Hussein Chouman
email: hussein.chouman@imt-atlantique.fr

Annie Gravey

Philippe Gravey

Youssouf Hadhbi

Hervé Kerivin

Michel Morvan

Annegret Wagler

Impact of RSA Optimization Objectives on Optical Network State

Keywords: RSA, Optical network, flexgrid, optimization

This paper focuses on off-line dimensioning of flexgrid networks. As any operator wishes to satisfy all traffic demands characterized by a traffic matrix, considering blocking as an optimization objective is not appropriate. We thus consider two other objectives, one of them related to path lengths and the other related to spectrum occupancy. We also define several indicators to characterize a mapping of demands on a given network, including some which are related to the remaining capacity of the network. We apply off-line dimensioning algorithms with the two different optimization objectives on a traffic matrix to be supported by a small flexgrid network and assess the impact of the selected optimization objective on the network level metrics in order to provide insight on how such an off-line dimensioning should be performed.

I. INTRODUCTION

The emergence of coherent transponders offering variable baud rate and modulation formats, together with the evolution of WDM from fixed to variable frequency grid, generated a great interest in the optical networking community on the Routing and Spectral Assignment (RSA) problem and related topics such as spectrum fragmentation.

Optical networks design presents two successive phases: first, in the "off-line" or dimensioning phase, which occurs before a network is deployed, a set of known traffic demands (expressed as a traffic matrix) are mapped on a (previously empty) network. Afterwards, in the "on-line" or operational phase, new traffic demands arrive, stay for sometime and eventually depart; these should be accommodated on top of the current load of the network.

The off-line network dimensioning process is addressed in the present work. A classical method for solving this problem is to derive exact or heuristic formulations of an optimization problem with a suitable objective function. The choice of optimization objective is not straightforward and there is no clear indications in literature which is the best optimization objective that yields the best network state.

Network cost seems a natural choice, but the way to assess this cost raises many questions. Should the cost only include CAPEX or should it take account of amortization and operational expenses over a given period? How to account for technology and cost evolution and traffic changes during that period? In other words, it is difficult to formulate pertinently the off-line problem without having an idea of the future evolution of the network, which corresponds to the on-line problem. Another well-known approach considered for (optical) network dimensioning consists in computing the blocking probability in presence of statistical variations of the global set of demands. This method has the merit to account for some demand variability but it has a major drawback, as it does not yield a resource allocation that ensures that all demands are satisfied. Therefore, it cannot be used for an operational deployment.

Thus we adopted the approach of satisfying a static set of demands while optimizing a given objective function and tried to reach a better understanding of the pros and cons of different objective functions. This approach was followed e.g. in [START_REF] Balon | How well do traffic engineering objective functions meet te requirements?[END_REF] for the off-line design of IP networks. Usually in literature only one metric is optimized and no comparison is done between solutions achieved with different optimization objectives. To perform such a comparison, we defined a set of optical "network health" metrics, in order to evaluate the network ability to accommodate new demands or some growth of the existing ones. Some of these metrics can be used as objective functions, other cannot, depending on the possibility of applying them to the formulation that we adopted. We will then use them with different objective functions and compare the values of the set of metrics for the different solutions corresponding to these objective functions. Network health metrics are expected to play various roles: firstly, to help analysing the characteristics of the network dimensioning obtained with different objective functions and secondly to evaluate the ability of the different designs to accept a "reasonable" amount of additional traffic demands. Ideally, some of these metrics could serve, when crossing a threshold value, as a decision tool for setting up additional resources in some parts of the network in order to prevent a possible congestion during network operation.

In the context of the RSA problem, the evaluation of how much the available spectrum has been fragmented has generated many studies. In particular, several "fragmentation metrics" have been proposed. It seems clear that increasing fragmentation contributes to reducing the optical spectrum availability and is worth to be considered when assessing network state. However, fragmentation metrics are generally measured at link level and it is not easy to aggregate them into network level indicators. This kind of problem concerns the construction of any network metric derived from elementary one (e.g. at link or path level). In particular, an important question is whether it is more relevant to average individual values or to consider worst case values.

The rest of the paper is organized as follows: Section II identifies relevant works related to network level metrics. Section III introduces the optimization algorithms used to map a given traffic matrix on a given network. In Section IV, several network level metrics are classified into some that characterize the state of the network and others that estimate the remaining capacity while Section V applies optimization algorithms to a practical dimensioning problem. Section VI concludes the paper.

II. RELATED WORKS

A metric can be used either as an optimization criterion or as a network health indicator.

A. Optimization metrics

In [START_REF] Ruiz | Column generation algorithm for rsa problems in flexgrid optical networks[END_REF], [START_REF] Klinkowski | Valid inequalities for the routing and spectrum allocation problem in elastic optical networks[END_REF], the proposed optimization objective metric aims to minimize the volume of unsatisfied traffic. In [START_REF] Żotkiewicz | Optimization models for flexgrid elastic optical networks[END_REF], the proposed optimization process minimizes the sum of lengths of the paths carrying the demands, which aims to route demands over shortest paths in order to minimize global spectrum occupancy and delay. In [START_REF] Amar | How problematic is spectrum fragmentation in operator's gridless network?[END_REF], the optimization process minimizes the total transponders cost necessary to satisfy all demands. Other works aim to limit fragmentation by minimizing the max slice-index of the network in [START_REF] Walkowiak | Optimization of multicast traffic in elastic optical networks with distanceadaptive transmission[END_REF], or to minimize the summation, over links, of the max slice-index of each link in the network [START_REF] Ghallaj | Optimizing spectrum allocation in flex-grid optical networks[END_REF].

B. Network State Metrics

One of the main aspects of the off-line design phase is to minimize links utilization (number of slices used) or maximize the residual spectrum (free slices) after serving the required demands of the traffic matrix. In this context, different works in literature propose metrics to characterize the state of the available spectrum. In [START_REF] Klinkowski | Solving large instances of the rsa problem in flexgrid elastic optical networks[END_REF], authors use the spectrum usage which represents the number of slices used such that a slice is said "used" if and only if it is used at least on one edge. In [START_REF] Agrawal | Spectrum efficient distance-adaptive paths for fixed and fixed-alternate routing in elastic optical networks[END_REF], the metrics used are the spectrum utilized in Hertz, and resource utilization ratio which is the spectrum utilized over the number of demands established. The available spectrum is also reduced by fragmentation resulting from RSA in the flexgrid network.

Several fragmentation metrics have been proposed in literature to monitor spectrum fragmentation quantitatively and qualitatively, including external fragmentation (EF), Shanon's entropy (SE) and access blocking probability (ABP). EF is derived by computing the ratio of the maximum number of available contiguous slices to the number of all available slices [START_REF] Amar | Spectrum fragmentation issue in flexible optical networks: analysis and good practices[END_REF]. SE is based on Shanon's entropy formula, it is computed as follows [START_REF] Wright | Simulation results of shannon entropy based flexgrid routing and spectrum assignment on a real network topology[END_REF]:

SE = -Σ N i=1 D i D ,
where D is the total number of frequency slices in the spectrum of a link or path, and D i is the size, in slices, of the ith block of contiguous used or unused spectrum (hence, N is the number of such blocks). ABP computes the ratio between the number of possible demands that can be placed on the available spectral voids in a certain link and the number that could be placed if all those free slices were contiguous [START_REF] Amar | Traffic forecast impact on spectrum fragmentation in gridless optical networks[END_REF].

