Aijing Lu

Marleny Jielin Wang

Caceres Najarro

Suming M Li
email: suming.li@umontpellier.fr

André Deratani

Andre Deratani

Synthesis and self-assembly of AB2-type amphiphilic copolymers from biobased hydroxypropyl methyl cellulose and poly(L-lactide)

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Micelles based on amphiphilic block copolymers have been extensively investigated as drug carrier due to their favorable advantages in molecular design, long circulation, enhanced drug loading and therapeutic effect, and biocompatibility [START_REF] Kataoka | Block copolymer micelles for drug delivery: design, characterization and biological significance[END_REF]. Thus, biocompatible and bioresorbable aliphatic polyesters such as polylactide (PLA), polyglycolide (PGA), poly(lactide-co-glycolide) (PLGA), and poly(ε-caprolactone) (PCL) have been commonly used as the hydrophobic block, whereas poly(ethylene glycol) (PEG), poly(N-vinyl-2-pyrrolidone) (PVP), poly(N-isopropylacrylamide) (PNIPAAm), poly(vinyl alcohol) (PVA), poly(amino acid), and more recently polysaccharides as the hydrophilic block to construct amphiphilic block copolymers [START_REF] Elsabahy | Design of polymeric nanoparticles for biomedical delivery applications[END_REF][START_REF] Guo | Preparation of cellulose-graft-poly(varepsiloncaprolactone) nanomicelles by homogeneous ROP in ionic liquid[END_REF][START_REF] Mi | Molecular Cancer Imaging with Polymeric Nanoassemblies: From Tumor Detection to Theranostics[END_REF][START_REF] Yi | Polyester micelles for drug delivery and cancer theranostics: Current achievements, progresses and future perspectives[END_REF]. PLA, as an FDAapproved biobased polyester, is widely used in biomedical and pharmaceutical fields in the form of drug carrier, tissue engineering scaffolds, bone fracture internal fixation devices, orthopedic screws and plates, etc, due to its good degradability, processability, and mechanical strength [START_REF] Lassalle | PLA nano-and microparticles for drug delivery: an overview of the methods of preparation[END_REF][START_REF] Rajendra | Biomedical Applications of Poly(Lactic Acid)[END_REF][START_REF] Ramzi | Recent applications of polylactic acid in pharmaceutical and medical industries[END_REF][START_REF] Rancan | Investigation of Polylactic Acid (PLA) Nanoparticles as Drug Delivery Systems for Local Dermatotherapy[END_REF][START_REF] Saini | Poly(lactic acid) blends in biomedical applications[END_REF][START_REF] Tsuji | Poly(lactide) stereocomplexes: formation, structure, properties, degradation, and applications[END_REF]. Interestingly, PLA stereocomplex has been used to construct colloidal systems such as mixed micelles and hydrogels for pharmaceutical applications (L. [START_REF] Yang | Novel biodegradable polylactide/poly(ethylene glycol) micelles prepared by direct dissolution method for controlled delivery of anticancer drugs[END_REF].

PEG is also an FDA-approved polymer widely used for biomedical and pharmaceutical applications. Nevertheless, PEG as a polyether is prone to peroxidation, which could adversely affect cells [START_REF] Barz | Overcoming the PEG-addiction: well-defined alternatives to PEG, from structure-property relationships to better defined therapeutics[END_REF][START_REF] Ishida | Accelerated blood clearance (ABC) phenomenon upon repeated injection of PEGylated liposomes[END_REF].

Therefore, great effort has been made to search for hydrophilic alternatives of PEG.

Polysaccharides have attracted increasing attention in recent years for uses as biomaterials to solve problems related to immunogenicity and toxicity associated with synthetic polymers because of their biodegradability, biocompatibility and bio-based nature [START_REF] Aminabhavi | Guar gum as platform for the oral controlled release of therapeutics[END_REF][START_REF] Ganguly | Polysaccharidebased micro/nanohydrogels for delivering macromolecular therapeutics[END_REF]. Certain polysaccharides present inherent bioactivity that can help in mucoadhesion, thus improving drug targeting and diminishing inflammatory response. Recent years have witnessed a rapid growth on polysaccharide based micelles as drug delivery system [START_REF] Aminabhavi | Guar gum as platform for the oral controlled release of therapeutics[END_REF][START_REF] Rudzinski | Chitosan as a carrier for targeted delivery of small interfering RNA[END_REF], such as pullulan [START_REF] Jeong | Adriamycin release from selfassembling nanospheres of poly(DL-lactide-co-glycolide)-grafted pullulan[END_REF], cellulose [START_REF] Guo | Self-assembly and paclitaxel loading capacity of cellulose-graft-poly(lactide) nanomicelles[END_REF], dextran [START_REF] Jeong | Doxorubicinincorporated polymeric micelles composed of dextran-b-poly(DL-lactide-co-glycolide) copolymer[END_REF][START_REF] Sun | Biodegradable micelles with sheddable poly(ethylene glycol) shells for triggered intracellular release of doxorubicin[END_REF], chitosan [START_REF] Huo | Somatostatin receptormediated tumor-targeting drug delivery using octreotide-PEG-deoxycholic acid conjugatemodified N-deoxycholic acid-O, N-hydroxyethylation chitosan micelles[END_REF], heparin [START_REF] Tian | Synthesis and characterization of Poloxamer 188-grafted heparin copolymer[END_REF], and hyaluronan (Y. L. [START_REF] Yang | Synthesis of diblock copolymers consisting of hyaluronan and poly(2-ethyl-2-oxazoline)[END_REF]. Among them, cellulose is of great importance because of its outstanding properties, but its applications are restrained due to its poor solubility. Thus, soluble cellulose derivatives have been studied as a hydrophilic building block to prepare amphiphilic copolymers which can self-assemble in micelles, such as ethyl cellulose, hydroxyl propyl cellulose [START_REF] Ostmark | Unimolecular nanocontainers prepared by ROP and subsequent ATRP from hydroxypropylcellulose[END_REF], hydroxyl propyl methyl cellulose (HPMC) [START_REF] Ostmark | Unimolecular nanocontainers prepared by ROP and subsequent ATRP from hydroxypropylcellulose[END_REF], hydroxyl ethyl cellulose [START_REF] Hsieh | Nano-Sized Micelles of Block Copolymers of Methoxy Poly(ethylene glycol)-Poly(ε-caprolactone)-Graft-2-Hydroxyethyl Cellulose for Doxorubicin Delivery[END_REF] etc. Graft copolymerization of PLA onto cellulose and cellulose derivatives have been reported [START_REF] Chen | New tissue engineering material copolymers of derivatives of cellulose and lactide: their synthesis and characterization[END_REF][START_REF] Teramoto | Cellulose diacetate-graft-poly(lactic acid)s: synthesis of wide-ranging compositions and their thermal and mechanical properties[END_REF][START_REF] Yuan | Syntheses, characterization, and in vitro degradation of ethyl cellulose-graft-poly(epsilon-caprolactone)-block-poly(L-lactide) copolymers by sequential ringopening polymerization[END_REF]. However, the grafting of PLA on the side chain hydroxyl group results in complex chain structure and the reaction is poorly controllable.

In our previous work [START_REF] Wang | Synthesis and Self-Assembly of Amphiphilic Block Copolymers from Biobased Hydroxypropyl Methyl Cellulose and Poly(l -lactide)[END_REF], linear HPMC-b-PLA diblock copolymers with various HPMC block lengths were synthesized by combination of ring-opening polymerization, amination reduction and thiol-click reaction. The resulted amphiphilic copolymers are susceptible to self-assemble into spherical micelles with a diameter of 60-120 nm. It has been shown that the geometrical structure of copolymers affect the morphology, stability and size of micelles because the hydrophobic and hydrophilic segments lead to complex spatial arrangements during the micellization process [START_REF] Cao | Effect of architecture on the micellar properties of poly (varepsilon-caprolactone) containing sulfobetaines[END_REF][START_REF] Li | Multicompartment micelles from ABC miktoarm stars in water[END_REF]. In this work, amphiphilic AB2-type PLA-(HPMC)2 copolymers with various HPMC molar masses were synthesized and characterized. The self-assembly properties of the copolymers were investigated from the morphology, size and size distribution, cloud point (Cp) and critical micelle concentration (CMC) measurements.

Materials and Methods

Materials

L-lactide was purchased from Purac Biochem (Goerinchem, Netherlands) and purified by recrystallization from ethyl acetate. HPMC (K4M) was kindly provided by Colorcon-Dow chemical (Bougival, France). Propynol, stannous octoate (Sn(Oct)2), cysteamine, sodium cyanoborohydride (NaBH3CN), 1,4-dithiothreito (DTT), 2,2-dimethoxy-2phenylacetophenone (DMPA), diethyl ether and dimethyl sulfoxide were purchased from Sigma-Aldrich, and used as received. Toluene was purchased from Sigma-Aldrich, dried over by calcium chloride (CaCl2) and distilled prior to use.

Synthesis of alkynyl terminated PLLA

Alkynyl terminated PLLA was synthesis by ring-open polymerization, of L-lactide using propynol as initiator and Sn(Oct)2 as catalyst. Typically, L-lactide (7.2 g), propynol (0.112 g),

and Sn(Oct)2 (0.2 g) were introduced into a dried Schlenk tube, and 30 mL anhydrous toluene was added under stirring. The solution was degassed by performing five freeze-pump-thaw cycles. The reaction then proceeded 24 h at 80℃. Finally the product was obtained by precipitation in methanol, followed by vacuum drying.

2.3 Enzymatic degradation of HPMC and determination of substitution degree HPMC was depolymerized to yield HPMC oligomers as reported in a previous work [START_REF] Wang | Synthesis and Self-Assembly of Amphiphilic Block Copolymers from Biobased Hydroxypropyl Methyl Cellulose and Poly(l -lactide)[END_REF]. Briefly, 1 g HPMC was added in 100 mL citrate phosphate buffer (pH = 5), and stirred overnight at 47 °C to ensure complete dissolution. Cellulase was added to the solution, and enzymatic depolymerization proceeded for various time periods. The reaction was stopped by heating the solution to 85 °C, and the suspension was hot filtrated. The collected solution was purified by dialysis for 3 days using a membrane of MWCO=3500 g mol -1 , followed by freeze drying.

