defined so as to minimize the linear regression error. Several comments can be made regarding those results:

(i) The stability margin is improved by increasing the chord length at the tip (h = 100%) (cf. FIGURE 14 bottom-right caption). This tendency makes sense to the extent that the section subject to lower massflow rate (because of the distortion) is located in the last 35% of relative height. Hence it is the most critical sector regarding stability. Thus, increasing locally the solidity in this region (via an increase of chord length) can improve the stability margin. However, it is not always possible to use this lever because of structural considerations. (ii) Decreasing the trailing edge metal angle β sq,2 enables to improve both stability and efficiency. This is not surprising as it also decreases the global fan work. (iii) Finally, leading edge metal angle variations ∆β sq,1 have an antagonist effect on the stability margin and the isentropic efficiency. Also, the fact that this parameter has more impact on the isentropic efficiency in the lower part of the relative height (h = 50%) can be explained as a higher propor- tion of the massflow rate gets through this region. Besides, the fact that an increase of β sq,1 applied in the distorted region improves the isentropic efficiency can be easily explained as profiles are thus re-aligned with the decelerated flow.

Finally, these previous tendencies could have been a priori guessed by fan designers. However, the methodology provides a quantification of these tendencies. They also highlight the fact that the body-force methodology provides consistent results in spite of its previously highlighted flaws (especially regarding the prevision of the isentropic efficiency). The results have also been used to build a meta-model by means of RBF. Then, a Non-dominated Sorting GA 2 (NSGA2) [START_REF] Deb | A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: Nsga-ii[END_REF] algorithm based on this meta-model has enabled to determine an optimized Pareto front represented in FIGURE 16. The meta-model mean and maximal errors are respectively 0.06% and 0.30% for the stability margin assessment and 0.03% and 0.11% for the isentropic efficiency 5 .

From the computed Pareto front, three candidates have been picked-up so as to illustrate geometric and performance changes on relevant designs (cf. large colored points in FIGURE 16). A first candidate has been selected in order to maximize the stability margin (∆SM = +3.9%, ∆η is = +0.0%) without decrease of the initial isentropic efficiency (pink point of FIGURE 16)). Oppositely, another one has been selected so as to maximize the isentropic efficiency (∆SM = +0.04%, ∆η is = +1.5%) without deterioration of the stability margin (blue point of FIGURE 16)). Finally, a third and last one has been selected as a trade-off (∆SM = +2.52%, ∆η is = +1.09%) between the previous ones (green point of FIGURE 16). FIGURE 17 shows the geometric variations in reference to the initial geometry that have been applied to these selected candidates. It shows that the candidate that maximizes the stability margin reaches this achievement by a strong increase of chord length c and a decrease of the inlet metal angle β sq,1 at high values of relative height h > 0.5. On the other hand, the candidate that maximizes the isentropic efficiency manages so by a strong increase of β sq,1 at mid-height and by a global decrease of the outlet metal angle β sq,2 . The "tradeoff" candidate shows variations somewhere in-between the two others. Moreover, these values are consistent with the global tendencies showed by FIGURE 14 and FIGURE 15. Body-force computations have confirmed the meta-model predicted values. This can be illustrated by FIGURE 18 which shows the orbits of FIGURE 18. Fan Orbits Comparison between the Selected Candidates circumferential working conditions of these three selected candidates in terms of stagnation pressure ratio p i /p i,0 (left caption) and isentropic efficiency (right caption). Stability limits have been added in the left caption in order to provide a reference for each orbit and thus to correctly assess the margins.

Finally, in order to assess these results, another URANS computation has been performed using the "trade-off" geometry. Table 2 summarizes the principal results extracted from both the optimization process and the URANS computation of this "trade-off" geometry. Compared to the NSGA2 prediction, the URANS computation estimates a far lower gain in stability margin ∆SM tradeo f f ,URANS = +0.43% < ∆SM tradeo f f ,NSGA2 = +2.52% and a higher gain in terms of isentropic efficiency ∆η is,tradeo f f ,URANS = +1.47% > ∆η is,tradeo f f ,NSGA2 = +1.09% while the stagnation pressure ratio remains well conserved (∆ (p i /p i,0 ) / (p i /p i,0 ) re f = -0.30%) as well as the massflow rate ∆ ṁ/ ṁre f = -0.28%. Figure 19 and Figure 20 respectively present a comparison of slices at h = 80% and at a station downstream of the fan colored by the normalized entropy s. In these figures, it can be qualitatively noted that the optimized geometry produces less entropy than the reference and shows thinner trailing edge wakes, which illustrates the isentropic efficiency gain.

Regarding the stability margin prediction, as stagnation pressure orbits are well reproduced by the body-force methodology, one would have expected a better match between the predicted and the computed results. Actually, this difference mostly comes from a poor estimation of the stability limit (which is, as a reminder, assessed by RBF interpolation over the extremities of the design domain). Indeed, an underestimation of the massflow rate condition of 5.3% has been noted compared to the RANS prediction, which calls the entire optimization process into question. Therefore, it underlines the importance of improving the model in order to intrinsically (and thus efficiently) capture the stability limit.

To conclude, even though some discrepancies remain between the computed results and the NASGA2 prediction, tendencies remain correct and the new design happens to be more adapted to the applied inlet distortion.

CONCLUSION, DISCUSSION AND PERSPECTIVES

In this paper, a body-force approach has been assessed and applied in order to efficiently explore a fan design domain in stagnation pressure distortion conditions. Presented results have highlighted the extend to which the body-force approach can reproduce a fan stage behavior in distorted conditions. Comparisons have been made with both "meshed blades" RANS and URANS computations which require additionnal computational effort by approximately two orders. The study has notably shown that the selected body-force formulation [START_REF] Hall | Analysis of civil aircraft propulsors with boundary layer ingestion[END_REF][START_REF] Thollet | Body-force modeling of fanairframe interactions[END_REF] globally underestimates the stagnation temperature ratio which leads, in distorted condition, to an overestimation of the isentropic efficiency. However, the global behavior is well reproduced. In particular, the stability margin agrees well with the URANS results. Moreover, the formulation has been able to reproduce the impact of geometric changes (in terms of inlet and outlet blade metal angles β sq,1 and β sq,2 and chord length c) when compared with mixing plane "meshed blades" computations. Improvements of the formulation could consist in improving the stagnation temperature ratio prediction. This requires to add modeling of what can increase the blade loading (in other words, what can locally induce additional deviation within the row). This can involve blade flow separations, shocks, tip gap effects, etc. Indeed, the formulation at stake only induces deviation following the blade metal angle β sq , which is a good assumption in subsonic conditions far from the hub and shroud walls, but becomes more hazardous in transonic conditions or when subject to secondary flows.

