Emilio Sassine
email: emilio.sassine@gmail.com

Yassine Cherif
email: yassine.cherif@univ-artois.fr

Joseph Dgheim

Emmanuel Antczak

Experimental and numerical thermal assessment of Lebanese traditional hollow blocks

Keywords: Hollow block, heat modelling, buoyancy in cavities, thermal resistance, heat flow

Hollow-core concrete blocks constitute the main method of construction for vertical walls in Lebanon where almost all walls are masonry and use the same block shape. The detailed investigation of the main parameters involved in the heat transfer in these blocks is thus of great importance for analyzing and understanding them and providing recommendations for their improvement.

This paper offers an experimental and numerical analysis of heat performance the Lebanese concrete hollow block. After validating the numerical model by comparing it to experimental results, a deep and detailed analysis is done for understanding the complexity of heat transfer phenomena inside the block. Then a parametric study is performed to understand the effect of various parameters on the overall thermal resistance of the block, these parameters include the concrete solid mixture, the cavities infill material, and the geometry configuration of the block.

Introduction

Energy savings in the building sector offer untapped potential in Lebanon because of the lack of national policies aimed at the energy efficiency of buildings on the one hand, and the lack of knowledge of good practices and know-how on the other.

Concrete hollow blocks offer a number of advantages compared solid blocks in terms of weight, thermal insulation, and acoustic insulation. Many research studies aim at developing new hollow blocks with improved thermal properties [START_REF] Caruana | Overview of testing methodologies for thermally improved hollow-core concrete blocks, 6th International Conference on Sustainability in Energy and Buildings[END_REF].

In emerging countries such as Lebanon, concrete hollow blocks are widely used due to their advantages such as lifespan, low cost, low maintenance, fire resistance, and simplicity in implementation. The Lebanese buildings are characterized by their reinforced concrete structures and their vertical walls in single or double wall block masonry. This mode of construction dominates although new technologies have difficulty finding a place in this sector characterized by its high resistance to modifications because of the low-skilled labor force and the low costs of the materials currently used in this sector. The thermal performances of these walls remain until now unknown for lack of scientific thermal characterization method at different scales (material, wall, building, etc.).

The ever growing international concerns related to energy and resource conservation have led to many scientific and industrial efforts towards the improvement of masonry units in order to meet energy saving requirements. However, these changes cannot occur without understanding the detailed thermal behavior of these units and their thermo-physical properties.

Many authors studied the influence of hollow blocks configuration on their thermal performance. [START_REF] Xamán | Thermal performance of a hollow block with/without insulating and reflective materials for roofing in Mexico[END_REF]) studied the thermal performance of four different configurations of hollow block materials for roofing in Mexico using a numerical approach based on the finite-volume method. The results showed the advantage of using the combination of insulating material and reflective coating on roofs with hot climates. [START_REF] Urban | Arranging Insulation for Better Thermal Resistance in Concrete and Masonry Wall Systems[END_REF]) investigated different commercially available Concrete Masonry Unit (CMU) geometries and concrete mixtures using finite difference modeling to show their impact on R-values using six basic CMU shapes. [START_REF] Fogiatto | Thermal transmittance evaluation of concrete hollow blocks[END_REF]) determined the best cavity configuration for predefined external dimensions providing the best thermal resistance by examining 10 different block configurations using the software ANSYS®; the contribution of each heat transfer mode was also evaluated. [START_REF] Diaz | Thermal design optimization of lightweight concrete blocks for internal one-way spanning slabs floors by FEM[END_REF]) performed a numerical thermal analysis to optimize the thermal performance the lightweight hollow block design for internal floors using the finite element method. The number the vertical and horizontal intermediate bulkheads were varied to create five different configurations with constant external dimensions (57 cm x 25 cm x 20 cm). Five different lightweight concrete mixtures have also been studied. The results showed that the main parameters influencing the thermal performance were the number of the horizontal intermediate bulkheads and thermal conductivity of the mixture. [START_REF] Kus | Hot box measurements of pumice aggregate concrete hollow block walls[END_REF]) conducted an experimental hygrothermal performance study on pumice aggregate concrete blocks by measuring temperature and humidity at the surfaces of the test wall and in the hollows of blocks. [START_REF] Wang | Analysis of heat transfer properties of hollow block wall filled by different materials in solar greenhouse[END_REF] found that modifying the thermal properties of hollow block walls used for solar greenhouses by filling soil and perlite in their cavities led to improving their thermal performance by increasing their thermal capacity and increasing their thermal resistance. [START_REF] Zhang | Impact factors analysis on the thermal performance of hollow block wall[END_REF]) used numerical and experimental approaches to evaluate the thermal resistance, decrement factor, and time lag of a hollow block wall. They found that the main factors influencing thermal performance, ranked in descending order, were: decreasing the thermal conductivity of block material, increasing the thermal capacity of block material, decreasing the equivalent thermal conductivity of the material inside holes, increasing block thickness (accompanying the increase of hole spacing), and increasing hole rows (maintaining small hole thickness).

Many studies discussed the potential of thermal, and/or mechanical improvement of building blocks by using recycled waste materials in the concrete mixture. [START_REF] Xu | Utilizing wood fiber produced with wood waste to reinforce autoclaved aerated concrete[END_REF]) studied experimentally the effect of fibers content on the physical properties of autoclaved aerated concrete. He found that adding wood fibers into AAC was better than adding of polyester fibers.

Increasing wood fibers content also led to increasing volume density and thermal conductivity, as well as increasing the mechanical properties and particularly the flexural strength. The Scanning Electron Microscope (SEM) observations revealed that the reinforcing mechanism of wood fiber was a physical interaction. [START_REF] Caruana | Overview of testing methodologies for thermally improved hollow-core concrete blocks, 6th International Conference on Sustainability in Energy and Buildings[END_REF]) provided an overview of testing methodologies for thermally improved hollow core concrete blocks by discussing testing methodologies for the load bearing as well as thermal characterization of hollow concrete blocks. [START_REF] Caruana | Determination of thermal characteristics of standard and improved hollow concrete blocks using different measurement techniques[END_REF]) also investigated the thermal properties of a new building block aiming at improving its thermal properties without compromising their compressive strength, physical dimensions or manufacturing process. Measurement techniques were applied to obtain comparative values of the thermal transmittance for standard and improved HCBs, using different EN and draft standards. (Fraile-Garcia et al. 2018) examined the thermal behavior of light concrete construction elements made with different amounts of rubber particles (0%, 10% and 20%) extracted from end-of-life tires. Three different closed test cells were built and subjected to several heating/cooling periods. The recordings of the temperature inside the cells and inside their enclosures (walls, ceilings, floors) showed that the thermal behavior depended on the percentage of rubber particles. [START_REF] Meng | Recycling of wastes for value-added applications in concrete blocks: An overview[END_REF]) provided a review on the use of various kinds of wastes (i.e. recycled concrete, crushed brick, soda lime glass, cathode ray tube glass, crumb rubber, ceramic and tile waste, etc.) in the production of concrete blocks. The results underlined a good potential for incorporating wastes as aggregate in concrete blocks without compromising the compliance of concrete blocks with standard requirements. [START_REF] Mohammed | Properties of crumb rubber hollow concrete block[END_REF] evaluated miscellaneous physical properties (compressive strength, thermal conductivity, acoustic absorption and transmission loss, and electrical resistivity) for crumb rubber hollow concrete blocks (39 cm x 19 cm x 19 cm) using different amounts (0%, 10%, 25% and 50%) of crumb rubber for replacing fine aggregates. [START_REF] Zhu | Study on thermal properties of recycled aggregate concrete and recycled concrete blocks[END_REF]) defined an influence coefficient "c" for better explaining the effect of recycled concrete aggregate (RCA) on thermal conductivity of the recycled aggregate concrete (RAC). A modified formula for calculating the thermal conductivity of RAC based on that of normal concrete was also proposed. The results showed that the recycled coarse aggregate has the greatest influence on thermal conductivity of RAC. This paper presents an experimental and numerical detailed description of the thermal performance of Lebanese traditional hollow block. The particularity of these blocks resides in their particular circular and elliptical hollow shapes.