One first notes that EF, SE and ABP are link-based metrics; therefore, the fragmentation of a given path is defined as the average of the fragmentation of each link of this path. This implies that the resulting information is quite imprecise because it fails to recognize misalignment of spectrum holes in a path. Furthermore, as the usability of a given spectrum hole depends on the granularity of the transponder (i.e. the number of slices that it used in a given configuration), a fragmentation metric should be expressed as a function of granularity. However, this is the case neither for EF nor for SE which are unable to distinguish the case when fragmented spectrum holes match the available granularities from the inappropriate fragmented cases. On the other hand, ABP, which does take into account transponders' granularity, provides only an average measure of fragmentation over transponders granularities. Lastly, the network state metrics mentioned in this section provide qualitative assessments of a given mapping, but provide no quantitative information on the remaining capacity on the network.

C. Network's Remaining Capacity [START_REF] Balon | How well do traffic engineering objective functions meet te requirements?[END_REF] proposes some strategies to roughly estimate networks' capability to host future demands. However, this paper deals with IP networks only which significantly differ from flexgrid networks as they present no spectrum fragmentation issue.

The authors in [START_REF] Pederzolli | Path-based fragmentation metric and rsa algorithms for elastic optical networks[END_REF] divide the residual spectrum resources or the "free" slices into "wasted", "usable" and "unusable" slices. The wasted slices are those that are only free on some, but not all, of the links of a path; unusable slices are those free on all links of a path but belong to a block of slots that is too small with respect to the number of slices requested by a transponder. Lastly, the remaining slices are the usable ones.

D. Contributions

In this work, we propose a new optimization objective that aims to reduce fragmentation and consequently, increases the capability of the network to host future demands. Unlike work in literature' where a single optimization metric is used without comparing it to other optimization objectives, we compare our proposed optimization objective to a traditional one in literature that aims to which aims to route demands over the shortest paths.

The comparison includes, but is not limited to, the remaining capacity (wasted and usable) and fragmentation metrics at the network level as a function of transponder's granularity.

III. OFF-LINE DIMENSIONING

In this section we propose a network model and then we explain how we set the number of slices ensuring a scenario without any blocking.

A. Network Model

An input of the off-line RSA problem is formally defined by

• an optical-network topology represented as an undirected, loopless, and connected graph G = (V, E), • an edge-length (in kms) vector ∈ R E + , • an optimum spectrum S = {1, • • • , s} os available frequency slices, and • a multiset K of traffic demands where each k ∈ K is specified by an origin node o k ∈ V , a destination node

d k ∈ V \ {o k }, a requested number w k ∈ N + of slices,
and a transmission reach ∈ R + (in kms). The off-line RSA problem aims at determining for each traffic demand k ∈ K, a lightpath composed of a route (i.e., an (o k , d k )-path P k in G) whose length does not exceed the transmission reach k and a channel (i.e., a subset S k ⊆ S of slices) such that

• the slices allocated to a traffic demand k (i.e., slices in S k )

are the same all along the route P k (spectrum continuity), • the slices allocated to a traffic demand k are contiguous (spectrum contiguity), • a same slice cannot be allocated to two different traffic demands on the same link (non-overlapping spectrum), • and some objective function is minimized. A new Integer-Linear Program (ILP) model for the off-line RSA problem was introduced in [START_REF] Hadhbi | A novel integer linear programming model for routing and spectrum assignment in optical networks[END_REF]. This model has a polynomial number of variables as it is based on the socalled edge-node formulation for the route determination and a non-polynomial number of constraints that can be handled in a polynomial time through the well-known cutting-plane approach [START_REF] Hadhbi | A novel integer linear programming model for routing and spectrum assignment in optical networks[END_REF]. This tractable model is the first complete ILP model for the off-line RSA problem as it precisely encodes the set of all feasible solutions to the RSA problem and therefore can be used to optimize any objective function used as quality measure by a network operator. In this paper, we only focus on two specific objective functions

• the minimization of the sum of the maximum slice used over all the demands, i.e., min k∈K s∈S

z k s , ( Objective 1)
where z k s is the binary variable that equals 1 if and only if the slice s ∈ S is the last slice assigned to demand k ∈ K and • the minimization of the sum of the route lengths over all the demands, i.e.,

min k∈K e∈E e x k e ( Objective 2)
where x k e is the binary variable that equals 1 if and only if the route P k of demand k ∈ K uses edge e ∈ E. The ILP model with (Objective 1) or (Objective 2) is solved through a branch-and cut framework (i.e., a cutting-plane algorithm embedded in a branch-and-bound approach) that was implemented using Cplex 12.8. The efficiency of a branchand-cut approach significantly depends on the strength of both the lower bounds (i.e., best possible objective) on the optimal value that the linear relaxations of the ILP model provide and the upper bounds (i.e., best found objective) obtained by feasible solutions encountered during the whole solution process. To improve the latter bounds, we also incorporated a greedy-based heuristic that is called frequently during the course of the branching tree. Such an approach may fail to find an optimal solution within a time limit but it usually provides a feasible solution whose quality can be assessed when put in perspective with the lower bound. This is known as the (relative objective) gap that is equal to the best found objectivebest possible objective best objective found ;

the smaller the gap is the better is the incumbent solution.

B. Admissible topology

To run a simulation, that uses the above objective metrics, we need to set the number of slices (N slices ). In a fiber of 4 THz spectrum, the nominal number of slices is 320 (12.5 GHz per slice). However, it is better to decrease this number in order to decrease the number of constraints in the ILP model and consequently, simulation period. Therefore, to find the admissible topology (no blocking and average link occupation (L occupation ) ≤ 80 %), we do the following. We start by N slices = 320 and then:

1) We run a simulation which minimizes the number of blocked demands (N blocked ) (we set the objective function to minimize N blocked ). 2) If N blocked = 0 and L occupation ≤ 80 %, we repeat the the previous step with N' slices = N slices -40. Otherwise, the number of slice is N slices . We keep on repeating these two steps until we reach the minimum number of slices that satisfies blocking and spectrum utilization conditions.

IV. NETWORK LEVEL METRICS

We propose network health metrics that will be used in analyzing the results. We classify these metrics into "network state" (qualitative) metrics and "remaining capacity" (quantitative) metrics.