The formulation has then been tested in an optimization process involving a design domain defined by six parameters: the inlet and outlet blade metal angles β sq,1 and β sq,2 and the chord length c at two relative heights (h = 50% and h = 100%). 320 points following a LHS have been computed in distorted conditions in a relative short amount of time (several days on 128 cores). This amount of evaluations was enough to build a metamodel on which a NSGA2 algorithm has enabled to identify optimum candidates (i.e. a Pareto front). However, the process has been slowed down by the necessity to assess the stability margin for each point as the body-force formulation at stake cannot predict the positive slope of the fan characteristic. It has been decided to interpolate these values from the domain extremities which have been assessed by "meshed blades" RANS computations. This process is quite a drawback, which highlights the necessity to improve the body-force formulation in order to intrinsically assess the stability limit. Indeed, through this process, several approximations and assumptions have been made and are summarized hereafter:

(i) The stability limits are extracted from RANS computations which are known to lack accuracy to detect stall inception compared to higher fidelity methods or experiments (ex. DES, LES) [START_REF] Gourdain | Simulation numérique des phénomnes de décollement tournant dans les compresseurs axiaux[END_REF][START_REF] Choi | Effects of fan speed on rotating stall inception and recovery[END_REF]. Besides, interpolations based on the domain extremities bring additional approximation and can bring discredit upon the stability margin prediction (as emphasized by the assessment of the selected optimized design). (ii) The stability margin defined by equation [START_REF] Hardin | Aircraft system study of boundary layer ingesting propulsion[END_REF] assumes that the stability limit does not depend on local inlet conditions and thus remains the same in the performance map (p i /p i,0 ) vs. ṁcorr i.e. S remains the same. The stability margin is therefore only affected by variations of local working conditions due to the inlet distortion (i.e. at specific circumferential locations) i.e. the displacement of M in the performance map (cf. FIGURE 10). Besides, this definition is very conservative as it implies that a blade only needs to reach stall conditions at a specific circumferential location for the entire row to be considered in stall conditions, which has been demonstrated not to be always the case [START_REF] Perovic | Stall inception in a boundary layer ingesting fan[END_REF].

Nevertheless, results have shown that geometric variation impacts are qualitatively consistent. An URANS simulation of an identified candidate showing an optimal combinaison of efficiency and stability margin has shown that strong discrepencies remain but that the design is indeed more robust to inlet distortion. Additional quantitative assessment are nonetheless required in order to completely assess the precision of the presented methodology. Moreover, several other assumptions have been made. In particular, the working conditions (massflow rate, stagnation pressure ratio) have not been conserved systematically between assessed points. In addition, multi-disciplinary constraints, such has mechanical robustness, have not been taken into account. Finally, this study has been focused solely on the fan blades. It could be interesting to extend this approach to nonaxisymetric OGV as their potential have been identified by many studies as good candidates to answer the issue of fan stage robustness to inlet distortion [START_REF] Hall | Analysis of civil aircraft propulsors with boundary layer ingestion[END_REF][START_REF] Gunn | Non-axisymmetric stator design for boundary layer ingesting fans[END_REF][START_REF] Kumar | Aerodynamic design system for nonaxisymmetric boundary layer ingestion fans[END_REF].

FIGURE 3 .

 3 FIGURE 3. Body-force Computational Domain

FIGURE 4 .FIGURE 5 .

 45 FIGURE 4. Design Parameters Variations

FIGURE 6 .

 6 FIGURE 6. Stagnation Pressure Ratio and Isentropic Efficiency Characteristics for Blade and Body-force Computations -Variations of β sq,2

FIGURE 7 .

 7 FIGURE 7. Stagnation Pressure Ratio and Isentropic Efficiency Characteristics for Blade and Body-force Computations -Variations of c

FIGURE 8 .

 8 FIGURE 8. Circumferentially averaged tangential force f θ comparison at h = 20% (subsonic behavior) and h = 90% (transonic behavior) of the body-force model and RANS (meshed blades) results

FIGURE 9 .

 9 FIGURE 9. Post-processing in Distorted Inlet Conditions

FIGURE 10 .FIGURE 11 .

 1011 FIGURE 10. Stagnation Pressure Ratio and Isentropic Efficiency Characteristics and Orbits in Distorted Conditions for Blade and Bodyforce Computations

FIGURE 12 .

 12 FIGURE 12. Flow fluctuations throughout the circumference upstream and downstream of the fan at h = 83%. Body-force and URANS comparison

FIGURE 13 .

 13 FIGURE 13. Domain border characteristics and interpolated stability limits

FIGURE 14 .

 14 FIGURE 14. Stability Margin Variations against the Design Variables

FIGURE 15 .

 15 FIGURE 15. Isentropic Efficiency Variations against the Design Variables

FIGURE 16 .

 16 FIGURE 16. Optimized and Raw Pareto Fronts

FIGURE 17 .

 17 FIGURE 17. Geometric Variations Comparison between the Selected Candidates

FIGURE 19 .FIGURE 20 .

 1920 FIGURE 19. Slices at h = 80% colored by normalized entropy. Comparison between the reference geometry (left) and the "trade-off" optimized design (URANS snapshots)

TABLE 2 .

 2 "Trade-off" geometry summarized results∆SM [%] ∆η is [%]

				∆(p i /p i,0) p i /p i,0	[%]	∆ ṁ ṁre f [%]
	NSGA2	+2.52	+1.09	-		-
	URANS	+0.43	+1.47	-0.30		-0.28

Angle or Mach number distortions can still remain because of the nacelle face droop angle and thus negatively affect the fan performances.

The normal n p of the flow plane is defined by f • n p = 0

The stability limit is defined by the point of maximum static pressure ratio of the characteristic (i.e. (p/p i,0 ) max )[START_REF] Dunham | Non-axisymmetric flows in axial compressors[END_REF]

In the case of swirl fluctuations, the value of reference α re f has been taken differently downstream and upstream so as to keep the fluctuations within similar amplitude range.

These errors are calculated by building a meta-model based on a domain which is relieved of a single point. The relative difference between the computed and the assessed values defines the local error and this process is then repeated for every computed point.

ACKNOWLEDGMENT

We would like to thank Safran Aircraft Engines for the funding, for the supporting and for sharing valuable data without which this research work would not have been possible. We are also greatly indebted to William Thollet and Guillaume Dufour who share with us their knowledge in body-force modeling methods. Finally, we would like to thank Itham Salah El Din for his technical support and advice on the optimization set-up.


text:
Sandrine Fauqueux
email: sandrine.fauqueux@onera.fr.

Renaud Davy
email: renaud.davy@onera.fr.