After validation of the cavity and block thermal numerical models, the heat thermal transfer inside the block is determined numerically by assessing the influence of each heat transfer mode in this process.

An experimental validation of the model as well as a thermal characterization of the concrete solid matrix is performed in the lab using the Heat Flux Method.

Once the numerical model of the heat transfer in the block is validated, the improvement of the thermal behavior of the block is discussed based on three main parameters: the concrete mixture thermal conductivity, the air gaps infill with insulation materials, and the blocks internal shape (bulkheads configuration).

Experimental aspect

Hollow blocks manufacture in Lebanon

Masonry hollow block is nowadays the most used construction technology in the Lebanese context. These blocks are manufactured in a traditional craft way with unknown thermal properties.

In order to understand the specificities and manufacturing method of cinderblock in Lebanon, a case study for an industrial plant in Deir Shomra, Kesrouane in Mount-Lebanon is described in the following section. The site is located at the bottom of a valley.

The blocks mixture consists of small size aggregates (25% of total aggregate volume) and powdered aggregates (75% of the total aggregate volume). The aggregates are dosed using a container that contains 200 liters which is filled 5 times to form 1 m 3 of aggregates. 50 kg of cement are added to the aggregates (1 m 3 ) from a silo and weighed using a hand-crafted system.

Finally, 50 liters of water are added using a bucket of capacity 20 liters (2.5 full). The whole is mixed using a mechanical mixing system and the mixture is then transported with a large wheelbarrow to the various artisans working on in-line production presses. The 10 cm blocks are currently the most requested. Each mold can simultaneously manufacture two blocks as shown in Fig. 2.

After their manufacture, the blocks are watered twice: the night of the same day and the next morning.

In sunny season (April -October) the blocks can be delivered two days after their manufacture, while in cold season this can last between 3 and 4 days so that the blocks dry and take their mechanical properties.

Experimental measurements

Experimental setup and samples

In this part the aim will be to evaluate the thermal performances of the concrete hollow blocks used in the Lebanese constructions.

Two samples were investigated, the first one consisting of half a hollow block (20cm x 20cm x 10cm), and the other sample prepared by gluing two flat concrete pieces from the same mixture to have a thickness of 55 mm (Fig. 4). A thermal characterization device of construction materials will determine the thermal properties (thermal resistance and equivalent thermal conductivity) of the concrete hollow blocks.

The guarded hot plate method was considered for the experimental thermal characterization of the sample (Fig. 5). An imposed temperature change on each face of the sample is set and simultaneous measurements of the temperature and exchanged heat flux variations during heating and cooling processes allow the determination of the thermo-physical properties of the sample, namely the thermal conductivity λ and the specific heat capacity cp [START_REF] Younsi | A novel technique for experimental thermophysical characterization of phase-change materials[END_REF]. The heat flux and the temperature on both sides of the sample are measured simultaneously. The application of Fourier's law for a unidirectional steady state heat transfer gives:

R T   1  and R T   2  (1)
The use of the generalized quantities Σφ and ΔT makes it possible to write:

R T    2  (2)
The advantage of using the generalized quantities Σφ and ΔT lies in the reduction of the measurement time. In fact, Fig. 6 shows that the sum of the heat fluxes (Σφ) stabilizes after 10 minutes while the steady state is reached after round 90 minutes for φ1 and for φ2.

The thermal resistance is thus given by:

    T R 2 and R e   (3) b.) Specific heat
During the transient phase, the sample stores or releases heat energy Q as its temperature increases or decreases. This energy is related to the heat fluxes difference () according to the relation:

   f i t t dt Q .  (4)
the temperatures on each face at the initial time ti, and ΣTf represents the sum of the temperatures on each face at the final time tf.

2 i f T T C Q     (5)
The heat capacity of the sample is thus given by:

i f i t t T T dt C f i       . 2  (6)
Knowing the density of the sample and its thickness, the specific heat can be deduced:

e C c p   (7)

Experimental results

Fig. 6 and Fig. 7 show the experimental measurements of heat flux and temperature for the concrete solid mixture and the concrete hollow block. Table 1 represents the thermophysical parameters of the concrete mixture that were determined by applying Eq. 3 and Eq. 6 using the experimental measurements (Fig. 6).

The thermal conductivity and the heat capacity can thus be determined: Similarly, the thermal resistance of the hollow block is determined using Eq. 3 and Fig. 7:

For t =
For t = 4h:

Numerical aspect

The simulations were performed using the software COMSOL® version 5.3a. In this twodimensional study, a structured mesh was considered using steady-state heat transfer.

Pre-test and model validation

Validation of the cavity model

Besides convection, thermal radiation was also considered using the discrete ordinates method (DO) due to its validity for radiation in any kind of gaseous medium. The air circulation inside the cavity promoting the natural convection, was considered as a laminar flow. By coupling the "Laminar Flow" model with the "Heat Transfer in Fluids" model in COMSOL Multiphysics, the continuity equations, the Navier-Stokes equations, and the energy equations were accounted. The air density was assumed to be dependent of pressure and temperature varying according to the ideal gas relation.

The validity of the methodology was verified by comparison with benchmark results [START_REF] Vahl | Natural convection f air in a square cavity, a benchmark numerical solution[END_REF] for a cavity flow with Rayleigh number Ra=10 5 .

For a square or rectangular-profile cavity, with four lateral insulated surfaces and two parallel faces with imposed temperatures, the Rayleigh number "Ra" is given by (Bergman et al. 2011):

   . ) 10 )( ( 3 3     H T T g Ra B A (8)
Where g [m/s 2 ] is the gravity acceleration, β [K -1 ] is the expansion coefficient, TA [K] and TB [K] are the imposed temperatures on the two parallel sides, H [mm] is the cavity length, ϑ [m 2 /s] is the kinematic viscosity and α [m 2 /s] is the thermal diffusivity.