A. Network State Metrics

The following metrics are considered: Total number of used slices: the summation of the number of used slices over edges. This reflects network congestion. Fiber's last empty space size: the size of the hole bordered by fiber's max used slice and the right edge of the spectrum. Fiber's max used slice index [START_REF] Walkowiak | Optimization of multicast traffic in elastic optical networks with distanceadaptive transmission[END_REF]: we plot maximum used slice index and fiber's last empty space size in each fiber vs fiber ID. Holes statistics: a spectrum hole is a set of contiguous free slices bordered by occupied slices. The larger the number of spectrum holes, the more the network is fragmented. Therefore, we will count their number, mean size and summation of spectrum holes over all links. Network's EF, SE and ABP: we find the fragmentation per link then, we calculate the average, over all links, to derive network level fragmentation. Metric 1: The sum of the maximum slice used over all the demands, which corresponds to the objective 1 of the previous section. Metric 2: The sum of the route lengths over all the demands (corresponding to objective 2).

B. Remaining Capacity Metrics

Network's remaining capacity: the following is inspired from [START_REF] Pederzolli | Path-based fragmentation metric and rsa algorithms for elastic optical networks[END_REF], however, here we compute the number of usable and non-usable slices as function of each transponder's granularity. For each (s -d) pair m, we have I m (n) feasible paths (a path that satisfies transponder's transmission reach and has at least n free slices). On each path p m,i , there are occupied and un-occupied (free) slices. Free slices are divided into two categories:

• Wasted slices W m,i which is the summation of all slices that are only free on some, but not all, of the links of a path p m,i . • Accessible A m,i slices: a set of slices, in which each slice of the hole satisfies continuity over the path p m,i and could be used to allocate traffic if it satisfies the transponder's granularity. For more ease, we can this kind of holes as path-hole. If each path-hole h m,i,j can accommodate x m,i,j (n) = | h m,i,j | DIV (n) and DIV is the integer division, and n is transponder's granularity. The total number of transponders of granularity n, that can be accommodated on path p m,i is:

X m,i (n) = Σ J j=1 x m,i,j (n) ( 1
)
where J is the number of path-holes in p m,i . At network level, the average number of transponders, of granularity n, that can be accommodated is:

X(n) = Σ M m=1 ( Σ I m,n i=1 X m,i (n) Im,n ) M (2)
Usable-to-Accessible-ratio (U AR): is the path p m,i fragmentation defined as follows:

U AR m,i (n) = U m,i (n) accessible m,i (3)
To find network's U AR, we use the cumulative distribution function (CDF)instead of the average value over all paths. Indeed, as U AR m,i is a ratio, averaging its value over all paths would be hard to analyze. Hence, for each pair, we find the least fragmented path's UAR. Then, all these values of all the pairs are saved in an array whose CDF is plotted.

V. SMALL TOPOLOGY RESULTS

A. Problem Specification

The transponder used has a capacity that varies between 100 Gb/s (with a 3000 km reach), 200 Gb/s (1500 km reach), and 400 Gb/s (600 km reach) which respectively requires 3, 5 and 6 slices. A set of 15 demands (total demand of 68 slices) is randomly generated and the highest possible rate format is allocated. The topology used is the 5 Node Spain topology which has 7 links of average length 367 km, average node degree 2.8 [START_REF] Mariño | Net2plan: a publicly-available open-source network planner[END_REF]. The number of slices/fiber is 30 as explained in Section III-B. We compare the network health metrics resulting of two optimization objectives mentioned in Section III-A using metrics detailed in Section IV. The simulation results shown here are optimal as Gap=0.

B. Network state

Using objective 2 aims at minimizing the sum of demands' route lengths, i.e. demands are routed over the shortest paths possible. Consequently, they traverse fewer links (which limits the total number of used slices), as shown in Tab. I. On the other hand, using objective 1 aims at minimizing the sum of demands' highest index, therefore, it forces demands to get closer to the left hand side of the spectrum. To achieve this goal, demands may be routed over longer paths (which increases metric 2) as they traverse more links (higher total number of used slices). However result, demands are "squeezed" and the spectrum load is distributed in a balanced way among links as illustrated in Fig. 1.

To better explain demands "squeezing" and load distribution, we plot fibers' max used-slice-index and last empty space size Vs. fibers IDs in Figs. 1 and 2 resulting of using objectives 1 and 2, respectively. These two plots show the distribution of slices in the fibers.The size of the empty space, from the lastslice used to left edge of the spectrum in most of the fibers (blue area), is up to 17 to 20 slices using objective 1, while it ranges between 1 to 5 slices for objective 2. This significant difference between the two plots clearly shows the advantage of squeezing obtained by using objective 1. It explains also the huge difference in the number of spectrum holes, between the two objectives as shown in Tab. I) where objective 1 yields 2 holes (Σ | holes | = 6 slices), and objective 2 yields 12 holes (Σ | holes |= 111 slices).

C. Network Fragmentation

Tab. I shows fragmentation metrics EF, SE, and ABP results for objectives 1 and 2. Results obtained by using the latter are higher, by a magnitude of 4 to 5 , than that of objective 1. Therefore, it again shows how objective 1 significantly reduces fragmentation compared to objective 2. The previous metrics are good in terms of comparison between network states. However, to better understand the global network fragmentation, we use the CDF(n) of the paths' U AR of each objective, which are plotted in Figs. 3 and4 for objectives 1 and 2, respectively. We observe that:

• For n = 2, using objective 1, all paths have UAR < 5 %, while using objective 2, only 50 % of the paths have UAR < 5 % and the remaining paths have UAR=5-12 %. • For n = 3, using objective 1, all paths have UAR = 0 %, while using objective 2, only 60 % of the paths have UAR = 0 % and the remaining paths have UAR=5-30 %. • For n = 4, using objective 1, all paths have UAR = 10-15 %, while using objective 2, only 60 % of the paths have UAR < 15 % and the remaining paths have UAR=15-38 %. • For n = 5, using objective 1, all paths have UAR = 15-20 %, while using objective 2, only 50 % of the paths have UAR < 20 % and the remaining paths have UAR=20-60 %. • Generally, using objective 1, all the paths have UAR < 20 %, while using objective 2, paths have UAR = 0-60 %. Using CDF (as function of n) for paths' UAR presents some advantages. Firstly, considering only average fragmentation over paths might hide relevant information about highly fragmented paths; on the other hand, using CDF of paths' UAR provides us quantitative information about the percentage of fragmented paths at a specific fragmentation level. Hence, this could be used as an alert by the operator: when a certain number of paths (say Z) reach a fragmentation level (α), the operator could decide that an action should be taken. Secondly, the CDF(n) figures show how fragmentation of the residual capacity depends on transponder's granularity (n). Moreover, it is clear how fragmentation increases as the granularity increases. This is why, considering fragmentation per granularity is better than considering only a fragmentation as average of fragmentation over different granularities, which might hide a lot of information especially if there is a large range of transponders granularities.

D. Network's Remaining capacity

After demonstrating the importance of the objective function on the network's fragmentation, we show, in this section, how it affects the usability of the residual capacity. Fig. 5 presents the network average wasted slices and accessible slice. It shows that the percentage of wasted slice decreases from 24 % (using objective 2) to 5 % (using objective 1) out of 30 slices per link. Consequently, the percentage of accessible slices increases from 39 % (using objective 2) to 56 % (using objective 1).