F Méry

Duct modal detection tool to characterize the noise source generated by an air pump

Aircraft air conditioning system must be compliant with the Noise Regulations of every countries in which it will be operated. In the Clean Sky 2 project IDEAS, the system of interest is composed of an air pump, developped by Liebherr Aerospace, that generates low and mid-frequency noise that may be mitigated by the use of innovative acoustic liners. The aim of this study is to assess the level and modal content of the generated noise source in two industrial operating conditions. The modal detection solution proposed by ONERA leads to the post-treatments of measurements obtained by a flush-mounted microphone array located downstream of the diffuser. As microphones will be implemented in a hard-wall region of the duct, the modal basis characterizing the measured acoustic field does not depend on the lined walls, located in the diffuser area.

The deconvolution method ARMADA leads to the minimization of the discrepancy, in the least-square sense, between measurements and an analytical model of the acoustic pressure field that propagates in cylindrical ducts. In the first part of this article, the analytical models in uniform and non-uniform flows are introduced, with underlying hypotheses. In the case of shear flow, the modal shapes are given by ONERA's MAMOUT code. Then, the deconvolution method is presented, considering that the modes are correlated, as wave reflections may occur at the end of Liebherr's bench. In the second part, the design of the IDEAS microphone array is defined, considering uniform flow only. Different sensor arrangements are compared. One design is retained and the associated estimation errors are assessed. In the third part, the impact of estimating modes under the uniform flow hypothesis despite the fact that shear flow occurs during measurements is evaluated. A potential misleading impact is highlighted at 3150 Hz on the m = 0 and n = 1 mode. Finally, the experimental setup is introduced and the analysis of the measurements performed in the facility without use of acoustic liners are detailed for two operating conditions.

I. Introduction

On modern aircraft, passengers and crew breathe a mixture of fresh and recirculated air. This combination rather than fresh only allows the regulation of temperature, pressure and humidity. The air is bled from the engines and supplied to air conditioning units. It is then ducted into the cabin, circulated and eventually drawn into the lower fuselage where it is sucked out by the pressurization outflow valve for the cycle to begin again. Besides creating a safe and comfortable environment, the aircraft air conditioning systems generate noise. This noise must be mitigated to comply with the local regulations on perceived noise for both the passengers and the airport workers when the aircraft is at the gate or is taxiing to the runway. As a consequence, any new generation of air conditioning system must be shown by the manufacturer to be compliant with the Noise Regulations of every countries in which it will be operated. The system of interest in the Clean Sky 2 project IDEAS, and developped by Liebherr Aerospace, is composed of an air pump that generates low and mid-frequency noise. This noise is conducted through several ducts to the exterior of the aircraft, thus contributing to ramp noise. Consequently, this new technology requires the use of innovative acoustic liners to mitigate the noise source. An accurate knowledge of the noise source is therefore needed.

ONERA recently developed a modal deconvolution method, called ARMADA [START_REF] Fauqueux | Modal deconvolution method in a finite circular duct, using flush-mounted microphones[END_REF], based on the iterative minimization of the discrepancy, in the least-square sense, between flush-mounted microphone measurements and an analytical model that represents the propagation between the sources and the measurements positions. A rigid and infinite cylindrical or annular duct of constant radius subjects to a uniform and subsonic flow along the revolution axis of the duct is currently considered. In the industrial operating conditions of the air-pump of interest, most of these hypotheses are satisfied, except that the flow may potentially be sheared. A modal detection network that ensures the efficiency of the modal detection process, and suited to the geometry and constraints of the installation, has also to be defined.

In the first part of this article, the modal deconvolution method under uniform flow hypotheses is briefly presented before being extended to sheared flow. The second part is dedicated to the definition of the position of the microphones that ensures the good efficiency of the deconvolution method at the operating points. In the third part of this article, the influence of shear flow in our modal detection tool at one specific operating point is studied. Finally, the measurements performed in the facility without use of acoustic liners are introduced. Measurements with acoustic liners will be performed in the near future.

II. Description of the deconvolution method under uniform and sheared flow hypotheses

The modal detection network will be implemented in a hard-wall region of the duct with circular and constant section, located downstream of the diffuser. Reflections may occur at the end of the duct, as termination is not equipped with acoustic treatment. The flow is perturbed by sound coming from inside the pipe but perturbations amplitudes are considered to be small enough for linearisation. The acoustic noise of interest is a broad band one, mostly energetic between 300 and 2000 Hz.

In the IDEAS project framework, two different flows are considered. An axisymmetric compressible RANS computation has been driven by Liebherr Aerospace under these two conditions. It is assumed that the mean flow is perfectly axisymmetric with only an axial component. The flow is clearly sheared in the radial direction, even if the shear is much less pronounced than what can be observed in the mixing region. The RANS flow solution was post-treated to assess the equivalent mean temperature, flow velocity and density at each axial location, by averaging over the cross-section. It has been ensured that the same mass flow is kept whatever the axial position is. This leads to the two equivalent uniform flow conditions:

• PT1: Speed of sound 352 m • s -1 and Mach 0.15 • PT2: Speed of sound 380 m • s -1 and Mach 0.16

We thereby consider a rigid and infinite cylindrical duct, of radius R = 86.5mm, subjects to a uniform or shear subsonic flow along the revolution axis of the duct (0x). The Fourier transform of the acoustic pressure field at frequency ν, defined by:

p (x, ν) = ∫ +∞ -∞ p (x, t) e -2 i π ν t dt (1)
and propagating inside such a duct can be expressed as an infinite sum of azimuthal and radial modes:

p (x, r, θ, ν) = +∞ m=-∞ n=+∞ n=0 ε=± A ε m,n (ν) f m,n (r) e -i m θ e -i k ε m, n (ν) x (2)
m ∈ Z and n ∈ N are called the azimuthal modes and radial modes respectively and A ε m,n (ν) is the mode amplitude. ε = + and ε =are related to waves propagating downstream and upstream respectively. (r, θ) is the polar coordinate system of the duct section (see Fig. 1). In practice, the sum is finite. N ω mode denotes the number of modes that are considered in [START_REF] Rienstra | An Introduction to Acoustics[END_REF], defining the modal projecting base.

A. Modal base in the case of uniform flow

In case of uniform flow of Mach number M 0 , f m,n and k ε m,n can be analytically expressed [START_REF] Rienstra | An Introduction to Acoustics[END_REF]:

f m,n (r) = J m µ m,n r R (3)
k ε m,n (ν) = 1 β 2 0       -M 0 k 0 (ν) + ε k 2 0 (ν) -β 2 0 µ 2 m,n R 2       (4)
where J m denotes the Bessel functions of the first kind and µ m,n is the (n + 1) th root of J m . β 0 and k 0 (ν) are defined as:

β 2 0 = 1 -M 2 0 and k 0 (ν) = 2 π ν c 0 (5)
and c 0 denotes the speed of sound. For mode of azimuthal order m and radial order n such that:

β 0 µ m,n ≤ R k 0 (ν) (6)
k ε m,n (ν) takes real value and the mode is cut-on. Otherwise, the imaginary part of k ε m,n (ν) takes non zero value and the mode is cut-off.