The air properties at 300 K can be obtained from reference tables [START_REF] Bergman | Fundamentals of heat and mass transfer[END_REF]) and are reported in Table 2 as well as the boundary temperatures, the cavity length, and the Rayleigh number obtained from Eq. ( 1). The pre-test was performed considering the same simulation conditions used for the blocks. The dimensionless temperature profile as well as the velocity fields (x and y components) are compared with the benchmark solution in Fig. 8910. the isotherms disperse more and more in the cavity, which shows the increase in the temperature of the air in the cavity. On the other hand, at the right wall (TB <TA) the isotherms take a reverse form compared to what has been said for the left wall. As the temperature of the right wall is lower than the left one, the air cools near this wall; the beginning of cooling begins in the upper zone and gradually progresses to cool the air further to the lower part of the left wall. In addition, as the rectangular cavity is isolated from the top and from the bottom, the isotherms have a symmetrical shape. These forms of isotherms translate the movement of convection in a clockwise direction. This recirculation is created due the air density difference at the hot wall vicinity. In fact, by receiving heat, the air becomes lighter and thereby ascends along this wall.

On the other hand, it cools and becomes heavier, and thereby it goes down near the cold wall.

Validation at the block scale

The validation of the model was done by comparing the results with the ones obtained by Fogiatto et al. [START_REF] Fogiatto | Thermal transmittance evaluation of concrete hollow blocks[END_REF] . The same thermo physical properties for concrete and air used for the model validation are reported in Table 3. The 10 models described in the literature [START_REF] Diaz | Thermal design optimization of lightweight concrete blocks for internal one-way spanning slabs floors by FEM[END_REF] were reproduced using the same assumptions and the equivalent thermal resistance was deduced by determining the total heat flux simulated for an imposed temperature difference of 20 °K.

The heat flux was determined numerically and the thermal resistance was deduced by the relation:

    T R
The comparison between the numerical model and the results of [START_REF] Fogiatto | Thermal transmittance evaluation of concrete hollow blocks[END_REF] is summarized in Table 4.

The relative deviation is less than 10% for all the block configurations.

3D numerical modeling and simulation of the block

The dimensions of the studied wall are reported in Fig. 8. A temperature gradient of 20 K is imposed on the long lateral faces of the block (273 K -293 K) while the two other faces are remaining adiabatic.

Numerical results

The numerical results for the 3D model allow a better understanding of the heat transfer and temperature distribution in the block in quantitative and qualitative manners. Fig. 12 shows that the temperature gradient within the block decreases progressively from the cold side to the hot side. The air velocity inside the cavities reaches maximum values of around 4 cm/s (Fig. 13) which shows that convective heat exchanges are taking place. The maximum heat flux is noticed at the edges of the block where there are no cavities preventing conductive transfer, the solid matrix is thus the domain of the main heat flow in the block (Fig. 14).

Comparison between experimental and numerical results

The simulated average heat flux crossing the block is 127.5 W.m -2 . The equivalent thermal resistance of the block is thus given by:

W K m T R / . 159 . 0 5 . 127 20 2     
The difference between experimental (0.151 m 2 .K/W) and numerical (0.159 m 2 .K/W) results is around 5%.

Parametric analysis of the block

Parametric analysis by modifying the solid matrix

One of the possible improvement scenarios of the hollow concrete block is to improve the thermal conductivity of concrete solid matrix by incorporating some materials to its composition like recycled solid wastes for example.

The relation between the solid material thermal conductivity and the global hollow block composition was obtained numerically and is presented in Fig. 15. The results show that reducing the thermal conductivity of the concrete mixture from 1.4 to 0.4 will increase the thermal resistance of the block by more than 3 times from 0.17 to 0.5 m 2 .K.W -1 . This can be practically done by reducing the thermal resistance of the concrete mixture by replacing the normal standard concrete mixture by lightweight concrete or by replacing a part of the aggregates by other aggregates having a better thermal resistance such as wood or Expanded polystyrene EPS.

Parametric analysis by filling the hollow part

Another possible way to improve the thermal performance of hollow blocks is to fill its air gaps by insulation materials. Fig. 16 shows that the minimum thermal conductivity needed to increase the thermal resistance of the blocks is below 0.2 W.m -1 .K -1 ; in other words, any material having  s , W.m -1 .K -1 a higher thermal conductivity will be useless for this application. It can also be noticed that an improvement of around 40% in thermal resistance can be achieved by filling the air gaps by an insulation material of around 0.04 W.m -1 .K -1 (which is the case of most insulation materials [START_REF] Bergman | Fundamentals of heat and mass transfer[END_REF])).

Parametric analysis by modifying the geometry

The last improvement that will be discussed is the effect of modifying the block internal geometry by maintaining the same external dimensions (Total Area: 400 cm 2 ), and the same concrete mixture quantity represented by the solid cross section As (Solid Area: 221.52 cm 2 ).

The Lebanese hollow block configuration (Model 0), will be compared with 6 different rectangular cavity shapes by maintaining a uniform concrete matrix thickness "t".

The comparison was made by assessing the 2D heat transfer for the different cases and comparing the equivalent thermal resistance for the different configurations.

 i , W.m -1 .K -1
The comparison of the equivalent thermal resistances of the different configurations presented in Fig. 17 shows that the best configuration is the one presenting two air cavities in series with respect to the heat flux direction (Model 3). On the other hand, the worst configuration is the one presenting two air cavities in parallel with respect to the heat flux direction (Model 2). These results are not surprising since it is commonly known that the parallel model provides the upper bound of effective thermal conductivity in heterogeneous media while the series model gives the lower bound [START_REF] Xu | Experimental study and modeling on effective thermal conductivity of EPS lightweight concrete[END_REF]). The air cavities play an important role of damming the heat flux through the block while the concrete solid matrix plays the role of thermal bridge allowing the heat to go through the block. Fig. 18 shows how the series configuration (Model 3) remains by far the best configuration for different thermal conductivities of the concrete mixture. This configuration is to be recommended when for reaching high thermal resistance values. The Lebanese configuration (Model 0) does not offer any particular advantage from a thermal perspective and its shape is mainly due to manufacturing considerations and implementation in construction site.

Conclusions and recommendations

The free convection model inside the block cavities was validated by comparing temperature and velocity profiles to benchmark values available in the literature. Then the validation of the block model was performed by comparing the results of 10 block models available in the literature [START_REF] Fogiatto | Thermal transmittance evaluation of concrete hollow blocks[END_REF].

A 3D block modeling and simulation was performed and a slight difference of nearly 5% was found between the thermal resistance of the experimentally determined block and the numerically simulated block. Then a numerical analysis was performed for understanding the main heat transfer phenomena inside the concrete hollow block. Finally, a parametric study is performed to understand the effect of various parameters on the overall thermal resistance of the block, these parameters include the concrete solid mixture, the cavities infill material, and the geometry configuration of the block. The conclusions drawn from this work will provide a good basis for further explorations for improving concrete hollow blocks by incorporating new materials in their composition and by preserving the same shape and dimensions of prototype block for ergonomic reasons as well as local workmanship techniques.