HPMC is a cellulose derivative whose hydroxyl groups are substituted by methyl and hydroxypropyl groups. The average number of substituents per repeat unit is characterized by the molar substitution or hydroxypropylaction (MSHP) and the degree of substitution or methylation (DSMe). The former can be higher than 3 and the latter varies between 0 and 3.

HPMC with a DSMe between 0.1 and 2.0 is water soluble as the substitution groups allow to decrease the intra-and inter-chain hydrogen bonding. In contrast, when the DS is above 2, HPMC becomes insoluble in water due to the presence of many hydrophobic methyl groups. [START_REF] Claes | New Methods for Enzyme Hydrolysis,Analysis, and Characterization of Modified Cellulose[END_REF]. The 1 H NMR spectra of acetylated HPMC oligomers were acquired in CDCl3 at 50 °C. The DSMe and MSHP values were then calculated according to the method reported by Fitzpatrik [START_REF] Fitzpatrick | NMR, Cloud-Point Measurements and Enzymatic Depolymerization: Complementary Tools to Investigate Substituent Patterns in Modified Celluloses[END_REF].

Synthesis of thiol terminated HPMC

Thiol terminated HPMC was obtained by reductive amination connecting the aldehyde end group of HPMC molecules and the amine group of cysteamine in the presence of sodium cyanoborohydride. Briefly, HPMC with DSMe of 1.43 and average molar mass of 7000 g mol -1 (1 g) and NaBH3CN (0.2 g) were solubilized in a mixture (3/1 v/v) of 50 mL dimethyl sulfoxide (DMSO) and 0.01 M NaCl at 60 °C. Cysteamine (0.1 g) was added, and the solution was stirred at 60 °C for 6 days under reflux. The solution was then dialyzed against deionized water for 3 days (MWCO 3500 g mol -1 ), followed by freeze drying. Afterwards, the product was dissolved in water, and stirred overnight in the presence of excess DTT under nitrogen atmosphere. Finally, the solution was dialyzed against deionized water and freeze dried.

Ellman assay was performed to determine the content of thiol group using 5,5'-dithio-bis-(2nitrobenzoic acid) (DTNB) as Ellman's reagent. DTNB is a versatile water soluble compound for quantitating free sulfhydryl groups as a measurable yellow-colored product is formed when DTNB reacts with sulfhydryl in solution.

2.5 Synthesis of (HPMC)2-b-PLLA Amphiphilic block copolymers were synthesized by thiol-yne click reaction connecting thiol terminated HPMC and alkynyl terminated PLLA. The reaction was initiated in a Dinics M3 UV chamber equipped with a PL-L 36 W/01/4P Hg Lamp. Typically, thiol terminated HPMC (1.0 g), alkynyl terminated PLLA (0.15 g), and DMPA (0.013 g) were dissolved in 10 mL DMSO. The solution was degassed for 30 min using purified argon, and then subjected to UV irradiation for 5 h. The crude product was collected by precipitation in acetonitrile, followed by centrifugation at 4500 rpm for 15 min. The above dissolution-precipitation cycle was repeated twice. Finally, (HPMC)2-b-PLLA copolymers were obtained as a white solid after vacuum drying.

Characterization

1 H NMR spectra were recorded on Bruker spectrometer operating at 300 MHz (AMX300) using CDCl3 and DMSO-d6 as solvent. Diffusion ordered spectroscopy (DOSY)

NMR was performed on Bruker Avance (AQS600) NMR spectrometer, operating at 600 MHz and equipped with a Bruker multinuclear z-gradient inverse probe head which is able to produce gradients in the z direction with the strength of 55 Gcm -1 . The DOSY spectra were acquired from Bruker topspin software (version 2.1) with the ledbpgp2s pulse program. The strength of the pulsed field gradients with respect to maximum 32 increments on a quadratic scale was logarithmically incremented from 2 to 95%. The diffusion sensitive period (Δ) of 200 ms and the gradient duration (δ) of 5 ms were optimized to allow the signals of interest to decrease by a factor of 10-20, in order to keep the relaxation contribution to the signal attenuation constant and ensure full signal attenuation for all samples. Data were processed using the MestRe Nova software.

Size exclusion chromatography (SEC), equipped with MALLS in combination with refractive index (RI) detection allows to determine the absolute molar mass of polymers. 5 mg of samples were added in the mobile phase (0.01 M NaCl containing 0.02% NaN3), and stirred for 24 h at room temperature. The solution was filtered using 0.45 µm polytetrafluoroethylene (PTFE) filter (Millipore). The filtrate was injected through a 100 µL loop (Rheodyne injector 7725), and eluted on a TSK-GEL GMPWXL 7.8×300 mm column (TosoHaas Bioseparation Specialists, Stuttgart, Germany) at a flow rate of 0.5 mL min -1 (Waters pump 515). MALS detector (Dawn DSP, Wyatt Technology Co, Santa Barbara, CA, USA) and RI detector (Optilab Wyatt Technology Co) were used to online determine the absolute molar mass for each elution fraction of 0.01 mL, which enables to calculate the weight average molar mass (Mw) and the dispersity (Ð=Mw/Mn). The refractive index increment (dn/dc) for calculation was 0.137 mL g -1 . Data analysis was realized using Astra 4

(Wyatt Technology Co).

Fourier transform infrared (FT-IR) spectra were recorded on a Nicolet NEXUS spectrometer with a DTGS detector at 4 cm -1 resolution. 64 scans were taken per sample in the frequency range from 400 to 4000 cm -1 . Sample pellets were prepared by mixing 2-2.5 mg of polymer with 200 mg of spectral grade potassium bromide, followed by compression using a press.

The cloud point (Cp) of the copolymers was estimated from transmittance changes of copolymer solutions in the temperature range from 30 to 90℃. The solution at 3.0 mg mL -1 was stirred for 24 h and kept at 5 °C overnight before analysis. Measurements were made at a wavelength of 600 nm with a Perkin Elmer Lambda 35 UV-visible spectrometer equipped with a Peltier temperature programmer PTP-1+1. The temperature ramp was 0.1 °C min -1 .

Temp Lab software was used for data treatment.

Dynamic light scattering (DLS) was performed at 25°C with 90° scattering angle using Zano-ZS (Malvern Instrument) equipped with a He-Ne laser (λ =632.8 nm). The aqueous solutions of copolymers at 1.0 mg mL -1 were filtered through a 0.45 µm PTFE microfilter before measurements. The correlation of the distribution of diffusion coefficient (D) was analyzed via the general purpose method (non-negative least squares). The apparent equivalent hydrodynamic radius (RH) was obtained using the cumulate method from the Stoke-Einstein equation.

The critical micelle concentration (CMC) of copolymers was determined using an autocorrelation function according to the methodology proposed by Muller et al [START_REF] Muller | Self-organization in water of well-defined amphiphilic poly(vinyl acetate)-b-poly(vinyl alcohol) diblock copolymers[END_REF]. Aqueous solutions of copolymers were prepared in deionized water at concentrations ranging from 0.006 to 1.0 g L -1 , and were incubated overnight. The scattering intensity of the solutions was measured at 25 °C with a Malvern Instrument Nano-ZS. The scattering intensity proportional to the size and number of objects present in the solution was recorded. The CMC was taken as the intersection of regression lines from the plots of the scattering intensity against the polymer concentration.

Transmission electron microscopy (TEM) was performed on JEOL 1200 EXII instrument, operating at an acceleration voltage of 120 kV. 5 µL of micellar solution at 1.0 mg mL -1 were dropped onto a carbon coated copper grid, and air dried before measurements.

Results and Discuss

3.1 Synthesis of amphiphilic block copolymers HPMC was depolymerized by using cellulase to yield oligomers with various molar masses. The weight average molar mass (Mw) of the resulted HPMC oligomers was 5000, 7000 and 10000 Daltons, respectively, as determined by SEC. The NMR spectra of acetylated HPMC were recorded in CDCl3 (Figure S1). The DSMe and MSHP were calculated according to the method reported by Fitzpatrik (Table S1). The DSMe varies from 1.17 Thiol terminated HPMC was obtained by reductive amination of HPMC with cysteamine. The molecular chain of cellulose has a non reducing group at one end, and a hemiacetal group at the other end. The latter is a reducing group which can be easily converted to aldehyde group [START_REF] Schatz | Polysaccharide-Containing Block Copolymers: Synthesis, Properties and Applications of an Emerging Family of Glycoconjugates[END_REF]. The amine functional group of cysteamine reacted with the unique aldehyde endgroup of HPMC through reductive amination to yield HPMC-S-S-HPMC together with small amounts of unreacted HPMC, HPMC-SH, and HPMC-S-S-(CH2)6-NH2. HPMC-S-S-HPMC was then reduced using DTT to yield HPMC-SH [START_REF] Wang | Synthesis and Self-Assembly of Amphiphilic Block Copolymers from Biobased Hydroxypropyl Methyl Cellulose and Poly(l -lactide)[END_REF]. The final product was characterized by using Ellman assay and SEC measurement. Ellman's assay showed that the content of thiol terminated HPMC in the final product was 88%. SEC confirmed that the final product is predominantly composed of HPMC-SH because its Mw is almost the same as that of the initial HPMC. These findings showed that thiol terminated HPMC was successful synthesized.

Finally amphiphilic block copolymers were obtained by UV-initiated thiol-yne click reaction between thiol terminated HPMC and alkynyl terminated PLLA. Three copolymers, namely (HPMC5K)2-PLA2K, (HPMC7K)2-PLA2K and (HPMC10K)2-PLA2K were synthesized using the same hydrophobic PLA block with Mn of 2000 g mol -1 and different hydrophilic HPMC-SH blocks with Mw of 5000, 7000, and 10000 g mol -1 , respectively.

