Dwinanto Sukamto
email: dwinanto.sukamto@insa-strasbourg.fr

Monica Siroux
email: monica.siroux@insa-strasbourg.fr

Francois Gloriant
email: francois.gloriant@insa-strasbourg.fr

Hot Box Investigations of a Ventilated Bioclimatic Wall for NZEB Building Façade

Keywords: thermal performance, ventilated bioclimatic wall, air space thickness, air flow rate, hot box

The building sector is the largest consumer of energy, but there are still major scientific challenges in this field. The façade, being the interface between the exterior and interior space, plays a key role in the energy efficiency of a building. In this context, this paper focuses on a ventilated bioclimatic wall for nearly zero-energy buildings (NZEB). The aim of this study is to investigate an experimental setup based on a hot box for the characterization of the thermal performances of the ventilated wall. A specific ventilated prototype and an original thermal metrology are developed. This paper presents the ventilated prototype, the experimental setup, and the experimental results on the thermal performances of the ventilated wall. The influence of the air space thickness and the air flow rate on the thermal performances of the ventilated wall is studied.

Introduction

The building sector is the industry mainly responsible for energy consumption in Europe [START_REF]Bâtiments à Énergie Positive et Réduction Carbone[END_REF]. Reducing the energy consumption and the emission of gases from buildings has become a recent priority in Europe. In this context, France is placing the construction sector at the center of its strategy to meet this challenge [START_REF]Evaluation of the Energy Performance of Building[END_REF]. Nowadays, buildings must ensure low energy consumption and comfortable environment quality. However, the building envelope has one of the largest impacts on building energy consumption. These aspects generate innovative façade design [START_REF] Pflug | Modeling of facade elements with switchable U-value[END_REF][START_REF] Pflug | Potential analysis of a new removable insulation system[END_REF]. In particular, a ventilated façade can be also a response to this approach [START_REF] Gregório-Atem | Opaque Ventilated Façade (OVF) Thermal Performance Simulation for Office Buildings in Brazil[END_REF].

Ventilated façades are used often by architects. In addition to the aesthetic aspect, the primary purpose of this type of façade is to protect the insulation materials by dissipating moisture. A ventilated façade can also provide energy savings. Nowadays, in Europe, the construction of zero-energy buildings (ZEB) and the transformation of existing buildings into nearly zero-energy buildings (NZEB) are expected. For these reasons, a ventilated façade can be an interesting solution for both existing and new buildings.

A ventilated façade is composed of two panes separated by a ventilated cavity. One possible definition of ventilated façade is "A ventilated double facade can be defined as a traditional single façade doubled inside or outside by a second, essentially glazed facade. Each of these two facades is commonly called a skin. A ventilated cavity-having a width which can range from several centimetres at the narrowest to several metres for the widest accessible cavities-is located between these two skins" [START_REF] Loncour | Classification & illustration of facade concepts[END_REF].

There are several different ventilated façade concepts and several classifications [START_REF] Gregório-Atem | Opaque Ventilated Façade (OVF) Thermal Performance Simulation for Office Buildings in Brazil[END_REF][START_REF] Loncour | Classification & illustration of facade concepts[END_REF]. Generally, classifications are based on façade geometry, the type of ventilation of the cavity, and the airflow concept [START_REF] Chan | Review of passive solar heating and cooling technologies[END_REF][START_REF] Lai | Solar façades: A review[END_REF][START_REF] Barbosa | Perspectives of double skin façades for naturally ventilated buildings: A review[END_REF]. The ventilated façade can be continuous or divided into modules. The air layer forms a thermal buffer zone. The façade geometry can be box window type, shaft box type, a corridor façade, or a multi-story double skin façade. The cavity ventilation that occurs through openings in the external and internal skin can be Energies 2021, 14, 1327 2 of 16 natural, mechanical, or hybrid. Possible airflow concepts are as follows: supply air, exhaust air, static air buffer, external air curtain, and internal air curtain [START_REF] Haase | Simulation of ventilated facades in hot and humide climate[END_REF]. The presence of integrated systems (photovoltaic, solar collector) can also be classification criteria. For example, Ibanez-Puy [START_REF] Ibanez-Puy | Opaque Ventilated Façades: Thermal and energy performance review[END_REF] classifies ventilated façades as double skin façades, building integrated photovoltaic, solar chimneys, Trombe wall, solar wall, façade solar collectors, and opaque ventilated façades.

For several years, publications on ventilated walls increased significantly [START_REF] Shameri | Perspectives of double skin façade systems in buildings and energy saving[END_REF][START_REF] Quesada | A comprehensive review of solar facades. Opaque solar facades[END_REF][START_REF] Khanal | Solar chimney-A passive strategy for natural ventilation[END_REF][START_REF] Saadatian | Trombe walls: A review of opportunities and challenges in research and development[END_REF]. Despite the high number of publications on ventilated walls, the influence of the air space thickness cavity on the thermal performances of the ventilated wall was rarely studied. Table 1 provides an overview of different studies on the ventilated façade from the point of view of air space thickness.

A hot box apparatus is used to measure the thermal behavior of large specimens of building materials. Two standard methods are given in the literature: guarded hot box (GHB) and calibrated hot box (CHB) [START_REF] Asdrubaldi | Thermal transmittance measurements with the hot box method: Calibration, experimental procedures, and uncertainty analyses of three different approaches[END_REF][START_REF] Lucchi | Definition of an experimental procedure with the hot box method for the thermal performance evaluation of inhomogeneous walls[END_REF]. The guarded hot box (GHB) is composed of three cells: a hot cell, a cold cell, and a guarded chamber used to minimize lateral heat losses. A calibrated hot box (CHB) is composed only of two cells: a hot cell and a cold cell [START_REF] Lucchi | Definition of an experimental procedure with the hot box method for the thermal performance evaluation of inhomogeneous walls[END_REF]. Different standard hot box experiments can be carried out: the European EN ISO 8990, the American ASTM C1363-05, and the Russian GOST 26602.1-99 [START_REF] Asdrubaldi | Thermal transmittance measurements with the hot box method: Calibration, experimental procedures, and uncertainty analyses of three different approaches[END_REF][START_REF] Lucchi | Definition of an experimental procedure with the hot box method for the thermal performance evaluation of inhomogeneous walls[END_REF]. Many publications deal with the thermal characterization of building materials using a hot box apparatus. However, few papers studied the thermal properties of air cavity wall using a hot box. Aviram [START_REF] Aviram | Thermal properties of a variable cavity wall[END_REF] studied the thermal behavior of a variable ratio cavity wall using a guarded hot box and CFD analysis. Lucchi [START_REF] Lucchi | Definition of an experimental procedure with the hot box method for the thermal performance evaluation of inhomogeneous walls[END_REF] introduces a hot box procedure to study the thermal performance and the air flux in a wall with an internal air cavity. More recently, a new hot box was developed to study the hydrothermal behavior of historic walls [START_REF] Andreotti | Design and Construction of a New Metering Hot Box for the In Situ Hygrothermal Measurement in Dynamic Conditions of Historic Masonries[END_REF].

This paper describes an experimental study on the thermal performance of a ventilated bioclimatic wall. The objective of this study was to develop an experimental setup able to characterize the influence of the air space thickness and the air flow rate on the thermal performances of the ventilated wall. A specific ventilated prototype and an original thermal metrology based on a hot box was developed in this study.

The overall structure of this paper takes the form of four parts, including this introduction. The second part, Experimental Setup, presents the ventilated bioclimatic prototype, the hot box, and the instrumentation involved in the test specimen. The third part presents the experimental results for ambient temperature, surface temperature, heat transfer coefficient, and pre-heating efficiency, which allows for the evaluation of the thermal performance of the ventilated wall. Finally, a conclusion gives a summary of the findings.

Experimental Setup

Bioclimatic Ventilated Wall and Hot Box

The ventilated wall prototype is presented in Figure 1. This prototype consists of two walls and a ventilated air chamber (a cavity between the two walls). The first wall is fixed, while the second one is mobile, allowing the thickness of the ventilated cavity to be varied. These two walls are made of an aluminum alloy and polyethylene. The dimensions are Energies 2021, 14, 1327 3 of 16 1 m high, 1 m wide, and 3 mm thick. Based on the manufacturer's data, the aluminum composite has a thermal resistance of 0.008 m 2 •K•W -1 and has low roughness, with the surfaces being smooth. The ventilated air chamber is a non-hermetically sealed cavity that participates in the supply of fresh air to the building. A convective exchange takes place in the cavity along the axis of the wall, which disturbs the radial heat flow between the interior and exterior environments (Figure 1).

Experimental Setup

Bioclimatic Ventilated Wall and Hot Box

The ventilated wall prototype is presented in Figure 1. This prototype consists of two walls and a ventilated air chamber (a cavity between the two walls). The first wall is fixed, while the second one is mobile, allowing the thickness of the ventilated cavity to be varied. These two walls are made of an aluminum alloy and polyethylene. The dimensions are 1 m high, 1 m wide, and 3 mm thick. Based on the manufacturer's data, the aluminum composite has a thermal resistance of 0.008 m 2 •K•W -1 and has low roughness, with the surfaces being smooth. The ventilated air chamber is a non-hermetically sealed cavity that participates in the supply of fresh air to the building. A convective exchange takes place in the cavity along the axis of the wall, which disturbs the radial heat flow between the interior and exterior environments (Figure 1). The experimental study was carried out in a hot box (Figure 2). This hot box is a device composed of two climatic chambers whose ambiances are controlled. The first climatic chamber reproduced indoor conditions, and the second reproduced outdoor conditions (Figure 2). The hot box is characterized by a range of temperature from -30°C to +60°C and a range of humidity from 10% to 98%. Each climatic chamber has its own refrigeration unit that allows for the production of cold with a temperature accuracy of ± 2 °C. Each climatic chamber also has 3 electric resistors of 2.5 kW that allow for the production of heat with a temperature accuracy of ± 3 °C. The hot box was connected to a computer to control the temperature and humidity of each climatic chamber. These two climatic chambers were separated by a sample holder (size of 1 × 1 × 0.4 m). One of the two chambers was mounted on slides, allowing access to the sample holder as well as to the inside of the two climatic chambers. The ventilated bioclimatic prototype was fixed in the specimen holder. The experimental study was carried out in a hot box (Figure 2). This hot box is a device composed of two climatic chambers whose ambiances are controlled. The first climatic chamber reproduced indoor conditions, and the second reproduced outdoor conditions (Figure 2). The hot box is characterized by a range of temperature from -30 • C to +60 • C and a range of humidity from 10% to 98%. Each climatic chamber has its own refrigeration unit that allows for the production of cold with a temperature accuracy of ±2 • C. Each climatic chamber also has 3 electric resistors of 2.5 kW that allow for the production of heat with a temperature accuracy of ±3 • C. The hot box was connected to a computer to control the temperature and humidity of each climatic chamber. These two climatic chambers were separated by a sample holder (size of 1 × 1 × 0.4 m). One of the two chambers was mounted on slides, allowing access to the sample holder as well as to the inside of the two climatic chambers. The ventilated bioclimatic prototype was fixed in the specimen holder. A ventilation system was used to impose air flow in the cavity. Moving the mobile wall can vary the air space thickness. Experiments were carried out for different configurations to provide variable air space thickness and variable air flow rate. The air space thickness varied from 5 mm to 85 mm. The air flow rate varied from 10 to 30 m 3 •h -1 . Fifty experiments were carried out. An overview of the fifty experiments is given in Table 2.