Figure 1 -

 1 Figure 1-Aggregate, cement, and water dosing

Figure 2 -

 2 Figure 2-Work area of the craftsman and mold

Figure 3 -

 3 Figure 3-Storage and drying of the blocks

Figure 4 -

 4 Figure 4-Experimented samples of the hollow block (left) and the solid mixture (right)

Figure 5 -

 5 Figure 5-Thermal characterization device for building materials

Figure 6 -

 6 Figure 6-Experimental results for the concrete mixture

  005 = 0.151 𝑚2 . 𝐾. 𝑊-1

Fig. 8

 8 Fig. 8 shows the isotherms obtained for Ra = 10 5 in a closed rectangular cavity. Regarding the shape of the isotherms, the isotherms in the vicinity of the left wall (TA> TB) are very tight in the lower part at y <H/2 showing the start of air heating by convection. Beyond this part (y > H/2)

Fig. 9 ,

 9 Fig.9, and Fig.10show the horizontal (X) and vertical (Y) velocity fields respectively. It can be clearly seen that the vertical speed components are much higher than the horizontal ones; this explains that the natural convection recirculation movement is gravitational and caused by the variation of the fluid density in the cavity. The vertical component of the speed becomes very significant near the walls (left and right) and takes maximum values. In the middle of the cavity, the fluid is almost at rest, the horizontal and vertical speeds are almost zero.

Figure 8 -Figure 9 -Figure 10 -

 8910 Figure 8-Temperature profiles: (a-left) this work; (b-right) benchmark

Figure 11 -

 11 Figure 11-Numerical model of the concrete hollow block

Figure 12 -Figure 13 -Figure 14 -

 121314 Figure 12-Temperature distribution inside the hollow block

Figure 15 -

 15 Figure 15-Relation between the thermal conductivity of the solid mixture and the equivalent thermal resistance of the hollow block

Figure 16 -

 16 Figure 16-Relation between the thermal conductivity of the air gap infill and the equivalent thermal resistance of the hollow block

Figure 18 -

 18 Figure 18-Equivalent thermal resistance vs concrete mixture thermal conductivity for different configurations

Table 1 -Thermal properties of the concrete mixture of the block

 1 Where the contact resistance was considered equal to 0.005 m 2 .K.W -1[START_REF] Tittelein | Identification of Thermal Properties and Thermodynamic Model for a Cement Mortar Containing PCM by Using Inverse Method[END_REF].

			1h, Eq. 3 gives:
	𝑅 =		2 × (22.11 -17.14) 224	-0.005 = 0.0394 𝑚 2 . 𝐾. 𝑊 -1
	𝜆 =	0.055 0.0394	= 1.40 𝑊. 𝑚 -1 . 𝐾 -1
	Eq. 6 between t=1.65h and t=4h gives:
	𝐶 =	2 × 416176 45.4 -39.3	= 136228 𝐽. 𝐾 -1
	Eq. 7 gives:
	𝑐 𝑝 =	136228 2240 × 0.055	= 1106 𝐽. 𝐾 -1
					Thermal conductivity	Density (kg.m -3 )	Specific Heat (J.kg -1 .K -1 )
					(W.m -1 .K -1 )
	Concrete mixture	1.4	2240	1106

Figure 7-Experimental results of the block

Table 2 -Values for calculation of Rayleigh number

 2

	Variable	Units	Value
	g	[m.s -2 ]	9.81
	β	[K -1 ]	0.0033
	TA	[K]	305
	TB	[K]	295
	H	[mm]	48
	ν	[m 2 .s -1 ]	1.59 x 10 -5
	α	[m 2 .s -1 ]	2.25 x 10 -5
	Ra	[-]	10 5

Table 3 -Thermo-physical properties of concrete and air

 3

	Property	Units	Air	Concrete
	Ρ	[kg/m 3 ]	Ideal gas	2300
	cp	[kJ/(kg.K)]	0.0033	0.88
	K	[W/(m.K)]	0.0263	1.50
	Ε	[ ]	-	0.9
	Μ	[N.s/m 2 ]	1.846 x 10 -5	-

Table 4 -Comparison between simulated results and bibliography results for the 10 cases

 4

	Bloc number	Bloc configuration	Thermal Resistance [6] (m 2 .K/W) [4]	Thermal Resistance from the present study (m 2 .K/W)	Relative deviation (%)
	#1		0.187	0.183	2.13
	#2		0.178	0.175	1.68
	#3		0.192	0.176	8.3
	#4		0.196	0.179	8.67
	#5		0.237	0.248	4.64
	#6		0.273	0.299	9.52
	#7		0.252	0.246	2.38
	#8		0.284	0.289	1.76
	#9		0.276	0.292	5.79
	#10		0.108	0.108	0

Shape

Cavity width (l)


Emilio Sassine
email: emilio.sassine@gmail.com

Yassine Cherif
email: yassine.cherif@univ-artois.fr

Joseph Dgheim

Emmanuel Antczak

Investigation of the mechanical and thermal performances of concrete hollow blocks

Keywords: Hollow block design, heat modeling, mechanical modeling, thermal resistance, principal stresses

In order to improve the thermal performance of external masonry walls, the impact of the blocks' internal configuration on both their mechanical strength and thermal properties is numerically investigated. Ten different configurations with approximatively similar void ratios of about 40% (38.25% -40.5%) and same external dimensions (40 cm x 20 cm x 10 cm) were compared thermally and mechanically in order to investigate the effect of bulkheads on their performance and to promote some design guidelines. The complexity of the heat transfer phenomena inside the blocks as well as the stress distribution for vertical and lateral loads are detailed, leading to a better understanding of their thermal and mechanical behaviors. The numerical results indicated that the longitudinal bulkheads improve the thermal resistance of the blocks and thus reduce the heat flux passing through the blocks, while the transversal bulkheads reduce the thermal performance of the hollow blocks by creating heat bridges inside their structure. On the other hand, the mechanical behavior of the investigated blocks varies very slightly between the investigated models in vertical compression in the z-axis direction which reduces the influence of this parameter in the selection of the best shape configuration. However, the mechanical resistance to side compression increases by increasing the transversal bulkheads and decreasing the number of longitudinal bulkheads.

Introduction

Masonry hollow blocks are widely used as construction materials for what they offer as advantages in terms of physical properties (mechanical resistance, thermal and sound insulation, fire resistance, life span), low cost, low maintenance, and ease of implementation. The ever growing international interest related to the energy efficiency in buildings has led to many scientific and industrial efforts towards the improvement of masonry units in order to meet nowadays' energy saving requirements. These improvements can be made either by modifying the concrete mixture for reducing its thermal conductivity and thus reducing the conductive heat flow across the blocks, by filling the air cavities by insulation material, or by modifying the block configuration (vertical and horizontal bulkheads and cavities shape). However, this modification alters the mechanical properties of the block, especially its resistance to compression. The purpose of this work is to explore the effect of the block configuration on its thermal and mechanical properties so as to better understand the relationship between these parameters and the selection criteria of a given configuration.

