Alexandre Altes

MD Nicolas Thellier

MD Yohann Bohbot

MD Wassima Marsou

MD Gagandeep Chadha

MD Camille Binda

MD Anne Ringle

MD Amandine Mailliet

Nathalie Marotte

Clemence Riolet

MD Christophe Tribouilloy

MD PhD Sylvestre Maréchaux
email: sylvestre.marechaux@yahoo.fr

Prognostic Impact of the Ratio of Acceleration Time to Ejection Time in Patients with Low Gradient Severe Aortic Stenosis and Preserved Ejection Fraction

Keywords: aortic stenosis, low gradient, ejection dynamics parameters, echocardiography, outcome

The clinical management of patients with low gradient severe aortic stenosis (LG-SAS) and preserved left ventricular ejection fraction (LVEF) remains challenging owing to their heterogeneity. The aim to this study was to evaluate the relationship between an ejection dynamic parameter linked to AS severity and outcome, the ratio of acceleration time (AT) to ejection time (ET), in a cohort of patients with LG-SAS and preserved LVEF. 356 patients with LG-AS (defined by AVA≤ 1 cm² and/or AVAi≤ 0.6 cm²/m² and mean aortic pressure gradient<40 mmHg) and preserved LVEF≥50% were studied. The relationship between AT/ET and all-cause and cardiac mortality during follow-up was studied. Median follow-up was 41 months (interquartile range, 35-47 months). Median AT/ET was 0.32 (IQR, 0.29-0.36). The 5-year estimates of all-cause and cardiac mortality were respectively 57±7%, 36±7% for patients with AT/ET > 0.36 versus 43±4%, 16±3% for patients with AT/ET ≤ 0.36 (p=0.024 and p<0.001, respectively). After adjustment on known predictors of outcome including aortic valve replacement (AVR) used as a time-dependent covariate, there was a significant increase in all-cause mortality risk for patients with AT/ET > 0.36 (adjusted HR 2.04 [95% CI, 1.32-3.13]; p=0.001) and cardiac mortality risk (adjusted HR 2.89 [95% CI, 1.54-5.43]) compared with patients with AT/ET ≤ 0.36. The association of AT/ET > 0.36 and all-cause or cardiac mortality risk was consistent in subgroups of patients with LG-SAS and preserved EF. In conclusion, an AT/ET ratio of more than 0.36 is an independent predictor of mortality in patients with LG-SAS and preserved EF.

INTRODUCTION

According to current guidelines, severe aortic stenosis (SAS) is classically defined by a transvalvular mean pressure gradient (MPG) > 40 mm Hg or peak aortic jet velocity > 4 m/s 1 , and/or an aortic valve area (AVA) ≤ 1.0 cm 2 or an AVA indexed to body surface area (AVAi) ≤ 0.6 cm 2 /m 2 . However, up to 40% of patients with severe AS show discrepancies in their Doppler-echocardiographic AS parameters, the most common of which being the association of a reduced AVA but a low transvalvular gradient (< 40 mmHg) [START_REF] Clavel | Cardiac Imaging for Assessing Low-Gradient Severe Aortic Stenosis[END_REF] . In this subgroup of patients with "low gradient severe aortic stenosis" (LG-SAS), it is still a matter of debate to determine who should be considered for aortic valve replacement (AVR) and who should be treated conservatively [START_REF] Reddy | Paradox of Low-Gradient Aortic Stenosis[END_REF] . The ratio of acceleration time (AT) to ejection time (ET) is a simple and reproducible parameter which may be used to assess native AS severity [START_REF] Gamaza-Chulian | Ratio of Acceleration Time to Ejection Time for Assessing Aortic Stenosis Severity[END_REF] , but its prognostic implications have not been specifically evaluated in patients with LG-SAS and preserved left ventricular ejection fraction (LVEF). The present study addressed this issue.

METHODS

Between 2012 and 2018, 356 patients ≥ 18 years of age diagnosed with severe AS [defined as AVA ≤ 1 cm² and/or AVA normalized to body surface area (BSA) ≤ 0.6 cm²/m²], a low transaortic gradient (defined as mean aortic pressure gradient (MPG) < 40 mmHg) and preserved LVEF ≥ 50% in the echocardiography laboratories of two tertiary hospitals in France (Lille and Amiens) were prospectively included in the present ancillary study from a larger registry. We excluded: (i) >mild aortic and/or mitral regurgitation; (ii) prosthetic valves, congenital heart disease, supravalvular or subvalvular AS, or dynamic LV outflow tract obstruction; (iii) mitral stenosis; and (iv) patients who refused to participate in the study.

Clinical and demographic baseline characteristics were collected. The Charlson comorbidity index, summating the patient's individual comorbidities, was calculated. [START_REF] Charlson | A new method of classifying prognostic comorbidity in longitudinal studies: development and validation[END_REF] Coronary artery disease was defined by the presence of documented history of acute coronary syndromes, coronary artery disease previously confirmed by coronary angiography (reduction of normal diameter ≥ 50% in the left main coronary artery and ≥ 70% in the right coronary artery, left anterior descending coronary artery, left anterior descending coronary artery, and circumflex coronary artery), or history of coronary revascularization. The study had been approved by an independent ethic committee and was conducted in accordance with institutional policies, national legal requirements, and the revised Declaration of Helsinki. Authorization for research participation was obtained for all patients.

All patients underwent a comprehensive Doppler-echocardiography study, using commercially available ultrasound systems by experienced echocardiographers. Aortic flow was recorded using continuous-wave Doppler, by imaging and non-imaging transducers, systematically in several acoustic windows (apical 5-chamber, right parasternal, suprasternal, epigastric). The highest aortic velocity was used to calculate aortic time-velocity integral and mean pressure gradient (MPG). As recommended by current guidelines [START_REF] Baumgartner | Recommendations on the Echocardiographic Assessment of Aortic Valve Stenosis: A Focused Update from the European Association of Cardiovascular Imaging and the American Society of Echocardiography[END_REF] , wall (high-pass) filters are set at a high level and gain is decreased to optimize identification of the velocity curve from the spectrogram envelope. LV stroke volume (LV-SV) was calculated by multiplying the LV outflow tract area with the LV outflow tract time-velocity integral obtained by pulsed Doppler in the apical 5-chamber view. The LV outflow tract diameter was measured in zoomed parasternal long-axis views in early systole at the level of aortic cusp insertion (inner-to-inner edge). Aortic valve area (AVA) was calculated using the continuity equation. AVA and LV-SV were indexed to BSA. Low flow was considered if LV-SV was < 35 mL/m². [START_REF] Hachicha | Paradoxical low-flow, low-gradient severe aortic stenosis despite preserved ejection fraction is associated with higher afterload and reduced survival[END_REF] Conventional echocardiographic measurements were performed according to current EACVI/ASE guidelines. [START_REF] Lang | Recommendations for cardiac chamber quantification by echocardiography in adults: an update from the American Society of Echocardiography and the European Association of Cardiovascular Imaging[END_REF] AT was defined as time from the start to the peak of flow through the valve by continuous-wave Doppler. ET was defined from aortic valve opening to aortic valve closure (Figure 1). Continuous-wave Doppler recordings were performed at a sweep speed of 100 mm/sec. The AT/ET ratio was then calculated. Interobserver variability of AT/ET ratio has been previously reported in our institution, with an Intra Class Correlation coefficient at 0.90 CI95% 0.78-0.96 and a coefficient of variation=7.3%. [START_REF] Griguer | Clinical Significance of Ejection Dynamics Parameters in Patients with Aortic Stenosis: An Outcome Study[END_REF] When patients were in sinus rhythm, three cardiac cycles were averaged for all measures. For patients in atrial fibrillation, five cardiac cycles were averaged. Echocardiograms were stored in Digital Imaging and Communications in Medicine format to allow subsequent offline analysis.