To show how the usability of the remaining capacity is enhanced by using objective 1, we plot, in Fig. 6, X(n) (the average number of transponders of granularity (n) that could be allocated) versus transponders' granularity (n), for both objectives. The result are as follows: Fig. 6. Network average X Vs. n resulting from objective 1 and 2.

• For n= 2, using objective 2, X(n)= 5, while using objective 1, X(n)= 8. Thus, the percentage of increase is 60 %. • For n= 5, using objective 2, X(n)= 1.8, while using objective 1, X(n)= 3. This is a 65 % increase Therefore, using objective 1, the number of transponders that could be allocated, generally, increases by 60 % w.r.t objective 2. As a conclusion, optimizing objective 1, which decreases the number of holes by squeezing demands, leads to a lower average wasted slices and consequently, increases the accessible and usable slices compared to objective 2.

VI. CONCLUSION

In this work we have designed a flexible-grid optical network assuming a static RSA. We have compared, using the remaining capacity concept, two different optimization objective: 1) our newly proposed metric minimizes the sum of the maximum slice used over all the demands (obj. 1). 2) the second metric minimizes the sum of the route lengths over all the demands (obj. 2). The comparison was done qualitatively using spectrum occupancy, number of spectral holes, and fragmentation metrics EF, SE and ABP. The quantitative comparison was done using the wasted slices, usable slices and UAR path fragmentation ratio. obj. 1 was able to squeeze demands, thus, it dramatically decreases fragmentation. As a result, using obj. 1, the no. of transponders that could be allocated, increases by 60 % w.r.t obj. 2. Moreover, using obj. 1, all the paths had UAR < 20 %, while using obj. 2, paths have UAR between 0 % to 60 %. As a future work, we plan to consider the on-line RSA problem and we will consider how to best use some of these network health metrics to decide when the network need to be partly or fully re-dimensioned in order to avoid blocking.

Fig. 1 .

 1 Fig. 1. Last empty space size and max used slice index vs fiber Ids (objective 1).

Fig. 2 .

 2 Fig. 2. Last empty space size and max used slice index vs fiber Ids (objective 2).

Fig. 3 .Fig. 4 .

 34 Fig. 3. Fragmentation CDF from objective 1.

Fig. 5 .

 5 Fig. 5. Network's average wasted and accessible slices using objective 1 and 2.

TABLE I DIFFERENT

 I NETWORK HEALTH METRICS FROM OBJECTIVE 1 AND 2.

	Metric	Objective 1 Objective 2
	Metric 1	117	260
	Metric 2	7040	6604
	Number used slices	80	77
	Number holes	2	12
	Σ | holes |	6	111
	EF	0.003	0.051
	SE	0.050	0.244
	ABP	0.001	0.006


Min Wei

Yilin Fan

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

Gilles Flamant

Design and Optimization of Baffled Fluid Distributor for Realizing Target Flow Distribution in a Tubular Solar Receiver

Keywords: Flow distribution, Tubular solar receiver, Perforated baffle, Optimization method, Pressure drop

come

Design and optimization of baffled fluid distributor for realizing target flow distribution in a tubular solar receiver

Introduction

Nowadays, climate change, diminishing world energy resource and increasing energy cost stimulate the diversification of renewable and sustainable energies and the innovation in the industry [START_REF] Olabi | Developments in sustainable energy and environmental protection[END_REF][START_REF] Olabi | State of the art on renewable and sustainable energy[END_REF][START_REF] Danielewicz | Experimental and analytical performance investigation of air to air two phase closed thermosyphon based heat exchangers[END_REF]. Solar energy, especially Concentrated Solar Power (CSP), is considered as one of the promising alternatives to conventional fossil sources. As a key component in current commercial or experimental CSP projects, tubular-type solar receiver is widely used because of its relatively low cost and long lifetime [START_REF] Grange | Thermal performances of a high temperature air solar absorber based on compact heat exchanger technology[END_REF][START_REF] Li | Micro-channel pressurized-air solar receiver based on compact heat exchanger concept[END_REF][START_REF] Baroutaji | Analysis and optimization of sandwich tubes energy absorbers under lateral loading[END_REF][START_REF] Boerema | High temperature solar thermal central-receiver billboard design[END_REF][START_REF] Wang | Experimental study on a coiled tube solar receiver under variable solar radiation condition[END_REF]. However, the problem of fluid flow maldistribution is frequently encountered in CSP solar receivers due to their multi-tubular or multi-level configurations.

In the literature, investigations have conclusively shown that the receiver's efficiency decreases when the flow distribution of heat transfer fluid among parallel tubes becomes less uniform [START_REF] Chiou | Thermal performance deterioration in crossflow heat exchanger due to the flow nonuniformity[END_REF][START_REF] Jones | Consideration of the heat-removal factor for liquid-cooled flat-plate solar collectors[END_REF][START_REF] Jones | Flow distribution in manifolded solar collectors with negligible buoyancy effects[END_REF][START_REF] Gunnewiek | Effect of wind on flow distribution in unglazed transpired plate collectors[END_REF][START_REF] Karwa | Effect of flow maldistribution on thermal performance of a solar air heater array with subcollectors in parallel[END_REF]etc). For example, [START_REF] Chiou | The effect of nonuniform fluid flow distribution on the thermal performance of solar collector[END_REF] found that the performance deterioration could be higher than 20% due to non-uniform flow distribution under some operation conditions. [START_REF] Wang | Analysis and performance of flat-plate solar collector arrays[END_REF] studied the performance of flat-plate solar collector arrays and also confirmed that the efficiency decreased about 15%-30% due to the flow non-uniformity. Fan and Furbo (2007) reported that non-uniform flow distribution can lead to boiling problems in the strips of flat-plate solar receiver, which may result in physical damage or acidification corrosion on the receiver. Moreover, another particular negative effect due to flow maldistribution is the presence of "hot spots" on the receiver walls. The high thermal-mechanical stress caused by local overheating will lead to the mechanical failure or reduced lifetime of solar receivers by a factor of 10 to 20 [START_REF] Fork | Life estimation of pressurized-air solar-thermal receiver tubes[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].

However, the heat flux distribution on the receiver wall is not always as uniform as assumed. Actually non-uniform heat flux has been observed both numerically and experimentally in laboratory PROMES, France. For example, a dish/Stirling system "EuroDish" which has a complete mirror of 53.1 m 2 had been tested experimentally [START_REF] Reinalter | Detailed performance analysis of a 10kW dish/Stirling system[END_REF] and numerically [START_REF] Caliot | Validation of a Monte Carlo integral formulation applied to solar facility simulations and use of sensitivies[END_REF], and a Gaussian-like heat flux distribution on the focal plane was observed. [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] took the pictures by a high resolution CCD camera for the distribution of concentrated solar flux on the absorber. As a qualitative study, 30 heliostats were used to reflect the full moon light which was similar to sun shape to a single centered aiming point. They reported that the flux density distribution delivered on the measurement target was also Gaussian-like.