Many studies discussed the thermal improvement potential of building blocks by either modifying the concrete composition as follows: the thermal conductivity, the thermal capacity, the thermal conductivity of the inserted material inside the cavities, the block thickness, and the number of hole rows. Martinez et al. [START_REF] Martínez | Thermo-fluid dynamic analysis of concrete masonry units via experimental testing and numerical modeling[END_REF] evaluated the effect of the thermal conductivity of the solid mixture, the geometry, and adding insulation materials on the heat flux, the distribution of temperatures, and the air velocity field inside the cavities, through experimental measurements and numerical simulations using ANSYS Fluent®.

In addition to the thermal studies related to concrete hollow blocks, their mechanical behavior was also examined at the block scale level in many research works. Barbosa et al. [10] analyzed numerically the behavior of hollow concrete blocks under uniaxial compression. Their numerical results were compared to the masonry unit experimental ones. The elastic and inelastic parameters (compressive strength, tensile strength, stress-strain relationships and fracture energy) were acquired from concrete samples that constitute the blocks. Ben Ayed et al. In this work, the mechanical and thermal behaviors of concrete hollow blocks are simultaneously studied by varying the blocks' internal shape aiming at determining the optimal hollow block design providing the optimal compromise between thermal resistance and mechanical resistance by examining ten different hollow block configurations. The heat flow inside the blocks is visualized numerically and the main parameters affecting the thermal performance are discussed. Heat conduction through the solid concrete matrix as well as convective and radiative heat flows inside the air cavities were visualized and discussed to highlight the effect of the block geometrical parameters on its thermal performance. The mechanical behavior of the block was investigated in two different load configurations: the first one consisting of a vertical compression in the z-axis direction and a lateral compression in the y-axis direction.

Scope of the study

Ten different configurations were considered with approximatively similar void ratios of about 40% (38.25% -40.5%). All the blocks have the same external dimensions (40 cm x 20 cm x 10 cm), they present hollows and are closed from the bottom with a 2 cm concrete thickness. They differ by the number and the shape of the cavities.

The dimensions of the studied blocks are reported in Table 1. In the first section of this paper the methodology is presented and validated for both thermal and mechanical In the second section, the thermal and mechanical results for the ten hollow blocks are presented in order to better understand, on the one hand, the effect of the blocks' internal shape on these parameters, and, on the second hand, to compare and promote some block configurations according to their mechanical and thermal performances.

The blocks are traditionally handcrafted and the mixture is made of 25% of fine gravel aggregates (4 -8 mm) and 75% of powdered stone dust (0.5 -4 mm). One cubic meter of aggregates is mixed with 50 kg of Portland cement and 50 liters of water to form the solid matrix mixture. The thermal and mechanical properties of the concrete used in this study are summarized in Table 2.

Table 2-Physical properties of the adopted concrete composing the blocks Density (kg.m -3 ) 2300

Heat Capacity at constant pressure (J.kg -1 .K -1 ) 880

Thermal conductivity (W.m -1 .K -1 ) 1.4

Young's modulus (Pa) 25 x 10 9

Compressive strength (Pa) 30 x 10 6

Poisson's ratio (-) 0.2

Figure 2-Experimental device for determination of thermophysical properties of building materials

The experimental device used for thermal characterization of building materials is described in Fig. The method consists in subjecting a sample of thickness "e" (m) to a temperature gradient, so as to impose a flux transfer from the hot side to the cold one. The heat flux φ (W.m -2 ) and the temperature on both sides of the sample are measured simultaneously. The Fourier's law applied in unidirectional steady state conditions gives [24]:

R T   1  and R T   2  (1)
Eq. 1 can be written in terms of the generalized quantities Σφ (W.m -2 ) and ΔT (K) as:

R T    2  (2)
The thermal resistance R (m 2 .K.W -1 ) and the thermal conductivity λ (W.m -1 .K -1 ) are thus given by:

    T R 2 and R e   (3)
b.) Determination of the specific heat Starting from a stable initial steady state, a temperature variation is imposed by changing the set point on one or both sample faces. The average initial temperature of the sample ΣTi (K) will change, as will the fluxes over each side. After the reestablishment of the steady state, the material has found a new stable state, associated with a new average final temperature ΣTf (K).

During the transient phase, the relation between the stored (or released) heat energy Q (J.m -2 ) and the heat fluxes difference  (W.m -2 ) is given by [24]:

   f i t t dt Q .  (4)
It can also be related to average temperatures ΣTi/2 and ΣTf/2, where ΣTi and ΣTf represents the sum of the temperatures on each face at the initial time ti (s) and at the final time tf (s).

2 i f T T C Q     (5)
The heat capacity of the sample C (J.m -2 .K -1 ) is thus given by:

i f i t t T T dt C f i       . 2  (6)
The specific heat cp (J.kg -1 .K -1 ) can be deduced knowing the density and the thickness of the sample:

e C c p   (7)

Experimental thermal results for the concrete mixture

Figure 3-Experimental measurement results for the concrete mixture [24]

Fig. 3 provides the experimental results for the concrete mixture. Using Eq. ( 1) and Eq. ( 3) and based on the experimental results presented in Fig. 3 in steady state conditions (t≈1.6h), the thermal conductivity of the concrete mixture can be determined.

Using Eq. ( 6) and Eq. ( 7) and based on the experimental results presented in Fig. 3 in transient conditions (1.7h<t<3.5h), the specific heat of the concrete mixture can be determined.

The determined thermal conductivity and specific heat of the concrete mixture are reported in Table 2.

Numerical thermal assessment method

In what follows, and after the validation of the heat transfer model, the different block configurations will be thermally compared based on the following criteria:

-Temperature distribution -Heat flux distribution -Dynamic and thermal convection phenomena inside the cavities A parametric study for different concrete mixture thermal conductivities was also presented to evaluate the effect of the concrete mixture's thermal conductivity on the overall thermal performances of the different block shapes.

The conductive heat transfer mode in the concrete mixture as well as the convective heat transfer inside the cavities, were separately investigated to understand the effect of each heat transfer mode.

The blocks are studied using a 3D geometry in steady state boundary conditions; a temperature gradient of 20 °C is imposed on the opposite faces of the block (0°C -20°C), the other faces remaining adiabatic as shown in Fig. 4.

This gradient was chosen because it is high enough to generate a noticeable heat transfer in the block for a better visualization of heat transfer phenomena and a better precision in the simulated results, while remaining in the temperature range encountered in real life situation in building energy simulation cases.

In the coupled CFD-thermal analysis of the hollow blocks, it is essential to account the three key mechanisms of the heat transfer (i.e., conduction, convection, and radiation). The conduction occurs in the solid concrete mixture while convection and radiation occur inside the block cavities.