Thermal Metrology

The instrumentation was composed of K-type thermocouples and PT-100 probes. The hot box ambient temperature was measured in the hot cell and the cold cell with K-type thermocouples and PT-100 sensors (hot box sensors). The hot box instrumentation is presented in Figure 3. The ventilated prototype consisted of two walls and an air cavity. The first wall was fixed while the second one was mobile, allowing the thickness of the ventilated cavity to be varied. The wall surface temperature was measured by 24 K-type thermocouples (12 thermocouples on the fixed wall and 12 thermocouples on the mobile wall). The position of the thermocouples is given in Figures 456. Thermocouple measurements were compared to those obtained using an infrared camera (FLIR T650 SC). The infrared camera was installed in the front of the wall. The area covered by the infrared camera had a surface area of 10 cm 2 . The thermal flux was measured with 8 Captec flux meters (4 flux meters on the fixed wall and 4 flux meters on the mobile wall), allowing for the estimation of heat flux on both sides of the ventilated cavity (Figure 4). The air cavity temperature was measured by 10 K-type thermocouples (4 thermocouples in the cavity, 3 thermocouples at the entrance of the cavity, and 3 thermocouples at the exit of the cavity). To control the airflow velocity inside the air cavity, a convergent plenum was installed (Figure 4). A fan was used to vary the airflow velocity inside the cavity. A CTV 210 hot wire anemometer from the manufacturer KIMO measured the airflow rate. All the Energies 2021, 14, 1327 5 of 16 sensors were connected to a Keithley 2700 data acquisition system. Data acquisition and processing were then carried out via the LabVIEW software (Figure 7). The characteristics of the sensors used are given in Table 3.

Table 3. Thermal metrology.

Type Metrologic Means Range Uncertainty

Air temperature PT-100

Experimental Results

Specific experiments were undertaken to test the hot box and the ventilated wall prototype (Figure 8). Experiments were carried out in steady state under the following conditions:

-Hot cell: ambient temperature conditions of 20 • C and relative humidity (55%); -Cold cell: temperature conditions of 0 • C and relative humidity (75%); -Variable airflow rate: 10, 20, and 30 m 3 •h -1 ; -Variable air space thickness: from 5 to 85 mm.

These specific experiments were carried out to investigate the ventilated wall prototype in terms of its capability to pre-heat the air in the cavity. Our experiments are distinguished from standard hot box methods used for the estimation of the thermal resistance of a wall in steady-state regime by using a calibrated hot box or guarded hot box such as EN ISO 8990.

Ambient Temperature

The ambient temperature was measured in both the hot cell and cold cell with K-type thermocouples and PT-100 sensors (hot box sensors). The temperature difference between the hot cell and the cold cell was 20 • C. This temperature difference allowed us to obtain a heat flux that could be measured correctly by the flux meters [START_REF] Sukamto | Mise en place d 'un dispositif expérimental pour la caractérisation énergétique d'une paroi bioclimatique ventilée[END_REF]. Under the specified conditions (20 • C in the hot cell and 0 • C in the cold cell), a significant difference (2.6 • C for the cold cell and 4.3 • C for the hot cell) was observed between the ambient temperature measured with the hot box sensor and the ambient temperature measured by our thermocouples. This difference can be explained by the fact that the cells are not watertight, since an artificial air flow was created between them. However, for standard use of the hot box, there should not be any air exchange between the cells; the regulation system of the device was thus disturbed.

Figure 9 presents the air temperature in the cold cell and the hot cell. The temperature was measured by six K-type thermocouples (three thermocouples (T42, T43, and T44) in the cold cell and three thermocouples (T39, T40, and T41) in the hot cell). It can be seen in Figure 9 that the air temperature inside the hot box was homogenous in both the cold cell and the hot cell. The mean air temperature was -4.6 ± 0.05 • C for the cold cell and 15.7 ± 0.18 • C for the hot cell.

Surface Temperatures

Measurement of surface temperatures was performed by thermocouples. Thermocouple measurements (T8 and T5 thermocouple measurements) were compared to data obtained using an FLIR 650 TS infrared camera (Figure 10). The area covered by the infrared camera had a surface area of 10 cm 2 . The T8 and T5 thermocouples were located on the upper and lower part of this surface. The mean value measured by thermocouples T5 and T8 and the infrared camera were 12.14 • C, 12.11 • C, and 12.18 • C, respectively. The measurements of the surface temperature performed by the thermocouples are in accordance with the measurements made by the infrared camera and confirm the reliability of the instrumentation of the walls (Figure 11).

The Convection Heat Transfer Coefficient

The convection heat transfer coefficient was estimated from Newton's law, involving the heat flux, the ambient temperature, and the wall temperature:

h = q/(T s -T a ) (1)
The heat flux, the ambient temperature, and the wall temperature were measured using thermal metrology describes before. We used two cavity thicknesses (0.005 and 0.01 m) and three airflow rates (10, 20, and 30 m 3 •h -1 ). Each sample wall was equipped with four flux meters and four thermocouples at different heights, making it possible to obtain four local estimates of the h-factor per cell. For the ambient temperature in each of the cells, a single air temperature was considered, corresponding to the average of the measurements of the three thermocouples installed in each cell. Figures 12 and13 show the local convection heat transfer coefficient h in the cold cell and in the hot cell. The values of the heat transfer coefficient varied between 15 and 25 W.m -2 •K -1 . The mean value of the heat transfer coefficient was 19.9 ± 1.3 W.m -2 •K -1 for the cold cell and 18.7 ± 2.3 W.m -2 •K -1 for the hot cell. This order of magnitude is consistent with the correlations that can be found in the literature [START_REF] Padet | Convection Thermique et Massique Principes Généraux[END_REF]. From Figures 12 and13, we also noted that the thickness of the cavity and the airflow rate had no significant influence on the convection heat transfer coefficient value. This means that these two parameters had no influence on the ambiances of the two climate cells. On the other hand, the coefficient h depends on the position from which it was calculated, and it can be seen that for a given height, the estimates of the coefficient h are relatively homogeneous. It can be thought that, in each of the cells, the air movements caused by the regulation system were not uniform near the walls of the sample, causing local variations in the coefficient h. As depicted in Figure 12, two values are apart for 10 m 3 /h_5 mm. This can be explained by a bad contact between the flux meters and the sample during this test. Finally, the uncertainties for the local convection heat transfer coefficient h were estimated (Table 4). An uncertainty of 5% for the flux meters and an uncertainty of 0.02 • C for the thermocouples were considered. As can be observed from Table 4, the uncertainties are smaller in the hot cell, as the fluxes and temperature differences involved are larger. It is in the cold cell, at the lower part of the wall, that the uncertainties are the greatest; at this point, the temperatures between the wall and the environment of the cold cell are very close, often of the order of one-tenth of a degree.

Thermal Performance of the Ventilated Wall

The objective of this study was to characterize the influence of the air space thickness and the air flow rate on the thermal performances of the ventilated wall. The thermal performance of the ventilated wall was evaluated through the pre-heating efficiency according to Equation (2) "which represent the potentiality of the ventilatd wall in terms of its capability to pre-heat the air in the cavity" [START_REF] Fantuccia | Dynamic insulation systems: Experimental analysis on a parietodynamic wall[END_REF].

E = (T inlet -T out )/(T in -T out ) (2)
where -T inlet : the supply air temperature measured in the top of the air cavity; -T out : the outside air temperature (cold cell); -T in : the inside air temperature (hot cell).

Figure 16 shows the pre-heating efficiency of the ventilated wall. Experiments were carried out for different air space thicknesses (5, 20, 50, and 85 mm) and different air flow rates (10, 20, and 30 m 3 •h -1 ). It can be seen that the pre-heating efficiency of the ventilated wall increased with the air space thickness. For an air flow of 10 m 3 •h -1 , the pre-heating efficiency of the ventilated wall increased from 57% (for an air space thickness of 5 mm) to 65% (for an air space thickness of 85 mm). It can be also seen that the pre-heating efficiency of the ventilated wall decreased with the air flow rate. For an air space thickness of 85 mm, the pre-heating efficiency of the ventilated wall decreased from 65% (for an air flow rate 10 m 3 •h -1 ) to 54% (for an air flow rate 30 m 3 •h -1 ).

Conclusions

This work describes an experimental study on the thermal performance of a ventilated bioclimatic wall. A ventilated wall prototype and an original hot box thermal metrology were developed. This paper presents the ventilated wall prototype, the experimental setup, and the experimental results. The ambient temperature, the surface temperature of the ventilated prototype, and the local heat transfer coefficients of the ventilated wall were measured. In addition, an uncertainty analysis on the local heat transfer coefficient is presented. Experimental results show that vertical position had a significant influence on the local heat transfer coefficient value. Finally, the influence of the air space thickness and the air flow rate on the thermal performances of the ventilated wall was studied. The results show the pre-heating efficiency of the ventilated wall increased with the air space thickness and decreased with the air flow rate. It is suggested that the increase in air space thickness of the ventilated wall can be used to obtain an interesting ventilated façade performance.

Figure 1 .

 1 Figure 1. Ventilated wall.

  A ventilation system was used to impose air flow in the cavity. Moving the mobile wall can vary the air space thickness. recovered by the air direction of air flow between the two cells

Figure 1 .

 1 Figure 1. Ventilated wall.

Figure 2 .

 2 Figure 2. Hot box.

Figure 3 .

 3 Figure 3. Hot box instrumentation. Two climatic chambers are separated by a sample holder. The ambient temperature is measured by 6 K-type thermocouples (3 thermocouples, T42, T43, and T44 in the cold cell and 3 thermocouples, T39, T40, T41 in the hot cell).

Figure 4 .of 16 Figure 5 .

 4165 Figure 4. Ventilated wall instrumentation: wall surface temperature is measured by K-type thermocouples, air temperature is measured by K-type thermocouples, thermal flux is measured by flux meters, and airflow rate is measured by a hot wire anemometer.

Figure 6 .

 6 Figure 6. Fixed wall instrumentation.

Figure 7 .

 7 Figure 7. Data acquisition system.

  -70 • C to + 200 • C 0.1 • C K-type Thermocouples -75 • C to + 250 • C 0.02 • C Surface temperature K-type Thermocouples -75 • C to + 250 • C 0.02 • C Infrared camera FLIR T650 SC -40 • C to + 150 •

Figure 8 .