Figure 2 shows the 1 H NMR spectra of (HPMC7K)2-PLA2K copolymer in D2O and DMSO-d6. The spectrum in DMSO-d6 shows the signal at 1.05 ppm (a) corresponding to the methyl protons in the hydroxypropyl group, and those in the range of 2.75-4.75 ppm (b) to the methyl protons adjacent to the oxygen moieties of the ether linkages, inner methylene and methine protons and HPMC backbone protons (Figure 2A). The signals at 1.49 (d) and 5.24 ppm (c) belong to the methyl and methine protons of PLLA blocks. Moreover, the signal detected at 2.3 ppm is assigned to the methylene group of the connecting unit. These results confirm the successful synthesis of copolymers by thiol-yne click reaction. The molar mass of the copolymers was determined by SEC in conjunction with online light scattering and refractive index (RI) detectors, as shown in Figure 5. RI detector is most commonly used as concentration detector whose response is proportional to the total solute concentration in the detector cell. Meanwhile, the response of multi-angle laser light scattering (MALLS) detector depends on the molar mass of a polymer in the detector cell [START_REF] Kostanski | Size-exclusion chromatography-a review of calibration methodologies[END_REF]. Combination of the two curves allows to determine the weight average molar mass of copolymers. The Mw and dispersity data of the three copolymers are summarized in Table 1. The Mw ranges from 12000 for (HPMC5K) 2 -PLA2K to 22 0000 g mol -1 for (HPMC10K) 2 -PLA2K. It is noted that the Mw of the copolymers well agrees with the sum of the molar masses of both components although the value of Mn is used for PLA. Meanwhile, the dispersity (Ð=Mw/Mn) ranges from 1.13 to 1.26, in agreement with narrow molar mass distribution of copolymers.

Table 1. Characterization of (HPMC)2-PLA copolymers Determined from SEC/MALS/RI measurement in water at 5.0mg mL -1 b)

Sample M w, SEC a) (g/mol) Ð a) Cp b) (°C) CMC c) (mg/mL) R h d) (nm) PDI d) (HPMC10K) 2 -PLA2K 22000
Determined from turbidimetry measurement in water at 20.0 mg mL -1 c)

Determined at 25°C by dynamic light scattering d)

Determined at 25°C by dynamic light scattering at 1.0 mg mL -1 3.2. Cloud point of (HPMC)2-PLA copolymers

The cloud point (Cp), also named lower critical solution temperature (LCST), is considered as the solubility limit of amphiphiles due to phase-separation as the temperature increases. In fact, the hydrogen bonding between polymer chains and surrounding water molecules weakens when the temperature approaches the cloud point, leading to decrease of the polymer solubility and phase separation. Thus the polymer precipitates out of solution as a consequence of equal chemical potentials between the two phases: one is rich in polymer, and the other rich in solvent [START_REF] Sardar | Solution Behavior of Nonionic Polymer Hydroxypropylmethyl Cellulose: Effect of Salts on the Energetics at the Cloud Point[END_REF].

Figure 6 shows the transmittance changes of (HPMC7K)2-PLA2K solutions at 3.0 mg/mL, in comparison with HPMC7K homopolymer. The two solutions initially exhibit a transmittance close to 100%. With increasing temperature, the transmittance of HPMC7K remained almost unchanged till 90°C. This indicates that HPMC oligomers are not thermosresponsive in this temperature range, in contrast to high molar mass HPMC [START_REF] Sardar | Solution Behavior of Nonionic Polymer Hydroxypropylmethyl Cellulose: Effect of Salts on the Energetics at the Cloud Point[END_REF]. Interestingly, (HPMC7K)2-PLA2K exhibits a decrease of light transmittance to nearly 20% from about 60°C up to 90 °C. The clouding at a given temperature results from the destruction of hydrogen bonding between water and HPMC molecules leading to phase separation [START_REF] Khan | Cloud point, fluorimetric and 1H NMR studies of ibuprofen-polymer systems[END_REF]. The Cp value was determined by extrapolation of the linear region of 100% of transmittance and tangent line of the inflexion point. A Cp value of 64.8°C is obtained for (HPMC7K)2-PLA2K. Similarly, Cp values of 65.2 and 64.1°C are obtained for (HPMC10K)2-PLA2K and (HPMC5K)2-PLA2K (Table 1).

Therefore, the length of HPMC blocks has little effect on the Cp value of the copolymers.

Similar results have been previously reported in literature for poly(lactide-b-Nisopropylacrylamide-b-lactide) (PLA-b-PNIPAAmb-PLA) triblock copolymers [START_REF] You | Preparation and characterization of thermally responsive and biodegradable block copolymer comprised of PNIPAAM and PLA by combination of ROP and RAFT methods[END_REF]. In fact, a Cp value of about 31°C was obtained for copolymers with PLA/PNIPAAm ratios of 1.0/1.9, 1.0/3.0 and 1.0/3.6. In our previous work, the phase separation behavior of linear HPMC-b-PLA diblock copolymers with different HPMC block lengths (HPMC5K-PLA2K, HPMC8K-PLA2K, and HPMC10K-PLA2K) was investigated [START_REF] Wang | Synthesis and Self-Assembly of Amphiphilic Block Copolymers from Biobased Hydroxypropyl Methyl Cellulose and Poly(l -lactide)[END_REF]. All copolymers exhibit a slow decrease of light transmittance from 55 to 80 °C. Beyond, a sharper decrease of transmittance down to 78% is detected. The three diblock copolymers exhibit almost the same Cp values of c.a. 80°C. Therefore, the length of HPMC blocks has little effect on the Cp, but the copolymer topology seems to significantly affect the phase separation behavior of HPMC-b-PLA copolymers. It has been reported that the cloud point was elevated by more than 40 °C through the linear-to-cyclic topological conversion of the polymer amphiphiles [START_REF] Honda | Topology-directed control on thermal stability: micelles formed from linear and cyclized amphiphilic block copolymers[END_REF].

3.3. Self-assembly of (HPMC)2-PLA copolymers

The self-assembly properties of (HPMC)2-PLA copolymers were investigated to evaluate their potential as drug carrier. NMR provides a means to analyze the self-assembly behavior of amphiphilic block copolymers in solvents which can dissolve one of the two blocks only. Similar to low molar mass surfactants, the self-assembly of the amphiphilic copolymers occurs when the concentration reaches the CMC. The latter is an important parameter of amphiphilic copolymers as it determines the stability of micelles. In the case of drug delivery systems, drug loaded micelles should remain stable with dilution after intravenous injection.

In this work, the CMC was determined from scattering intensity changes of copolymer solutions as a function of concentration. At low concentrations, the scattered intensity is very low because of the absence of nano-objects. With increasing concentration, the intensity strongly increases due to the formation of nano-micelles. The CMC value is obtained from the crossover point of the two regression lines. As shown in Table 1, the CMC of (HPMC10K)2-PLA2K, (HPMC7K)2-PLA2K, and (HPMC5K)2-PLA2K copolymers is 0.16, 0.15, and 0.14 g L -1 , respectively. These values well agree with the reported CMC value 0.13 mg/mL for cellulose-g-polylactide [START_REF] Guo | Self-assembly and paclitaxel loading capacity of cellulose-graft-poly(lactide) nanomicelles[END_REF]. Obviously the CMC increases with the hydrophilic ratio or the HPMC block length as the self-assembly of neutral block copolymers is mainly determined by the hydrophilic/hydrophobic balance [START_REF] Larue | Effect of the soluble block size on spherical diblock copolymer micelles[END_REF].

The self-assembled (HPMC)2-PLA micelles are composed of a PLA core surrounded by a HPMC corona. The average diameter of (HPMC5K)2-PLA2K, (HPMC7K)2-PLA2K, (HPMC10K)2-PLA2K micelles was 66, 90 and 120 as determined by DLS, as shown in Table 1. All the micelles exhibited a unimodal size distribution, and the PDI was in the range of 0.12 to 0.21. Thus, the micelle size of copolymers increases with the increase of hydrophilic HPMC chain length, in agreement with literature. In our previous work, the average micelle size of HPMC5K-PLA2K, HPMC8K-PLA2K, HPMC10K-PLA2K diblock copolymers was 62, 96, 120 nm, respectively [START_REF] Wang | Synthesis and Self-Assembly of Amphiphilic Block Copolymers from Biobased Hydroxypropyl Methyl Cellulose and Poly(l -lactide)[END_REF]. Thus, micelles of (HPMC5K)2-PLA2K had a smaller size compared to HPMC10K-PLA2K with the same hydrophilic/hydrophobic balance but a Y-type topology. It has been reported that the chain architecture affects the micellization properties, including the aggregation number, size, polydispersity, and micelle density [START_REF] Liu | Syntheses and micellar properties of welldefined amphiphilic AB2 and A2B Y-shaped miktoarm star copolymers of ɛ-caprolactone and 2-(dimethylamino)ethyl methacrylate[END_REF]. Generally, the micelle size of multiple-arm copolymers is smaller than that of linear ones. Meanwhile, micelles with the same HPMC block length but different topologies had almost the same size. This finding could be attributed to the fact that the core is composed of the same PLA blocks, and the corona of similar HPMC chains although the chain density should be higher for Y-type copolymers. The morphology of (HPMC)2-PLA micelles was characterized by using TEM. Figure 7 shows that the micelles were spherical in shape and uniformly distributed. The observed micelle size increased with the increasing of the HPMC chain length, which well agreed with the results obtained by DLS.

  Scheme 1. Synthesis route of (HPMC)2-b-PLA block copolymers

Figure 1 .

 1 Figure 1. NMR spectrum of alkynyl-terminated PLA in CDCl3

Figure 2

 2 Figure 2. 1 H NMR spectra of (HPMC)2-PLA in DMSO-d6 (A) and in D2O (B)

Figure 3 .