B. Modal base in the case of shear flow

In order to take into account the shear flow effect on the acoustic propagation, let us consider an inviscid perfect gas flowing along an infinitely long cylindrical duct of radius R. Let U x , U r and U θ be the axial, radial and azimuthal components of the velocity vector U, ρ the density and s the entropy per unit mass. γ = C p /C v represents the ratio of specific heat capacities at constant pressure and constant volume respectively. The pressure, the density and the absolute temperature T satisfy the equation of state p = ρ C p -C v T. Then, if D/Dt stands for the convective derivation ∂/∂t + (U • ∇), the Euler equations read:

∂ ρ ∂t + ∇ • (ρ U) = 0 (7) ρ DU Dt + ∇p = 0 (8) Ds Dt = 0 (9)
The flow field is written as the sum of a steady base flow and a small-amplitude unsteady perturbation:

(U, p, ρ) = (U, p 0 /γ, ρ 0 ) + (u , p , ρ ) (10)
The base flow is supposed to be streamwise-invariant and axisymmetric, which means that:

U x = U x (r) , U r = 0, U θ = 0, ρ 0 = ρ 0 (r) (11)
RANS results show that these assumptions are quite well verified in the cylindrical duct downstream of the diffuser. In order to obtain the linearized Euler equations, small amplitude disturbance fields is sought for under harmonic formation:

u x , u r , u θ , p , ρ = U x , U r , U θ , P , Σ e i (2 πνt-k x-mθ) (12)
where ω = 2πν is the pulsation, k the complex wave number and m the azimuthal wave number. The velocity, pressure and density disturbance U x , U r , U θ , P , Σ are unknown functions of r. The entropy equation simplifies the system and assumes that P = Σ, the linearised Euler equations for mass and momentum can be written as follows:

L (k, ω) • X = 0 ( 13
)
where L is a linear operator and X = P , U x , U r , U θ . m is fixed and ω and k must be fit to consider either a spatial or a temporal eigenvalue problem (usually, for acoustics issues, a spatial eigenvalue problem is considered, i.e. ν is a given real frequency). Concerning the boundary conditions, the duct wall can be constituted of either a locally reacting liner of complex specific impedance Z or a rigid wall (Z → ∞ or U r = 0 at the wall). This boundary satisfies Ingard-Myers condition [START_REF] Ingard | Influence of fluid motion past a plane boundary on sound reflection, absorption, and transmissions[END_REF][START_REF] Myers | On the acoustic boundary condition in the presence of flow[END_REF]. Solving this linear problem with the boundary condition for a fixed azimuthal wavenumber m, a known mean velocity and density profile and impedance Z implies solving a dispersion relation:

D (k, ω, m) = 0 (14)
This generalized eigenvalue problem can be solved thanks to the MAMOUT code [START_REF] Boyer | Theoretical investigation of hydrodynamic surface mode in a lined duct with sheared flow and comparison with experiment[END_REF], in which the equations are discretized along the radial direction through a spectral collocation method (Tchebychech polynomials) or through a compact finite-difference scheme (Gamet scheme, of order 6 on a regular mesh). ω is a real number fixed by the considered frequency. k is obtained thanks to the dispersion relation solution, and is a complex number. For each eigenvalue k, the associated X eigenvector is also computed. The resolution of such an eigenvalue problem enables to compute cut-on modes but also cut-off modes, in both directions of propagation (downstream and upstream).

In the present configurations, shear flow has only significant effect on the wave numbers for left-running cut-on modes at high frequencies (over 2500 Hz). Consequently, the simplified uniform flow framework may be sufficient for correctly catching the acoustic propagation within the IDEAS duct, and therefore for designing the modal detection ring. However, this point will be checked in section IV.

C. ARMADA deconvolution method

In the following, only cut-on modes are considered, composing the modal projecting base. N ω mode denotes the number of cut-on modes at the frequency ν. The analytical model of the Fourier transform of the acoustic pressure field being defined, either under uniform or shear flow, we can now briefly introduce our deconvolution method, called ARMADA (Azimuthal and Radial Modal Analysis for in-Duct Acoustics). See [START_REF] Fauqueux | Modal deconvolution method in a finite circular duct, using flush-mounted microphones[END_REF] for more details. We consider an antenna composed by N flush-mounted microphones of coordinates (x i , R, θ i ) i=1..N in the coordinate system represented in Fig. 1. The power cross-spectral density is defined as the Fourier transform of the temporal cross-correlation function. The signal being centred, one can introduce the power cross-spectral matrix (PCSM) of the measured acoustic pressure field, estimated through Welch's method [START_REF] Welch | The use of Fast Fourier Transform for the estimation of power spectra : A method based on time averaging over short, modified periodograms[END_REF]:

Γ mesure i, j (ν) = p (x i , R, θ i , ν) p * x j , R, θ j , ν (15)
where z * is the complex conjugate of z and z denotes the expected value on the different temporal data segments.

In the same way, one can introduce the PCSM of the modes amplitude:

S [m i ,n i ,ε i ] [mj,nj,εj ] (ν) = A ε i m i ,n i (ν) A ε j * m j ,n j (ν) (16)
By using the theoretical expression of the measured acoustic pressures given by equation ( 2), an analytical model of the measured power PCSM is derived:

Γ model (ν) = G (ν) S (ν) G H (ν) (17)
G is the modal propagation matrix, defined as:

G i,[m j ,n j ,ε j ] (ν) = f m j ,n j (R) e -i m j θ i e -i k ε j m j , n j (ν) x i (18)
and G H is the conjugate transposed matrix of G. Our modal deconvolution consists in estimating S by solving the following minimization problem in the least square sense:

S (ν) = arg min S(ν) Γ mesure (ν) -G (ν) S (ν) G H (ν) 2 F (19)
with • F the Frobenius matrix norm. S (ν) being a power cross spectral matrix, it has to be a Hermitian, positive-semi definite matrix. The quadratic minimization problem (19) is iteratively resolved using a gradient descent algorithm considering that modes are uncorrelated (i.e: S (ν) is a diagonal matrix at each frequency ν), or correlated (i.e : S (ν) is a full matrix at each frequency ν). See [START_REF] Fleury | Analysis of jet-airfoil interaction noise sources by using a microphone array technique[END_REF] for more details on the iterative process. Non-linear constraint is imposed by projection during the iterative process, and the minimization algorithm is performed frequency by frequency, independently. None regularization scheme is currently used. In Liebherr Aerospace's facility, the deconvolution method is applied under the correlated modes assumption, as reflections may occur at the end of the duct that imply correlation between downstram and upstream modes at least [START_REF] Fauqueux | Modal deconvolution method in a finite circular duct, using flush-mounted microphones[END_REF].