 8 Figure 8. Experimental setup.

Figure 9 .

 9 Figure 9. Ambient temperature in the cold cell and in the hot cell.

Figure 10 .

 10 Figure 10. Surface temperature measurement by infrared camera.

Figure 11 .

 11 Figure 11. Surface temperature. Thermocouple and infrared camera measurements.

Figure 12 .

 12 Figure 12. Local convection heat transfer coefficient in the cold cell.

Figure 13 .

 13 Figure 13. Local convection heat transfer coefficient in the hot cell.

Figures 14 and 15 17 Figure 14 .

 151714 Figures 14 and 15 show the global heat transfer coefficient in both the hot cell and the cold cell. The global heat transfer coefficient was estimated from temperature measurements (thermocouples) and heat fluxes (flux meters measurements). As we can see in Figures14 and 15, there are small differences between the measurements, which can be explained by the uncertainties of the convection heat transfer coefficient.Energies 2021, 14, x FOR PEER REVIEW 13 of 17

Figure 14 .

 14 Figure 14. Global heat transfer coefficient (cold cell).

Figure 15 .

 15 Figure 15. Global heat transfer coefficient (hot cell).

Figure 16 .

 16 Figure 16. Pre-heating efficiency of the ventilated wall. Experiments were carried out for variable air space thicknesses (5, 20, 50, and 85 mm) and variable air flow rates (10, 20, and 30 m 3 •h -1 ).

Table 2 .

 2 Overview of 50 experiments.

	Test Number	Air Space Thickness (mm)	Air Flow Velocity (m 3 •h -3 )	Cold Cell Temperature T a1 ( • C)	Hot Cell Temperature T a2 ( • C)	Temperature Difference between Hot Cell and Cold Cell ( • C)
	1	5	9.94	-4.34	15.56	19.9
	2	5	19.61	-4.64	15.63	20.27
	3	5	29.99	-4.45	15.82	20.27
	4		9.94	-4.64	15.58	20.23
	5		19.96	-4.72	15.61	20.32
	6		29.99	-4.47	15.8	20.27
	7		10	-4.61	15.58	20.19
	8		19.99	-4.63	15.66	20.29
	9		30.09	-4.41	15.87	20.28
	10		9.98	-4.67	15.62	20.29
	11		20.43	-4.66	15.64	20.29
	12		29.84	-4.41	15.86	20.27
	13		10.02	-4.67	15.61	20.28
	14		20.08	-4.63	15.68	20.31
	15		29.81	-4.43	15.87	20.3
	16		9.9	-4.65	15.66	20.31
	17		19.85	-4.62	15.71	20.33
	18		29.98	-4.32	15.9	20.22
	19		10.22	-4.65	15.63	20.28
	20		19.89	-4.64	15.63	20.27
	21		30.039	-4.32	15.95	20.28
	22		8.98	-4.67	15.66	20.33
	23		20.07	-4.61	15.74	20.35
	24		30.05	-4.29	15.86	20.15
	25		9.04	-4.66	15.64	20.3
	26		20.03	-4.62	15.6	20.22
	27		30.07	-4.40	15.91	20.31
	28		10.01	-4.71	15.59	20.31
	29		20.04	-4.65	15.62	20.27
	30		29.94	-4.45	15.82	20.27
	31		9.99	-4.62	15.66	20.28
	32		20.08	-4.61	15.51	20.11
	33		30.01	-4.37	15.77	20.14
	34		10.01	-4.63	15.59	20.22
	35		19.73	-4.58	15.7	20.28
	36		29.9	-4.19	15.85	20.04
	37		10.01	-4.65	15.52	20.17
	38		20.03	-4.69	15.64	20.32
	39		30.05	-4.45	15.85	20.3
	40		10.07	-4.72	15.58	20.29
	41		20.03	-4.64	15.53	20.16
	42		30.37	-4.37	15.78	20.15
	43		9.98	-4.68	15.47	20.15
	44		20.01	-4.70	15.57	20.27
	45		30.62	-4.38	15.71	20.09
	46		9.98	-4.75	15.47	20.22
	47		20.49	-4.66	15.59	20.25
	48		30.13	-4.46	15.72	20.18
	49		9.99	-4.67	15.49	20.16
	50		20.27	-4.72	15.5	20.22

Table 4 .

 4 Uncertainties for the local convection heat transfer coefficient.

	Vertical Position (m)	Cold Cell	Hot Cell

Cavity of 5 mm for 10, 20, 30 Flow Rate (m 3 •h -1 ) Cavity of 10 mm for 10, 20, 30 Flow Rate (m 3 •h -1 ) Cavity of 5 mm for 10, 20, 30 Flow Rate (m 3 •h -1 ) Cavity of 10 mm for 10, 20, 30 Flow Rate (m 3 •

													h -1 )
		10	20	30	10	20	30	10	20	30	10	20	30
	0.8	1.5	1.5	1.7	1.5	1.4	1.7	1.5	1.5	1.4	1.8	1.8	1.6
	0.56	2.7	2.3	2.7	2.2	2.0	2.4	1.2	1.2	1.1	1.3	1.3	1.2
	0.35	4.6	3.0	3.0	3.1	2.7	3.2	1.5	1.6	1.5	1.6	1.6	1.4
	0.12	5.1	2.8	2.7	3.1	2.9	3.4	1.4	1.3	1.5	1.4	1.5	1.2

Acknowledgments:

The authors acknowledge the financial contribution of the Arconic Foundation and the Ministry of higher education of Republic of Indonesia.

Funding: This research was funded by Arconic Foundation and by the Ministry of higher education of Republic of Indonesia.

Data Availability Statement:

The data presented in this study are available on request from the corresponding author.

Conflicts of Interest:

The authors declare no conflict of interest.

Nomenclature

E

pre-heating efficiency of the ventilated wall

Indices a ambiance of the cell in relative to the inside air temperature (hot cell) inlet relative to the supply air temperature measured in the top of the air cavity out relative to the outside air temperature (cold cell) s wall surface


Madalina Barbu

Monica Siroux
email: monica.siroux@insa-strasbourg.fr

George Darie

Numerical model and parametric analysis of a liquid based hybrid photovoltaic thermal (PVT) collector

Keywords:

A hybrid photovoltaic thermal (PVT) panel is a module in which the photovoltaic (PV) layer is not only producing electricity, but also operates as a thermal absorber. As a result, thermal and electrical energy are being produced simultaneously, operating as a micro-cogeneration equipment. As in any cogeneration system, there is tight link between the electrical and thermal performance and it is dependent on multiple parameters: climate conditions, thermo-physical, geometrical and electrical properties. This paper investigates the effect of the variation of several of these parameters on the electrical and thermal performance, as well as on the global output. In order to achieve this, a dynamic numerical model is proposed, which simulates the heat exchange between the layers of the PVT panel. The model was applied to two different climatic conditions: Bucharest, Romania and Strasbourg, France, in order to assess and compare their behavior and performance. The simulation computes the temperature of each layer at any particular time, and a slightly higher outlet temperature of the working fluid can be observed in Bucharest during the summer, and in Strasbourg during the winter. The model is validated against data from the literature and can be applied to any climatic conditions and adapted for multiple geometrical and thermo-physical configuration. Next, a one-factor-at-a-time parametric analysis is carried out in order to assess the impact of various parameters on the electrical, thermal and global efficiency. The results showed in most cases a compromise between the electrical and thermal performance: in terms of wind speed and insulation, the thermal benefits of low wind and high insulation overcome the decrease in electrical efficiency. The packing factor was found to be optimum when maximized, as the electrical benefits are more significant than the thermal loss. The width of the channels in the heat exchanger should also be maximized as far as technologically possible for best performance.

Introduction

Solar energy is among the most promising renewable sources, playing an essential role in the energy mix of the future. Energy can be extracted from the sun in the form of heat or electricity, by means of the established solar technologies currently available on the market: solar thermal (ST) panels and photovoltaic panels (PV). Several comprehensive reviews on the latest solar technologies has been carried out [START_REF] Ahmadi | Solar power technology for electricity generation: A critical review[END_REF][START_REF] Khatibi | Generation and combination of the solar cells: A current model review[END_REF] comparing and contrasting some of the technologies in terms of economic feasibility and CO 2 emissions.

Hybrid photovoltaic-thermal (PVT) collectors combine and maximize the benefits of these two types of energy extraction technologies into one single equipment, and can produce simultaneously electrical and thermal energy. This allows for a heat collector, the PV cells in PVT systems can have a 4%-12% better performance compared to the stand-alone PV module [START_REF] Ramos | Solar-Thermal and Hybrid Photovoltaic-Thermal Systems for Renewable Heating[END_REF]. By combining the electrical and thermal components, the global efficiency of the system is improved, since the global efficiency η GL can be written the sum between the electrical efficiency η EL and the thermal efficiency η TH . This solution is particularly interesting and cost-efficient for small scale applications, where heat and electricity are required simultaneously, and roof and facade space are limited, especially in urban locations. In addition, several studies (Barbu et al., 2019c;[START_REF] Herrando | Hybrid PV and solar-thermal systems for domestic heat and power provision in the UK: Techno-economic considerations[END_REF] investigate the economic benefits of the PVT systems compared to individual side-byside PV and solar-thermal panels, showing that the PVT panels may achieve an improved payback period and return on investment. The economic benefits are, however, highly sensitive to the energy market price fluctuations and the government incentives available for renewable energy (Feed in Tariffs, Green Certificates).

The PVT panels have been first introduced in the 1970s' [START_REF] Kern | Combined photovoltaic and thermal hybrid collector systems[END_REF] and have been slowly developing ever since, with increased research interest in the last decade. However, the PVT technology is not yet well-established in neither academic research field or on the commercial market. In recent years, the research on PVTs is focused on technology development, modeling, HVAC integration, and optimization. On one hand, research can focus on the development of PVT collectors in order to improve the design solutions and to increase the efficiency of solar energy conversion [START_REF] Papoutsis | Numerical simulation and parametric study of different types of solar cooling systems under mediterranean climatic conditions[END_REF][START_REF] Bombarda | Thermal and electric performances of roll-bond flat plate applied to conventional PV modules for heat recovery[END_REF][START_REF] Guo | Numerically study on a new hybrid photovoltaic thermal (PVT) collectors with natural circulation[END_REF]. On the other hand, the research can investigate the integration of PVT technologies in buildings to assess the operative performances and actual energy-economy benefits [START_REF] Chow | Energy performance of a solar hybrid collector system in a multistory apartment building[END_REF][START_REF] Prabhakant | Return on capital and earned carbon credit by hybrid solar Photovoltaic-wind turbine generators[END_REF][START_REF] Agrawal | Life cycle cost assessment of building integrated photovoltaic thermal (BIPVT) systems[END_REF]. This study falls in the first category, aiming to develop an explicit PVT model that can assess the suitability of various design parameters in any ambient conditions, which can consequently lead to improving performance of the panel.