After the initial medical management, treatment was conservative or surgical, as deemed appropriate by the patient's personal physician. The majority of patients were followed by clinical consultation and echocardiography in the outpatient clinics of the 2 tertiary centres. The others were followed in public hospitals or private practices by referring cardiologists working together with the tertiary centres. Information on follow-up was retrospectively obtained. Events were ascertained by direct patient interview and clinical examination and/or by repeated follow-up letters, questionnaires, and telephone calls to physicians, patients, and (if necessary) next of kin. Medical reports and death certificates were consulted for attribution of causes of death. Eighty-four percent of patients were followed for at least 2 years or until death. The outcome variable of the study was all-cause mortality.

Cardiac mortality was also studied. Clinical decisions regarding medical management and referral for surgery were made by the heart team with the approval of the patient's cardiologist in accordance with current practice guidelines.

The study population was divided in two groups according to AT/ET threshold of 0.36 as previously reported. The relationship between baseline continuous variables and the 2 groups was explored using an unpaired two-sided Student's t-test (for normally distributed variables) or Mann-Whitney U test (for non-normally distributed variables). Pearson chisquare test or Fisher exact test was used to examine the association between the two groups and baseline categorical variables.

Median follow-up time was obtained using the reverse Kaplan-Meier method. Event rates +/-SEs of the overall population and of the two groups were estimated according to the Kaplan-Meier method and compared using log-rank tests. Univariate and multivariate analyses of time to events were performed using Cox proportional-hazards models. We did not use model building techniques; covariates were entered in the models which were considered of potential prognostic impact on an epidemiologic basis. Models were adjusted for age, gender, Charlson comorbidity index (not including age), hypertension, coronary artery disease, atrial fibrillation, symptoms (New York Heart Association Class III-IV dyspnea, angina, or syncope), LVEF, low flow, aortic valve area, and aortic valve replacement (AVR). The effect of AVR on outcome was analyzed as a time-dependent covariable using the entire follow-up. The proportional hazards assumption was confirmed using statistics and graphs based on the Schoenfeld residuals. For continuous variables, the assumption of linearity was assessed by plotting residuals against independent variables. To evaluate whether AT/ET > 0.36 provides incremental prognostic value over clinical and echocardiographic variables, the differences between models were tested by calculating the overall difference in -2 log likelihood χ2 between models. We conducted subgroup analyses to determine the homogeneity of the association of AT/ET ratio and all-cause mortality. First, we estimated the effect of AT/ET on all-cause mortality in each subgroup using a Cox univariate model and then formally tested for first-order interactions entering interaction terms, separately for each subgroup. All p-values are the results of two-tailed tests. For all analyses, a P value of < 0.05 was considered statically significant. Data were analyzed with R version 3.4.4 (R Foundation for Statistical Computing, Vienna, Austria) and SPSS version 20.0 (IBM, Amonk, New York).

RESULTS

Three hundred and fifty-six patients were enrolled in the present study. Patient's clinical characteristics, overall and according to AT/ET threshold of 0.36 are detailed in Table 1. Median AT/ET was 0.32 (IQR, 0.29-0.36). Seventy-five patients (21%) had a value of AT/ET > 0.36. Patients with AT/ET > 0.36 were similar to patients with an AT/ET≤0.36 in terms of clinical profile (Table 1). The echocardiographic characteristics of the study population and according to AT/ET≤ or > 0.36 are detailed in Table 2. Briefly, patients with AT/ET > 0.36 had lower AVA, AVAi, higher peak aortic jet velocity and transaortic mean pressure gradient. They had a higher LV mass index but showed no other difference for LV function, volumes, or cardiac output.

During a median follow-up time of 41 months (interquartile range, 35-47 months), 135 patients died, 53 of them from cardiac causes. AVR was performed in 116 patients (surgical AVR in 72 patients, and TAVR in 44 patients), Among patients who underwent surgical AVR, 22 patients had at least one associated coronary artery bypass graft at the time of surgery. Overall, the 1-, 3-and 5-year estimates of all-cause mortality were 19+/-2%, 35+/-3%, 47+/-3%. The 5-year estimate of all-cause mortality was 57+/-7% for patients with AT/ET > 0.36 and 43+/-4% for patients with AT/ET ≤ 0.36 (p=0.024, figure 2A). Overall, the 1-, 3-and 5-year estimates of cardiac mortality were 19+/-2%, 14+/-2%, 21+/-3%. The 5-year estimate of cardiac mortality was 36+/-7% for patients with AT/ET > 0.36 and 16+/-3% for patients with AT/ET ≤ 0.36 (p<0.001, figure 2B). When patients were analyzed under medical treatment (i.e censored at the time of AVR if performed), AT/ET > 0.36 remained significantly associated with an increased risk of all-cause and cardiac mortality (log-rank p=0.009 and p=0.002, respectively).

After adjustment for covariates of prognostic importance, AVA, low flow status, and AVR as a time-dependent covariable, the association between AT/ET > 0.36 and all-cause mortality remained unchanged (adjusted HR 2.04 [95% CI, 1.32-3.13]; p=0.001, Table 3 4). Similar results were found for cardiac mortality (adjusted HR 2.89 [95% CI, 1.54-5.43], Table 3, Figure 3B).

The association of AT/ET > 0.36 and all-cause mortality risk was consistent in subgroups of patients with LG-AS (Figure 5). There were no significant interactions between AT/ET > 0.36 and any of the subgroups.

DISCUSSION

In the present study based on patients with low gradient SAS and preserved LVEF managed in clinical routine practice, we observed that an increase of AT/ET > 0.36 is associated with worse outcome. Accordingly, patients with AT/ET > 0.36 experienced a significant increase of the risk of all-cause and cardiac mortality during follow up after adjustment on known predictors of outcome including AVR used as a time-dependent covariate compared with patients with AT/ET ≤ 0.36. Hence, these results demonstrate the interest of assessing AT/ET in routine echocardiographic practice to identify patients with a poor outcome who may derive benefit from AVR among the challenging group of patients with LG-SAS and preserved EF.