III. Microphone array design

The operating condition and deconvolution method being defined, we now have to design a microphone network adapted to the installation constraints. The acoustic noise of interest is a broad band one, mostly energetic between 300 and 2000 Hz. The deconvolution method is applied at central frequency of third octave bands, from 200 Hz to 3150 Hz. A number of approximatively 30 microphones are owned by Liebherr Aerospace to design the flush mounted microphone array.

In this frequency bandwidth and under the operating condition, the azimuthal order of cut-on modes varies from -3 to 3, with a maximum of 2 radial modes. First of all, two typical microphone array designs are considered (see 2). The first one is composed of one ring and a linear array. The second one is composed of two rings and a linear array, each part being discretized with a constant step. We use the numerical evaluation procedure presented in [START_REF] Fauqueux | Modal deconvolution method in a finite circular duct, using flush-mounted microphones[END_REF], §2, to determine the minimum amount of microphones and the space step of the linear array that enables a satisfying estimation of the modal content. For frequency greater or equal to 2000 Hz, only the second design gives good estimation results. The linear array is composed of 7 microphones, with a space step of 0.025 m, surrounded by 2 rings. Only 7 microphones would be needed on the ring if modes were uncorrelated but, as we assume that modes are correlated, we decided to over-sample the ring by using 12 or 13 microphones.

Fig. 2 Typical microphone array designs.

Based on these preliminary design, five additional microphone arrays are defined (see figure 3):

• Design 1: 2 rings of 13 microphones -1 linear array of 7 regularly spaced microphones.

• Design 2: 1 ring of 13 microphones -1 helix of 13 microphones.

• Design 3: 2 rings of 13 microphones -1 helix of 13 microphones.

• Design 4: 2 rings of 13 microphones -half an helix of 6 microphones.

• Design 5: 1 ring of 13 microphones -1 helix of 13 microphones -1 ring of 6 microphones.

• Design 6: 1 ring of 12 microphones -1 helix of 11 microphones -1 ring of 6 microphones.

One way to choose the optimal design is to re-apply the numerical evaluation procedure to every design at each frequency. Another way is to evaluate the condition number of the matrix that has to be inversed when the deconvolution problem is written as a linear one (see [START_REF] Fauqueux | Modal deconvolution method in a finite circular duct, using flush-mounted microphones[END_REF] for mode details). Indeed, the problem to solve, which is formulated by equation ( 19), can be seen as searching the solution of:

Φ (ν) S (ν) = Γmesure (ν) (20)
where, for every matrix M, M corresponds to a vectorial shape of M, and Φ (ν) can be expressed as a function of G (ν). Φ (ν) depends on the modal basis as well as on the microphones position. Thus, its condition number gives us information about the ill or well-posed property of the problem formulated by equation ( 19), even if neither the constraint, nor the algorithm take part in the study. Such condition analysis were exploited by U. Tapken and L. Enghardt in [START_REF] Tapken | Optimization of sensor arrays for radial mode analysis in flow ducts[END_REF] to compare the suitability of different sensor arrangements, and choose their optimum number of axial and radial measurements positions. The condition number associated to each microphone array are represented in figure 4. One first can notice that the condition number is improved by using an helix microphone array instead of a linear one. It is also better to design a microphone array with 2 rings (design 2 vs. design 3, 4, 5 and 6). The 3 rd , 5 th and 6 th designs give the best condition numbers. As the amount of microphones of the 6 th design met the Liebherr Aerospace's request, this microphone array is selected. Finally, the numerical evaluation procedure is applied at the central frequency of third octave band between 200 Hz and 3150 Hz, considering the microphone array design 6 and the two operating points condition. A statistical information is obtained with 100 realizations of the simulated power-cross spectral matrix of the amplitude of modes (see [START_REF] Fauqueux | Modal deconvolution method in a finite circular duct, using flush-mounted microphones[END_REF], §2, for mode details). The estimation error is always smaller than 1% regardless the frequency.

IV. Influence of shear flow on the evaluation of the modal content of the acoustic pressure field by the deconvolution method

Our deconvolution method needs the knowledge of the modal base characteristics (azimuthal, radial and longitudinal dependence) of the acoustic propagation encountered during the measurements. Axisymmetric compressible RANS simulations have shown that the flow is clearly sheared in the radial direction. As presented in section II, the modal base may be obtained under uniform as well as shear flow (under certain assumptions). However, getting the characteristics in the shear flow case requires the knowledge of the flow profile and a pre-computation by using MAMOUT code. Thus, it is of interest to determine the configurations (frequency among other) where the approximation by a uniform flow leads to acceptable errors on the modal estimation.

In this section, we evaluate the influence of shear flow in our modal detection tool at one specific operating point 18) is slightly different as the radial dependence is normalized. Let us denotes S uniform and S shear as the power cross spectral matrix of the amplitudes B ε m,n of the modes given by ARMADA under the uniform and the shear flow hypotheses respectively.

Fig. 5 Wave numbers associated to operating condition PT2 and obtained under the uniform (green) and shear (blue) flow assumptions.

To evaluate the impact of estimating modes under the uniform flow hypothesis despite the fact that shear flow occurs during measurements, the following procedure is applied:

• A power cross-spectral matrix of the modes amplitude, denoted S shear sim , is randomly simulated. • The modal propagation matrix G shear is calculated and the power cross-spectral matrix from wall-flush mounted microphones is evaluated using: Γ shear = G shear S shear sim G shear H . It models the measured power cross-spectral matrix considering a shear flow.

• The quadratic minimization problem ( 19) is iteratively resolved, considering Γ mesure = Γ shear , with the estimated power cross-spectral matrix of the modes amplitude Suniform est , as a result.

• The global estimation error (%) under uniform flow hypothesis is evaluated by:

= Suniform est -S shear sim F S shear sim F × 100 (22)
• For each azimuthal mode m and radial mode n, the error percentage between the simulated mode amplitude and the estimated one, under uniform flow hypothesis is evaluated by:

[m,n,ε] = Suniform est [m,n,ε],[m,n,ε] -S shear sim [m,n,ε],[m,n,ε] F S shear sim [m,n,ε],[m,n,ε] F × 100 ( 23
)
The quantification is made with the microphone array design 6. We first check that the antenna is well suited for modal detection under the shear flow hypothesis, by applying the same numerical evaluation procedure as presented in section III. The estimation error is always smaller than 1% regardless the frequency.

The mean, minimum and maximum global estimation errors ( 22) are represented in Figure 6. The mean values of the mode amplitude error at each frequency are represented in Figure 7. For frequency smaller or equal to 1000 Hz, the mode amplitude error varies from 0% to 15%, with a mean value always smaller than 1%. The mean value of the mode amplitude error is always smaller than 5% at 1600 Hz, 2000 Hz and 2500 Hz but become more important at 3150 Hz (44%). At this frequency, the n = 1 radial mode is cut-on. This mode appears to be very sensitive to refraction effects, which are only taken into account when the modal basis is computed with the shear flow approach. As the statistical properties of the power cross spectral matrix of the modes that will occurs during the IDEAS measurements are not known, it is not possible to generalize this concluding remarks, but the potential misleading impact of considering an uniform flow modal basis on the characterization under shear flow conditions should be kept in mind at 3150 Hz.