Various configurations of PVT have been developed, and their performance varies according to multiple parameters: optical, geometrical and thermo-physical properties, type of fluid, wind speed and direction, solar radiation and ambient temperature. A number of models have been identified in the literature for various configurations and climatic conditions. [START_REF] Rejeb | Parameters effect analysis of a photovoltaic thermal collector: Case study for climatic conditions of monastir[END_REF] develops a sheet-and-tube PVT model and simulates its performance in Monastir, Tunisia. [START_REF] Aste | Design, modeling and performance monitoring of a photovoltaic-thermal (PVT) water collector[END_REF] proposes a model to evaluate the daily and annual yield of a thin film roll-bond PVT collector in Milan, Italy. [START_REF] Chow | Energy performance of a solar hybrid collector system in a multistory apartment building[END_REF] develops a seven-node model and simulates a sheet and tube panel.

A study by [START_REF] Brottier | Optimisation Biénergie D'un Panneau Solaire Multifonctionnel : Du Capteur Aux Installations in Situ[END_REF] follows 28 PVT panels installations in France, Portugal and Switzerland. The instantaneous response of a PVT collector to various climatic conditions is analyzed in a paper by [START_REF] Armstrong | A thermal model for photovoltaic panels under varying atmospheric conditions[END_REF]. Some parametric studies have been carried out for establishing the influence of mass flow rate [START_REF] Chow | Energy performance of a solar hybrid collector system in a multistory apartment building[END_REF], thickness of insulation, wind speed and tube spacing [START_REF] Rejeb | Parameters effect analysis of a photovoltaic thermal collector: Case study for climatic conditions of monastir[END_REF]. A box-channel model was developed and the influence of the channel dimensions was assessed in a study by [START_REF] Han | Effect of flow channel dimensions on the performance of a box-frame photovoltaic/thermal collector[END_REF]. Most of these models are based on the same theoretical background, starting from the classical numerical model developed by [START_REF] Florschuetz | Extension of the Hottel-Whillier model to the analysis of combined photovoltaic/thermal flat plate collectors[END_REF], based on the Hottel-Whillier-Bliss equations with various improvements, and the methodology of calculating the temperatures in the PVT layers is based on the exchange of energy between the layers and with the exterior.

A novel approach to simulating the operation of PVT panels is by using machine learning methods of artificial neural networks, least squares support vector machines and neuro-fuzzy. By comparing the performance of the three methods it was found that the least squares support vector machines method performed better than the other two, and is quite suitable for cases when experimental measurements are difficult or costly [START_REF] Ahmadi | Evaluation of electrical efficiency of photovoltaic thermal solar collector[END_REF][START_REF] Zamen | Optimization methods using artificial intelligence algorithms to estimate thermal efficiency of PV/T system[END_REF].

Another parametric study on a solar cooling system was done for the weather conditions of Athens, Greece [START_REF] Tzinnis | Parametric analysis of a solar cooling system designed for athens climate[END_REF], and after optimization it achieved a solar coverage ratio up to 89.85%. An interesting research [START_REF] Nhut | A parametric study of a solar-assisted house heating system with a seasonal underground thermal energy storage tank[END_REF] performed a parametric study of a solar-assisted house heating system coupled with a seasonal underground thermal energy storage tank, evaluating the annual energy production as a function of some of the main system parameters, such as the insulation thickness of the tank and the size of the tank.

Another parametric study was carried out on a grid-connected PV-battery systems investigating the effects of the weather, price of electricity, feed-in tariff, system cost, and electrical specifications from a technical-economical point of view [START_REF] Khalilpour | Technoeconomic parametric analysis of PV-battery systems[END_REF]. Another study [START_REF] Arsalis | Parametric study and cost analysis of a solarheating-and-cooling system for detached single-family households in hot climates[END_REF] investigates a solar-heating-and-cooling system for a single-family household in terms of cost savings.

This paper proposes an explicit dynamic numerical model of a particular configuration of PVT panel (water-based roll-bond collector with Mono-Crystalline Silicon PV cells). The thermal model is able to evaluate the temperatures at each layer inside the collector, taking into account all the heat transfer processes between the layers or with the environment. Based on the output temperatures, the system performance indicators can then be assessed for various operational and constructive parameters. As a case study, the model is applied to the climatic conditions of Bucharest, Romania and Strasbourg, France. The novelty of this paper stands in the easy to follow and replicate methodology for simulating the operation of the panels, along with the comparison of its performance in two climatic conditions. The one-factor-ata-time (OFAT) analysis carried out based on the numerical model reveals some useful design information for improving the performance of the collectors. Other aforementioned parametric studies have been carried out previously [START_REF] Chow | Energy performance of a solar hybrid collector system in a multistory apartment building[END_REF][START_REF] Aste | Design, modeling and performance monitoring of a photovoltaic-thermal (PVT) water collector[END_REF][START_REF] Han | Effect of flow channel dimensions on the performance of a box-frame photovoltaic/thermal collector[END_REF], but the novelty of this research stands in the application of the model in various climatic conditions, besides from the OFAT analysis, and the analysis of not only the electrical and thermal performance, but also of the global and primary energy saving (PES) efficiencies.

Description of PVT collector

The PVT collector can take various configurations: liquid, air or dual liquid-air based, flat or concentrated. The thermal collector can also have multiple geometries (harp, direct flow, fractal) and manufacturing methods (sheet and tube, roll bond, rectangular channels). A typical PVT system is obtained by coupling a solar thermal collector at the back of a PV panel. A comprehensive review is carried out by [START_REF] Aste | Water flat plate PV-thermal collectors: A review[END_REF], which summarizes the various geometrical configurations, thermo-physical properties and manufacturing solutions for PVT systems. One of the most common configurations, which was also used in this study, is described below.

The PVT chosen panel is liquid based, with a direct flow geometry of the collector and roll bond channels (Fig. 1). The liquid-based exchanger was chosen due to its better heat transfer capacity compared to the air-based systems, in spite of its higher level of complexity. In addition, the liquid systems can be utilized directly for preheating domestic hot water (DHW). The direct flow channel was chosen for their simplicity and well proved experimental performance. The roll bond fabrication method is easy to achieve, very flexible in terms of geometry and relatively cheap to produce [START_REF] Aste | Water flat plate PV-thermal collectors: A review[END_REF]. The PV cell is Mono Crystalline Silicon, with a reference electrical efficiency (η STC ) of 15% at the standard testing conditions (STC) -operating temperature (T ref ) of 25 • C, solar radiation of 1000 W/m 2 and wind speed 0 m/s. The temperature coefficient (β T ) is 0.5%/K. The relationship between the instantaneous electrical efficiency and the reference electrical efficiency is given below.

η EL = η STC [1 -β T ( T c -T ref ) ] (1)
where T c is the temperature of the cell.

The collector fluid is a 30/70% glycol-water mix due to the fact that it provides better heat transfer parameters than plain water, and it has a lower freezing point, which is important in continental climates where winters are defined by negative temperatures. The size of the panel is 1 m x 2 m, and the diameter of the channel is 0.008 m, with a thickness of insulation of 5 cm. Table 1 summarizes the main PVT collector properties.

The cross section of the panel is shown in Fig. 1, with all the composing layers and the corresponding subscripts -glass (g), Air gap, PV layer (p), thermal absorber (a), insulation, fluid (f). The PV layer is made of PV glass, ethylene vinyl acetate (EVA) film, Silicon cell, Tedlar layer and adhesive.

The assumptions made for developing the model are:

• edges are well insulated; no heat is lost at the edges;

• the layers are very thin thus the vertical heat conduction can be assumed negligible;

• the optical properties of materials remain constant along the area;

• no dust effects or partial shading are taken into account;

• the flow rate of the fluid is constant, ṁ = 0.05 kg/s.

• the initial fluid inlet temperature at time t = 0 is T i0 = 15 • C. At every consecutive time step, the new T fi is the average value between T i0 (assumed to be the cold-water inlet main temperature) and T fo from the previous time step (the temperature of the fluid at the outlet of the panel).

The layers analyzed in this model are: glass, PV, thermal absorber and fluid. Some models [START_REF] Chow | Energy performance of a solar hybrid collector system in a multistory apartment building[END_REF][START_REF] Pierrick | Dynamic numerical model of a high efficiency PV-T collector integrated into a domestic hot water system[END_REF] also analyze the behavior of the air gap layer and the insulation layer, but those temperatures were found not to be relevant for the purpose of this study, and thus the model is simplified to only compute the four layers of interest.

The thermal energy circulates through the layers of the PVT panel and to/from the exterior through three types of heat transfer processes: radiation, convection and conduction. Radiation is the process of heat transfer without any physical contact and it does not heat the transfer medium. Any physical body of a temperature greater than 0 K emits a level of radiation through electromagnetic waves. When radiation reaches a body, it can be either be absorbed, reflected or transmitted. Conduction is the movement of heat through a solid material or at the interface of two materials/layers that are in contact due to molecular collisions. Conduction that occurs on the solar panel is caused by the thermal gradients and its direction is from high temperature to low temperatures. Convection is the heat transfer in which energy moves within a fluid or gas through flow of matter from higher to lower temperature regions. It can be natural, occurring due to difference in densities, or forced by an external physical movement, such as a pump or the wind.

The heat transfer processes that take place inside a PVT panel are shown in Fig. 2; radiation with a yellow arrow, convection in green and conduction in red. The glass layer receives direct and indirect solar radiation from the sun, a fraction of which is reflected, another part absorbed and the rest transmitted to the PV layer. It also receives reflected radiation and radiative losses from the PV layer. It loses some heat to the environment through forced convection caused by the wind. The PV layer receives a fraction of the solar radiation, of which some is reflected back to the glass layer and some is absorbed. The PV layer also loses heat through radiation to the glass. In addition, natural (free) convective heat transfer between the glass and the PV takes place inside the air gap. Out of the incident radiation, a fraction is converted to electricity through the photovoltaic effect. The absorber layer exchanges heat with the PV layer, with the insulation and with the fluid through conduction, and the insulation loses heat to the environment through forced convection from the wind. The heat transfer in the fluid occurs also through forced convection generated by the pumping system.

Implementation of numerical model

The numerical model was developed in a previous paper (Barbu et al., 2019b) and is presented in the next section. [START_REF] Rejeb | Parameters effect analysis of a photovoltaic thermal collector: Case study for climatic conditions of monastir[END_REF][START_REF] Aste | Design, modeling and performance monitoring of a photovoltaic-thermal (PVT) water collector[END_REF][START_REF] Pierrick | Dynamic numerical model of a high efficiency PV-T collector integrated into a domestic hot water system[END_REF]. a Depends on the temperature.