Numerical and experimental thermal properties of the Lebanese traditional hollow block

The heat transfer simulation in hollow blocks is based on the thermal properties of the concrete mixture presented in Table 2. The air circulation inside the cavity, promoting the natural convection, was considered as a laminar flow. It is coupled with the radiation model using the discrete ordinates method (DO) and the simulation was performed using the implicit solver of COMSOL Multiphysics®. The air density was assumed to be dependent on

Validation of the mechanical model

A common representation of the stress-strain curve for concretes with strengths up to about 40 MPa is provided by the "Hognestad stress-strain modified model" [START_REF] Macgregor | Reinforced concrete mechanics and design[END_REF] shown in Fig. 7. The curve is composed by a second-degree parabola for a strain varying between zero and ε0=1.8f"c/Ec where f"c=0.9f'c, then followed by a downwardsloping line terminating at a limiting strain of 0.0038. In the adopted numerical model used for describing the block behavior, the concrete is considered as a nonlinear elastic material following the uniaxial data model provided by the Hognestad modified relation.

Numerical mechanical assessment method

In the numerical assessment, the blocks were subjected to a constant load assumed to be in their elastic loading margin aiming at comparing the stress distribution for each block configuration. At first, a vertical uniform boundary load of 100 kN was applied to the block in the z-axis direction (Fig. 9) and the stresses induced by this load were numerically simulated using the implicit solver of COMSOL Multiphysics® Modeling Software.

Figure 9-Boundary load for the compression in the z-axis direction

Analytical model

Numerical model

The failure criteria used in the study are the ones used for brittle solids and known as the maximum stress/strain criteria. The maximum stress criterion assumes that a material fails if the maximum principal stress σ1 in a material element exceeds its tensile strength σt , or alternatively, if the minimum principal stress σ3 is less than its compressive strength σc. The safe region for the material is thus: σc<σ3<σ1<σt.

Note that the convention that tension is positive has been used in the above expression.

Then, another parameter that is interesting to investigate for the mechanical behavior of the blocks is their compressive strength on their lateral surfaces. In fact, during transportation, storage, and implementation, the blocks experience some harsh conditions in the factory and on site that make their mechanical strength essential for their practical use. Thus, the mechanical resistance of the blocks was also studied according to the y-axis by applying a uniform boundary load of 10 kN on one face (Fig. 10), the other face being subjected to a fixed constraint.

Figure 10-Boundary load for the compression in the y-axis direction

Mesh independence analysis

The mesh independence is verified for Model 1 for both the thermal and mechanical simulation models. Also, the simulation time is evaluated in order to choose the optimal meshing configuration. It is important to mention that the meshing was performed automatically using the "Physics-controlled mesh" module of COMSOL Multiphysics® which allows a better accuracy of the results; this meshing is done depending on the physical property settings, the boundary conditions, and the geometry of the tested model. Fig. 11 shows the heat flux and the simulation duration for five different mesh configurations ("Extremely coarse", "Extra coarse", "Coarser", "Coarse", and "Normal"). The heat flux in the y-direction (perpendicular to the external block faces) stabilizes for an "Extremely coarse" mesh configuration (7436 elements). The simulation time increases from 25s for the "Extremely coarse" meshing (2632 elements) to 2324s (around 40 minutes) for the "Normal meshing" (103109 elements). "Finer" meshing configurations, beyond the "Normal" meshing, were not investigated to avoid very high simulation durations especially that the solution converges for lower meshing.

Thereafter, the "Extra coarse" meshing is adopted for its low simulation time (less than two minutes) and its good accuracy. simulation duration for nine different mesh configurations ("Extremely coarse", "Extra coarse", "Coarser", "Coarse", "Normal", "Fine", "Finer", "Extra fine", and "Extremely fine"). The Maximum First Principal Stress and Minimum Third Principal Stress require an "Extremely fine" meshing (142473 elements) to stabilize. The simulation time is relatively low compared to the heat transfer model and does not exceed four minutes for the "Extremely fine" meshing. In what follows, the "Extremely fine" meshing is adopted. A thermal conductivity of 1.4 W.m -1 .K -1 was considered for the concrete mixture.

Fig. 13 clearly shows the effect of longitudinal and transversal bulkheads on the conductive heat transfer in the different hollow block configurations. The longitudinal bulkheads improve the thermal resistance of the blocks and thus reduce the heat flux passing through the blocks. This can be clearly observed by comparing Models 1 and 3, Models 2 and 6, Models 4 and 7, and Models 5 and 8 respectively. Indeed, the external face of the blocks presenting a central longitudinal bulkhead (3, 6, 7, and 8) is darker than the face of the ones without a central longitudinal bulkhead (1, 2, 4, and 5) which means that the heat flux in the block with longitudinal bulkheads is lower than that in the blocks with transversal bulkheads. On the other hand, the transversal bulkheads reduce the thermal performance of the hollow blocks by creating heat bridges inside their structure, this can be observed by comparing Models 1, 2, 4, and 5.

Fig. 14 shows the temperature fields in the different studied blocks shapes and for a temperature difference of 20°C. Unlike the heat flux distribution that is highly dependent from the cavities number and shapes, the temperature distribution is very similar for the studied blocks. When the blocks possess one longitudinal row of cavities, the color of the cavities is mostly orange with one yellow colder side and one red warmer side, this is the case of the block models 1, 2, 4, and 5. However, when the blocks possess two longitudinal rows of cavities, the first row is mostly yellow with one blue colder side and one yellow warmer side, while the second row is mostly red with one yellow colder side and one red warmer side. Two facts can be confirmed from these observations:

1.) the temperature drop takes place mostly in the cavities which means that the cavities present the main barrier for the heat transfer in the hollow blocks. Furthermore, the average air velocity magnitude inside the cavities, reflecting the occurrence of natural convection, depends mainly on the number of longitudinal cavities. The hollow blocks presenting one row of cavities (Models 1, 2, 4, and 5) have higher velocity magnitudes as shown in Fig. 15 with average velocities exceeding 3 cm/s; while the remaining Models, presenting 2 rows of cavities, have lower average velocities less than 3 cm/s. On the other hand, adding adjacent cavities reduces slightly the air velocity in the cavities; this can be observed by comparing the average velocities in Models 1, 2, 4, and 5. The lower air velocities are inside the cavities, the lower are convective heat transfer currents, and the more insulating are the hollow blocks. Moreover, the comparison between models 9 and 10 shows that the elliptical cavity shape increases the air acceleration at the edges of the cavity leading to an increase in convective heat transfer inside the hollow block.

Fig. 16 shows the streamlines (left) and the isotherms (middle) for a vertical section in each block (right). For the streamlines, one can notice that the blocks presenting a single longitudinal series of the cavities, the curves form very dispersed and clearly visible elliptical recirculation zones occupying the entire volume of the cavities, these curves record maximum speeds of around 7cm/s at the edges of the cavities, which promotes and accelerates the convective heat transfer inside the blocks. One can also notice that the air velocity decreases while moving away from the edges of the cavities and the center of the cells remains almost inert. The remaining models presenting two longitudinal series of the cavities, the average maximum speed is around 5cm/s or less; the effect of convection is less favored in these models, the streamlines are very tight and take the form of a baguette shape.

The analysis of the isotherms allowing the distribution of the curves is divided into two families of shape.