Based on the fact of non-uniform heat flux on the solar receiver reported above, some studies [START_REF] Boerema | High temperature solar thermal central-receiver billboard design[END_REF]Wei et al., 2015a) reported that uniform fluid flow distribution among parallel tubes of solar receivers may not be the optimal option. [START_REF] Boerema | High temperature solar thermal central-receiver billboard design[END_REF] proposed an optimality criterion to optimize the fluid flow distribution among parallel tubes, suggesting that the peak temperature on the heated surface can be reduced by equalizing the outlet fluids temperatures from parallel tubes. The analytical results obtained showed that the ideal (optimized) fluid flow distribution can reduce the peak temperature by more than 200 °C. However, no clue indicated that the peak temperature was minimized. In our earlier study (Wei et al., 2015a), an original CFD (Computational Fluid Dynamics) -based evolutionary algorithm has been developed for the determination of the optimal fluid flow distribution in a tubular solar receiver subjected to a non-uniform net heat flux, so as to minimize the peak temperature on the receiver wall. Different shapes of optimal fluid distribution which are not uniform can be determined corresponding to different filling materials (varied thermal conductivity). The peak temperature on the heated surface can be reduced by about 240 K when Inconel 600 is used as the filling material, simply by optimizing the fluid flow distribution among parallel tubes. Nevertheless, our earlier study (Wei et al., 2015a) focused on how to determine the optimal fluid flow distribution while how to realize such optimal distribution still remained as an open question.

Therefore, one interesting issue with practical significance is the design and optimization of fluid distributor (and/or collector) for providing the already determined target (optimal) fluid distribution among parallel tubes. The fundamental design question is how to configure the distributor/collector so as to satisfy several important features at the same time:

• Target flow distribution which is not necessarily uniform;

• Small pressure drop increase which is related to the mechanical power consumption and the performance of downstream thermodynamic cycles;

• Compact for space saving;

• Flexible and adaptable to various geometries of the solar receiver and different operational conditions;

• Easy manufacturing and reasonable cost.

In the past two decades, different types of flow distribution management devices have been proposed [START_REF] Rebrov | Single-phase fluid flow distribution and heat transfer in microstructured reactors[END_REF][START_REF] Luo | Heuristic shape optimization of baffled fluid distributor for uniform flow distribution[END_REF], including the use of fluid distributor/collector with optimized geometrical parameters (e.g., [START_REF] Commenge | Optimal design for flow uniformity in microchannel reactors[END_REF][START_REF] Tondeur | Flow and pressure distribution in linear discrete "ladder-type" fluidic circuits: An analytical approach[END_REF][START_REF] Pistoresi | Numerical study on the improvement of flow distribution uniformity among parallel mini-channels[END_REF], the use of nature-inspired tree-like structures (e.g., [START_REF] Luo | Integration of constructal distributors to a mini crossflow heat exchanger and their assembly configuration optimization[END_REF][START_REF] Yue | Flow distribution and mass transfer in a parallel microchannel contactor integrated with constructal distributors[END_REF][START_REF] Tarlet | Entropy generation analysis of a mini heat exchanger for heat transfer intensification[END_REF], and adding "packings" as additional hydraulic resistance inside the distributor/collector (e.g., [START_REF] Rebrov | Header design for flow equalization in microstructured reactors[END_REF][START_REF] Chen | Experimental study on the two phase flow behavior in PEM fuel cell parallel channels with porous media inserts[END_REF]. As an improvement of "packings" method, the insertion of a perforated baffle in the inlet fluid distributor is also widely employed, as summarized in [START_REF] Luo | Heuristic shape optimization of baffled fluid distributor for uniform flow distribution[END_REF]. However, all these methods, either cost-effective or expensive, are only capable of approaching the uniform distribution among parallel channels or tubes (e.g., [START_REF] Lalot | Flow maldistribution in heat exchanger[END_REF][START_REF] Jiao | Experimental investigation of header configuration on flow maldistribution in plate-fin heat exchanger[END_REF][START_REF] Wen | An experimental and numerical investigation of flow patterns in the entrance of plate-fin heat exchanger[END_REF]. To the best of our knowledge, the only attempt in the literature for realizing non-uniform but target fluid flow distribution was reported in our recent paper (Wei et al., 2015b), by installing a geometrically optimized perforated baffle at the distributing manifold. Results showed that different target curves (either uniform or non-uniform) could be successfully reached, but the numerical examples investigated were restricted to a 2D domain without considering heat transfer. Although some clues for the extension of the optimization method to 3D components may be found in [START_REF] Luo | Heuristic shape optimization of baffled fluid distributor for uniform flow distribution[END_REF], the application with actual physical background and systematic investigations are still lacking.

As an important follow-up work of our previous study (Wei et al., 2015a), this paper presents an original study on the design and optimization of baffled fluid distributor for the realization of optimal fluid flow distribution in a CSP tubular receiver. Based on the practical background of CSP engineering application, this study attempts to demonstrate a complete approach in this field, by considering coupled heat transfer and fluid flow in real 3D geometries. As our optimization objective, the peak temperature on the receiver wall could be minimized accordingly by using the geometrically optimized baffle.

In the rest of this paper, we will first introduce the geometry of tubular solar receiver equipped with an inlet fluid distributor and an outlet fluid collector. Then two optimal fluid flow distributions already determined in (Wei et al., 2015a) using copper and Inconel 600 as the filling materials will be selected as the target curves to be realized by optimizing the configurations of perforated baffles. Detailed discussions will be devoted to the reduced peak temperature on the heated surface and the increased total pressure drop due to the insertion of the optimized perforated baffle. Finally, remarks and main conclusions will be summarized.

Solar receiver device and optimization method

In this section, the tubular solar receiver system equipped with the distributor and the collector will be briefly introduced. The optimization principles for the perforated baffle and the CFD simulation parameters will be presented as well.

Geometry of tubular solar receiver with distributor and collector

Figure 1 shows the geometry of the integrated tubular solar receiver system used in this study. It is composed of three parts: inlet rectangular baffled fluid distributor, tubes-in-matrix receiver and outlet rectangular fluid collector. The core part of the system is a cuboid solid monoblock, with the overall dimensions of 405 mm in length (z direction), 202 mm in width (x direction) and 49 mm in height (y direction). It is based on the tube-in-matrix concept, consisting of the external shell, the filling material and a bunch of parallel straight tubes. The external shell and the walls of tubes are made of Inconel 600 while the filling material could be different (copper for example). In more detail, 45 straight tubes with 6 mm in internal diameter and 405 mm in length are arranged symmetrically in three aligned layers. For the convenience of description, these tubes are indexed by i as layer A to C from top to bottom, and by j from 1 to 15 along the x direction. Detailed configuration and dimensions of the tubes-in-matrix part can be found in Wei et al. (2015a). The inlet baffled fluid distributor has a single inlet port in the center with 20 mm in diameter and 40 mm in length, as shown in Fig. 2. Connected to the inlet port is a rectangular distributing manifold with 202 mm in width, 49 mm in height and 30 mm in length. A perforated baffle is inserted in the distributing manifold, with the thickness (e) of 8 mm for sufficient mechanical strength. The overall dimension of the perforated baffle corresponds to the distributing manifold, while the configuration of orifices on the baffle is to be optimized for providing target fluid flow distribution among the downstream 45 parallel tubes. The endpoints of parallel tubes are connected to the outlet fluid collector which has the same geometry and dimension of the distributor, but without baffle insertion.