 3 Figure 3. DOSY NMR spectrum of (HPMC7K)2-PLA2K copolymer in DMSO-d6 at 298 K

Figure 4 .

 4 Figure 4. FT-IR spectra of (a) alkynyl terminated PLLA, (b) thiol terminated HPMC, and (c)

Figure 5 .

 5 Figure 5. Refractive index and light scattering curves of (HPMC7K)2-PLA2K (left), and SEC

Figure 6 .

 6 Figure 6. Transmittance changes of HPMC7K (a) and (HPMC7K)2-PLA2K (b) solutions at

Figure 2 (

 2 Figure 2(B) shows the 1 H NMR spectrum of (HPMC7K)2-PLA2K copolymer in deuterated

Figure 7 .

 7 Figure 7. TEM images of (HPMC)2-PLA micelles in water: (A) (HPMC5K)2-PLA2K; (B)

  amphiphilic block copolymers (HPMC)2-PLA with three different HPMC block lengths were synthesized by ring opening polymerization, reductive amination and click reaction. The copolymers were characterized by using various methods, including NMR, DOSY-NMR, SEC and FT-IR. The cloud point of AB2-type (HPMC)2-PLA are nearly 64 ℃, which is obviously lower than HPMC oligomers and linear HPMC-b-PLA diblock copolymers. The HPMC block length has little effect the Cp value. (HPMC)2-PLA copolymers can self-assemble to form spherical micelles with narrow distribution. The size of copolymer micelles increases with the increase of HPMC fraction. So does the CMC of (HPMC)2-PLA. It is thus concluded that biobased and biodegradable HPMC-PLA copolymers could be promising as nano-carrier of hydrophobic drugs. Keywords: Hydroxypropyl methyl cellulose; Poly(L-lactide); Topology; Self-assembly; Micelle

Acknowledgements

This work is supported in part by the scholarship from China Scholarship Council (CSC) under the Grant CSC N°201606240124.


Amine Laaouatni

Nadia Martaj

Rachid Bennacer
email: rachid.bennacer@ens-cachan.fr

Mohammed Lachi

Mohamed El Omari

Mohammed El Ganaoui

Thermal building control using active ventilated block integrating phase change material

Keywords: Phase change material (PCM), Thermal building, Modelling, Thermal modelling, Thermal inertia

In the context of improving energy efficiency and thermal comfort in the building, the use of phase change materials (PCMs) is one of the suggested solutions. The proposed integration solutions concern the building envelope as well as the applications related to its operation.

The study of the incorporation of PCM in the walls of the building was the subject of numerous works. However, the antisymmetric character of storing/recovering energy management in the walls is less controlled and do not fit the optimal conditions. In this study, a solution based on the direct integration of a stabilized PCM (gel) in an envelope including ventilation channels, was proposed to overcome this problem of antisymmetry storing/recovering and fitting with different fixed ambient conditions. The final aim is to develop the optimization strategy of a wall combining the heavy inertia offered by the PCM, intra-ventilation control and the contribution to air renewal energy demand.

In this context, an experimental study of a concrete block system is conducted to test the thermal response of this configuration by the application of cyclic solicitations. The comprehension of this integrated constructive solution essentially passes by the possession of validated numerical tools. For this, two models have been developed. The first model using the electrical-thermal analogy, based on an RC equivalence, is distinguished by its relative simplicity and weak time computing demand. Each of these two factors, R and C summarise the system properties and has a direct influence on the building transient simulations over a year. The second one is based on direct numerical simulation DNS of the energy and fluid flow equations using commercial code COMSOL Multiphysics. Such DNS is time consuming and could not be used to simulate over a year but aims only to validate the first RC circuit approach. A comparison of these two models with the idealised experimental data was carried out and allowed the validation of the thermal behaviour of the solution based on the integration of the PCM with core ventilation.

Introduction

Energy consumption in the building sector is gaining increasing interest as it is directly related to energy economics and sustainable development. The design and the choice of building materials, as well as the energy and thermal systems evolve very rapidly. In the energy challenge, the building is among the largest consumers of energy. Efficiency and optimization of energy systems remain among the main axes studied in order to reduce energy consumption and increase system performance. The need to balance available energy with occupant comfort involves the use of new innovative and effective technological solutions. In the area of housing, the cost and optimization of space are the two main reasons that require the decrease of the thickness of walls in new constructions; however, this reduction greatly affects the thermal inertia of the frame and makes it insufficient to effectively damp the oscillations due to the outdoor temperature variation. Moreover, the management of thermal comfort by control systems acting on ventilation poses the problem of increasing the electrical consumption energy and the drawbacks related to the operation of these devices (noise, maintenance, etc.).

The integration of phase change materials (PCMs) in the building envelope is an interesting solution to optimize energy consumption. When integrated into building elements, their energy storage capacity using latent heat is manifested. A thermal phase shift is performed during the energy storage phase, then comes the restitution phase with a damping, which reduces the needs according to the outdoor climatic conditions of the considered construction. This topic has been the subject of numerous studies aimed at answering the various difficulties that oppose the optimal use of this solution [START_REF] Zhou | Review on thermal energy storage with phase change materials (PCMs) in building applications[END_REF][START_REF] Khudhair | A review on energy conservation in building applications with thermal storage by latent heat using phase change materials[END_REF][START_REF] Kuznik | A review on phase change materials integrated in building walls[END_REF][START_REF] Cabeza | Materials used as PCM in thermal energy storage in buildings: A review[END_REF]. Among these constraints, one can cite the choice of the appropriate method of incorporation of these products but also the limits imposed by the use of such a material which changes phase.

Direct incorporation, impregnation of building materials, incorporation of capsules filled with PCM are part of the used methodologies. Thus, in order to minimize the risk of leakage during the melting of this type of product, mixtures are used which aim at reinforcing the PCM matrix, for example, the use of polymers which make it possible to have mechanically stable gels during the process liquid phase [START_REF] Royon | Thermal energy storage and release of a new component with PCM for integration in floors for thermal management of buildings[END_REF]. Similarly, the efficiency of the systems developed requires an optimization study on the types of PCM to be used, the dimensions of the system and the possible combinations in order to correctly manage the energy charged/discharged in these products [START_REF] Royon | Optimization of PCM embedded in a floor panel developed for thermal management of the lightweight envelope of buildings[END_REF][START_REF] Alawadhi | Thermal analysis of a building brick containing phase change material[END_REF].

Two categories of integration can be distinguished, when incorporated into the building envelope or in applications related to the thermal comfort regulation during the occupation of the building. In the first type of integration, the following cases can be mentioned: integration in floors [START_REF] Lin | Modeling and simulation of under-floor electric heating system with shape-stabilized PCM plates[END_REF][START_REF] Ansuini | Radiant floors integrated with PCM for indoor temperature control[END_REF][START_REF] Figueiredo | Mechanical and thermal characterization of concrete with incorporation of microencapsulated PCM for applications in thermally activated slabs[END_REF], walls [START_REF] Castell | Experimental study of using PCM in brick constructive solutions for passive cooling[END_REF][START_REF] Cabeza | Use of microencapsulated PCM in concrete walls for energy savings[END_REF][START_REF] Zhang | Development of a thermally enhanced frame wall with phase-change materials for on-peak air conditioning demand reduction and energy savings in residential buildings[END_REF], masonry elements [START_REF] Zhang | Thermal response of brick wall filled with phase change materials (PCM) under fluctuating outdoor temperatures[END_REF][START_REF] Principi | Thermal analysis of the application of PCM and low emissivity coating in hollow bricks[END_REF][START_REF] Wang | Experimental assessment on the use of phase change materials (PCMs)-bricks in the exterior wall of a full-scale room[END_REF] and roofs [START_REF] Pasupathy | Experimental investigation and numerical simulation analysis on the thermal performance of a building roof incorporating phase change material (PCM) for thermal management[END_REF][START_REF] Elarga | Experimental and numerical analyses on thermal performance of different typologies of PCMs integrated in the roof space[END_REF][START_REF] Guichard | A thermal model for phase change materials in a building roof for a tropical and humid climate: Model description and elements of validation[END_REF]. This integration category of PCM also includes its use in windows and glazing [START_REF] Goia | Improving thermal comfort conditions by means of PCM glazing systems[END_REF][START_REF] Li | Energy investigation of glazed windows containing Nano-PCM in different seasons[END_REF][START_REF] Li | Simulation research on the dynamic thermal performance of a novel triple-glazed window filled with PCM[END_REF] as well as in shading systems [START_REF] Alawadhi | Using phase change materials in window shutter to reduce the solar heat gain[END_REF][START_REF] Ismail | Thermally effective windows with moving phase change material curtains[END_REF][START_REF] Komerska | Experimental analysis of an external dynamic solar shading integrating PCMs: First results[END_REF]. Concerning the use of PCMs in comfort conditioning systems within the building, the free cooling method based on heat exchangers was developed according to several approaches [START_REF] Nagano | Study of a floor supply air conditioning system using granular phase change material to augment building mass thermal storage-Heat response in small scale experiments[END_REF][START_REF] Waqas | Phase change material (PCM) storage for free cooling of buildings -A review[END_REF][START_REF] Iten | A review on the air-PCM-TES application for free cooling and heating in the buildings[END_REF][START_REF] Akeiber | A review on phase change material (PCM) for sustainable passive cooling in building envelopes[END_REF]. These different studies have shown that the contribution of PCMs increases the thermal inertia and on the other hand improves the air quality in buildings.