V. Experimental setup

The test bench simulates a new concept of air conditioning system constituted by a jet pump. There is composed of an inlet, a mixing part where the jet is and a diffuser part (Figure 8 left). After the diffuser outlet, the duct is cylindrical (diameter of 173 mm -Figure 8 center) up to the anechoic chamber where the termination is flush to the wall (Figure 8 right). The ONERA modal detection ring take place 2.41 m from the jet exhaust, after about 1.66 m of cylindrical part. The ring has been instrumented with 29 1 /4" condenser microphones Brüel&Kjaer. As they are flush mounted with grid, pressure-field microphones type 4938, were chosen. Two sections of the ring (upstream and downstream) have been also instrumented with each three pressure taps in order to check the aerodynamic variation along the ring axial extent.

Fig. 8 Jet-pump geometry (left) -Photography of Liebherr's test bench (center) and termination in the anechoic chamber (right).

The signals are acquired with a B&K data acquisition system LANXI that allows synchronization accuracy between channel less than 20 ns. The sampling frequency was fixed at 32768 Hz and a high-pass filter was set at 22.4 Hz. Thirty seconds of signals were recorded, that allows to perform spectral analysis every 8 Hz with 240 averages. Different jet operating conditions of the pump are studied during the test, with c 0 varying from 335 to 355 m • s -1 and M 0 from 0.05 to 0.34. We focus on the ones that are closest to the specification conditions: Those values are provided by Liebherr Aerospace, based on temperature, pressure and mass flow rate measurements in the area of the microphones array. Unfortunately, the conditions that were considered in the preliminary study were not reached. Nevertheless, the suitability of the antenna design for modal deconvolution is preserved (see Figure 9). Let us notice that the condition number diverges in the vicinity of the modes cut-on frequency. Reproducibility test points are done.

Fig. 9 Condition number obtained under the operating conditions met during the test.

Azimuthal decomposition using the twelve microphones of the upstream ring of the array was process in real time as well as a first analysis of the complete radial and azimuthal decomposition using the deconvolution method. This work was done for the twelve central frequency of the third octave band between 200 Hz and 3150 Hz which is the band of interest for the liner characterizations. A post-processing of the data is also done for the frequencies of the entire bandwith between 200 Hz and 3150 Hz, with a frequency step of 8 Hz. Azimuthal Fourier transform allows us to determine the azimuthal mode amplitudes as a function of x and ν, for azimuthal order m varying from -5 to 5:

1 2π ∫ 2π 0 p (x, R, θ, ν) × e im 0 θ dθ = A m 0 (x, R, ν) (24)
The azimuthal mode amplitudes can also be recalculated from the power cross-spectral matrix of the amplitudes of the mode obtained by modal deconvolution:

A m 0 (x, R, ν) 2 +∞ n 1 ,n 2 =0 ε 1 ,ε 2 =± S [m 0 ,n 1 ,ε 1 ],[m 0 ,n 2 ,ε 2 ] (ν) J m 0 µ m 0 ,n 1 J m 0 µ m 0 ,n 2 e -i k ε m 0 , n 1 (ν)-k ε m 0 , n 2 (ν) x (25)
Azimuthal mode amplitudes obtained by those two methods are very closed (see Figure 10), except in the vicinity of the cut-on modes frequencies. The part of the measured cross-spectral matrix that is not explained by the estimated PSCM of the amplitudes of the mode is rather small, as represented in Figure 11. This leads us to have confidence in the results obtained by ARMADA on the basis of the measurements made with the microphones array designed in the preliminary study. In a rigid cylindrical duct, the sound intensity carried by the azimuthal mode m and radial mode n propagating in the direction given by ε may be expressed by [START_REF] Morfey | Acoustic energy in non-uniform flows[END_REF]:

I ε m,n (ν) = 1 ρ 0 c 0 A ε m,n (ν) 2 Y m,n k 0 (ν) β 2 0 k ε m,n (ν) + M 0 k 0 (ν) k 0 (ν) -M 0 k ε m,n (ν) 2 ( 26
)
where ρ 0 is the air density and Y m,n is given by:

Y m,n =              1 2 if m = n = 0; 1 2 1 - m 2 µ 2 m,n J 2 m,n µ m,n otherwise (27)
The upstream and downstream sound intensity are given by adding up the modal intensity of all modes propagating upstream and downstream respectively. Let's take a look at the mean, maximum and minimum values of the upstream modal intensity obtained on the different measurement points related to PTMes1 and PTMes2. Results, normalized by the upstream sound intensity, are given in Figures 12 and13. The two jet conditions PtMes1 and PtMes2 present very similar results. The m = 0 and n = 0 mode is predominant. The m = 0 and n = 1 mode, which appeared to be the most sensitive to refraction effects (see section IV), is not very energetic. The values are relatively constant from one measurement to another, except for the azimuthal mode 3. One can also notice that the positive and negative value of the same azimuthal order (m = ±1, ±2 and ±3) have equivalent intensity, which could be understandable as for jet noise application no preferential rotating sense propagation is expected. The positive and negative azimuthal mode are correlated. The reflexion coefficient associated to azimuthal modes are represented in Figures 14 and15. It is defined as the ratio between the downstream and the upstream azimuthal intensity, where I ε m 0 is calculated by summation of all I ε m 0 ,n . In the vicinity of the cut-on frequency, upstream and downstream mode intensities are equivalent, which exhibit important reflection. When the frequency increases and therefore deviates from the cut-on frequency, the upstream contribution strongly decreases. Once again, the two jet conditions PtMes1 and PtMes2 present very similar results, with a reflection coefficient slightly higher in the PTMes2 condition. Such a reflection implies non-zero correlation coefficients between upstream and downstream modes, that also justify the correlated mode assumption. Finally, this first experimental campaign validates the capability of the completed methodology for the characterization of the modal content of the jet pump noise. Measurements with acoustic liners will be performed in the near future. At that time, it would be interesting to also make measurements without acoustic liners in such a way that we can apply our deconvolution method under the shear flow hypothesis, in order to validate (or not...) the theoretical sensitivity analysis presented in section IV.

VI. Conclusion

In the first part of this article, a modal detection tool in presence of uniform as well as shear flow has been introduced. Based on the ONERA's deconvolution method ARMADA developed for infinite circular rigid duct of constant section in presence of uniform flow, it uses modal shapes given by MAMOUT code in case of shear flow. Only cut-on modes are considered. The modal detection network will be implemented in a hard-wall region of the duct with circular and constant section, located downstream of the diffuser, far from the lined walls. Hypotheses of circular rigid duct of constant radius are, then, verified. Correlations are taken into account in the estimation tool, as wave reflections may occur at the end of Liebherr's bench.