In actual operation, pressurized air at 15 bar with a constant mass flow-rate will be introduced to the inlet port of distributor, spread by the perforated baffle with optimized configuration, then heated up by passing inside the parallel tubes and finally resembled by the outlet fluid collector. Note that the rectangular-shape distributor and collector are proposed as a study example for compactness consideration. The optimization method for orifices' configuration is fully compatible with other possible geometries (conic or curvature shape) of the distributor/collector.

Optimization principles for perforated baffle

The basic idea of the optimization method is to adjust the size of orifices on the perforated baffle so as to approach the target mass flow-rate in every downstream parallel tube. For 3D devices, the baffle is divided into a number of virtual control groups, each control group corresponding to one downstream straight tube, as shown in Fig. 3. In each control group, N orifices are evenly distributed on the baffle. We annotate the mass flow-rate passing through the ijth tube as mij and its target value as m'ij. Based on mass conservation, we obtain:

15 15 1 1 C C ij ij i A j i A j m m m = = = = ′ = = ∑∑ ∑∑ (1) m m M = ( 2
)
where m is the total mass flow-rate of heat transfer fluid, m the mean mass flow-rate among all parallel tubes, and M is the total amount of parallel tubes (M=45 in this study).

The fluid mass flow-rate in every orifice and in every straight tube may be obtained by running the CFD simulation of the entire solar receiver system equipped with fluid distributor and collector. Then the diameter of orifices in ijth control group d ij will be varied simultaneously by comparing m ij and m' ij in the corresponding ijth tube. More precisely, if m ij is higher than its target m' ij , the sizes of the orifices d ij become smaller to reduce the flow-rate passing through the ijth tube, and vice versa. According to the repetitive nature of the variation process, an automatic iteration program is developed to realize the evolutionary procedure until the objective is achieved. The implementation of the algorithm is described in detail in Wei et al. (2015b).

In the present study, the increment or decrement of the orifice's diameter dij from t to t+1 iteration is determined by a variation rule presented in Eq. ( 3) which is a specific form of Ref. [START_REF] Luo | Heuristic shape optimization of baffled fluid distributor for uniform flow distribution[END_REF], based on the difference between mij and its target value m'ij:

, , 2 , 1 , ij t ij t ij t ij t m m d d m γ + ′ -   = +     (3)
γ is a relaxation factor always greater than zero introduced to control the speed of variation. Following this variation rule, the global porosity of the baffle Φ defined as follows is kept constant during the iterations:

15 2 1 4 C ij i A j N d S π = = Φ = ∑∑ (4)
where S is the surface area of baffle.

To quantify the degree of closeness between the target distribution curve and the final flow distribution achieved by optimization, the parameter maldistribution factor (MF) is defined as follows:

2 15 1 1 MF 1 C ij ij i A j m m M m = = ′ -   =   -   ∑∑ (5)
Note that Eq. ( 5) could be considered as a generalized form of "maldistribution factor" initially used in the literature (e.g., Fan et al., 2008;[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] to quantify the flow non-uniformity. The main difference lies in that m is replaced by ij m′ in the numerator of the summation items in Eq. ( 5), to evaluate the deviation between the current mass flow-rate and its target value in each tube. The target flow distribution is considered to be completely reached when the values of MF approach 0. In the current study, the optimization process will be terminated when the value of MF becomes smaller than 0.1.

Numerical parameters for CFD simulation

In this study, simulations were performed under steady state and the gravity effect was considered. Note that half of the real object was actually simulated for the purpose of lessening the computational burden. Structured meshes were generated for the fluid zone using software ICEM (version 12.1) whereas unstructured meshes were constructed for the solid zone. The implemental mesh involved about 4 255 980 cells and was applicable to our simulations after a grid independence test.

The working fluid used in this study was pressurized air at 15 bar, with constant total inlet mass flow-rate of 135 g s -1 . Two materials (copper and Inconel 600) were used as the filling materials for comparison. The temperature dependent physical properties of fluid and solid under the tested conditions may be found in Wei et al. (2015a).

Coupled fluid flow and heat transfer were simulated using a commercial code FLUENT (version 12.1.4), by solving Navier-Stokes equations and heat transfer equations. The fluid flow was calculated by the COUPLED method for pressure-velocity coupling, and a second order upwind differential scheme was applied for the spatial discretization. For the fluid zone, the inlet boundary condition was set as mass flow-rate of air at 300 K normal to the boundary of the distributor's inlet port whereas the pressure outlet boundary was set for the outlet port of collector with the gauge pressure value being zero. Standard k-ε model was used to predict turbulent flow. For the solid zone, a Gaussian-shape net heat flux was defined for the heated surface of the solar receiver. Details for this net absorbed heat flux distribution may be found in Fig. 3 of Ref. (Wei et al., 2015a), with the maximum local heat flux being 650 kW m -2 and the average heat flux of the whole surface being 284.6 kW m -2 . All the tube walls were defined as non-slip conditions with negligible thermal contact resistances. Other walls of the system were considered as adiabatic with negligible heat loss to the environment.

At each optimization step, MATLAB was used to take the computed flow field results from FLUENT, to perform calculations of the coordinates of simulation domain and to pass the data required to ICEM to regenerate the mesh for the updated geometry.

Initial baffle design and target curves

The initial orifices' configuration of the baffle subjected to optimization is shown in Fig. 4. 45 orifices (N=1) with identical initial diameter of 4 mm are evenly distributed on the baffle surface, the global porosity thus being 5.71%. The distance between the perforated baffle and the inlets of parallel tubes is 2 mm. The target curves to be realized by optimizing the baffled fluid distributor have already been determined in Wei et al. (2015a), with the purpose of minimizing the peak temperature (Tpeak) of the receiver wall. Here two target distributions with Inconel 600 or copper used as the filling material are selected as examples. Due to different thermal conductivities of the filling material, the shapes of these two target distributions are differed from each other, as shown in Figs. 6 and9 of the present article.