Management of patients with LG-SAS and preserved EF still represents a challenge mainly due to the heterogeneity of this specific population [START_REF] Baumgartner | Low Gradient Aortic Stenosis: Who Benefits From Intervention?[END_REF] . Among these patients, the identification of those who would take benefit of surgical or percutaneous AVR is of particular importance [START_REF] Mo | Aortic valve replacement improves survival in severe aortic stenosis with gradient-area mismatch[END_REF][START_REF] Osman | Meta-Analysis of Outcomes of Transcatheter Aortic Valve Implantation Among Patients With Low Gradient Severe Aortic Stenosis[END_REF][START_REF] Fischer-Rasokat | 1-Year Survival After TAVR of Patients With Low-Flow, Low-Gradient and High-Gradient Aortic Valve Stenosis in Matched Study Populations[END_REF] . Considering the currently available published data, it becomes clear that among patients with LG-AS and preserved EF, careful evaluation is needed to differentiate patients with "true-severe" AS that lead to discuss AVR from patient with pseudo-severe AS, actually moderate AS that should be managed conservatively [START_REF] Gomez | Severe aortic stenosis patients with preserved ejection fraction according to flow and gradient classification: Prevalence and outcomes[END_REF] . Use of multi-modality imaging permits to improve risk stratification [START_REF] Messika-Zeitoun | Low-Gradient Aortic Stenosis: Solving the Conundrum Using Multi-Modality Imaging[END_REF][START_REF] Delgado | How Do We Reconcile Echocardiography, Computed Tomography, and Hybrid Imaging in Assessing Discordant Grading of Aortic Stenosis Severity?[END_REF] . Measurement of the degree of aortic valve calcification (AVC) using multi-detector computed tomography (MDCT) is an accurate, flow-independent method to assess AS severity including LG-AS [START_REF] Pawade | Computed Tomography Aortic Valve Calcium Scoring in Patients With Aortic Stenosis[END_REF] . Stroke volume assessment by MDCT and Doppler data may be considered to avoid misclassifications in patients with LG-SAS due to the elliptical shape of the LVOT [START_REF] Stahli | Impact of stroke volume assessment by integrating multi-detector computed tomography and Doppler data on the classification of aortic stenosis[END_REF] . Recent data have suggested that calculation of projected AVA derived from preload or dobutamine/exercise stress echocardiography may be useful to predict adverse events in patients with LG-AS and preserved EF [START_REF] Kusunose | Preload Stress Echocardiography Predicts Outcomes in Patients With Preserved Ejection Fraction and Low-Gradient Aortic Stenosis[END_REF][START_REF] Clavel | Stress echocardiography to assess stenosis severity and predict outcome in patients with paradoxical low-flow, low-gradient aortic stenosis and preserved LVEF[END_REF] . However, access to multi-modality imaging is not widely open and calculation of projected AVA may be cumbersome in daily clinical practice. Beyond conventional indices of AS severity, new echocardiographic parameters, easy-to-measure and reproducible, are needed to guide routine practice.

The AT/ET ratio reflecting ejection dynamics through the valve, is a convenient, angle-independent, and reproducible parameter. Landmark AS studies introduced before the emergence of two-dimensional echocardiography have shown that moderate AS has fast upstroke and slow down-stroke; in contrast, severe calcified AS has slower up-stroke resulting in an aortic flow with rounded contour. Rapid early-systolic opening of normal the aortic valve on Doppler spectrograms is replaced by a slow end-systolic opening of the stenotic aortic valve [START_REF] Bermejo | Flow dynamics of stenotic aortic valves assessed by signal processing of Doppler spectrograms[END_REF] . However, although guidelines suggest that the aortic waveform shape could be useful to assess severity of native AS, until recently a few attention had been given to ejection dynamics parameters in the setting of these patients. We recently shown that prolonged AT/ET is a valuable parameter to predict mortality in unselected patients with at least moderate AS, with a prognostic cut-off value of 0.36 9 . Sato et al. have recently shown that prolonged time between left ventricular and aortic systolic pressure peaks measured by left cardiac catheterization is associated with SAS according to aortic valve calcification scoring, including a subgroup of patients with LG-SAS [START_REF] Sato | Association of Time Between Left Ventricular and Aortic Systolic Pressure Peaks With Severity of Aortic Stenosis and Calcification of Aortic Valve[END_REF] . To the best of our knowledge, no study has specifically studied the relationship between AT/ET ratio and outcome in patients with LG-SAS and preserved EF. In the present study, we observed that patients with LG-SAS and an AT/ET > 0.36 had a 2 fold increase of all-cause mortality and nearly 3 fold increase of cardiac mortality compared to patients with AT/ET ≤ 0.36. Interestingly, the AT/ET ratio performed well in subgroups of patients with LG-SAS and preserved EF, with no interaction between its prognostic power and sex, low flow status (defined by a stroke volume index <35 mL/m²), presence of AS-related symptoms, hypertension or atrial fibrillation.

AT/ET measurement is probably less angle-dependent than Doppler velocities. Hatle et al demonstrated that AT/ET well correlates with peak pressure gradient obtained by catheterism. [START_REF] Hatle | Non-invasive assessment of aortic stenosis by Doppler ultrasound[END_REF] Interestingly, in Hatle's report, some AS patients had an underestimation of peak pressure gradient compared with cardiac catheterism, but had a prolonged AT/ET ratio, suggesting that AT/ET may be effectively measured if the maximal velocity cannot be obtained by Doppler. However, as AS assessment primarily relies on the highest aortic jet velocity and gradients, we recommend to measure AT/ET on the corresponding acoustic window.

Our study has several limitations. Whereas echocardiograms were prospectively collected, follow-up data were obtained retrospectively; hence, our study presents inherent limitations of this type of analysis. The specific indications for surgery during follow-up were not collected in our database. However, diagnosis and follow-up were performed by cardiologists with expertise in valvular disease, and the surgical decisions were made by the heart team with the approval of the patients' physicians in accordance with current practice guidelines. Although it has been previously shown that assessing aortic valve calcification by multidetector computed tomography was of interest to stratify risk of patients with AS, this information was not available in our study.

In conclusion, our study shows that AT/ET ratio is independently predictive of allcause and cardiac mortality in patients with LG-SAS and preserved left ventricular ejection fraction and permits to identify a high risk subgroup of patients with a very poor outcome.

Figure 1 .

 1 Figure 1. Systolic time interval measurement on a continuous-wave Doppler recording of

Figure 2 .Figure 3 .Figure 4 .

 234 Figure 2. Kaplan Meier estimates for all-cause (A). and cardiac (B) mortality according to

Figure 5 .