Property

Heat balance

The numerical model is based on the heat transfer processes described in the previous section. A heat balance equation is developed for each of the significant layers of the system. This is based on the equation for the variation of internal energy in a physical body:

Mc dT dt = dU dt (2)
where M [kg] is the mass, c [J/(kg K)] is the specific heat capacity, T [K] the temperature, dt [s] is the chosen time step and dU [J] is the change in internal energy of the layer.

Glass layer (g)

For the thermal balance of the glass layer 'g' we considered the convective and radiative losses from the glass to the environment (Q g-e,CV and Q g-e,RD ), the convective and radiative heat transfer from the glass to the PV layer (Q g-p,CV and Q g-p,RD ) and the heat absorbed by the glass (Q g ).

M g c g dT g dt = Q g-e,CV + Q g-e,RD + Q g-p,CV + Q g-p,RD + Q g = h g-e,CV A g ( T e -T g ) + h g-e,RD A g ( T sky -T g ) + h gap A g ( T p -T g ) + h g-p,RD A g ( T p -T g ) + A g α g G irr (3)
where h g-e,CV is the coefficient of forced convection between the glass and environment [W/(m 2 K )], A g is the area of glass [m 2 ], T e is the temperature of the environment [K], T g is the temperature of the glass layer [K], h g-e,RD is the coefficient of radiative heat loss between the glass and environment [W/(m 2 K)], T sky is the equivalent radiative temperature of the sky [K], h gap is the thermal coefficient of the gap [W/(m 2 K)], T p is the temperature pf the PV layer [K], h g-p,RD is the coefficient of radiative heat loss between the glass and the PV layer [W/(m 2 K)], α g is the absorbance of the glass [-] and G irr is the incident irradiation on the absorber [W/m 2 ].

Multiple reviews were carried out on the coefficient of forced convection due to wind (h g-e,CV ) [START_REF] Rejeb | Parameters effect analysis of a photovoltaic thermal collector: Case study for climatic conditions of monastir[END_REF][START_REF] Aste | Water flat plate PV-thermal collectors: A review[END_REF], which can be expressed through various empirical correlations. The correlation proposed by [START_REF] Mcadams | Heat Transmission[END_REF] was chosen for this study. It applies to winds speeds (v w ) from 0 to 10 m/s, gives good agreement between experimental and simulation data, and has been widely used in the literature for numerical PVT models [START_REF] Rejeb | Parameters effect analysis of a photovoltaic thermal collector: Case study for climatic conditions of monastir[END_REF][START_REF] Lämmle | Thermal Management of PVT Collectors -Development and Modelling of Highly Efficient Glazed, Flat Plate PVT Collectors with Low-E Coatings and Overheating Protection[END_REF]. The coefficient of radiative heat loss (h g-e,RD ) is calculated by using Stefan-Boltzmann constant (σ = 5.67×10 -8 W m -2 K -4 ), the emissivity of glass (ε g ) and the equivalent radiative temperature of the sky (T sky ). There are multiple equations available for defining the sky temperature, most commonly it can be expressed as a linear function of the ambient temperature (T e ) and the sky cloud coverage in octaves (N) [START_REF] Aste | Design, modeling and performance monitoring of a photovoltaic-thermal (PVT) water collector[END_REF]. If clear sky conditions are assumed or there is no data available on the cloud coverage, Eq. ( 6) can be further simplified to Eq. ( 7), with less that 1% effect on the thermal and electrical output of the system [START_REF] Guarracino | Dynamic coupled thermal-and-electrical modelling of sheet-and-tube hybrid photovoltaic/thermal (PVT) collectors[END_REF][START_REF] Touafek | Theoretical and experimental study of sheet and tubes hybrid PVT collector[END_REF].

h g-e,RD = ε g σ (T g 2 + T sky 2 )(T g + T sky ) (5)
T sky = 0.0552T e 1.5

+ 2.652N (6)
T sky = 0.0552T e 1.5

(7)

The thermal coefficient of the gap (h gap ) takes into account the conduction in the top layers of the PV sandwich (PV glass and EVA) and the convective heat transfer in the air gap.

1

h gap = 1 h g-p,CV + H PVg k PVg + H EVA k EVA (8) h g-p,CV = Nu air k air H gap ( 9
)
where H is the height of the layer (measured at the middle of the layer) [m], k is the thermal conductivity [W/(m K)] and Nu air is the Nusselt number for air [-].

The Nusselt number correlation for inclined plates, tilt angles from 0 to 60, is given by [START_REF] Hollands | Free convective heat transfer across inclined air layers[END_REF]. The radiative coefficient between the glass and the PV (h g-p,RD ) is expressed below:

h g-p,RD = 1 1 εg + 1 εp -1 σ (T 2 g + T 2 p )(T g + T p ) (10)
where ε p is the emissivity of the PV layer [-] and ε g is the emissivity of the glass [-].

PV layer (p)

The PV layer 'p' includes glass, two layers of EVA, cell and Tedlar. The thermal balance of the PV layer considers the convective and radiative heat transfer to the PV layer (Q p-g,CV and Q p-g,RD ), the conductive heat transfer to the absorber layer (Q p-a,CD ), the heat absorbed by the PV layer (Q p ) and electricity production (E).

M p c p dT p dt = Q p-g,CV + Q p-g,RD + Q p-a,CD + Q p -E = h gap A p ( T g -T p ) + h p-g,RD A p ( T g -T p ) + h p-a,CD A p ( T a -T p ) + A p (ατ ) p G irr (1 -η EL(T ) ) (11)
where A p is the area of the PV layer [m 2 ], h p-a,CD is the conduction coefficient from the PV layer to the absorber [W/(m 2 K)], T a is the temperature of the thermal absorber layer [K], and η EL(T ) is the time dependent electrical efficiency of the cell [-].

The heat transfer coefficient in the air gap and the radiation coefficient are calculated in Eqs. ( 8)-( 10). The conduction coefficient through the layers of EVA, Tedlar and adhesive is:

1 h p-a,CD = H EVA k EVA + H Ted k Ted + H adh k adh (12)
The conversion efficiency of the cell decreases with the increase in the operating temperature T p and depends on the temperature coefficient β PV .

η EL(T ) = η STC [1 -β PV (T p -T ref )] (13)
where η STC is the reference electrical efficiency measured at the standard operating temperature

T ref of 25 • C [-].

Thermal absorber (a)

The thermal balance of the absorber layer 'a' considers the conductive heat transfer to the PV layer (Q a-p,CD ), the heat transfer to the fluid (Q a-f ) and the heat loss to the exterior through the insulation (Q a-e,CD ).

M a c a dT a dt = Q a-p,CD + Q a-f + Q a-e,CD = h a-p,CD A a ( T p -T a ) + h a-f A ch ( T f -T a ) + h a-e,CD A a (T e -T a ) (14)
where h a-p,CD is the heat transfer coefficient from the absorber to the PV layer [W/(m 2 K)], A a is the area of the thermal absorber

[m 2 ], A ch is the area of the channel [m 2 ], h a-f is the heat transfer coefficient of the fluid [W/(m 2 K)], T f is the temperature of the fluid [K], h a-e,CD is the conductive heat loss coefficient to the environment [W/(m 2 K)].
The heat transfer coefficient to the PV layer (h a-p,CD ) is the same as in Eq (12). The heat transfer coefficient of the fluid depends on the type of flow (laminar or turbulent) [START_REF] Incropera | Fundamentals of Heat and Mass Transfer[END_REF].

h a-f = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 4.36 k f D H , for Re < 2300 [-] 0.023 k f D H Re 0.8 Pr 0.4 for Re ≥ 2300 [-] 2 k f D H , for ṁ = 0 kg s (15)
where

Fluid (f)

The energy balance equation for the fluid 'f' considers the thermal energy coming from the absorber (Q f-a ) and the heat accumulated by the fluid (Q f ).

M f c f dT f dt = Q f -a + Q f = h a-f πDL 2 ( T a -T f ) + ṁc f ( T fi -T fo ) (16)
The heat transfer coefficient of the fluid is calculated in Eq. T f = 0.5T fi + 0.5T fo (17)

System performance

The energy performance of the system can be computed in terms of thermal, electrical and overall efficiencies (η TH , η EL , η overall ). The electrical efficiency is calculated as the ratio between the power generated by the system (P EL ) and the amount of solar radiation incident on the surface of the collector (AxG irr ):

η EL = P EL AG irr (18)
The standard equation for thermal efficiency is represented by the ratio between the amount of thermal energy generated by the system (Q TH ) and the solar radiation incident on the surface of the collector (AxG irr ):

η TH = Q TH AG irr = ṁc f (T fo -T fi ) AG irr ( 19
)
where ṁ is the mass flow rate of the fluid and c f is the specific heat capacity.

Another definition for the thermal efficiency was proposed by [START_REF] Bombarda | Thermal and electric performances of roll-bond flat plate applied to conventional PV modules for heat recovery[END_REF], where distinction is made between the electrical and thermal power of the system, and thus the electrical power production is subtracted from the total incident irradiation. This definition allows for a more accurate comparison between the thermal performance of a PVT and stand-alone solar thermal system.

η TH * = Q TH AG irr -P EL = ṁc f (T fo -T fi ) AG irr -P EL = η TH 1 -η EL (20)
From the point of view of the first law of thermodynamics, the global efficiency of the system (η GL ) can be calculated as the sum of the thermal and electrical efficiencies. This is also known as the first law efficiency:

η GL = η EL + η TH (21)
Another approach is to consider the difference in grade between the two types of energy, i.e. electrical energy is a higher form of energy compared to the thermal. Thus, due to the fact that a kW el cannot be directly compared to a kW th , some authors [START_REF] Aste | Design, modeling and performance monitoring of a photovoltaic-thermal (PVT) water collector[END_REF][START_REF] Huang | Performance evaluation of solar photovoltaic/thermal systems[END_REF] propose the primary energy saving efficiency (η PES ) as a more accurate way of assessing the system performance:

η PES = η TH + η EL η Tpower (22)
where η Tpower is the average efficiency of producing electrical power, and it depends on the country of reference. On average, it can be assumed to be 0.4 [START_REF] Aste | Design, modeling and performance monitoring of a photovoltaic-thermal (PVT) water collector[END_REF], and this value was used for the calculations. For Romania, the average efficiency of producing electrical power in 2016 was 0.532, while in France it was 0.601 (European Environment Agency, 2018).

MatLAB implementation

By rearranging the energy balance equations (3), ( 11), ( 14) and (16), a system of coupled equations is obtained. The system of equations is then solved in MatLAB with the ordinary 'fsolve' function. The coefficients C and D are defined by rearranging the free terms of the energy balance equations above.