Results and discussion

Copper used as the filling material

Using the proposed optimization method, the optimized configuration of perforated baffle can be obtained, as shown in Fig. 5. The orifices located in the peripheral areas of the baffle are small, corresponding to low flow-rates in downstream tubes. Although the target flow-rate in the center is high, the size of corresponding orifice should be reduced because it faces directly to the inlet port of the distributor. After 4 steps of optimization process, the MF value decreases from 0.579 (without baffle) to 0.095. The variation of MF value along the optimization steps is indicated in Table 1. The final flow distribution by insertion of the optimized baffle is in good agreement with the target curve, as shown in Fig. 6. Figure 7 shows the fluid flow distribution and the corresponding temperature field on the heated surface with insertion of the optimized baffle (Fig. 7b) compared to those without any baffle insertion (Fig. 7a). When no baffle is used, the highest mass flow-rates are located in central tubes (A8, B8, C8, etc.), the corresponding peak temperature (Tpeak) of the heated surface being 753.2 K and a temperature difference of 324.8 K between the maximum and the minimum. Obviously, the temperature field is still Gaussian-like due to the Gaussian-shape net heat flux absorbed by the receiver. By providing the optimal fluid flow distribution among parallel tubes using the optimized baffle, the isothermal lines on the heated surface are generally perpendicular to the direction of fluid flow, implying a minimized Tpeak value of 747.8 K. The max-min temperature difference is also reduced to 317.5 K. In fact, the target (optimal) flow distribution was determined by the optimality criterion of identical temperatures on the centerline perpendicular to the fluid flow direction. Note that the optimized temperature distribution on the heated surface and the minimized Tpeak value are very close to those presented in Fig. 8 of Ref. (Wei et al., 2015a), indicating the effectiveness of the optimized baffle insertion method for realizing the target flow distribution.

Meanwhile, the total pressure drop (∆ptotal) of the receiver system increases by 21.7% from 17442 Pa (without baffle) to 21230 Pa (with optimized baffle). In order to modulate the initial fluid flow distribution (Fig. 7a) to its optimal shape (Fig. 7b), extra mechanical power should be consumed due to the additional hydraulic resistance of the baffle. Nevertheless, its increase due to the baffle insertion is relatively small (<0.04 bar). The pressure drop values at different parts of the solar receiver are presented in Fig. 8. Note that the subscripts dis, tubes and col represent the distributor part, the tubes-in-matrix part and the collector part, respectively. It can be observed that the pressure drop in the distributor part (∆pdis) increases from 46 Pa to 5865 Pa due to the baffle insertion. The ratio of ∆pdis with respect to ∆ptotal augments from 0.26% to 27.5%. Meanwhile, ∆ptubes (average value among parallel tubes) decreases from 1982 Pa to 310 Pa because of relatively more uniform flow distribution among parallel tubes. Another observation is that the pressure drop in the collector (∆pcol) seems to be the major contribution to ∆ptotal for both cases with or without baffle insertion. In fact, the specific volume of pressurized air increases when it is heated up inside the parallel tubes. The volume flow-rate (thus the velocity) of air in the collector part is much higher than that in the distributor part, resulting in much higher value of ∆pcol. This also implies that the simple rectangular-shape collector used in this study is clearly not an optimal design and can be further improved. Nevertheless, the optimization method is fully compatible with other possible shapes of the fluid collector. Different configurations of the optimized baffle can be obtained regarding different geometries of collector, while the target fluid flow distribution can always be reached. It should be noted that due to relatively high thermal conductivity of copper (about 380 W m -1 K -1 ), the reduction of Tpeak by optimizing the fluid flow distribution is not significant (from 753.2 K to 747.8 K). To further verify the optimization method, another case with Inconel 600 used as the filling material (thermal conductivity about 20 W m -1 K -1 ) is tested and will be presented in the following sub-section.

Inconel 600 used as the filling material

As shown in Fig. 9, the target curve for this case is much less regular: almost all fluids should be allocated in layer A which is the closest to the heated surface. This implies a large flow gradient between the layer A and the layer B, as well as a number of empty tubes (m'ij=0) in layers B and C. These features of the target curve set higher requirements for the optimized baffle so that the results obtained by performing the standard optimization procedure (such as for the copper case) may be less satisfactory. Therefore, some special treatments should be added as useful complementary to the proposed optimization method.

For the actual optimization of the perforated baffle, the tubes with zero flow-rate are firstly blocked in the CFD simulation and the corresponding orifices are removed. Then, a partition wall is added in the middle of layer A and layer B, in order to prevent the transversal flow flux. The height of partition wall is 2 mm, corresponding to the distance between the perforated baffle and the inlets of parallel tubes, as shown in Fig. 10b.

By performing the optimization method, the target flow distribution (shown in Fig. 9) could be reached, indicated by the reduced MF value from 1.565 (without baffle) to 0.098 in 5 steps (Table 1). Fig. 10a shows the optimized configuration of the perforated baffle. It may be observed that the orifices located in layer A are significantly larger than those in layer B, corresponding to higher flow-rates of the target curve on layer A. For the filling material of Inconel 600 with relatively low thermal conductivity (about 20 W m -1 K -1 ), the shape of fluid flow distribution among parallel tubes could have a great influence on the temperature distribution (also Tpeak) on the heated surface of the solar receiver. The temperature field on the heated surface presents an obvious Gaussian-shape without baffle insertion, while it is much more uniform after the insertion of the optimized baffle. Comparing to the case of distributor without baffle, the Tpeak value is reduced from 1322.8 K to 1086.4 K by using the optimized baffled distributor, as shown in Fig. 11. The reduction on Tpeak (about 240 K) is rather significant by providing the optimal flow distribution among parallel tubes. Meanwhile, the temperature difference between the maximum and the minimum is also reduced from 950.8 K to 725.3 K. For the case of empty distributor without baffle insertion, the total pressure drop ∆ptotal of the system using Inconel 600 as the filling material reaches 17440 Pa, very close to that using copper (17442 Pa). However, the ∆ptotal increases from 17440 Pa to 41007 Pa due to insertion of the optimized baffle. This moderate increase of pressure drop (about 0.24 bar) is mainly due to the highly irregular character of the target distribution curve. As a result, the benefit of Tpeak reduction is at the cost of higher total pressure drop of the system.

The pressure drops at different parts of the receiver system are indicated in Fig. 12. It may be clearly observed that the increase of pressure drop in the distributor part (∆pdis from 60 Pa to 24417 Pa) is the main reason of the increased ∆ptotal, i.e., the ratio of ∆pdis to ∆ptotal augments from only 0.34% to 59.5%. In fact, to realize the highly irregular target curve, a number of orifices on the baffle have to be blocked, leading to a smaller global porosity on the baffle (2.79%). Meanwhile, ∆pcol (about 15 kPa) also contributes a lot to ∆ptotal for both cases with or without baffle insertion, mainly due to the non-optimized design of the fluid collector.

Discussions

The present study concerns the realization of the optimal fluid flow distribution in a tubular solar receiver, while the optimal distributions (target curves) have already been determined in our earlier study (Wei et al., 2015a). The temperature fields on the heated surface and the Tpeak values obtained in this study are generally in good agreement with those in Wei et al. (2015a). Nevertheless, small differences can still be observed between the two studies, e.g., minimized Tpeak value of 747.8 K instead of 762.8 K for the copper case and 1086.4 K instead of 1094.0 K for the Inconel 600 case. This is mainly due to the fact that only the tubes-in-matrix part has been considered in Ref. (Wei et al., 2015a) while the entire solar receiver system equipped with the distributor and the collector is taken into account in the present study. A small amount of the heat absorbed by the heated surface will inevitably be transferred to the distributor and the collector parts, and then to the pressurized air passing through. Hence, slightly smaller minimized Tpeak values are obtained in this study than those in Ref. (Wei et al., 2015a), notwithstanding the identical net heat flux absorbed by the heated surface of the tubes-in-matrix part.