 5 Figure 5. Hazard ratio and 95% confidence interval for risk of all-cause mortality associated

,

  Figure 3A).When replacing AT/ET > 0.36 by quartiles of AT/ET (AT/ET< 0.29, 0.29 to 0.32, 0.32 to 0.36 and > 0.36) in this multivariate model, only patients with an AT/ET > 0.36 had a significant increased risk of all-cause mortality (adjusted HR 2.44 [95% CI, 1.44-4.02]; p<0.001) compared to patients in the first quartile (AT/ET < 0.29). Adjusted all-cause mortality was similar in the other quartiles (AT/ET 0.29 to 0.32 and 0.32 to 0.36) compared with patients in the first quartile (p=0.204 and p=0.369 respectively). Replacing AVA by MPG or aortic peak velocity in this multivariate model did not alter the strength of the relationship between AT/ET > 0.36 and all-cause mortality (adjusted HR 1.92 [95% CI, 1.26-

	2.92]; p=0.002, and adjusted HR 1.93 [95% CI, 1.27-2.94]; p=0.002, respectively). AT/ET >
	0.36 provided incremental prognostic value over clinical and echocardiographic variables
	(Figure

Table 1 .

 1 Baseline demographic and clinical characteristics of the study patients with low gradient severe aortic stenosis according to AT/ET threshold ≤ or > 0.36

	Variable		AT/ET threshold
		All	≤ 0.36	> 0.36	P-value
		(N=356)	(N=281)	(N=75)
	Age (years)	81.0 [74.8;87.0] 81.0 [74.0;87.0]	83.0 [76.5;87.0]	0.397
	Women	206 (58%)	168 (60%)	38 (51%)	0.197
	Body mass index (kg/m 2 )	27.3 [24.2;31.7] 27.5 [24.1;32.3]	26.9 [24.2;29.4]	0.318
	Body surface area (kg/m 2 )	1.85 [1.69;2.00] 1.85 [1.68;2.02]	1.84 [1.73;1.95]	0.737
	Systolic blood pressure (mm Hg)	140 [121;150]	140 [123;150]	130 [120;149]	0.098
	Diastolic Blood Pressure (mm Hg)	73.0 [61.0;80.0] 75.5 [62.2;80.0]	70.0 [60.0;80.0]	0.332
	Heart rate (beats/min)	75.0 [66.0;87.0] 74.0 [65.0;86.0]	75.0 [68.0;87.0]	0.553
	Aortic Stenosis-related symptoms	103 (31%)	75 (28%)	28 (38%)	0.136
	History of hypertension	268 (75%)	213 (76%)	55 (73%)	0.772
	Diabetes mellitus	130 (36%)	107 (38%)	23 (31%)	0.294
	Dyslipidemia	189 (53%)	152 (54%)	37 (49%)	0.546
	Documented coronary artery disease	112 (31.5%)	81 (29%)	31 (41%)	0.053
	History of atrial fibrillation	128 (36.0%)	101 (36%)	27 (36%)	1.000
	Charlson comorbidity index	2.00 [1.00;3.00] 2.00 [1.00;3.00]	2.00 [1.00;3.00]	0.926
	Data are expressed as median (IQR) or number (percentage).
	Dyslipidemia was considered in case of hypercholesterolemia (defined as cholesterol level greater than 200
	mg/dL and/or LDL-C level greater than 100 mg/dL), hypertriglyceridemia (triglyceride level greater than 150
	mg/dL) or in case of hypolipidemic treatment

Table 2 .

 2 Echocardiographic characteristics of the study patients with low gradient AS according to AT/ET threshold of 0.36

	Variable

*: data available in 269 (76%) patients

Table 3 .

 3 Results of Cox multivariate analysis: relationship between AT/ET ratio > 0.36 and all-cause and cardiac mortality.

	All-cause mortality	Cardiac mortality


Xusheng Hu

Feng Zhu

Xiaolu Gong
email: gong@utt.fr

Experimental and numerical study on the thermal behavior of phase change material infiltrated in low porosity metal foam

Keywords: Phase change material, Low porosity metal foam, Numerical simulation, Modified Kelvin model, Thermal behavior

In this paper, a comprehensive experimental and numerical investigation is conducted to study the thermal behavior of phase change material (PCM) infiltrated in low porosity metal foam (LPMF). A visible experiment is designed to trace the temperature field and solid-liquid melting evolvement of PCM with and without metal foam. The experimental method and pore-scale numerical simulation are employed to extract temperature histories at different location during the melting process. Besides, the effective thermal conductivity of composite PCM is calculated by numerical simulation and compared with the theoretical models. Experimental results indicate that the thermal behavior of PCM can be visibly enhanced by embedding LPMF, e.g., by comparing with pure paraffin, the melting time of composite PCM is reduced by 45% and the maximum temperature difference is decreased by 83.3%. The porosity has influences on the thermal characteristics of composite PCMs. Numerical results exhibit good agreement with experimental data, which indicates that modified Kelvin model can be employed to predict the thermal performance of composite PCM. Also it can be found from numerical results that effective thermal conductivities of composite PCMs are distinctly heightened, e.g., the effective thermal conductivities of composite PCM with 67% porosity is about 108 times as much as that of paraffin. Numerical model is of great significance for study and design of thermal management system using composite PCM.

Introduction

In recent years, the global energy crisis is increasingly severe. The renewable energy resource and energy saving technology attract extensive attention, such as solar energy [START_REF] Tian | A review of solar collectors and thermal energy storage in solar thermal applications[END_REF], energy efficient building [START_REF] Borreguero | Polymeric-SiO2-PCMs for improving the thermal properties of gypsum applied in energy efficient buildings[END_REF], electric vehicles [START_REF] Tie | A review of energy sources and energy management system in electric vehicles[END_REF], etc. Thermal management system (TMS) plays an important role in the new energy utilization [START_REF] Rao | A review of power battery thermal energy management[END_REF]. Phase change material is widely applied in passive or hybrid passive thermal management system owing to the high latent heat and steady phase change temperature during the energy storage [START_REF] Shi | Non-steady experimental investigation on an integrated thermal management system for power battery with phase change materials[END_REF][START_REF] Wu | Experimental investigation on the thermal performance of heat pipe-assisted phase change material based battery thermal management system[END_REF]. However, the disadvantage of PCM is the lower thermal conductivity, which reduces heat transfer rate and hinders its application in TES [START_REF] Yang | Solidification of fluid saturated in open-cell metallic foams with graded morphologies[END_REF]. Many methods are developed to heighten its thermal behavior, such as inserting metal fins and pins [START_REF] Ji | Non-uniform heat transfer suppression to enhance PCM melting by angled fins[END_REF][START_REF] Arshad | Thermal performance of phase change material (PCM) based pin-finned heat sinks for electronics devices: Effect of pin thickness and PCM volume fraction[END_REF], encapsulated PCMs [START_REF] Zhao | Review on microencapsulated phase change materials (MEPCMs): Fabrication, characterization and applications[END_REF][START_REF] Memon | Utilization of macro encapsulated phase change materials for the development of thermal energy storage and structural lightweight aggregate concrete[END_REF], adding high thermal conductivity additive [START_REF] Li | Enhancement of heat transfer for thermal energy storage application using stearic acid nanocomposite with multi-walled carbon nanotubes[END_REF][START_REF] Li | Heat transfer characteristics of phase change nanocomposite materials for thermal energy storage application[END_REF] and embedding metal foam [START_REF] Yao | Pore Scale Investigation of Heat Conduction of High Porosity Open-Cell Metal Foam/Paraffin Composite[END_REF][START_REF] Ren | Investigation of the effect of metal foam characteristics on the PCM melting performance in a latent heat thermal energy storage unit by pore-scale lattice Boltzmann modeling[END_REF][START_REF] Hu | Pore-scale numerical simulation of the thermal performance for phase change material embedded in metal foam with cubic periodic cell structure[END_REF]. It is proved that the embedding metal foam is an effective approach for the improvement of thermal performance of PCM.