1.) The first family includes the models presenting one longitudinal row of cavities (1, 2, 4 and 5) where the isotherms present a sinusoidal shape tendency. The temperature shape results from the movement of convection recirculation in the air cavities with an upward movement near the hot wall and a downward movement on the cold wall side. A relatively high thermal gradient is observed between the two opposing cavity walls reaching 14.5°C. The isotherms present blue tendency in the bottom of the cavities and red tendency at the top, which is due to thermal stratification where cooler and denser air sinks, and warmer and thinner air rises creating layers of air with a gradient of temperatures.

2.) On the other hand, the second family of curves which groups models (3, 6, 7, 8, 9 and 10), the shape of the curves gives a moderately linear tendency remaining very tight and confined in the spaces where the boundary layers are very close to the cavity walls. As these models present two longitudinal series of the cavities, we observe the formation of two cells presenting symmetrical isotherms shapes (since the blocks are symmetrically heated) and different colors, red and yellow colors are observed inside the cavities near the hot block side, and green and blue colors are observed inside the cavities near the cold block side. The thermal gradient in each cell (hot or cold) is around 8.5 ° C.

In summary, the convective transfer is widely present in concrete hollow blocks whatever their shape is. From a dynamic and thermal perspective, one can found that the nature of the thermal transfer differs from one block to another at both quantitative and qualitative levels. Furthermore, the thermal conductivity of the concrete mixture was varied between 0.2 W.m -1 .K -1 and 1.8 W.m -1 .K - 1 in order to study the effect of the concrete mixture on the overall thermal performances of the different block shapes.

The parametric study for different concrete mixture thermal conductivities represented in Fig. 17 for the ten investigated models, shows that the best thermal configuration is obtained by 3 (two parallel cavities) and the worst case is for Model 5 (four adjacent cavities). A thermal resistance difference of 40% to 60% is noticed between these two models depending on the thermal conductivity of the concrete. The results of Fig. 17 confirm the observations of Fig. 13 showing the effect of longitudinal and transversal bulkheads (by maintaining a same void ratio). The critical zone for the third principal stress (compressive stress) is at the bottom of the blocks and more specifically at the corners (Fig. 19). The critical zone for the first principal stress (tensile stress) is at the bottom of the blocks inside the cavities as shown in Fig. 20.

Mechanical behavior, compression y-axis

The side compression of the blocks (y-axis) has an important impact on the adequate selection of the blocks since the blocks are designed to support various loads and often undergo various loads during their transport, storage, and implementation.

( The mechanical behavior of the investigated blocks varies very slightly in compression in the z-axis which reduces the influence of this parameter in the selection of the best shape configuration. This is mainly due to the initial assumption of selecting blocks having approximatively similar void ratios of about 40%, and thus implies comparing blocks with same active area in compression test.

) (2) (3) (4) (5) (6) (7) (8) 1

Final results & discussion

The mechanical and thermal results are summarized in Table 4. Knowing that the compressive strength for concrete varies from less than 10 N/mm 2 for lean concretes to more than 55 N/mm 2 for special concretes, and the tensile strength of concrete is usually considered about one-tenth of its compressive strength [34], it can be clearly concluded from Table 3 that the first principal stress is the main criterion of failure that needs to be verified when studying hollow block concrete samples. Indeed, the maximum first principal stress is found to be always higher than one-tenth of the minimum third principal stress (in absolute value). Models 1, 2, 8, 9, and 4 are less interesting, since there exists a better block configuration on both thermal and mechanical levels.

Conclusion

A numerical analysis was performed for understanding the main heat transfer phenomena and the mechanical behavior of the concrete hollow block. A parametric study is performed to understand the effect of the geometry configuration on the overall thermal and mechanical resistances of the block. The ten investigated blocks present different inner shape configurations and preserve the same external dimensions (40cm x 20cm x 10cm) for ergonomic reasons as well as local workmanship techniques. The simulation results indicated that the longitudinal bulkheads improve the thermal resistance of the blocks and thus, reduce the heat flux passing through the blocks, while the transversal bulkheads reduce the thermal performance of the hollow blocks by creating heat bridges inside their structure. On the other hand, the mechanical behavior of the investigated blocks varies very slightly between the investigated models in vertical compression in the z-axis direction which reduces the influence of this parameter in the selection of the best shape configuration. However, the mechanical resistance to side compression increases by increasing the transversal bulkheads and decreasing the number of longitudinal bulkheads.

Five optimal models can be considered for having interesting mechanical and/or thermal performances (Model 3, 5, 6, 7 and 10). Model 3 possesses the best configuration from a thermal point of view (lowest U-value), while model 5 is the most performant in lateral mechanical resistance. Models 6, 7, and 10 present an optimum compromise between thermal and mechanical performances. The existing Lebanese hollow block shape (Model 10) presents therefore a good compromise between thermal performance and mechanical strength. This configuration also offers many other practical advantages such as:

-Ease of craftwork demolding -Ease of implementation by being easy to break to obtain fraction of blocks (half block or quarter block) -Ease of creating passages for the pipes and electric cables in masonry walls without crumbling the blocks

The conclusions drawn from this work provide valuable data for improving concrete hollow blocks design by highlighting the main parameters that influence the mechanical and thermal performance of concrete masonry hollow blocks.

[ 1 - 3 ]

 13 , or hollow block's configuration. Urban et al. [4] compared different hollow blocks geometries available in the market by using numerical simulation methods. Fogiatto et al. [5] examined the best thermal resistance for ten different blocks by varying the cavity configuration using the same external dimensions using ANSYS®. Kanellopoulos et al. [6] performed a transient thermal analysis using the finite element method (FEM) to determine the thermal response of a perforated clay brick masonry wall. Coz Diaz et al. [7] optimized lightweight hollow blocks used in internal floors by varying the number of vertical and horizontal intermediate bulkheads preserving constant external dimensions. He found that the horizontal intermediate bulkheads and the thermal conductivity of the concrete mixture were the main parameters influencing the thermal performance. Zhang et al. [8] also ranked the principal factors influencing the dynamic thermal performance in descending order

[ 11 ]

 11 carried out compressive tests on superposed Interlocking Stabilized Earth Block (ISEB) that are laid dry without grout mortar. These tests showed that the contact area and the clearance between the blocks decreased the masonry's compressive strength. A simplified spring model was considered for estimating the apparent Young's modulus of the wall.Arun et al. [12] explored the compressive and flexural characteristics of concrete composite and expanded polystyrene (EPS) block under standard loading condition as well as their thermal performance using R-value. The proposed system helps in sustainable building construction by affording high thermal insulation with improved structural strength. Javadi et al. [13] used recycled concrete aggregates from demolition wastes and compared their durability and mechanical strength to bio-blocks produced with natural aggregates. The impact of the block design on its mechanical properties was studied by Coz Diaz et al. [14] who used the finite element method and a combined Drucker-Praguer and Willam-Warnke material models for concrete to compute the compressive resistance of different concrete blocks and determine the optimum hollow block from the handling and structural points of view. Javidan et al. [15] performed a numerical research for an optimum shape of hollow blocks based on the ratio the compressive strength and the weight. The interrelation between thermal and mechanical performance study of masonry blocks based on their mixture composition has been the subject of various works. Caruana et al. [16] optimized a hollow block mix design searching for improving its thermal properties without compromising its physical (compressive strength) and geometrical properties (external dimensions). Mohammed et al. [17] evaluated miscellaneous physical properties (thermal conductivity, compressive strength, acoustic absorption and other…) concrete blocks mixed with a crumb rubber concrete mixture. Su et al. [18] determined the mechanical strength and thermal properties of an optimized hollow block consisting of an extruded polystyrene layer fixed between an internal hollow concrete block part and an external concrete decoration layer. Experimental measurements and numerical simulations were performed for this purpose. Sutcu et al. [19] studied the mechanical strength and the thermal properties for hollow clay bricks made up with different paper waste concentrations. Xu et al. [20] studied experimentally the effect of wood fibers content on the thermal and mechanical properties of autoclaved aerated concrete.