The vertical elements of the building are the most studied, the wall panels that can be fixed on the base walls are among the solutions proposed. In this context, many products have been tested and marketed because this solution makes it possible to minimize the level of the peak temperature during hot periods and to save energy. The damping of the peaks of temperature obtained can go up to 4°C [START_REF] Ahmad | Experimental investigation and computer simulation of thermal behaviour of wallboards containing a phase change material[END_REF][START_REF] Kuznik | Experimental investigation of wallboard containing phase change material: Data for validation of numerical modeling[END_REF][START_REF] Hasse | Realization, test and modelling of honeycomb wallboards containing a Phase Change Material[END_REF]. Concrete and coatings PCM based are part of the solutions applied to the walls. They are found under several compositions with different percentages of PCM. The tests carried out in a real scale on cells, show the increase of the storage capacity as a function of the volume ratio of the PCMs, as well as the difference in indoor surface temperature with respect to a reference cell [START_REF] Cabeza | Use of microencapsulated PCM in concrete walls for energy savings[END_REF][START_REF] Zhang | Development of thermal energy storage concrete[END_REF][START_REF] Lee | Control aspects of latent heat storage and recovery in concrete[END_REF].

The solutions applied to the walls also include integration into the masonry elements. In this configuration Alawdhi et al. [START_REF] Alawadhi | Thermal analysis of a building brick containing phase change material[END_REF] conducted an optimization study of the amount of PCM to be introduced into bricks containing cylindrical cavities. By means of a 2D finite element model, they show that the best configuration reduces the inward heat flux on the wall by 17.55%.

Zhang et al. [START_REF] Zhang | Thermal response of brick wall filled with phase change materials (PCM) under fluctuating outdoor temperatures[END_REF] numerically studied the filling of holes in traditional bricks. As results, the study shows a 10% decrease in the amplitude of the surface temperature of the wall. Principi et al. [START_REF] Principi | Thermal analysis of the application of PCM and low emissivity coating in hollow bricks[END_REF] presented an experimental and theoretical study of the impact of a brick wall with PCM. As result, they showed that the presence of PCM affects the peak of heat and it generates a phase shift of 6 hours with a decrease of 25% of temperature amplitude.

In all these works on construction materials, the interest is on the behaviour of the PCM mainly in relation to the quantity used and the different combinations of filling in order to obtain the best possible configuration [START_REF] Kant | Heat transfer studies of building brick containing phase change materials[END_REF]. However, these applications do not allow the control and management of the stored/recovered energy in the PCM other than the change in quantities or the type of PCM. The PCM imposes its behaviour on the output indicators, which reflect the response of the system. A wall containing PCM is solicited from the outside by an oscillating temperature between two, maximum and minimum values influenced by the solar heat flux, while the interior is restricted by a comfort temperature generally set by conditioning systems that operate for compensating or eliminate the heat flux crossing the walls. In summer, the heat loads crossing the wall can be considered negligible if compared to the internal heat gains. Such heat loads are shifted during the night hours. The increase in apparent heat capacity and a fast discharge will act and enhance the indoor comfort as it is equivalent to efficient free cooling.

Often, the chosen phase change temperature is generally close to the comfort set point, this remains valid as long as the PCM part is close to the inner side. Depending on the season and weather conditions, charging and discharging a PCM-based wall does not occur at the same time. In this situation, a rebalancing is necessary in order to accelerate or slow down the charging period or to shift the discharge.

As part of this study, a new strategy is proposed for using a wall with PCM, adding internal ventilation through tubes located within the PCM contained in the wall. Fig. 1 shows the wall which consists of concrete blocks filled with PCM and ventilated channels. This solution allows both increasing the thermal inertia thanks to the inertial capacity of the PCM and to control the charge/discharge while enhancing the exchanges by injecting the air from the outside to the inside. It thus allows contributing to the conditioning of the renewal of the local air and the temperature change. The diagram presented in Fig. 1 also shows the utility of such a system in the management of energy needs. It allows, for example, the storage of the energy released during the night from heating [START_REF] Zhang | Application of latent heat thermal energy storage in buildings: State-of-the-art and outlook[END_REF]. Ventilation control through the system helps to meet some of the morning's energy needs when occupants wake up. The study of thermal solutions in the building usually goes through an experimental study which serves as a basis for validation of the models and optimization on the numerical level. The development of the numerical model is essential to show the usefulness of the solutions and quantify the thermal behaviour of the systems. However, the resolution of the studied physical system requires techniques based on equations describing the physical behaviour of heat transfer. Generally, three main thermal building models are currently used in modelling: multi-zone, zonal and CFD (Computational Fluid Dynamics) methods [START_REF] Foucquier | State of the art in building modelling and energy performances prediction: A review[END_REF].

In the multi-zone or nodal approach, modelling by electrical analogy is a technique often used. It simplifies the physical problem by linearizing the equations and thus reduces the calculation time. The equivalent electric circuit, formed of resistances and capacitances (RC), represents the fundamental physical relations which translate the studied thermal phenomena. It is based on the combination of a thermal resistance R and a thermal capacity C for each studied thermal element. The RC model is distinguished by its speed, which is an important factor in simulations related to the building where the heat exchanges require substantial time intervals where calculations generally become very heavy [START_REF] Davies | Optimal RC networks for walls[END_REF][START_REF] Goyal | A method for model-reduction of non-linear thermal dynamics of multi-zone buildings[END_REF][START_REF] Fraisse | Development of a simplified and accurate building model based on electrical analogy[END_REF][START_REF] Kämpf | A simplified thermal model to support analysis of urban resource flows[END_REF][START_REF] Park | Modeling of a building system and its parameter identification[END_REF][START_REF] Bueno | A resistance-capacitance network model for the analysis of the interactions between the energy performance of buildings and the urban climate[END_REF]. In addition to the phenomena often modelled, namely conduction and convection, a modelling of the phase change of paraffin with integrated ventilation system will be presented in this study. The features of the Simscape library integrated in the Matlab/Simulink environment are used. The method of resolution, thus adopted makes it possible to work on a circuit constituted of variable resistances and capacitances. This article is part of the studies on the direct integration of a phase change material into a concrete block [START_REF] Laaouatni | Study of improving the thermal response of a construction material containing a phase change material[END_REF][START_REF] Laaouatni | Phase change materials for improving the building thermal inertia[END_REF]. In this context, a structural element of "hollow concrete block" filled with a polymer-paraffin composite and provided with ventilation tubes allowing the circulation of air will be studied. The used mixture of PCM is mechanically stable and that avoids the problem of leakage. This study will be presented in four main axes:

• The first one is developed in approaching the actual operating conditions of the system and represents the experimental study which is used to compare the effect of PCM filling and the influence of tube condition (open, closed and ventilated).

• The second part is dedicated to modelling of the system under COMSOL Multiphysics.

• After validating the used model, the third part will be devoted to reducing the time calculation request by using the electrical analogy modelling and the equivalent RC model.

• Finally, a validation was performed by comparing the two models with the experimental data.

System description

Fig. 2(a) presents a diagram and an illustration of the developed system which is based on a commercial hollow Concrete Block having the following length x height x width dimensions: 50 cm x 20 cm x 9.5 cm. The concrete block consists of 3 parallelepiped cavities. A PVC tubes are used, with a diameter of 2 cm and a thickness of 1 mm. Two holes are carried out in the base of each cavity to place the six tubes. The PCM is incorporated directly in the molten state in the cavities (temperature higher than the temperature of the solid gel). The used PCM is a mixture of paraffin and a styrene-type polymer. The choice of paraffin removed both hydrated salts, known by their corrosive tendency, as well as unsuitable fatty acids because in our case the PCM is directly put in contact with the concrete having a basic pH. The use of a percentage of polymers in solution makes it possible to obtain a gel with acceptable stability in order to avoid problems of leakage in the liquid state. Currently, paraffin manufacturers offer a wide range that varies with melting temperature as well as latent heat. Normally, the choice will have to take into consideration the antisymmetry between the external conditions and the temperature of comfort fixed indoor. The melting temperature chosen is important to create a wall temperature stabilization bearing close to that of comfort and also offer a sufficient storage that allows the incoming air to be conditioned through the tubes. The realized mixture has the same characteristics as that used in the work of Royon et al. [START_REF] Royon | Thermal energy storage and release of a new component with PCM for integration in floors for thermal management of buildings[END_REF]. Base paraffin is a commercial product having a melting temperature of 27 °C and a latent heat of 179 kJ/kg. This choice is also based on experimental laboratory manipulation conditions, which requires a difference between the melting temperature and the ambient temperature to avoid the onset of melting. The percentage of the used polymer is of 15%. The latent heat after mixing is equal to 110 kJ/kg. Table 1 summarizes the characteristics of the mixture as well as the materials constituting the used system. Table 1. Thermo-physical properties of the materials constituting the system.

Experimental study

This part is devoted to the description of the used experimental device as well as the procedure explaining the progress of the realized tests.

The obtained results of the experimental study are presented and a comparative study of the different mounting configurations chosen is carried out to show the system behaviour.

Experimental device and procedure

Using a real scale system imposes constraints to fully adapt the experimental study and the actual operation of the considered element as a representative part of a sunny wall. The experimental device consists of an electrical resistance which allows imposing a power heat on the side surface of the concrete block. K-type thermocouples are arranged on both side surfaces, they are placed at mid-height and on the same horizontal plane (Fig. 2(b)). Linked to the acquisition system, they make it possible to follow the evolution of the temperature.

Small fans are used to inject air through the tubes. The speed and temperature of the air are checked with a hot-wire anemometer. In order to avoid thermal losses at the edges of the system, a layer of polyurethane foam surrounds the other surfaces is used. The second face of the block is in contact with the ambient air of the laboratory room which varies slightly around 21 °C.

A preliminary study of thermal stresses allowed choosing an adequate heating power so as not to exceed a temperature of 50 °C on the heated surface. This maximum temperature is chosen relative to the limit of the stability of the used gel. The solicitation chosen has three cycles with a duration of 24 hours for each one. Each cycle consists of applying a constant heat power for 5 to 6 hours, except for the concrete block without PCM which its inertia is too low and this duration of cycle is reduced in this case. After stopping the thermal solicitations, a normal cooling is applied for the rest of the cycle for all configurations, except in the case with forced ventilation; the air injection is activated directly after halting. The experimental device as presented in Fig. 3, allows simulation the operating principle of each studied element. Also, this installation would serve as a basis for the development and validation of a complete numerical tool.