The high porosity metal foam (HPMF) is usually made by the electro-deposition method and investment casting method, which possesses high porosity (the porosity ranges from 80% to 99%) and permeability [START_REF] Liu | Introduction to porous materials[END_REF]. Bhattacharya et al. [START_REF] Bhattacharya | Thermophysical properties of high porosity metal foams[END_REF] performed a study on thermophysical properties of HPMF by using the analytical and experimental approach. Lafdi et al. [START_REF] Lafdi | Experimental study on the influence of foam porosity and pore size on the melting of phase change materials[END_REF] designed an experiment to study the influences of porosity and pore size on the melting evolvement of PCM saturated in aluminum foam with porosities ranging from 88.4% to 96%. Yang et al [START_REF] Yang | Role of porous metal foam on the heat transfer enhancement for a thermal energy storage tube[END_REF][START_REF] Yang | Thermal response of annuli filled with metal foam for thermal energy storage: An experimental study[END_REF]. carried out the experimental and numerical investigation for influences of PCM saturated in metal foam on the thermal energy storage unit, and they demonstrated that embedding metal foam can distinctly strengthen heat transfer and improve temperature uniformity. Xiao et al. [START_REF] Boomsma | On the effective thermal conductivity of a three-dimensionally structured fluid-saturated metal foam[END_REF] prepared composite PCM which is made of metal foam and paraffin adopting the vacuum impregnation method and performed an experimental test on the thermal characterization of composite PCM with porosities larger than 95%.

The low porosity metal foam (LPMF) can be manufactured by the infiltration casting method. Its porosity is lower and usually ranges from 60% to 80%, but it possesses the high effective thermal conductivity and large specific area. Fernández-Morales et al. [START_REF] Fernández-Morales | Thermal and permeability properties of metal aluminum foams for functional applications[END_REF] implemented experiments of heat transfer and pressure drop to determine the permeability and thermal conductivity of metal foam with 63.3-67.6% porosities, which indicates LPMF has the potential for application in the heat exchangers, electronic cooling systems, etc. The permeability of metal foam was obtained using the experimental method and "bottleneck" model by Furman et al. [START_REF] Furman | Permeability of Aluminium Foams Produced by Replication Casting[END_REF]. They found that theoretical model exhibited well with experimental data. Whereas relative to the HPMF, study for LPMF are still very few.

Many models were developed for predicting the thermal behavior of metal foam. Krishnan et al. [START_REF] Krishnan | Simulation of Thermal Transport in Open-Cell Metal Foams: Effect of Periodic Unit-Cell Structure[END_REF] and Annapragada et al. [START_REF] Annapragada | Permeability and thermal transport in compressed open-celled foams[END_REF] performed the numerical simulation of thermal transport for open-cell metal foam by using face-centered-cubic (FCC) model. Krishnan et al. Boomsma et al. [START_REF] Boomsma | On the effective thermal conductivity of a three-dimensionally structured fluid-saturated metal foam[END_REF] developed tetrakaidecahedron model which is employed to calculate the effective thermal conductivity of a saturated metal foam. [START_REF] Krishnan | Direct Simulation of Transport in Open-Cell Metal Foam[END_REF] proposed the body-centered-cubic (BCC) according to the intricate geometry of metal foam and carried out the numerical simulation for effective thermal conductivity, pressure drop and local heat transfer coefficient. Whereas, these models are mainly used to numerically simulate thermophysical behavior of HPMF not for LPMF.

From the above literature, it can be found that HPMF has been subjected to intensive study. However, the study of LPMF is not enough. Particularly, the thermal behavior for LPMF filled with PCM is rarely studied at present, and the model used to predict the thermal performance of LPMF is also rarely proposed and investigated. Thus we combine the experimental and numerical approach to investigate the thermal behavior of PCM infiltrated in LPMF including the solid-liquid interface, temperature field, temperature history and effective thermal conductivity. The modified Kelvin model used for LPMF is developed and validated, which is of great significance for the study and design of TMS using composite PCM.

Description of experiment

Sample

In the present study, the low porosity aluminum foam is manufactured in University of Technology of Troyes by the infiltration casting method and selected as skeleton material. The infiltration casting method mainly consists of three steps: (1) the molten aluminum is poured into the mold which is filled with salt particle; (2) the liquid aluminum is infiltrated into the space of salt particle with the negative pressure condition; (3) the aluminum/salt composite is immersed in the water to remove the salt from the aluminum foam after the composite is cooled to room temperature. The rectangular aluminum foam samples of dimensions 60 45 25 × × mm are prepared for experiments, as displayed in Fig. 1. Aluminum foam with 67% and 75% porosities and 10 PPI (pores per inch) pore density is used as experimental samples, in which the porosity is determined by the mass-volume direct calculation = 1 ) 100%

f f m V ε ρ - × (
[17] and pore density is determined by the particle size of salt [START_REF] Ashby | Metal foams: a design guide[END_REF]. The paraffin R56 (provided by Merck Millipore) is used for experiment owing to the low melting point and high latent heat. The latent heat and transition temperature of paraffin are tested by differential scanning calorimeter (DSC) with the heating and cooling rate of 5 ℃/min. The DSC curve is displayed in Fig. 2. By analyzing the DSC curve, the melting range and latent heat of paraffin are determined. The thermophysical porosities of paraffin and aluminum are shown in Table 1.