  models. The validation of the heat transfer numerical model is realized by comparing the experimental measurements with the simulation results, for the heat fluxes at the boundaries of the block. The thermal properties of the concrete material are determined experimentally using a solid concrete sample. The mechanical numerical model based on the "Hognestad stress-strain modified model" is validated by being compared to the experimental measurements of Álvarez-Pérez et al. [21].

  Before performing the numerical heat simulations, the numerical model was validated by comparing the numerical results to the experimental ones for a particular hollow block shape used very frequently in the Lebanese constructions (Model 10). The tangential gradient flux sensors used in this work use printed circuit technology; they are used in many building thermal applications (Cherif et al. [22] and Zalewski et al. [23]). The schematic construction principle of the sensor is shown in Fig. 1a. These fluxmeters present the advantage of providing a good sensitivity (~100 µV/W/m 2 ).

Figure 1 -

 1 Figure 1-Schematic drawing of the used heat flux sensors: (a) Heat flux sensor's sketch (b) Calibration apparatusThe zero-flux method was adopted for the calibration of the fluxmeter for determining its sensitivity and response to thermal solicitations[START_REF] Cherif | Superficial heat transfer by forced convection and radiation in a horizontal channel[END_REF][START_REF] Zalewski | Experimental and numerical characterization of thermal bridges in prefabricated building walls[END_REF]. The fluxmeter to be calibrated is covered with a heating resistor which dissipates a known amount of electrical heat by Joule effect as shown in Fig.1b. The calibration results provide a relation

  2. It is composed of two thermostatic baths related to two heating plates allowing to impose the temperature boundary conditions on the experimented building material. The heat fluxes and the temperatures on both sides of the sample were simultaneously measured using two thermocouples of T-type and two tangential gradient fluxmeters, having an active surface of 0.15 x 0.15 m². The lateral faces of the sample are covered with an insulation material to impose unidirectional heat transfer conditions. a.) Determination of the thermal conductivity

Figure 4 -

 4 Figure 4-Imposed temperatures and adiabatic surfaces

  pressure and temperature varying according to the ideal gas relation [25]. The validity of the heat transfer model inside the cavities of the blocks has been realized in previous works [24, 26]. The same temperature boundary conditions applied to the block in the experimental test were imposed to the simulated block on the opposite faces, the other faces remaining adiabatic. The typical Lebanese hollow Block (Model 10) was used for the validation of the numerical model.

Figure 5 - 10 ) [ 24 ]Fig. 5 ,

 510245 Figure 5-Experimental measurements and numerical results for the Lebanese traditional concrete hollow block (Model 10) [24]Fig.5, shows the evolution of the imposed temperatures and the measured and simulated heat fluxes on the block faces. The wall is first subjected to a 10°C temperature gradient by imposing about 14°C and 24°C a temperature conditions on its boundary edges. A steady state is reached after about 3 hours. Then, at t~4.2h, the block face subjected to a temperature of 14°C was heated to reach a temperature close to the other face (24°C). The comparison between the numerical results and the experimental measurements for the heat fluxes evolution at the boundaries of the block gives very similar results, except for the first part (between t=0h and t=2h) where the heat flux is affected by the material history (initial conditions) and the heat stored in the material before starting the tests.In order to define how well the measured and the simulated heat fluxes are identical, the Nash-Sutcliffe Efficiency coefficient (NSE)[START_REF]Online Calculators[END_REF] was determined for φ1 and φ2 as shown in Fig.6. The results indicate a good match between the simulation results and the measurements with NSE values close to 1 for both φ1 and φ2.

Figure 6 -

 6 Figure 6-Nash Sutcliffe model for φ1 (a) and φ2 (b)

Figure 7 -

 7 Figure 7-Hognestad modified stress-strain model

Figure 8 -

 8 Figure 8-Comparison between the numerical Hognestad stress-strain modified model and the experimental results obtained by Álvarez-Pérez et al. [21]

Figure 11 -Figure 12 -

 1112 Figure 11-Variations of the Total Heat Flux in the y-direction, and the total simulation time for different numbers of meshing elements

Figure 13 - 2 )Figure 14 -

 13214 Figure 13-Total heat flux in the y-direction (W.m -2 )

2 .Figure 15 -

 215 Figure 15-Air velocity magnitude inside the cavities (m/s)

Figure 16 -Figure 17 -

 1617 Figure 16-Air velocity field contour (cm.s -1 ) and isotherms (°C) inside vertical cross planes for the ten investigated models

Figure 18 -Figure 19 -Figure 20 -

 181920 Figure 18-First principal stress distribution for z-axis compression (MPa)Fig.18and Fig.19show a very similar stress distribution for the first and third principal stresses in the ten different investigated hollow blocks. The visualization of the results shows no favor for a particular block configuration when it comes to mechanical resistance to vertical compression in the z-direction; this is mainly due to the fact that the blocks were chosen so as to have similar void ratios and therefore similar active area against compression.

( 10 )Figure 21 -Figure 22 -

 102122 Figure 21-First principal stress distribution for y-axis compression (MPa)

Figure 23 -

 23 Figure 23-Pareto chart of the first principal stress distribution for y-axis compression and the U-value for the ten investigated block configurations

  W.m -2 .K -1

Table 1 -Configurations and dimension of the 10 tested cases

 1

	Block Model	Block configuration	Block name	Cavities dimensions (cm x cm)
	#1		1 cavity	34 x 5
	#2		2 cavities,1 row	17 x 5.2
	#3		2 cavities, 2 rows	36 x 2.5
	#4		3 cavities, 1 row	10 x 6
	#5		4 cavities, 1 row	7.5 x 6
	#6		4 cavities 2 x 2	17 x 2.6
	#7		6 cavities 3 x 2	11 x 2.7
	#8		8 cavities 4 x 2	8 x 2.8
	#9		8 cavities + alternation	3.2 x 2.8 / 8 x 2.8
	#10		Lebanese hollow block	R 2.8 / a 8.4, b 3.6

Table 3 -Summary of the mechanical and thermal results

 3

	z-compression (100kN)	y-compression (10kN)

Acknowledgement

This work is supported by a Grant from the Lebanese University and the LGCgE Laboratory.

Compliance with Ethical Standards

The authors declare that they have no conflict of interest.