⎧ ⎪ ⎨ ⎪ ⎩ C 1 T g + C 2 T p -D 1 = 0 C 3 T g + C 4 T p + C 5 T a -D 2 = 0 C 6 T p + C 7 T a + C 8 T f -D 3 = 0 C 9 T a + C 10 T f -D 1 = 0 (23)
where

C 1 = M g c g dt + h g-e,CV A g + h g-e,RD A g + h gap A g + h g-p,RD A g (24)
C 3 = -h gap A p -h p-g,RD A p (27) C 4 = M p c p dt + h gap A p + h p-g,RD A p (28) C 5 = -h p-a,CD A p (29) D 2 = M p c p dt T p + A p (ατ ) p G irr (1 -η EL(T ) ) (30) C 6 = -h a-p,CD A a (31) C 7 = M a c a dt + h a-p,CD A a + h a-f A ch + h a-e,CD A a (32) C 8 = -h a-f A ch (33) D 3 = M a c a dt T a + h a-e,CD A a T e ( 34
)
C 9 = -h a-f π DL 2 (35) C 10 = M f c f dt + h a-f πDL 2 + 2 ṁc f (36) D 4 = M f c f dt T f + 2 mc f T fi (37)
The fsolve function solves the equations for each individual timestep and gives a vector solution for [T p T g T a T f ]. The results are plotted on a graph and shown in Section 4. For the parametric study, a 'for' function is introduced to vary the variables in a chosen range, while keeping the rest constant.

Model validation

The developed model was validated by comparing the performance results to experimental data found in the literature. In order to achieve this, the PVT model described by [START_REF] Mcadams | Heat Transmission[END_REF] is recreated using this model, with the same configuration and operating conditions used in the experiment. The variation of the thermal and electrical efficiency is plotted against the reduced temperature for both the simulated and experimental model. The characteristics of the experimental model are summarized in Table 2.

The system performance can be assessed by computing the electrical and thermal efficiencies, according to Eqs. ( 18)-( 19). These are plotted against the reduced temperature, defined in Eq. ( 38) below, to account for the fact that the thermal efficiency is, in fact, dependent on ambient temperature, inlet temperature and global irradiation.

T * = T fi -T amb G irr (38)
Fig. 3 shows a good resemblance between the reference experimental data and the results obtained by the numerical model developed in this paper. It shows that under different operation conditions and constructive parameters of the collectors, the result of the experimental study from the literature can be recreated with the present numerical model.

Results and discussion

Case studies

The developed numerical model was applied for a PVT panel in two different climate conditions: Bucharest and Strasbourg, which enables a comparison of the performance of the PVT system in both regions. The climate of Romania is temperate continental, with hot and sunny summers with occasional showers and thunderstorms and cold, cloudy winters and frequent snow and fog. In the southern region, where Bucharest is located, the winters are milder and the summers hotter. France's climate is temperate, with four specific regions. Alsace, where Strasbourg is located, is characterized by a continental climate which harbors cold winters, hot summers, and strong winds. The average summer months temperature in Bucharest is 29.1 • C, while in Strasburg it is 24.2 • C. For the winter months, Bucharest has an average of 4 • C, and Strasbourg 5.2 • C. The monthly summer sun hours are 278.8 and 222.0 for Bucharest and Strasbourg respectively, while the winter sun hours are 72.6 and 54.5 (WMO, 2019).

The variation of the weather during the day was used as an input into the numerical model in order to assess in real time the temperatures inside the panel. The input is: solar irradiation (G [W/m 2 ]), ambient temperature (Tamb [ • C]) and wind speed (W [m/s]). The yearly profile with hourly resolution for each parameter was obtained from Meteonorm Database for both Bucharest, Romania and Strasbourg, France. The meteorological conditions of a typical day are presented in the figures below. Fig. 4a andb show the daily weather conditions for Strasbourg and Bucharest respectively from 8:00 to 20:00, with an hourly time resolution, while Fig. 5a andb show the winter weather conditions. Given the fact that the simulations were run for one day, it is important to note the sensitivity of the chosen data on the results of the numerical simulations. This means that the specific day which was chosen from the Meteonorm yearly database is of crucial importance and given the significant day to day variation of the weather, a random day can be misrepresenting the seasonal climate. In this case, the same data was chosen for both cities, making sure no extreme condition occur and the two weather conditions are comparable at the given time. The representative winter day is 3rd of January and for the summer is 12th of July. Daily simulations were carried out for a typical summer and winter day in Bucharest, Romania and Strasbourg, France, from 8:00 to 20:00, with the weather data including ambient temperature, wind speed and irradiation, discussed in the previous section. Given these conditions, the time dependent temperature variation of the layers in the panel is shown below. Fig. 6a andb show the temperature evolution of the layers, as well as the ambient temperature, during a summer day in Strasbourg and Bucharest respectively. Fig. 7a andb show the temperature evolution of the layers, as well as the ambient temperature, during a winter day in Strasbourg and Bucharest respectively. As expected, the layers with the higher thermal conductivity reach the highest temperatures. In all cases, the PV layer is the most heated, reaching 52 • C for a summer day in Bucharest. The next layer is the thermal absorber, followed by the fluid. The glass layer is always the coolest. The temperature profile of all the layers follows closely the profile of the ambient temperature and global irradiation. The effect of the wind speed is not easily noticeable in the simulations. It can also be observed that the working fluid reaches slightly higher temperatures in Bucharest during the summer and in Strasbourg during the winter, due to their specific weather conditions. As a result, assuming that the system would have a water storage tank, it can be expected that the system would have a better thermal efficiency in Bucharest compared to Strasbourg during the summer, and higher in Strasbourg during the winter.

Table 3

Range of variation of parameters for OFAT parametric analysis.

Wind speed 0-10 m/s Packing factor 75%-95% Thickness of insulation 1-10 cm Width (thickness) of tube 5-10 mm

Parametric analysis

Next, a OFAT analysis was carried out for some of the main constructive parameters of the panel, in order to evaluate their impact on the system efficiency. This is achieved by varying the parameters according to Table 3, while keeping the rest of them to the default values reported in Table 1. The reference electrical efficiency of the panel is 15%, while the thermal efficiency is dependent on ambient temperature, inlet temperature and global irradiation.

The effect of the two main meteorological conditions, ambient temperature and solar radiation, on the PVT behavior are easily observable in the case studies above. However, the influence of the third component, wind speed, is not so obvious, therefore this relationship is further analyzed by varying its value from 0 to 10 m/s and assessing the system performance. Increasing the wind speed results in a higher forced convection coefficient, according to Eq. ( 4). The results in Fig. 8 indicate a 2% decrease in the thermal efficiency, which can be explained by an increased forced convection heat loss from the glass to the ambient. On the other hand, a 0.1% increase in the electrical efficiency can be observed. A higher forced convection leads to a better cooling of the PV cell and a lower operating temperature, which leads to a slightly improved electrical output. However, the effect of high wind speeds on the decrease in thermal performance is much more significant than the beneficial increase in electrical performance. Overall, the global efficiency decreases with higher wind speeds. It is also noticeable that the slope of electrical efficiency decreases gradually and tends towards flattening, which indicates that the operating temperature is close to reaching the STC value of 25 • C, after which cooling no longer has an effect on the electrical output.

Apart from the convective losses caused by the wind, heat losses also take place at the back of the panel through the layer of insulation, and their magnitude is directly influenced by the thickness of the layer. The effect of the insulation losses on the system performance can be assessed by a parametric variation of the insulation thickness, from 0.01 to 0.1 m. Fig. 9 shows a steep effect on both efficiencies at first, followed by a flattened curve, which indicates the fact that after a certain thickness, the increase in insulation has an insignificant effect on the panel performance. In the case of the electrical performance, this occurs at 0.05 m of insulation, where the PV cells reach the STC temperature. For the thermal performance, the curve is not completely flat even at 0.1%, but the slope decreases gradually. Again, as in the case of wind speed, the effect on the thermal performance is much more significant than on the electrical one, with a variation of 3.5% compared to 0.1%. Overall, the global efficiency increases with thicker insulation.

The effect of the packing factor of the PV cell on the system efficiency is illustrated in Fig. 10. Typical commercial cells have packing factors of 75% up to 95% for the highest performing ones. It can be observed that the increased packing factor has a significant positive effect on the electrical performance, with more than 2% improvement, due to a larger surface area of the panel being covered with PV cells, and implicitly, a larger amount of incident solar radiation being converted to electricity. The 20% decrease in thermal efficiency with higher packing factors is explained by the fact that a higher fraction of the solar energy is converted to electricity, and thus a smaller fraction remains to be converted into heat. Thus, the variation of the two efficiencies appears equivalent; However, depending on the consumer profile, it might be beneficial to choose a high packing factor for the PVT design, as the electrical energy is more valuable both economically and exergetically than the thermal energy. A high packing factor typically involves increased costs of the PV cells, thus a cost-benefit analysis should be carried out for optimum design. Overall, the global efficiency increases with a higher packing factor.

Finally, the width of the absorber channels is varied from 5 to 10 mm for a constant flow rate of 0.05 kg/s. A wider diameter channel results in a larger cross section area and thus a slower flow rate of the fluid in the channels, but also a larger area of heat transfer between the absorber and the fluid. A better performance of both components is observed with the increase in channel diameter, shown in Fig. 11. The electrical efficiency increases by 0.6% due to better cooling of the PV, while the significant increase of 20% in the thermal efficiency is explained by a better heat transfer between the absorber and the fluid, and thus an improved thermal energy output. However, there are technical constrains on the fabrication methods of roll bond absorbers that limit the width to 8-10 mm depending on manufacturing method, materials and distance between the channels. Overall, the global and PES efficiencies increase with a higher tube thickness.

The results of the sensitivity analysis mostly indicate that, as in any cogeneration system, there is a compromise between the production of electrical and thermal power of the system. The design can lean towards improving one of the performances of the system. In most cases, the electrical performance will be more important, due to a higher value of electrical energy compared to thermal energy.

As a result of this analysis, a number of technical recommendations can be formulated for the efficient operation of PVT collectors. First of all, the roof installation of the panels is important, not only in terms of South-facing positioning (for the Northern hemisphere) and tilting it to a suitable angle according to the latitude of the location, but also in terms of wind protection. In some cases, space will be limited and there is no flexibility with the mounting, which can lead to high winds and a decrease in global efficiency of the collector. Where there is flexibility in the location, a rose wind is a useful tool for ensuring there is wind protection for the system of collectors, thus improving the overall efficiency. Wind protection is not only beneficial for the efficiency, but also decreases the amount of dust collected and reduces the maintenance costs. In terms of insulation, it was observed that the benefits of increasing its width are significant. However, this can add to the cost of production and can increase the bulkiness and weight of the equipment. Another technical solution is investigating alternative materials for the thermal insulation, with lower coefficient of thermal loss, so that both the thickness and thermal losses can be kept to a minimum. In terms of choosing a packing factor for the PV layer, it is important to keep in mind the end consumer and the applications. Some manufacturers have even started to produce two types of collectors, thermally driven and electrically driven, depending on the needs of the end user Solar Solutions, width of the channels can be maximized but keeping in mind that there will be an increase in the pumping power for maintaining the required flow rate.