Experimental apparatus

The schematic diagram of experimental setup used in this research is presented in Fig. 3. The setup is designed to obtain the temperature variation and capture phase transition evolvement. The rectangular aluminum foam ( 60 45 25 × × mm) is used for the experiment. A container is made by using transparent acrylic glasses and aluminum plate in order to hold the composite PCM and pure paraffin. The bottom of the container is the aluminum plate, which is attached with an electric heating film by using silicon paste to reduce the thermal contact resistance. Electric heating film used as heat source can provide the constant and uniform heat flux by controlling DC power. The top of the container is a thin aluminum sheet exposed to ambient air, which presents actual convective boundary condition in engineering applications. An observation window is designed by using optical glass as the front face of container to obtain infrared images, which is owing to that optical glass can availably transmit infrared of wavelength range of IR camera. The other faces consist of transparent acrylic glasses of thickness 2 mm. The front-back (z axis) faces of experimental setup are exposed to the environment, which conduces to the visualization of the melting evolution by using IR and HD camera. The polyurethane foam as insulation block is pasted on the left-right (x axis) faces of experimental setup to reduce the heat loss. Three T-type thermocouples are calibrated and fixed within composite at three different location (TC#1 (x=30 mm, y=10 mm), TC#2 (x= 30 mm, y=20 mm) and TC#3 (x=30 mm, y=30 mm)) along the symmetry axis of z direction to trace the temperature variation within samples during the melting process. The data acquisition is used to automatically trace the temperature data from thermocouples and send the temperature data to the computer at regular time intervals. The snapshots of temperature field and solid-liquid melting evolution is extracted using the infrared camera and high definition camera. To investigate the thermal behavior enhancement of composite PCM, the rectangular pure paraffin sample of dimensions ( 60 45 25 × × mm) is also tested for comparison.

In this experiment, the uncertainty analysis is performed. The uncertainty is mainly attributed to the errors of power, thermocouple and filling volume. The uncertainty of power ( E ∆ ) caused by DC power and heating film is 3.2%, which is provided by the manufacturer. The uncertainty of temperature ( T ∆ ) associated with thermocouples is 0.1 ℃. The uncertainty of paraffin filling volume ( V ∆ ) is 4.1%, which is caused by the void within composite PCM. It can be calculated by [START_REF] Xiao | Preparation and thermal characterization of paraffin/metal foam composite phase change material[END_REF]

: = 1 ) 100% f p m V V ε ρ ∆ ∆ - × ( (1)
Where, m ∆ is the mass difference between composite PCM and aluminum foam.

Numerical investigation

Numerical model

The reconstruction of the intricate metal foam structure plays an important role in the pore-scale numerical simulation. There are several familiar models used to numerically simulate the thermal behavior of metal foam, such as tetrakaidecahedron, FCC and BCC etc. However, FCC and BCC models are not completely open-cell when porosity is below a certain limit, e.g., minimum porosity of experimental sample is 67%, and it is difficult to build by using FCC and BCC models. The modified Kelvin model was used to analyze the mechanical property of metal foam by Su et al [START_REF] Su | Structural modeling and mechanical behavior of Metal-Porous-Polymer-Composites (MPPCs) with different polymer volume fractions[END_REF]. It is proved that the modified Kelvin model can represent the property of metal foam. Also, the modified Kelvin model can remain the open-cell structure as long as the porosity is greater than 50%. In present study, the modified Kelvin model is employed to simulate the thermal performance of LPAF. A tetrakaidecahedron is modeled to represent the salt particle. A unit cell of modified Kelvin model is created by cutting the tetrakaidecahedron from a solid cube, as presented in Fig. 4(a). To analyze the thermal behavior of metal foam using the pore-scale numerical method, the periodic distributed modified Kelvin cell is used to build the topology structure of aluminum foam. The dimensions of topology structure of aluminum foam are 60 45 25 × × mm, which is equal to the dimensions of experimental sample, as shown in Fig. 4(b). The side length of solid cube is l and the distance between two opposite faces of the tetrakaidecahedron is a. The porosity of metal foam can be obtained by the following equation: Numerical model is modeled according to the actual size of experimental samples to be used for simulation, which has a high requirement for computer performance and will consume too much computing time. Hence we conduct a study on the influence of number of cells in the x, y and z axis on thermal performance simulation. It is noted that the number of cells in the x axis has no influence on numerical results, which owes to the periodicity and symmetry of geometry and heat boundary condition. The number of cells in the y and z axis is decided according to the actual size of experimental samples, e.g., the length of b and d are 25mm and 45 mm respectively in this paper. Therefore, the computational zone consists of a single slice composite PCM model in the left-right (x axis) direction to save the computing time, and paraffin (white) is filled into the cave of modified Kelvin cell to simulate the thermal behavior of composite PCM, as presented in Fig. 5

3 3 2 9 3 = 100% 16 l a l a ε   --     ×       (2)
+ ( )=0 p p u t ρ ρ ∂ ∇⋅ ∂ r (3)
Momentum equation:

( ) ( ) ( ) ( ) ) T p p p p p m u u u pI u u g T T Su t ρ ρ µ µ ρ γ ∂   + ⋅∇ = ∇ ⋅ -+ ∇ + ∇ + - +   ∂ r r r r r r r (4)
Where, p is the pressure within composite PCM, I r is a unit vector in x, y, and z directions.

The S in Eq. ( 4) is the parameter of damping force term for momentum equation, S can be defined as [31]:

( ) 2 mush 3 1 l l f S A c f - = + (5)
Where, mush A is the mushy zone constant, which can be employed to measure the amplitude of the damping and set to ( ) ( )

1 2 1 0 / 1 l f T T T T   = - -    1 1 2 2 > T T T T T T T < < < (6)
The heat transfer within composite PCM includes heat conduction, convective heat transfer and natural convection during the phase change process. Energy equation of heat transfer is given as follows:

The convective heat transfer for liquid PCM can be defined by: ( )

p p p p p p p p p T f c c u T k T L t t ρ ρ ρ ∂ ∂ + ⋅ ∇ = ∇ ∇ - ∂ ∂ r (7)
For heat transfer within metal foam, energy transport can be defined by: ( )

f f f f f T c k T t ρ ∂ = ∇ ⋅ ∇ ∂ (8)
Conjugate heat transfer of the contact surface between PCM and metal foam can be written as follows:

p f T T = (9) p f p f T T k k n n ∂ ∂ = ∂ ∂ (10)

Initial and boundary conditions

The initial temperature of composite PCM is the constant, which is the same as the ambient temperature ( 0 20 T = ℃). It is written as follow:

0 , p f T T T = = 0 ≤ x ≤l, 0 ≤ y ≤ d, 0 ≤ z ≤ b (11)
The boundary conditions of composite PCM can be observed in Fig. 5. The bottom face of model is imposed with constant heat flux q. The top and front-back faces of model are the convective boundary condition. The other two faces are the symmetry boundary condition. The boundary condition used for governing equations can be defined as follows:

, T k q x ∂ - = ∂ y = 0, 0 ≤ x ≤l, 0 ≤ z ≤ b (bottom face) (12) 1 0 ( ), w T k h T T x ∂ - = - ∂ y = d, 0 ≤ x ≤l, 0 ≤ z ≤ b (top face) (13) 2 0 ( ), w T k h T T x ∂ - = - ∂ z = 0 or d, 0 ≤ x ≤l, 0 ≤ y ≤ d (
Where, Tw is the face temperature of composite PCM, h is the heat transfer coefficient of natural convection caused by the temperature difference between the face and ambient air, which can be estimated by the empirical correlations [START_REF] Yang | Heat transfer[END_REF]:

( )

1/5 air 1 0.62k h GrPr D = (16)
( )

1/4 air 2 0.59k h GrPr D = (17)

Computation setup

CFD Fluent 18.0 package is employed in numerical calculation. PISO algorithm is selected to couple the velocity-pressure term. Governing equations of momentum and energy are discretized employing second order upwind schemes and pressure equation is discretized adopting PRESTO approach. The high-performance computer and parallel processing are used for the present model considering a large number of grids. The computational domain of composite PCM is divided by ICEM CFD 18.0. The tetrahedron element is employed to mesh the domain of metal foam and PCM. The tetrahedron unstructured grid is conducive to the division of intricate structure for composite PCM. The mesh independence analysis is implemented to verify the precision of numerical results under the same boundary conditions. Three different numbers of grids (coarse grid of 3126k cells, medium grid of 5243k cells and fine grid of 9266k cells) are tested. The numerical results indicate that the difference of three grids is within 1%. Considering the accuracy of simulation and reducing the computation time, 5243k grids are selected, which can ensure mesh independence.

Results and discussion

Experimental results

To investigate melting evolution, the infrared camera is used to record the temperature variation of PCM. The infrared images of pure paraffin are displayed in Fig. 6, in which white presents the liquid phase and red presents the solid-liquid interface. The bottom of pure paraffin is heated by the horizontal heat source with heat flux 3500 W/m 2 . At the initial stages of melting t = 800 s, the temperature of pure paraffin near heat source is higher than other location, while the most zones of pure paraffin are close to room temperature 20 ℃. This phenomenon is because a large amount of heat concentrates on the bottom owing to the low thermal conductivity of pure paraffin. The solid-liquid interface is parallel to the heat source, which indicates that the heat conduction dominates heat transfer at the beginning of melting. As time progresses, the interface gradually thickens and appears wavy owing to the effect of natural convection. The hot liquid near heat source flows upward and impinges on the interface owing to the action of buoyancy, which is conducive to the heat exchange between liquid paraffin and solid paraffin. Natural convection gradually dominates the melting evolvement of pure paraffin. It is noted that the upper middle domain of pure paraffin melts fastest as shown in Fig. 6(d). While the left and right boundary zones of pure paraffin melt slower and temperature is lower, which is due to the heat loss from transparent acrylic glasses. Heat conduction and natural convection have combined effects on the melting process. To distinctly display the melting evolvement of low porosity composite PCM, infrared images of composite PCM at difference time are presented in Fig. 7. Fig. 7(a) shows composite PCM which consists of the low porosity aluminum foam and pure paraffin. The temperature of composite PCM rapidly rises, and the average temperature is about 35℃ at t=500 s, which is owing to high thermal conductivity of LPMF. Also it is observed that there is temperature difference between aluminum skeleton and paraffin (observed by the black circle), which can demonstrate the existence of local thermal non-equilibrium phenomenon. As the melting progresses, the heat is quickly transferred between aluminum foam and paraffin, and the temperature difference between the bottom and top domain within composite PCM is about 20 ℃ at t = 750 s, which is smaller than 38 ℃ for pure paraffin at t = 800 s. To study the influences of porosity on thermal behavior, the snapshots of melting evolution for different porosities composite PCMs are obtained by directly capturing from the transparent acrylic glasses, as shown in Fig. 8. It is found that there is no obvious solid-liquid interface within composite PCM with 67% porosity. This is because effective thermal conductivity of composite PCM increases as the porosity decreases, which conduces to quick heat exchange between aluminum ligament and paraffin within the whole domain of composite PCM. Conversely, there is a relatively distinct interface within the composite with 75% porosity, which results from the combined action of natural convection and heat conduction. It is found from pictures that the melting time becomes longer as the increase of aluminum foam porosity, e.g., the paraffin within composite with 67% porosity has completely melted compared to composite with 75% porosity at 2460 s. By comparing with pure paraffin, the melting time of composite PCMs with 67% and 75% porosity is reduced by 45% and 36%, respectively. The temperature difference between TC#1 and TC#3 during the melting process is presented in Fig. 9. The temperature difference curves of composite PCM for 67% and 75% porosities have a similar trend. The temperature difference firstly rises and then decreases. Moreover, it rises and then decreases again. The temperature difference of composite PCM gradually rises until it reaches the first maximum value, which is because a large amount of heat is accumulated at bottom domain in the vicinity of heat source in the initial stage of heating. Then the heat is transferred quickly from the bottom domain to top domain of composite PCM through the high conductivity metal skeleton, which results in that temperature difference reduces a few degrees and reaches a minimum value at the solid phase for paraffin. When the paraffin at the location TC#3 starts to melt, its temperature keeps a steady value. However, the temperature at the location TC#1 still continues to rise. Hence, the temperature difference between TC#1 and TC#3 reaches the second maximum value. Furthermore, it is observed that the temperature difference of 67% porosity composite is higher than that of 75% porosity composite in time ranging from 1690 s to 2000 s. This phenomenon is due to that paraffin within 67% porosity composite at TC# 3 location melts earlier, which results in that the increase rate of its temperature difference is larger than that of 75% porosity composite. Thus temperature difference of 67% porosity composite is higher. The temperature difference presents a slight decline after paraffin at the top domain melts. The maximum temperature difference of pure paraffin is 22.8 ℃ during the melting process, which is caused by low thermal conductivity of pure paraffin. The heat cannot be transferred quickly within pure paraffin, which results in the bigger temperature difference. Whereas the maximum temperature difference of composite PCM with 67% porosity is 3.8 ℃ and is reduced by 83.3% compared with pure paraffin. It can be confirmed that the temperature field within composite PCMs is the more uniform and thermal performance of PCM can be heightened by embedding low porosity aluminum foam.