In the second part, the design of the IDEAS microphone array has been defined, considering uniform flow only. In the frequency bandwidth of interest and under the operating conditions, the azimuthal order of cut-on modes varies from -3 to 3, with a maximum of 2 radial modes. Different sensor arrangements have been defined and compared in terms of condition number associated to the linear problem our deconvolution method have to solve. Composed by 29 microphones, the retained design allows an accurate theoretical estimation of the correlated modal content of the acoustic pressure field for the two operating points, at the central frequency of third octave bands between 200 and 3150 Hz.

In the third part, the influence of shear flow in our modal detection tool under the second operating point condition is evaluated. The microphone array defined in section III is well suited for modal detection under the shear flow hypothesis. The impact of estimating modes under the uniform flow hypothesis despite the fact that shear flow occurs during measurements has been evaluated. As the statistical properties of the power cross spectral matrix of the modes that will occur during the IDEAS measurements are not known, it is not possible to generalize this study, but considering an uniform flow modal basis to characterize the modal content of the acoustic pressure field produced under shear flow conditions has a minor impact in most cases for frequencies smaller than 2500 Hz but can become important at 3150 Hz.

Finally, the experimental setup has been introduced. Analysis of the measurements performed in the facility without use of acoustic liners has been detailed for two operating conditions. The part of the measured cross-spectral matrix that is not explained by the ARMADA estimated modes is rather small. Furthermore, azimuthal mode amplitudes are very closed from the one obtained by azimuthal Fourier transform. This leads us to have confidence in the results obtained by ARMADA on the basis of the measurements made with the microphones array designed in the preliminary study. The two jet conditions present very similar results. The m = 0 and n = 0 mode is predominant. Positive and negative values of the same azimuthal order have equivalent intensity, which could be understandable as for jet noise application no preferential rotating sense propagation is expected. Important reflection coefficients have been exhibit in the vicinity of the associated cut-on frequency, that strongly decreases as the frequency increases. Correlation occurs between upstream and downstream modes as well as positive and negative azimuthal modes, that justify the correlated mode assumption. Measurements with acoustic liners will be performed in the near future.

Fig. 1

 1 Fig. 1 In duct orthonormal coordinate system.

Fig. 3

 3 Fig. 3 Microphone array designs defined in this study.

Fig. 4

 4 Fig. 4 Condition number obtained under the two operating point conditions, considering each microphone array design defined in this study.

Fig. 6

 6 Fig. 6 Mean, minimum and maximum estimation errors (equation (22)) obtained with 100 simulations at the central frequency of third octave band between 200 Hz and 3150 Hz (left) and between 200 Hz and 1000 Hz (right).

Fig. 7

 7 Fig. 7 Mean value of the mode amplitude error (equation (23)) obtained with 100 simulations at the central frequency of third octave band between 200 Hz and 3150 Hz.

•

  PTMes1: Speed of sound 352 m • s -1 and Mach 0.25 • PTMes2: Speed of sound 346 m • s -1 and Mach 0.19

Fig. 10

 10 Fig. 10 Discrepancy between the azimuthal mode amplitude obtained by equations (24) and (25) under the operating conditions PTMes1 (left) and PTMes2 (right).

Fig. 12

 12 Fig. 12 Normalized mean, maximum and minimum values of the upstream modal intensity obtained under the operating condition PTMes1.

Fig. 14

 14 Fig. 14 Normalized mean, maximum and minimum values of the azimuthal mode reflection obtained under the operating condition PTMes1.

Acknowledgement

This project has received funding from the Clean Sky 2 Joint Undertaking under the European Union's Horizon 2020 research and innovation programme under grant agreement No 716499. This study was conducted in collaboration with Liebherr Aerospace, and with the participation of ATECA for the manufacturing of the ONERA modal detection ring. Finally, the authors would like to thank Laurent Coste from ONERA and Mathieu Albert from Liebherr Aerospace for their involvement during the experimental campaign.


Julien Eymard
email: julien.eymard@cea.fr

Vincent Barth

Leo Basset

Eric Gerritsen

Adrien Danel

Mathieu Hebert

Raphael Clerc

Topic 4: Photovoltaic Modules and BoS Components Subtopic 4.1: PV Module Design, Manufacture, Performance and Reliability EUPVSEC-19 -Optimising SHJ solar cell bifaciality towards a monolithic module architecture

Aim and approach

According to the ITRPV roadmap, 10 % of the module market will be truly bifacial (glass/glass design) by 2020 [START_REF]ITRPV -Results 2017[END_REF]. PV manufacturers are now largely focused on this technology, as it is a straightforward way to increase the output of a PV power plant by exploiting the part of the solar spectrum arriving at the module rearside [START_REF]Bifacial Solar Module Technology[END_REF]. When placed vertically, bifacial modules also allow east-west configurations without multiplying the number of panels.

Bifacial cells can also be assembled in a monolithic architecture that consists of cells, whose front (FS) and rearsides (RS) are alternately placed up and down (Fig. 1), which then allows a planar interconnection between the cells, connecting frontsides to frontsides and rearsides to rearsides. We have studied the performance of such modules in relation to the bifaciality of the cells.

Scientific innovation and relevance

Such a monolithic architecture, as already proposed by Buck et al [START_REF] Buck | Large Area Screen Printed N-Type MC-SI Solar cells With B-Emitter: Efficiencies Close to 15% and Innovative Module Interconnection[END_REF], avoids the "top/bottom" passage of cell interconnect ribbons. This observation implies several benefits [START_REF] Zarmai | A review of interconnection technologies for improved crystalline silicon solar cell photovoltaic module assembly[END_REF]: (i) an increased reliability thanks to the reduction of thermomechanical stresses on interconnections and cells. (ii) Simplification of string and module manufacturing, notably when using wire interconnection. (iii) A reduction in the intercell space with two consequences: a decrease in the associated series resistance, and a gain in useful surface area and therefore in module efficiency. (iv) The possibility of using different ribbons types or sizes on front-and rearsides.