Performance maximization

The PVT panel can be optimized in order to maximize the production of either thermal or electrical energy, depending on the scope and building applications:

• electrical-driven -maximizing the electrical efficiency • thermal-driven -maximizing the thermal efficiency • compromise solution -maximizing the global efficiency

The results are summarized in Table 4. In the case of most parameters, the compromise solution coincides with the thermal-driven solution, except for the packing factor.

Conclusion and perspectives

This study proposed an explicit numerical model of a PVT collector based on the heat balance equations between each layer. The model has been validated numerically recreating a panel that has been experimentally studied in the literature, and the results show good accordance. Next, the model was applied in two case studies for the climates of Bucharest, Romania, and Strasbourg, France. The input consists of ambient temperature, wind speed and global irradiance, which were obtained for both locations from the Meteonorm Database. Simulations were run for representative summer and winter days, from 8:00 to 20:00. The results show the temperature of each layer at any particular time, and it can be observed a slightly higher temperature of the working fluid in Bucharest during the summer, and in Strasbourg during the winter.

The OFAT parametric analyses carried out in this study showed in most cases a compromise between the electrical and thermal performance of the PVT system. In terms of wind speed and insulation, it was observed that the thermal benefits of low wind and high insulation overcome the decrease in electrical efficiency. For optimum design, the insulation thickness should be maximized and the wind speed minimized, which can be achieved by strategic placement of the panels in areas protected from wind. The packing factor was found to be optimum when maximized, as the electrical benefits are more significant than the thermal ones. The width of the channels in the heat exchanger should also be maximized as far as technologically possible for best performance.

Future work will be focused on setting up an experimental stand for a more direct validation of the model and of the analysis, and the dynamic coupling with residential consumers. Also, an algorithm for optimizing the design can be developed, taking into account the exergy of the system, the global performance, and the cost benefits.

Nomenclature

Fig. 1 .

 1 Fig. 1. Cross section of the PVT collector(Barbu et al., 2019a).

Fig. 2 .

 2 Fig. 2. Schematic heat transfer processes in the layer of a PVT panel.. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

  h g-e,CV = ⎧ ⎪ ⎨ ⎪ ⎩ 5.7 + 3.8v w for v w < 5 m s 6.47 + v 0.78 w for v w ≥ 5

  k f is the thermal conductivity of the fluid [W/(m K)], D H is the hydraulic diameter [m], Re is the Reynolds number [-], Pr is the Prandtl number [-] and ṁ is the mass flow rate [kg/s].

(

  15), D is the diameter of the pipe and L the length [m], ṁ [kg/s] is the mass flow rate of the fluid in the channels and the inlet and outlet temperatures are represented by T fi and T fo respectively [K].

Fig. 3 .

 3 Fig. 3. Comparison of simulation and experimental results of thermal and electrical efficiencies as a function of the reduced temperature.

Fig. 4 .

 4 Fig. 4. Temperature, wind speed and solar radiation for a summer day in (a) Strasbourg (b) Bucharest.

Fig. 5 .

 5 Fig. 5. Temperature, wind speed and solar radiation for a winter day in (a) Strasbourg (b) Bucharest.

Fig. 6 .

 6 Fig. 6. Time dependent temperature evolution of PVT layers for a summer day in (a) Strasbourg (b) Bucharest.

Fig. 7 .

 7 Fig. 7. Time dependent temperature evolution of PVT layers for a winter day in (a) Strasbourg (b) Bucharest.

Fig. 8 .

 8 Fig. 8. Variation of thermal and electrical performance as a function of wind speed.

Fig. 9 .

 9 Fig. 9. Variation of thermal and electrical performance as a function of the thickness of insulation.

Fig.

  Fig. Variation of thermal and electrical performance as a function of the packing factor.

Fig. 11 .

 11 Fig. 11. Variation of thermal and electrical performance as a function of tube width.

Table 1

 1 PVT properties

Table 2

 2 Parameters of validation model[START_REF] Bhattarai | Simulation and model validation of sheet and tube type photovoltaic thermal solar system and conventional solar collecting system in transient states[END_REF].

	Collector size	2 m 2	Tube diameter	9 mm
	Cell efficiency	17.3%	Mass flow rate	0.002 kg/m 2 s
	Packing factor	80%	Solar radiation	1000 W/m 2
	Temperature coefficient -0.405%/K Wind speed	1 m/s
	No. of tubes		10	Ambient temperature 30 • C
	Tube spacing		10 cm	Tilt angle	30 •
	C 2 = -h gap A g -h g-p,RD A g	(25)
	D 1 =	M g c g dt	T g + h g-e,CV A g T e + h g-e,RD A g T sky + A g α g G irr	(26)

Table 4

 4 Efficiency maximizing solutions.

		Electrical-driven	Thermal-driven	Compromise
		solution	solution	solution
	Wind speed (m/s)	10	0	10
	Insulation thickness (cm) 1	10	10
	Packing factor (%)	95	75	95
	Tube width (mm)	30	30	30

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Jean-Baptiste Courbot
email: jean-baptiste.courbot@uha.fr

Vincent Mazet
email: vincent.mazet@unistra.fr

Pairwise and Hidden Markov Random Fields in Image Segmentation

Keywords: Bayesian Image Restoration, Pairwise Markov Random Fields, Hidden Markov Random Fields

The purpose of this paper is to identify the similarities and differences between two image restoration approaches based on Markov field modeling. The first one is the wellknown Bayesian approach which models the unknowns with a Markovian prior. In the second approach, as proposed by Pieczynski and Tebbache [1], the pair unknowns-observations as a whole is considered Markovian. The two approaches are compared based on their posterior distribution, synthetic results and real examples, when applied to the segmentation of degraded images.

I. INTRODUCTION

Hidden Markov random fields (HMRF) are a widespread tool for image segmentation introduced in the 1980's [START_REF] Besag | On the statistical analysis of dirty pictures[END_REF], [START_REF] Geman | Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images[END_REF]. This approach models the observation y by a likelihood p(y|z) where z are the (hidden) labels, and the prior p(z) of the latter is a Markov random field so as to model spatial interactions between pixels. HMRF have been successfully applied in many works [START_REF] Besag | On the statistical analysis of dirty pictures[END_REF]- [START_REF] Li | Markov Random Field Modeling in Image Analysis[END_REF].

Besides, Pairwise Markov random fields (PMRF) have been introduced by Pieczynski and Tebbache in 2000 [START_REF] Pieczynski | Pairwise Markov random fields and segmentation of textured images[END_REF], [START_REF] Pieczynski | Pairwise Markov random fields and its application in textured images segmentation[END_REF]. The PMRF model considers the Markovianity on the couple (z, y) instead of the labels z only as in the HMRF model. Neither y nor z are necessarily Markovian, but z|y and y|z are Markovian. PMRF models have been introduced to avoid approximations when estimating the labels from the observations in textured images.

We and colleagues have often employed the PMRF model in the past, see e.g. [START_REF] Benboudjema | Parameter estimation in pairwise Markov fields[END_REF]- [START_REF] Courbot | Pairwise Markov fields for segmentation in astronomical hyperspectral images[END_REF]. However the PMRF model has been rarely used in the community, contrary to the HMRF model. Nevertheless, both share strong similarities: this is quite intriguing and invites comparison of the two models.To the best of our knowledge, no comparison has been made in the literature, even in [START_REF] Pieczynski | Pairwise Markov random fields and segmentation of textured images[END_REF]. Therefore, our goal in this paper is to investigate the links and differences between the two aforementioned models, so as to identify the limits and advantages of the models.

The problem is formalized in section II. The two approaches are mathematically detailed in terms of their posterior distribution in section III and their results are compared on synthetic and real images in section IV. A discussion is finally given in section V.

II. SCOPE OF THE STUDY

We consider the problem of joint deconvolutionsegmentation [START_REF] Ayasso | Joint NDT image restoration and segmentation using Gauss-Markov-Potts prior models and variational Bayesian computation[END_REF], [START_REF] Kim | Joint deconvolution/segmentation of microscope images of materials[END_REF]- [START_REF] Zhao | Joint segmentation and deconvolution of ultrasound images using a hierarchical Bayesian model based on generalized Gaussian priors[END_REF], in which a blurry observation y ∈ R S×B with S pixels and B bands has to be segmented in K classes. Introducing a blur within the segmentation task allows to consider more general problems.

The goal is to estimate the label image z ∈ K S×B that gathers the labels of each pixel in the observation, where K = {1, . . . , K} denotes the set of classes. The observation y is modeled as a blurry and noisy observation of an unknown image x ∈ R S×B :

y = Hx + e (1)
where H ∈ R S×S and e ∈ R S×B . H is the Toeplitz matrix of the point spread function (PSF), which introduces the same spatial blur for each band, without blur between bands. The PSF is supposed to be known and symmetric, so H is a known and symmetric matrix. The special case where H is the identity matrix yields the classical segmentation problem. e is an additive white Gaussian noise with zero mean and known variance σ 2 . In addition to this, the unknown image x is directly related to the label image z such that each pixel x s with class z s equals a signal w k ∈ R B where k = z s : ∀s, x s = w k if and only if z s = k.

(

The classes are considered to follow a Potts model [START_REF] Pereyra | Estimating the granularity coefficient of a Potts-Markov random field within a Markov chain Monte Carlo algorithm[END_REF] so as to favor the grouping of pixels of the same class:

p(z) ∝ exp   β (s,t)∈C δ(z s , z t )   ( 3
)
where β is an hyperparameter, a.k.a. the "granularity" coefficient and C is the set of neighboring pair of sites in the images (or set of cliques).

III. DESCRIPTION OF THE TWO APPROACHES

The HMRF approach usually works with the posterior p(z|y) whereas the PMRF approach describes the joint distribution p(z s , y s |z -s , y -s ) in one site s (the notation -s means the whole set of sites but s). To compare the two approaches we choose to write their conditional posteriors p(z s , y s |z -s , y -s ).