Numerical results

The numerical results of modified Kelvin model are validated with the experiment data, in which aluminum foam with 75% porosity is impregnated with paraffin wax R56. The same constant heat flux 3500 W/m 2 and boundary conditions as the experiment are supplied to the numerical model. The temperature comparison of numerical results and experimental data at three measurement points (TC#1, TC#2 and TC#3) is presented in Fig. 10. It is noted that the temperature variation of numerical simulation compares well with experimental data before the paraffin begins to melt. The numerical results are slightly larger than experimental data during the melting process. Also it is observed that the temperature rise rate of composite PCM is slower than before when the paraffin starts to melt, which is attributed to a large amount of energy stored in paraffin by the latent heat. It is demonstrated that modified Kelvin model can be applied to theoretically predict thermal characteristic of composite PCM with LPMF. The computational zone of numerical simulation is made up of a single slice composite PCM model in the x axis direction, as mentioned in section 3.1. To show the comparison of temperature distribution and infrared image, the model of composite PCM is cut into two part along the symmetry plane of x direction. The comparison of temperature distribution and infrared image at t = 1000 s is displayed in Fig. 11, in which the porosity of aluminum foam is 67%. It is found that the numerical result of temperature distribution is in good agreement with infrared image. This indicates that the modified Kelvin model can be used for pore-scale numerical simulation of composite PCM infiltrated in LPMF. Accurate information on heat transfer and temperature distribution within composite PCM is critical for the design of TMS. The effective thermal conductivity is an important parameter and required in numerical models of thermal behavior for composite PCM. In this paper, the effective thermal conductivities of composites with different porosities are evaluated by the pore-scale numerical simulation in the steady state, based on the above successful demonstration of modified Kelvin model. The top and bottom walls are applied to the constant temperature 290 K and 310 K, respectively. All other walls are insulation, as displayed in Fig. 12(a). In present numerical simulation, two assumptions are recommended:

(1) the heat transfer in composite is only heat conduction and one dimension; (2) the heat flux for any cross section is constant and equal along the direction of heat conduction (y direction).

The effective thermal conductivity λ is calculated according to Fourier's equation:

d d T Q A x λ = - (18)
Upon convergence of the numerical solution, the heat flux over model under the constant temperature gradient can be extracted by integrating heat flux through top face or bottom face. As the heat flux and temperature gradient are obtained, the effective thermal conductivities of composite PCMs are evaluated based on Fourier's law. The comparison of effective thermal conductivity is displayed in Fig. 12(b). The thermal conductivity of paraffin is 0.2 W/(m K) ⋅ . Whereas, the effective thermal conductivities of composite PCMs for 67% and 75% porosities are 21.67 W/(m K) ⋅ and 11.59 W/(m K) ⋅ , which are 108 and 58 times that of pure paraffin. It is proved that thermal conductivities can be obviously heightened by embedding LPMF in paraffin. It is noted that the numerical results of effective thermal conductivities compare well with the theoretical model developed by Boomsma et al, especially for the porosity 70%. It is also indicated that this theoretical model is proposed not only in the high porosity range but also in a certain low porosity range. There is a certain deviation between numerical results of effective thermal conductivities with the model prediction from Bhattacharya et al, which may be because the model prediction is recommended for use in the higher porosity range.

Conclusion

The thermal behavior of PCM with and without LPMF is experimentally and numerically investigated in this paper. The modified Kelvin model is developed to represent composite PCM and numerical results are compared to experimental data to verify the validity of model. Compared to that of pure paraffin, the melting time of composite PCM with 67% porosity can be reduced by 45%. The melting time of PCM infiltrated in LPMF can be significantly shortened, which is conducive to the improvement of efficiency for energy store in thermal management system. Besides, the temperature field is more uniform within composite PCM, e.g., the maximum temperature difference within composite PCM can be reduced by 83.3% compared with pure paraffin. It is found that the porosity has a distinct effect on the thermal performance. For instance, the melting time is extended as the increase of metal foam porosity and solid-liquid interface is not distinct for relatively lower porosity during the melting time. The numerical results of temperature histories and temperature distribution compare well with experimental data, which indicates modified Kelvin model can be used to predict the thermal performance of composite PCMs with LPMF. Also, it can be noted from results that effective thermal conductivities of PCM infiltrated in LPMF are drastically enhanced and numerical results are good in agreement with previous theoretical models. It is demonstrated form above results that the thermal performance of PCM is improved by embedding LPMF and modified Kelvin model is of great significance for study on the thermal behavior of composite PCM.

Fig. 1 .

 1 Fig. 1. Low porosity aluminum foam

Fig. 2 .

 2 Fig. 2. DSC tested result of paraffin wax

Fig. 3 .

 3 Fig. 3. Schematic diagram of experimental setup

Fig. 4 .

 4 Fig. 4. Numerical model of (a) a modified Kelvin unit cell (b) metal foam with periodic modified Kelvin cell

  . The boundary condition of numerical model is the same as the experimental design. The bottom face of model is imposed with constant heat flux. Due to the symmetry of model, the left-right faces are the symmetry boundary condition. A convective boundary condition is applied on the top and front-back (z axis) faces of model to simulation cooling situation by the air convection.

Fig. 5 .

 5 Fig. 5. Pore-scale numerical model of composite PCM with the boundary conditions 3.2. Numerical method The pore-scale numerical model is established based on the following assumptions: (1) The thermophysical characteristics of PCM and metal foam are considered as homogeneous and constant during the phase change process; (2) The liquid phase paraffin is assumed to be incompressible and Newtonian; (3) The volume change of paraffin is ignored within the enclosed container and natural convection caused by buoyancy is simulated based on Boussinesq approximation. Numerical model of composite PCM is enclosed, and there is no material exchange with the external. Hence, continuity equation is represented in Eq. (3):

  l f is the liquid fraction of PCM during the phase change process, as defined by:

  or l, 0 ≤ y ≤d, 0 ≤ z ≤ b (left and right faces)

Fig. 6 .

 6 Fig. 6. Infrared images of pure paraffin at different time (a) t = 800 s (b) t = 1800 s (c) t = 2600 s (d) t = 3300 s

Fig. 7 .

 7 Fig. 7. (a) Experimental sample and infrared images at different time (b) t = 500 s (c) t = 750 s (d) t = 1000 s

tFig. 8 .

 8 Fig. 8. Comparison of melting evolution for different porosities composite PCMs (a) = 67% ε

Fig. 9 .

 9 Fig. 9. Comparison of temperature difference between TC#1 and TC#3 during the melting process

Fig. 10 .

 10 Fig. 10. Comparison of modified Kelvin model and experiment

Fig. 11 .

 11 Fig. 11. Comparison of temperature distribution and infrared image

Fig. 12 .

 12 Fig. 12. (a) Numerical model in steady state (b) comparison of effective thermal conductivity of composite PCM

Table 1 Thermophysical

 1

		characteristics for paraffin and aluminum
	Material	Property			Value
	Paraffin	Specific heat ( J/(kg K) ⋅	)	2100
		Density (	3 kg/m )		840
		Thermal conductivity ( W/(m K) ⋅ )	0.2
		Viscosity ( kg/(m s) ⋅ )		0.003
		Thermal expansion coefficient ( 1 K -)	0.0004
		Latent heat ( kJ/kg )		120.7
		Solidus temperature ( K )	323.75
		Liquidus temperature ( K )	339.65
	Aluminum	Specific heat ( J/(kg K) ⋅	)	963
		Density (	3 kg/m )		2680
		Thermal conductivity ( W/(m K) ⋅	)	160

Acknowledgement

This research is supported by the Chinese Scholarship Council (CSC).