The main challenge in such a concept is to avoid electrical mismatch: the module will be limited by the photocurrent generated by the rear surfaces of the cells if bifaciality does not equal 100 %. Buck et al. [START_REF] Schulte-Huxel | Flip-Flop Cell Interconnection Enabled by an Extremely High Bifacial Factor of Screen-Printed Ion Implanted n-PERT Si Solar Cells[END_REF] have studied this 'flip-flop' architecture with two kinds of mc-cell: a p-bulk P-emitter and an n-bulk B-emitter. As the silicon heterojunction solar cell (SHJ) technology is intrinsically symmetrical, Augusto et al. [START_REF] Augusto | Series connection front-to-front and back-to-back of silicon heterojunction solar cells[END_REF] used it with n and p-emitter. Using such design with two different kinds of solar cells allows to match the current of the front side, but increase the complexity of the cell integration into modules. Thus, Schulte-Huxel et al. [START_REF] Walter | Evaluation of Solder Resists for Module Integration of MWT Solar Cells[END_REF] studied the possibility to use a single cell type (n-PERT) with a high bifaciality factor. In the present study, we combine both approaches: using the intrinsic bifaciality of SHJ cells (n-CZ bulk, rear emitter) to make a monolithic module with a single type of solar cells.

It is possible to make SHJ cells with 100% bifaciality by adapting the metallization on the front-and rearside. From a STC cell performance point of view, it is preferable to tend towards medium (or 85 % to 90 %) bifaciality (Fig. 2) [START_REF] Danel | Bifaciality optimization of silicon heterojunction solar cells[END_REF]. However, for a monolithic module, it is preferable to aim for the highest bifaciality to minimize mismatch between neighbouring cells. Therefore, the purpose of this transverse study is to determine the best compromise on cell bifaciality for maximum performance of a monolithic module.

Preliminary results and conclusions

SHJ solar cells with different rearside metallization pitch have been made, leading to cell bifacialities from 78 % to 94 %. In standard test condition (STC -1000 W/m² on the front side) increasing pitch and bifaciality leads to a fall in efficiency due to increasing series resistance as shown with experimental SHJ cell data and confirmed by modelling (Fig. 2).

CTM perspective.

When embedded into a monolithic module, cells with bifaciality lower than 100 % evoke electrical mismatch. This has been quantified with Spice-like modelling on the front-up and rear-up cells, described by 2-diode model (with different photo generated currents). The gain in efficiency offered by the monolithic design comes from the reduction of inactive surface (intercell space). By reducing the intercell space of a 72-cell module from 3mm to 1mm, the module surface area can be reduced by 1.25 % (206 cm²) for a similar power. This percentage increases proportionally if half-cut (+2.46 %) or quarter-cells (+4.74 %) are used, as is often proposed to further increase efficiency of bifacial modules. The reduced intercell spacing also results in a small decrease in series resistance (0.1% gain in 𝑃 𝑚𝑝𝑝 ). The threshold bifaciality beyond which a monolithic module is interesting from a CTM efficiency point of view is evaluated numerically at 92 %. This threshold will now be compared to the experimental values (Fig. 3).

Efficiency perspective. Higher bifaciality was found to decrease STC cell efficiency (FF losses, Fig. 2), but increases module performance (smaller mismatch). From a transversal cell/module perspective, it reveals an optimum in bifaciality (controlled here by rearside finger pitch) that maximizes the monolithic module efficiency. Numerical experiments indicate an optimal pitch of 1.2 mm, which seems to be confirmed by the manufacture and characterization of first monolithic minimodule (Fig. 5). CELLS. Various cells with different rear side pitches have been made, leading to cell bifaciality from 78 % to 94 %. The numerical modelling of the cells involve: series resistance breakdown (bulk, a-Si, lateral conduction, TCO, contact, finger, ribbons) [START_REF] Basset | Series Resistance Breakdown of Silicon Heterojunction Solar Cells Produced on CEA-INES Pilot Line[END_REF], two-diodes model parameters (I01, n1, I02, n2), and a photo generated current affected by metallisation shadowing and absorptions.

MODULE.

The bifaciality to be taken into account when calculating the electrical mismatch is that of the module. It is therefore necessary to study how module bifaciality differs from cell bifaciality. In a first part, we will thus investigate how the module assembly and its bill of materials (BOM) will influence cell bifaciality.

The cell bifaciality in photo generated current is expressed as: 𝐵𝐹 𝑐𝑒𝑙𝑙-𝐼𝑠𝑐 = 𝐼𝑠𝑐 𝐹𝐴𝑅 /𝐼𝑠𝑐 𝐹𝐴𝑉 . When looking at how the photo generated current is modified by the moduling step, we have to focus on several points [START_REF] Haedrich | Unified methodology for determining CTM ratios: Systematic prediction of module power[END_REF]: With monolithic design, a symmetrical bill of materials will be employed: the same glass, encapsulant, and ribbons are used for front-and rearside of the module. These components therefore have no effect on module bifaciality. The difference between frontside metallization and rearside metallization changes the bifaciality of the module. This is due to the effective shading of 1-pass screen-printed fingers [START_REF] Rodríguez | Analysis and performance of dispensed and screen printed front side contacts at cell and module level[END_REF] differing in cell (95 %) and encapsulated module (72 %) as shown in Figure 3.


CTM perspective: Focusing on the CTM value, the better the bifaciality, the better the cell-to-module ratio. In Figure 4, absorption losses in optical stack and additional resistance effects have been included in the cell performance, in such a way that CTM only represents electrical mismatch losses. Efficiency perspective: Focusing on the global efficiency of the module, which includes cell performance, mismatch losses, optical stack and ribbon resistance, we obtain an optimal value of the rearside pitch of 1.2 mm, corresponding to a solar cell bifaciality of 93% (Fig. 5).

Three monolithic mini-modules (2*2 full-cells) have been made with three different rearside pitch: 0.3, 1.05 & 2.1 mm. Measurements -with sun simulator uncertainty of 0.3% absshow a good agreement with numerical modelling. These results will be statistically consolidated with a larger number of modules.

Topic 4 : 4 . 1 :

 441 Photovoltaic Modules and BoS Components Subtopic PV Module Design, Manufacture, Performance and Reliability Explanatory Page DESIGN. A monolithic module diagram is presented in Figure 1. The absence of the elbow made by the cell interconnection ribbons is an additional factor of reliability and simplifies the module assembly: there are only front-to-front and back-to-back connections.

  Optical stack o Air/Glass & Glass/encapsulant reflection o Glass & encapsulant absorption o Refractive index adaptation encapsulant / cell  Effective shading of ribbons cell interconnect  Coupling between finger and air / glass interface

Table 1 : 9 1Figure 1 :Figure 2 :Figure 3 :Topic 4 : 4 . 1 :

 19123441 Figure 1: Schematic diagram of a classic interconnected module (a) and a monolithic module with planar interconnect (b)

Figure 5 :

 5 Figure 5: Monolithic module efficiency as a function of rear side pitch of HJT solar cell. Varying the rear side pitch directly affects cell bifaciality. Experimental mini-modules show good agreement with modelling.

Figure 4 :

 4 Figure 4: Losses in CTM efficiency caused by electrical mismatch as a function of module bifaciality (black). The horizontal lines (blue, magenta & red) show the efficiency gain caused by the reduction of inactive area for full-cells, half-cells, and quarter-cells respectively.