A. HMRF Approach

The likelihood results from the model of an additive white Gaussian noise:

p(y|z) = 1 (2πσ 2 ) (SB)/2 exp - y -Hx 2 F 2σ 2 , (4)
where x s = w k if and only if z s = k, and • F denotes the Frobenius norm. Then, the joint distribution is:

p(z, y) ∝ exp   - y -Hx 2 F 2σ 2 + β (s,t)∈C δ(z s , z t )   . ( 5
)
To get the conditional posterior of (z s , y s ), Eq. ( 5) has to be written by distinguishing the terms with z s or y s from the others (which will be denoted "ct"). So,

y -Hx 2 F = y s -(Hx) s 2 + i =s y i -(Hx) i 2 (6)
where • denotes the Euclidean norm. Note that the vectors are row vectors since they correspond to a site in the images. For example, (Hx) i is the ith row of matrix Hx. Because of the PSF H, the term in the sum in Eq. ( 6) may depend of z s (through x). For all i = s, this terms expands as:

y i -(Hx) i 2 = y i 2 -2y i (Hx) T i + (Hx) i 2 (7)
where:

y i 2 = ct y i (Hx) T i = b y ib (Hx) ib = b y ib H is x sb + ct (Hx) i 2 = b ((Hx) ib ) 2 = b n H in x nb 2 = b H is x sb + n =s H in x nb 2 = b (H is x sb ) 2 + 2H is x sb n =s H in x nb + ct
So, Eq. ( 7) yields:

y i -(Hx) i 2 = b -2y ib H is x sb (8)
+ (H is x sb ) 2 + 2H is x sb n =s H in x nb + ct.
In addition, the prior on z is defined in Eq. ( 3), and:

(s,t)∈C δ(z s , z t ) = t∈Ns δ(z s , z t ) + ct (9)
where N s is the neighborhood of s, to be defined by the user. Putting Eqs. ( 6), ( 8) and ( 9) in Eq. ( 5), one can show that the conditional posterior of the HMRF approach is the product of two Gaussians and a Potts distribution:

p(z s , y s |z -s , y -s ) ∝ exp - 1 2σ 2 y s -(Hx) s 2 (10) × exp - 1 2ρ s x s -m s 2 exp β t∈Ns δ(z s , z t )
where m s ∈ R S×B and ρ s ∈ R such that

m s = 1 i =s H 2 is i =s   y i H is - n =s H is H in x n   , ρ s = σ 2 i =s H 2 is .

B. PMRF Approach

With the PRMF approach, the pair (z, y) is assumed Markovian with respect to a given neighborhood [START_REF] Pieczynski | Pairwise Markov random fields and segmentation of textured images[END_REF]. The assumption of white noise implies that the variables in y are independent given z, so the joint distribution in a site s can be written:

p(z s , y s |z -s , y -s ) = p(y s |z s , z -s , y -s )p(z s |z -s , y -s ) = p(y s |z s , z -s )p(z s |z -s , y -s ) (11)
Assuming a white Gaussian noise,

p(y s |z s , z -s ) ∝ exp - 1 2σ 2 y s -(Hx) s 2 . (12)
We assume, furthermore, that z s and z -s are independent given y -s , and that z is Markovian. So, the second term of ( 11) is written as:

p(z s |z -s , y -s ) = p(z s |z Ns ). ( 13
)
Following the Potts model (3), we obtain that

p(z s |z Ns ) ∝ exp β t∈Ns δ(z s , z t ) (14)
Using Eqs. ( 12) and ( 14) in [START_REF] Kim | Joint deconvolution/segmentation of microscope images of materials[END_REF], we have:

p(z s , y s |z -s , y -s ) ∝ exp - 1 2σ 2 y s -(Hx) s 2 (15)
× exp β t∈Ns δ(z s , z t )
Note that there are two neighborhood systems embedding the Markovianity of the pair (z, y): in the first term, the considered neighborhood covers the whole lattice, while in the second it covers the closest neighbors of s, denoted N s . This choice ensures that the comparison between PMRF and HMRF does not depend on a particular neighboring system. Let us also remark that since pairwise Markov fields are a general class of models, specific assumptions have been made to tailor it to the considered problem. Hence, it should be noted that Eq. ( 15) describes a particular PMRF model.

C. Comparison of the two models

The posterior distributions in a site for the two approaches are given in Eqs. [START_REF] Courbot | Pairwise Markov fields for segmentation in astronomical hyperspectral images[END_REF] and [START_REF] Olmos | A biologically inspired algorithm for the recovery of shading and reflectance images[END_REF]. Identifying the three exponential terms in [START_REF] Courbot | Pairwise Markov fields for segmentation in astronomical hyperspectral images[END_REF] as proportional to three densities f , g and h let us rewrite [START_REF] Courbot | Pairwise Markov fields for segmentation in astronomical hyperspectral images[END_REF] as

p(z s , y s |z -s , y -s ) ∝ f (y s ,z s , z -s )g(y -s , z s , z -s )h(z s , z -s ) (16)
and [START_REF] Olmos | A biologically inspired algorithm for the recovery of shading and reflectance images[END_REF] as

p(z s , y s |z -s , y -s ) ∝ f (y s , z s , z -s )h(z s , z -s ). (17)
Then, the only difference between the two models lies in the density g, which adds, with respect to Eq. ( 15), a novel dependency between y -s and z. The graphical representations of the dependencies, conditioned on a site s, are given in Figure 1.

Noteworthy, in the PMRF approach, the relation between two variables from z and y depends on the conditioning site. Indeed, let s and t be two sites in the image: if we condition by the s site, the dependency between z s and y t is embedded in the g function, whereas if t is the conditioning site, the dependency is contained in the f function.

Let us also remark that the dependency embedded in the g density is also related to the assumption, for PMRF, that z -s and z s are independent given y -s .

Finally, we can see that if the blur is missing, i.e. H is the identity matrix, then (y i -(Hx) i ) 2 = (y i -x i ) 2 so the corresponding term in [START_REF] Pieczynski | Pairwise Markov random fields and its application in textured images segmentation[END_REF] does not depend on s. In this case g ∝ 1 and the two approaches are equivalent.

IV. NUMERICAL RESULTS

The theoretical difference between the two approaches highlighted in the previous section is hard to interpret in practice. So we present in this section a numerical comparison of the two methods for the retrieval of noisy, blurred images. Using the posterior distributions ( 10) and ( 15), the segmentation is obtained by estimating the Maximum A Posteriori (MAP) [START_REF] Geman | Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images[END_REF]: The MAP is estimated by using a Gibbs sampler (50 iterations) with a simulated annealing, and obtained in a supervised fashion: the model parameters are assumed to be known. We first consider the estimation of z from synthetic images y with S = 100 × 100 pixels, for a segmentation in 3 classes using B = 3 colors1 . The Potts parameter β, wich is not involved in the differences between the two approaches, is fixed to β = 0.6 in all experiments.

ẑMAP = arg max z∈K S p(z|y). ( 18
)
We consider a Gaussian PSF blur, with its standard deviation σ PSF . Since most of its energy is contained in its center, it is usual to truncate the PSF to its central values. We represent this truncation as a ratio r between the truncation radius and σ PSF , such that r = 0.5 means that 48.5 % of the PSF energy is kept, and r = 3 means 99.9% of the PSF energy is kept. Summing up, the parameters considered in the numerical comparison are σ, σ PSF , and r, and we evaluate error rate between the ground truth z and its MAP estimates ẑHMRF and ẑPMRF .

An example of synthetic image, together with its segmentations, is depicted in Fig. 2, and the complete results are given in Fig. 3. Several comments can be made regarding these results:

• The two models behave as expected: when the image is harder to process (higher σ or σ PSF ) the results worsen and are more scattered, while accounting for more PSF coefficient yields better results. • In average the HMRF model yields better results than the PMRF model. Hence, the additional term g in Eq. 16 does play an efficient part in the segmentation. • Unexpectedly, the main differences between the models are driven by changes in σ, as seen in Fig. 3. On the contrary, for a given σ, varying σ PSF or r does not change the performance gap between the methods, as depicted in Figs. 3b and3c. In consequence, the additional term is weakly influenced by the PSF parameters. • When either σ or σ PSF are low, the two models tend to provide similar results. This was theoretically expected for σ PSF but it is also the case for σ. To sum up, the difference between the two approaches is visible at low signal-to-noise ratio: clearly, the HMRF model yields better results than the PMRF model at high noise level.

We also tested the two models on real images extracted from the McGill calibrated colour image database [START_REF] Olmos | A biologically inspired algorithm for the recovery of shading and reflectance images[END_REF] (http://tabby. vision.mcgill.ca/) in the "flower" category. That these 100×100pixel images are blurred with a Gaussian PSF (σ PSF = 5), add Gaussian white noise (σ = 0.05), and consider r = 1 and β = 0.6. The means in {w 1 , w 2 , w 3 } defined in Eq. ( 2) were obtained by using the Kmeans algorithm [START_REF] Macqueen | Some methods for classification and analysis of multivariate observations[END_REF] on Hx. The results on five real images are depicted in Fig. 4:

• On the most homogeneous images (#1 and #2), there is no major differences between the two models. Hence, most of the classification is carried out in the f and h densities common to the two models. • Clear differences appear on images #3, #4, and #5: the HMRF model follows more closely the observed y than the PMRF model, and the latter tolerates gaps from y that enforces spatial regularity. • The parameter set is the same for the two methods, but may not be optimal, as we do not know the "true" parameters of the images. In addition, the optimal parameters may differ between the PMRF and HMRF models. Unsupervised parameter estimation should handle this point, but is beyond the scope of this paper.

V. DISCUSSION

In the context of image segmentation, we have compared the Markovian approach commonly used in the Bayesian framework with an instance of pairwise Markov field proposed by Pieczynski and Tebbache [START_REF] Pieczynski | Pairwise Markov random fields and segmentation of textured images[END_REF].

We highlighted that these approaches differ by a single term in their site-wise posterior distribution. This term is written as a Gaussian on x s , whose mean and variance depend through the PSF on the observations and classes in sites other than s. While it is difficult to grasp the theoretical impact of this term, numerical simulations show that it does affect the results, in favor of the HMRF model. Surprisingly, this clear difference is driven by the noise level instead of the PSF, although the main difference between the two models is a term modeling the spatial dependence through the PSF. The next step of this research will be dedicated to the interpretation and understanding of this observation.

Further works on this topic may compare further these models in the context of unsupervised parameter estimation, with non-stationary processes, or in the image restoration problem for instance.

Fig. 1 .

 1 Fig. 1. Graphical representation of the statistical dependencies embedded by the two models, conditioned on a site s (colored). No link between two variables means that they are independent given the variables in the s site. For clarity, only one dimension of the lattice is depicted. The neighborhood considered for the Markov process contains only the two immediate neighbors, while the PSF covers 5 sites.

Fig. 2 .

 2 Fig.2. Example of synthetic image with its MAP estimation using the HMRF model and the PMRF model. The parameters are: σ = 0.2, σ P SF = 5 and r = 1 (cf. dotted line in Fig.3).

  Error rate for a varying σ with σ PSF = 5 pixels and r = 1. Error rate for a varying σ PSF with σ = 0.1 and r = 1. Error rate for a varying r with σ = 0.1 and σ PSF = 5 pixels.

Fig. 3 .

 3 Fig.3. Numerical comparison of the two models studied in this paper. The dots represent the average error rates over 30 trials, and the shaded contours depict the 1 st and 9 th deciles of the results. For each studied parameter, the two others are fixed, so that the dashed vertical line is common between all graphs.

Fig. 4 .

 4 Fig. 4. Example of real image with the 3-class MAP estimations using the HMRF model and the PMRF models. The results are depicted for x in order to indicate the mean color vectors used for each class. The parameters are β = 0.6, σ = 0.1, σ PSF = 5 and r = 1.

Note that an extension to B > 3 colors could be considered to handle multi-and hyper-spectral images, together with cross-band noise correlations.