Experimental results

The adopted experimental methodology makes it possible to follow the evolution of the temperatures by the application of three identical cycles of heat fluxes. The use of cyclic stresses allows testing the thermal behaviour of the PCM over a more prolonged period. The finding of each configuration is presented in Fig. 4. The curves show the temperature evolutions recorded on the lateral surface in contact with the ambient air. The heat flux is imposed in the same way for all configurations. However, the experimental protocol makes that the operating intervals are different according to the configuration tested. The threshold temperature of 50 °C is the basis for the control of the electrical resistance which simulates the sunshine and not the duration of application of the thermal stresses, which is the result of the inertia and the thermal behaviour of each configuration. Two systems are studied here; empty concrete block without PCM and system with PCM and tubes. For the second system, three cases can be distinguished according to the state of the tubes which can be, closed, opened or ventilated. The analysis of temperature evolution from the third cycle as represented in Fig. 5, shows the effect of PCM filling. On the one hand, it contributes to the increase in the phase shift from 0.5 hour in the case without PCM to 3.8 hours; on the other hand, the PCM allows the reduction of recorded temperature peaks of about 3.4 °C. The study of the state of the tubes on the response of the wall shows that the decrease in the recorded maximum temperature is a function of the transfer mode imposed through the tubes.

In the case where tubes are closed, the response of the system is explained by the predominance of conduction due to the diameter of the tubes that does not promote the establishment of the convection heat transfer. When the tubes are open, in this case, they favour a convective heat exchange thanks to the difference in density of the air between the inside and the outside of the tubes; this small contribution explains the decrease in the temperature evolution. Forced convection induces by the air injection further enhances system response. The ventilation allows lowering the temperature level quickly on the inner surface; a decrease of temperature from 3.4 to 4.7 °C compared to the case without PCM. This gain in temperature is accompanied by a decrease in phase shift time from 3.8 to 2.9 hours which is a normal result because the application of ventilation improves the discharging by a withdrawal of energy and thus induces a cooling of the system too. This behaviour also explains the ability of the ventilated system to store more energy, which therefore increases the necessary time to reach the threshold temperature on the heated surface. The curves show (Fig. 5) that it takes 6.5 instead of 4.5 hours for the system without ventilation, while only 2.4 hours are needed for the concrete block without PCM. The temperature of the outgoing air measured by the hot-wire anemometer shows that the maximum increase thereof reaches about 10 °C during the exchange of the discharged energy through the tubes. This operating scenario makes it possible to heat the incoming air from the outdoor by supplying the energy recovered on the PCMs during the renewal of air.

The response of the complete system with ventilated tubes shows that the discharging of the energy is not sufficient to reach the same temperature at the end of the cycle in the case without PCM. These results clearly demonstrate the role of PCMs in energy storage and the efficiency of the recovery system by the ventilated tubes. However, an optimization study on the quantity of PCM and the characteristics of the tubes will allow finding the best configuration with the experimental used conditions.

Numerical simulation

As part of phase change problems solving, the use of COMSOL Multiphysics has demonstrated good correlations with experimental results [START_REF] Lamberg | Numerical and experimental investigation of melting and freezing processes in phase change material storage[END_REF][START_REF] Dhaidan | Experimental and numerical study of constrained melting of n-octadecane with CuO nanoparticle dispersions in a horizontal cylindrical capsule subjected to a constant heat flux[END_REF][START_REF] Al-Saadi | Modeling phase change materials embedded in building enclosure: A review[END_REF]. It allows the solving of problems governed by partial differential equations by the finite element method. The formulation using apparent specific heat approach is used to model the phase change [START_REF] Ahmad | Thermal testing and numerical simulation of a prototype cell using light wallboards coupling vacuum isolation panels 28 and phase change material[END_REF].

Three-dimensional modelling (3D) has been developed in this study. The partial differential equation that governs heat transfer in the model is given by Eq. ( 1).

(

)( ) ( )

P T C u T k T t ρ ∂ + ∇ = ∇ ∇ ∂ (1)
where ρ , p C and k are the intrinsic characteristics of the concerned material, u is the velocity field. The main characteristics of the PCM are evaluated experimentally. The different thermo-physical properties of the materials (Table 1) are introduced into the modelling. The thermo-physical properties of the PCM are governed by Eqs. ( 2)-( 6).

l v s v pcm k f k f k ) 1 ( - + = (2) l v s v pcm f f ρ ρ ρ ) 1 ( - + = (3) ( ) (1 )( ) v p s v p l m p pcm pcm f C f C C L T ρ + - ρ ∂α = + × ∆ ρ ∂ (4) 1 ; T 2 T
T 0.5 ; 2 2 0 ; T 2 sl sl sl sl T T T Tsl T T T T T T < -∆   - ∆ ∆  θ = + - ≤ ≤ +  ∆  > + ∆   (5)
( )

1 2 ; T 2 1 T T ; 2 2 2 1 2 ; T 2 sl v l v s m sl sl pcm sl T T f f T T T T - < -∆  -ρ -ρ ∆ ∆  α = - ≤ ≤ +  ×ρ   + ∆  (6)
where v f is the volume fraction of the PCM solid phase, T ∆ is the phase transition interval, sl T is the phase change temperature and m α is the mass fraction defined from s ρ , l ρ and v f .

The considered boundary conditions are as follows: on the heated side surface, a homogeneous temperature corresponding to the average temperature recorded experimentally was imposed. On the other side surface, a convective exchange with the ambient environment was used. The convective transfer coefficient, conv h between the lateral surface of the block and the ambient air, is calculated by Eq. [START_REF] Alawadhi | Thermal analysis of a building brick containing phase change material[END_REF].

air conv w Nu k h L = (7)
where Nu is the Nusselt number, air k is the air thermal conductivity and w L is the characteristic length of the surface of the concrete block. In our case, the flow is laminar and Nu is calculated from the correlation on a vertical plane [START_REF] Mcadams | Heat Transmission[END_REF] as:

1 4 0.59 Nu Ra = × (8)
The convection coefficient conv h calculated with the correlation is equal to 3.3 W/m 2 .K. In addition to the convective effects, the lateral surface is subject to the phenomenon of radiation where the radiation heat flux is written as:

( )

4 4 amb w rad T T Q - =εσ (9)
where ε is the emissivity of the concrete block which takes as value 0.9.

For modelling the case without ventilation, it is considered that the diameters of the tubes do not allow the appearance of a significant convective mode. However, the injection of ambient air, with the fans through the tubes, is activated with a measured speed of 1.3 m/s.

The calculation of the Reynolds number R e shows that the regime is laminar. The studied model has short diameter pipes; which imposes to take into account the effects of the inlet region, the correlation of Seider and Tate [START_REF] Incropera | Fundamentals of heat and mass transfer[END_REF]

    µ = × ×     µ     (10)
Dynamic viscosity µ and the Prandtl number r P are evaluated at the average temperature of the air between the inlet and the outlet of the tube. Thus, wt µ is the viscosity of the air determined for the wall temperature of the tube wt T . i D and t L represent respectively the inside diameter and the length of the pipe. As of the correlation, the forced convection coefficient through the tube is set equal to 11.7 (W/m 2 .K). On the other surfaces of the concrete block; a zero heat flow condition is used (adiabatic surface).

Modelling by electrical analogy

It becomes a classical approach in order to simplify a thermal system and predict the temperature evolution over time under the control-command strategy. First, the principle of thermo-electric analogy and the decomposition of the system into a unitary element will be presented. After that, the modelling of the system will be detailed.

Principle of thermal modelling by electrical analogy

Thermal modelling by electrical analogy is based on the analogy with Ohm's law in electricity. The temperature and the heat flux play respectively in this problem the same role as the potential and the current in electricity. It is, therefore, possible to adopt for the thermal resistances the same representation as for the electrical resistances. As the electrical and thermal systems are similar, going from one system to another is possible while respecting the equivalences existing between the different components (Fig. 6). ( )

wo wi w k S Q T T e = - (11)
In electricity, the application of a potential difference V ∆ to a similar element is governed by Ohm's law, the electric current I is equivalent to the heat flux and it is expressed as:

V R V e S I elec w ∆ = ∆ = 1 σ (12)
According to the two equations ( 11) and ( 12), the similarity with the electrical resistance allows to define the thermal resistance of conduction R (K/W):

w e R k S = (13)
It quantifies the resistance that the system opposes to the passage of heat flux. Its expression depends on the considered geometry, k is the thermal conductivity and S is the surface of the layer.

In the same way, the stored energy in Joules by a mass m, having a mass capacity m c at the temperature ( )

K T is written: T mc dt Q U m∆ = = ∫ (14) dt dT mc Q m = (15) 14
Analogously, the equation of the intensity I which passes through a capacitance elec C with a tension V is written:

dt dV C I elec = (16)
The thermal capacity C (J/K) represents the storage of the heat in the layer of the wall and it is given by:

m w C c e S =ρ (17)
The notion of thermal resistance allows the modelling of other phenomena, for example, the convective exchanges between a solid wall of temperature w T and a fluid at a temperature T . Indeed, through Newton's law expressed in Eq. ( 18), a convective resistance conv R is defined by Eq. ( 19).

( ) The thermal/electrical equivalence makes it possible to derive the following properties:

conv conv w Q R T T = - (18) S h R conv conv 1 = (19)
An imposed temperature is identical to a voltage generator.

A heat flow is identical to an electric power generator.

A body of mass m and heat capacity m c ; heated to the temperature T , is identical to a capacitor having a capacity m mc related to the mass and the imposed voltage source.

Physical module decomposition

The passage from a purely thermal system to a model based on electrical phenomena requires an understanding of the physics that governs the system. In our configuration, there are 3 parts: concrete, phase change material and air tube. The study of the symmetry of the concrete block, in a wall or as an isolated element, allows us to adopt the decomposition of the system into 6 identical elements as represented in (Fig. 7).

Fig. 7. System decomposition into unitary elements.