The optimized baffle insertion in the fluid distributor can effectively reduce Tpeak of the solar receiver, but at the cost of higher ∆ptotal of the system. In fact, extra mechanical energy has to be consumed to modulate the initial fluid flow distribution toward the target shape. However, a great proportion of total pressure drop is generated in the rectangular fluid collector whose geometry is not yet optimized. By simply using conic or curvatured shape collector, the total pressure drop of the system could be further reduced. Note that the proposed optimization method is compatible with other shapes of distributor/collector; only the final configuration of the optimized baffle will be varied correspondingly. In summary, realizing a certain target flow distribution with the minimum increase of pressure drop is certainly an interesting topic of practical significance, by introducing multi-criteria optimization methodologies. By examining Figs. 6 and9, it could be seen that the flow distributions provided by optimized baffles are globally in nice agreement with the target curves. Small deviations may be observed for several individual tubes, mainly due to the termination criterion selected in this study (MF<0.1) for the optimization method. When better agreement to the target curve is requested, smaller values of MF could be considered as the termination criterion but more optimization steps (thus longer computational time) will be needed. For some extreme cases when highly irregular target curves are involved, some actions should be taken as useful complementary to the optimization method, including the partition walls or a second header. Moreover, the perforated baffle to be optimized may also be integrated as inlet openings of the parallel tubes.

It should be noted that the proposed optimization method is based on the results of CFD simulation. Therefore, the difficulties occurred in simulation, such as for highly turbulent or complex fluid (e.g. non-Newtonian), will influence inevitably the effectiveness of our method. Selecting appropriate and adaptive CFD codes or models is essential for precise calculation of the flow and temperature fields. Besides common commercial CFD codes, we may turn to some in-house codes or advanced numerical techniques such as the lattice Boltzmann method [START_REF] Wang | Lattice Boltzmann method for shape optimization of fluid distributor[END_REF]. Deep understandings on fundamental physical phenomena and on conjugated heat and mass transfers should be firstly acquired for the implementation of the optimization method.

In this study, some technical problems such as fabrication tolerances, fouling, contact thermal resistances are ignored to reduce the complexity of physical problems. However, the factors having great influences could be considered in the CFD simulation or in the optimization process. If we already know the variation of tube diameter due to fabrication tolerances, we could take the exact dimensions into account at the beginning when creating and meshing the object. As far as the fouling or contact thermal resistances are concerned, boundary conditions for the relative tube might have to be modified accordingly. For the variation in fluid composition, the physical properties of working fluid should be well defined to reflect the reality.

Conclusion and perspectives

In this article, an original study on the design and optimization of baffled fluid distributor is presented for the realization of optimal fluid flow distribution in a CSP tubular receiver. The optimization method proposed adjusts the sizes of orifices on the baffle inserted in the distributing manifold so as to approach the target flow-rate among parallel tubes. Two optimized baffles are then obtained, with copper or Inconel 600 used as the filling material. Based on the results obtained, main conclusions can be summarized as follows:

 The optimization method is effective for reaching the optimal configurations of the perforated baffle. The final fluid distributions provided by the optimized baffles are in good agreement with the target curves, with the values of maldistribution factor (MF) smaller than 0.1.

 The peak temperature on the receiver wall can be minimized accordingly by optimizing the flow distribution. Tpeak on the heated surface can be reduced from 753.2 K to 747.8 K for the copper case, and from 1322.8 to 1086.4 for the Inconel 600 case.

 The total pressure drop of the solar receiver system with insertion of optimized baffle is about 0.21 bar for the copper case, and about 0.41 bar for the Inconel 600 case. The pressure drop increase due to the baffle insertion is relatively moderate (0.04 bar for the copper case; 0.24 bar for the Inconel 600 case).

Our future work will focus on the experimental tests of a real tubular solar receiver which is an essential step for the validation of the optimization method and the numerical results. The tubular solar receiver equipped with optimized baffled flow distributors will be fabricated and tested. The measurement of temperature distribution on the heated surface will indicate the improvement of temperature uniformity and the reduction of peak wall temperature. Also, a series of perforated baffles can be designed under different typical working conditions for replacement. The evaluations on global thermal performance and total pressure drop will illustrate the real benefits of optimized flow distribution.

The present study illustrates that the insertion of a geometrically optimized baffle is generally a practical solution with various advantages: capable of realizing non-uniform target distribution; small pressure drop increase; compact geometry; flexible and adaptive, easy fabrication with a reasonable cost, etc. Its application could be extended to various engineering devices involving fluid maldistribution problem, including heat sinks for cooling of electronic components, fuel cells, reactors with coupled heat and mass transfer, absorption or distillation columns. This is another direction of our future work.

Acknowledgement

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

Figure 1 :

 1 Figure 1: Schematic view of the tubular solar receiver equipped with an inlet fluid distributor and an outlet fluid collector.

Figure 2 :

 2 Figure 2: Geometry and dimensions of the baffled inlet fluid distributor (unit: mm).

Figure 3 :

 3 Figure 3: Division of control groups.

Figure 4 :

 4 Figure 4: Geometry and dimensions of the initial (uniform) perforated baffle for optimization (unit: mm).

Figure 5 :

 5 Figure 5: Optimized configuration of the perforated baffle for solar receiver using copper as the filling material (diameter, unit: mm).

Figure 6 :

 6 Figure 6: Comparison between the final flow distribution with optimized baffle and the target curve (copper as the filling material).

Figure 7 :

 7 Figure 7: Fluid flow distribution among parallel tubes and the corresponding temperature field of the heated surface (copper as the filling material): (a) without baffle insertion; (b) with insertion of optimized baffle.

Figure 8 :

 8 Figure 8: Pressure drop values on different parts of the solar receiver system (copper as the filling material): (a) without baffle insertion; (b) with optimized baffle insertion.

Figure 9 :

 9 Figure 9: Comparison between the final flow distribution with optimized baffle and the target curve (Inconel 600 as the filling material).

Figure 10 :

 10 Figure 10: Optimized baffled fluid distributor for the tubular solar receiver when Inconel 600 is used as the filling material: (a) optimized baffle configuration (diameter, unit: mm); (b) distributor geometry.

Figure 11 :

 11 Figure 11: Fluid flow distribution among parallel tubes and the corresponding temperature field of the heated surface (Inconel 600 as the filling material): (a) without baffle insertion; (b) with insertion of optimized baffle.

Figure 12 :

 12 Figure 12: Pressure drop values on different parts of the solar receiver (Inconel 600 as filling material): (a) without baffle insertion; (b) with optimized baffle insertion.

N

  total amount of orifices in one control group p pressure Pa

Table 1 : MF values in the optimization process.

 1

	Case	No baffle	Initial baffle	Step 1	Step 2	Step 3	Step 4
	Copper	0.579	0.599	0.159	0.145	0.095	-
	Inconel 600	1.565	1.093	0.377	0.247	0.182	0.098