Similarly, as a precaution, a study was conducted for the validation of the chosen geometric layout. Numerical simulations based on the finite element model were carried out in the case without ventilation. Configuration (c) shows the 3D complete system. It reveals that numerical results obtained from the 3D configurations are close to the experimental test then that obtained from 2D configuration. This is can be explained by the effect of the rib of the neglected concrete in the geometry (a). In addition, finite element flow modelling with geometry (a) is complicated to transform the outflow into an exchange coefficient. Therefore, configurations (b) and (c) can be considered to perform numerical validation. The configuration (c) has been adopted. According to the analysis carried out, the physical problem to be modelled is presented in Fig. 9; the diagram represents the chosen revealing zone. Physically, the influence of the tube is done in three dimensions. The RC circuit approach is based on the average temperatures of each modelled block; the horizontal flow between the two faces represents the main heat exchange in our system, the integration of ventilation adds a second exchange on the vertical. The modelling of these two exchanges with the electrical analogy requires a good understanding of the system from the two geometries (a) and (b) (Fig. 8). Thus, the hypothesis of a unidirectional transfer allows distinguishing three parallel levels that represent the succession of the elements constituting each part. Fig. 9. The configuration of the revealing zone studied according to thermal exchanges.

System modelling

The RC circuit modelling, proposed in this work, is based on the use of the Matlab Simscape tool. Its library proposes several models for the direct integration of electrical, thermal, mechanical and hydraulic phenomena which allow to quickly creating physical system models in the Simulink environment.

The electrical model enables the creation of the circuit from the physical connections to obtain block diagrams. In the sense of electrical and thermal analogy, every part of our physical model will be replaced by an electrical circuit of resistances and capacitances, as well as for imposed boundary conditions. Before arriving at the equivalent diagram representing the studied element, simple models were used to translate the behaviour of a single block of PCM. Thereafter, the other elements were added progressively; the concrete parts as well as the air tube. The diagram in Fig. 10 represents the equivalent thermal model with its boundary conditions. In the experimental module, the lateral surface is subjected to convective exchanges; the convection coefficient conv h is calculated in the like in section 4. The convection resistance with the ambient air is written as follows:

S h R conv conv 1 = (22)
For taking into account the heat transfer by radiation from a surface to inside a building, there are several methods that estimate, in a more or less approximate way, the radiative exchanges between the surfaces. A linear equation expressing the heat flux between a wall of temperature T and all the other walls is used [START_REF] Davies | Optimal designs for star circuits for radiant exchange in a room[END_REF]. The radiative heat flux on the lateral face is given by: ( )

w rad rad T T S h Q - = (23)
The expression of rad h depends on the average temperature of the walls w T . This temperature is considered equal to the ambient temperature: The radiative resistance rad R on the lateral surface of our model is:

S h R rad rad 1 = (25)
The elements containing the paraffin/polymer mixture constitute a system that stores the heat in a sensible and latent form, which results in a storage capacity and a thermal resistance that vary according to the solid and liquid state of the mixture. This is why in order to encompass the entire phenomenon of phase change by electrical analogy; each volume of PCM is replaced by an electrical block consisting of a resistance and a variable capacity depending on the temperature.

The variable capacity pcm C for a PCM block having a volume pcm V is given by Eq. [START_REF] Nagano | Study of a floor supply air conditioning system using granular phase change material to augment building mass thermal storage-Heat response in small scale experiments[END_REF].

pcm pcm p pcm pcm V c C ρ = (26) with pcm p c
is the variation of specific capacity of the mixture adopted for thermo-electric modelling.

The density of the phase change material is defined by Eq. ( 27).

( )

       > ≤ ≤ + < = ; ; 2 ; s s sl l sl om l om pcm T T T T T T T ρ ρ ρ ρ ρ ( 27
)
where om T is the temperature that marks the start of the fusion obtained from the DSC experimental measurements and sl T is the solid-liquid phase transition temperature.

The specific capacity modelling is based on the experimentally obtained curve, the work of Alisetti and Roy [START_REF] Alisetti | Forced Convection Heat Transfer to Phase Change Material Slurries in Circular Ducts[END_REF] shows that the equivalent function shape ( ) The adopted equivalent thermal capacity is, as simple as possible according to the announced indications; it respects the conservation of the amount of involved energy. Fig. 11 shows the evolution of the involved thermal capacity; its equation is written as: [START_REF] Iten | A review on the air-PCM-TES application for free cooling and heating in the buildings[END_REF] Likewise, as given in relation [START_REF] Akeiber | A review on phase change material (PCM) for sustainable passive cooling in building envelopes[END_REF], the resistance of each block of the PCM is a variable resistance depending on the liquid or solid conductivity according to the temperature:

T
pcm pcm pcm pcm S k e R = (29)
where pcm e is the thickness of each part of the PCM and pcm k is the variable conductivity according to the state of the PCM solution which is expressed by Eq. (

;

om pcm l om sl l sl k T T k k k T T T k T T <   = + ≤ ≤   >  (30)
•

Tube and airflow

Taking into account the tube in the modelling of the studied thermal system begins with its thermal resistance. The assumption of a unidirectional heat transfer and the behaviour of the module impose the consideration of two heat flows which exchange to the right and to the left of the tube. Subsequently, splitting the thermal resistance in half is necessary to differentiate the total resistance on each side.

The thermal resistance of the tube t R is written as:

t t i o t L k D D R π 2 log       = (31)
where where air k is the thermal conductivity of the air and air S is the lateral surface of the air volume.

In the case of forced flow with fans, the problem evolves in the three directions; taking into account the passage of air using the exchange coefficients is not sufficient for modelling electrical-thermal equivalence. The injection of air through the tube is characterized by two elements, the speed and the temperature of the incoming air. In the sense of electrical analogy modelling, a temperature or a voltage will model the interaction of the flow of air entering with the flows exchanged horizontally across the system.

In order to identify this temperature, it is necessary to express the heat balance equation at the midpoint of the tube which is the mean temperature over the height. Fig. 12(a) describes the flows exchanged in the horizontal and vertical direction. The heat balance equation ( 33) can be written and then resolved for the air temperature in the middle of the system air T : The resolution of the air temperature, as a function of the travelled distance, in the case where the convection coefficients through the tube are equal [START_REF] Dermardiros | Modelling of an active PCM thermal energy storage for control applications[END_REF], gives the expression [START_REF] Lee | Control aspects of latent heat storage and recovery in concrete[END_REF].

) ( ) ( ) ( 2 1 air amb T air f T out in air p air T T R T T R T T C m - + - = - & (33)
( ) (  ) discretization with more nodes will be applied to each element [START_REF] Stathopoulos | Air-PCM heat exchanger for peak load management: Experimental and simulation[END_REF].

      + + -         + - + + + = 2 1 2 1 2
1 1 log / 1 4 pcm o i c o c pcm t v e D D e H D k k k h -   = + + +     π   (35) ( ) ( ) 1

Validation results and discussion

Fig. 13 presents the comparison between the results obtained from the FEM/RC simulations and those obtained by experimentation for the two cases without and with ventilation. The FEM was developed primarily as a reference for examining the accuracy of the model with RC network. For this purpose, the first comparison concerns the results of the FEM with the experiment; Table 2 represents the values of the relative errors calculated with respect to the experimental data. The average relative error does not exceed 1.4% and the maximum relative error is about 4.6% in the case without ventilation. This small discrepancy between these data is probably due to three factors: the degree of uniformity of the thermal power given by the heating resistor, the influence of thermal insulation, which is not ideal for eliminating all heat losses. For modelling the PCM, the used effective thermal capacity, only involves the measurement performed during the fusion.

Indeed, during solidification, the curve is slightly different. Taking into account this hysteresis between fusion and solidification during numerical validation has been the subject of numerous studies. Baghban et al. [START_REF] Baghban | Numerical Simulation of a Building Envelope with High Performance Materials[END_REF] indicated that it is essential to use the melting and freezing characteristics of PCM to achieve good agreement between experimental and numerical results. Kośny [START_REF] Kośny | PCM-Enhanced Building Components[END_REF] stated that the use of simplified algorithms that do not include hysteresis can lead to significant inaccuracies for a large number of non-paraffinic organic and inorganic PCMs. Goia et al. [START_REF] Goia | Modelling and experimental validation of an algorithm for simulation of hysteresis effects in phase change materials for building components[END_REF] demonstrated that for paraffin organic PCM, the available building energy simulation software is fairly accurate even if the effect of hysteresis is neglected.

Thus, according to these different studies, the fact that hysteresis is not taken into account does not affect significantly our results. Despite the noted slight difference, there is a good agreement between the simulation and the experiment, which makes possible to validate the finite element model used in the simulation. For comparison of the RC model with the experimental data, Fig. 13 illustrates a similar trend of variation between the simulated and measured values, in both cases with and without ventilation. Relative errors calculated with respect to the experiment are acceptable for direct validation. The difference between the two numerical models is mainly due to the modelling of the phenomenon of phase change with the electrical analogy which is based on the use of variable resistances and capacitances. Thus, the simplifying assumptions used in the RC model do not allow capturing the simulated complexity in 3D with the CFD model.

These discrepancies have no significant effect on the accuracy of the obtained simulation results. The temperature on the lateral surface of the system is very well predicted by the RC network model and the maximum relative error is less than 4.0%. It can be concluded that the developed RC model can be directly used to replace a fairly heavy CFD simulation in computing time.

Table 2. Results of relative error between experimental and numerical data.

Following the analysis previously conducted on the experimental results, Fig. 13(b) clearly illustrates that ventilation occurs in the heat transfer and that its influence even acts on the heated side surface. The temperature evolution of this face gives more information on the state of the PCM inside. For both cases, a surface can be distinguished, which evolves between the horizontal line that shows the phase change temperature of the mixture sl T and the ambient temperature amb T . The temperature variation on the internal face evolves in this surface; it approaches of sl T in the case without ventilation and tends towards amb T with ventilation.

In order to make a link against a real situation, this difference with respect to the ambient temperature defines the heat flux passing through the wall and consequently the energy requirement of air conditioning to reach the comfort temperature replaced here by the ambient temperature. The gradient of the applied thermal heat flux acts progressively, the concrete part rapidly reaches the steady state and all the energy transmitted is stored by PCM. Gradually, the temperature of the internal face exceeds the ambient temperature and a heat flux begins to leave this face towards the ambient environment. After stopping the thermal solicitations, the temperature decreases and the PCM become a source that releases energy. In Fig. 13(a), the stabilization bearing below sl T marks the solidification with a temperature close and parallel to sl T . The air injection at amb T by ventilation after stopping of the thermal power gives rise to a vertical heat transfer which is added to that established horizontally. The presence of the insulation on the heated side explains that the descent, noted on the heating curve is mainly due to the evacuation of energy through the tubes (Fig. 13(b)), the influence is significant because the average temperature on the internal face goes from 25 to 23.8 °C. Thus, decreasing the energy requirement is also accompanied by a significant capacity to increase the temperature of the air entering through the tubes.

Conclusions

In this work, a study of a thermal system developed to regulate the indoor temperature of a building have been proposed. The combination of the inertia effect of PCM introduced in the wall elements coupled to a ventilation system leads to a better use of the energy stored in a building wall. This solution uses building materials as a basic element and aims to improve the thermal response of PCM building walls and to demonstrate the feasibility of such an energy solution.

An experimental device was made based on a hollow concrete block filled with mechanically stable paraffin and equipped with ventilation tubes. The results have confirmed that the insertion of paraffin with tubes allows the creation of a thermal phase shift and the improvement of the inertial capabilities of the studied element. The ventilated tubes play the role of the regulator to unload the system and thus heat the injected air. The operation principle of this type of system is confirmed. However, the small scale of the experiment and the influence of the air flow at the entry of the tubes constitute limitations for the generalization of the experimental results in the real scale of a wall. In this framework, it is important to specify that there are many optimization parameters that can be used to operate the system at the wall scale as well as adaptation to climate conditions and desired operation. These parameters are: tube dimensions compared to the amount of PCM, flow velocity, the geometry of the tubes and their arrangement in the cavities, the phase change temperature as well as the type of the PCM and the materials of the constructions used.

On the numerical section, an RC network model has been developed, it proposes a simplified approach to models the thermal heat transfer by using the Simscape tool under Matlab, it allows both to get closer to the performance of CFD models in taking into account the evolution of the physical phenomena and it offers a great speed of the execution of the simulations, a parameter required in the calculations related to the building. Similarly, the model proposes a new approach to modelling the phase change by using grey boxes with their resistances and varying capacitances. The finite element model is proposed as a reference for comparing the accuracy of the RC model.

The analysis of the obtained results has confirmed the validity of the RC model for predicting the temperature evolutions of the system surface. The model developed in this study will be used to define the quantity of the PCM and the optimal dimensions of the elements of the system according to scenarios closer to reality. This study has made possible to test the feasibility and the performance of this proposed solution, and the validation of numerical tools allows the passage from elementary reasoning to a larger scale, this will be the basis for achieving the final aim of obtaining an optimal wall based on PCM filled elements and controlled by ventilation pipes.

Density

eT

  Element thickness (m) h Convective heat transfer coefficient (W/m 2 K) k Thermal conductivity (W/m K) Onset temperature of the melting (K, °C) out T Air outlet temperature (K, °C) sl T Phase change temperature (K, °C) u Velocity field (m/s) Greek letters L ∆ Latent heat (J/kg) ε Emissivity µ Dynamic viscosity (kg/m s) ρ Density (kg/m 3 ) σ Stefan-Boltzmann constant (W/m 2 K 4 ) ϕ Solar radiation heat flux (W/m 2 ) Subscripts amb Ambient

Fig. 1 .

 1 Fig. 1. Principle and utility of the tubes in the developed solution.

Fig. 2 .

 2 Fig. 2. Description of the developed system: (a) Schematic of the ventilated hollow concrete block integrating PCM (b) Schematic diagram of the experimental setup.

Fig. 3 .

 3 Fig. 3. Photography of the experimental bench.

Fig. 4 .

 4 Fig. 4. Evolution of the temperature on the lateral surface of the concrete block for different configurations.

Fig. 5 .

 5 Fig. 5. Temperature analysis on the lateral surface of the concrete block for the different configurations starting from the third cycle.

Fig. 6 .

 6 Fig. 6. Equivalent electrical circuit (1R1C) in the unsteady case.

convQ

  is the convection heat flux and conv h is the convective exchange coefficient between the wall and the fluid.

Fig. 8

 8 Fig. 8 represents comparative study between 2D and 3D (configurations (b) and (c)) simulations of the lateral surface temperatures of the complete system and the unit element as well as the experimental data. (1) Configuration (a) illustrates a section of the unitary element at horizontal level on the tube. (2) Configuration (b) represents 3D unit element. (3)

Fig. 8 .

 8 Fig. 8. Evolution of the average temperature on the inner surface for different geometric approaches in the case without ventilation.

Fig. 10 .c

 10 Fig. 10. Equivalent system diagram with the boundary conditions. In our model, each concrete element would be replaced by an electrical circuit consisting of a resistance c R and a capacitance c C . According to the dimensions of each block, the formulas used in the calculation are:

•••Fig. 11 .

 11 Fig. 11. Adopted modelling of the specific capacity of the PCM.

iD

  and o D are the inner and outer diameters of the tube, t L is the length of the tube and t k is the thermal conductivity of the tube. The modelling of the flow is different according to the state of the tubes: open or ventilated. The dimensions of the tube do not allow the appearance of a significant convective mode in the open configuration case. The air part through the tube is modelled with a thermal resistance of heat conduction expressed by Eq. (32).

Fig. 12 .

 12 Fig. 12. Airflow modelling through the tube: (a) The schematic diagram of modelling parameters, (b) Equivalent electrical diagram of the middle part containing the tube.

where 1 T R and 2 TR

 12 represent the total thermal resistance respectively on the left between wo T and air T , and on the right between air T and amb T . in T and out T are respectively the inlet and outlet air temperature across the tube.

Fig. 12 (

 12 Fig. 12(b) shows the equivalent electrical diagram used in the middle part of the model (see Fig.10). The arrows on the components indicate that these elements are variable; it concerns the parts of the PCM as well as the temperature used in the modelling of the airflow. In the RC model thus developed, the objective was to evaluate the response of the surface of the system; a 1R1C circuit was used to model each part. The spatial discretization used is fine enough to accurately determine the temperature of the output surfaces. If the objective is to follow the detailed evolution of the temperature in a specific material, a

Fig. 13 .

 13 Fig. 13. Temperature of the surface of the concrete block, comparison of the experimental data with the two models RC and FEM: (a) without ventilation, (b) with ventilation.

Fig. 1 .

 1 Fig. 1. Principle and utility of the tubes in the developed solution.

Fig. 2 .

 2 Fig. 2. Description of the developed system: (a) Schematic of the ventilated hollow concrete block integrating PCM (b) Schematic diagram of the experimental setup.

Fig. 3 .

 3 Fig. 3. Photography of the experimental bench.

Fig. 4 .

 4 Fig. 4. Evolution of the temperature on the lateral surface of the concrete block for different configurations.

Fig. 5 .

 5 Fig. 5. Temperature analysis on the lateral surface of the concrete block for the different configurations starting from the third cycle.

Fig. 6 .

 6 Fig. 6. Equivalent electrical circuit (1R1C) in the unsteady case.

Fig. 7 .

 7 Fig. 7. System decomposition into unitary elements.

Fig. 8 .

 8 Fig. 8. Evolution of the average temperature on the inner surface for different geometric approaches in the case without ventilation.

Fig. 9 .

 9 Fig. 9. The configuration of the revealing zone studied according to thermal exchanges.

Fig. 10 .

 10 Fig. 10. Equivalent system diagram with the boundary conditions.

Fig. 11 .

 11 Fig.11. Adopted modelling of the specific capacity of the PCM.

Fig. 12 .Fig. 13 .

 1213 Fig. 12. Airflow modelling through the tube: (a) The schematic diagram of modelling parameters, (b) Equivalent electrical diagram of the middle part containing the tube.

  is used to calculate the Nusselt number

								t Nu
	inside the tube:
	N	ut	1.86	Re Pr	i t D L	0.33	wt	0.14

  through the tube during ventilation, the coefficient of radiation on the lateral surface and the convection coefficient with the ambient air. These coefficients are calculated previously in the description of the models.

	1 -H are two coefficients that translate the heat transfer by convection and 1 1 2 --+ + = conv rad h h H H (36) H and 2 1 conduction on both sides of the tube. Thus, v where h , rad h and conv h are respectively the T depends on the ambient temperature amb T and the temperature wo T that simulates the thermal solicitations on the lateral surface. When integrated into the RC model, this temperature is equivalent to a variable voltage generator according to the temperature of the PCM which reflects the variations of the resistances coefficient of convection The expression of the temperature air expressed in the coefficients 1 H and 2 H .

Table 1 .

 1 Thermo-physical properties of the materials constituting the system.

			Thermal conductivity	Specific heat
		(kg/m 3 )	(W/(m K))		capacity
					(J/(kg K))
	PCM
	T < 27.5 °C	850	0.28		2800
	T > 27.5 °C	780	0.18		2500
	concrete (at T=25	1000	0.9		1000
	°C)
	PVC	1400	0.16		1046
			Relative error	Relative error
			(average) (%)	(maximum) (%)
	without	RC-model	1.32		2.61
	ventilation	FEM	1.14		4.58
	with	RC-model	1.37		3.38
	ventilation	FEM	1.32		3.03

Table 2 .

 2 Results of relative error between experimental and numerical data.

Acknowledgements

The authors would like to thank the Grand Troyes Community in France for partially funding this work. The LMT/ENS-Paris Saclay for affording the first version set-up.