This work presents the results of a set of steady-state numerical simulations about heat transfer in hollow blocks in the presence of coupled natural convection, conduction and radiation. Blocks with two air cells deep in the vertical direction and three identical cavities in the horizontal direction are considered (typically used for building ceilings). Moreover, their outside horizontal surface is subjected to an incident solar flux and outdoor environment temperature while the inside surface is exposed to typical indoor environment conditions. The flows are considered laminar and two-dimensional over the whole range of parameters examined. The conservation equations are solved by means of a finite difference method based on the control volumes approach, relying on the SIMPLE algorithm for what concerns the coupling of pressure and velocity. The effects of the number of cells in the horizontal direction and the thermal conductivity on the heat transfer through the alveolar structure have been investigated. The results show that the number of holes has a significant impact on the value of the overall heat flux through the considered structure.

The hollow blocks are generally the main component of the construction of building roofs in Morocco. These hollow blocks were designed to increase the thermal resistance of the roofs construction and for then help reduce the energy consumption requirements for heating or cooling. In general, each hollow block is formed by empty cavities surrounded by concrete partitions, and the heat transfer within these types of structures is complex and done by conduction, natural convection and radiation. Therefore, obvious non-linear characteristics exist within the heat transfer process of these perforations. So, the problems of coupling between the three modes of heat transfer are of great importance in recent years. Citing here some examples: Balaji et al. [

Sambou et al. [

Recently, a detailed numerical study that takes into account the coupling between the three modes of heat transfer in a three-cavity hollow block used in the construction of building roofs was presented by Najjaoui et al. [

This literature review shows that most studies concerning the coupling between the three modes of heat transfer (conduction, convection and/or radiation) are devoted to simple configurations consisting in rectangular cavities with or without conducting walls, subjected to a vertical temperature gradient. where the hallow block are heated from below or from above. The objective of this work is to study numerically in steady state, the heat transfer coupled by natural convection, conduction and radiation in an hallow block has two cells in the vertical direction but subject to real boundary conditions: such as incident solar flux, and heat exchange with the ambient environment via internal exchange coefficients h_{i} and external exchange coefficients h_{e}. The results will be presented in terms of streamlines and isotherms, temperature profiles and global heat transfer. moreover, the effect of thermal conductivity, the incident solar flux and the number of cells in the horizontal direction will be presented and examined.

_{i} and horizontal ones of thickness e’_{j}. The upper horizontal surface is exposed to an incident solar flux (G) and to an external ambient temperature T_{e} with a surface exchange coefficient h_{e}. The inside surface is submitted to indoor environment temperature, T_{i} and with a surface exchange coefficient h_{i}. The vertical faces are considered adiabatic.

By using the Boussinesq approximation, the obtained dimensionless governing equations are [

The dimensionless equation for heat conduction in solid walls is:_{s} is the dimensionless solid temperature and α_{f} and α_{s} are the fluid and the solid thermal diffusivities, respectively.

The boundary conditions are expressed as follows:

Adiabaticity condition on the vertical sides:

For the external horizontal face:

For the inner horizontal face:

The continuity of the temperature and the heat flux at the fluid–solid interfaces gives:

The dimensionless radiative heat flux can be expressed as [

The dimensionless average heat flux across the structure is given by:

The numerical code performed using FORTRAN software was validated by comparing the results with those of Lakhal et al. [

Ra | Strada et al. | Lakhal et al. | Ait-taleb et al. | Present code |
---|---|---|---|---|

10^{4} |
Nu = 2.153 | ------- | Nu = 2.164 | Nu = 2.161 |

10^{5} |
Nu = 3.888 | Nu = 4.00_{max} = 25.9 |
Nu = 3.917_{max} = 25.1 |
Nu = 3.913_{max} = 25.0 |

10^{6} |
------- | Nu = 6.91_{max} = 75.0 |
Nu = 6.702_{max} = 73.2 |
Nu = 6.698_{max} = 72.4 |

The results presented in this study are obtained for hollow block charactezied by the geometric dimensions given in _{s} varies between 0.5 and 1 W/mK. The incident solar flux G on the outside face of the structure is fixed at the value 1000 W/m^{2}, the fluid that reigns in the cavities is the air with the thermal conductivity k_{a} = 0.0262 W/mK and the Prandtl number Pr = 0.71, The emissivity of the internal surfaces of the cavities is ε

h | l | e_{i} |
e’_{j} |

7 | 13 | 2.5 | 2 |

_{s} = 0.5 and k_{s} = 0.8 W/mK there is an appearance of small vortices at the corners. This behavior indicates that the fluid heated from above or cooled from below tends to circulate just in the vicinity of the active walls. for k_{s} = 1 and k_{s} = 0.8 W/mK the isotherms are linear in the solid and in the fluid. And for ks = 1 W/mK the distortion of the isotherms in the solid walls separating the different cavities indicates a two-dimensional character of the heat transfer, whereas there are almost parallel lines in the cavities indicating a state of thermal stratification of the fluid.

_{s} increases, Indeed, the average value of the total flux increases from value 132.59 W/m² for k_{s} = 0.5 W/m.K to 217.75 W/m² for k_{s} = 1 W/m.K. This corresponds to an increase of approximately 64%. This can be explained by the conductive transfer rate which increases with increasing thermal conductivity according to Fourier’s law. The profile of this behavior remains the same for the three values of k_{s} considered. the heat flow presents a maximum at the level of the solid walls and inside the cavities there is a moderation of the heat transfer marking the absence of the convective transfer for the three values of k_{s} (heating from above).

In order to examine the effect of incident solar flux G on the temperatures of the upper and lower surfaces of the studied system. we represent on

The objective of this paragraph is to study the effect of the number of holes in a horizontal row on the heat flow through the hollow block. the ^{2}) crossing the alveolar structure as a function of the incident solar flux G, for thermal conductivity k_{s} = 1 W/m.K. The results of this figure show that this flux increases when the incident flux G increases, and varies almost linearly as a function of G especially for (Ny = 1). It can be noticed that the heat flux crossing the structure with one air cell deep in direction vertical (Ny = 1) is distinctly upper to the one found for (Ny = 2). Indeed, The structure having 2 holes in a vertical row decreases the heat transfer by about 35% with respect to the type with 1 hole. Indicating that this structure contributes to the improvement of the thermal performance of the roof of the building.

This work was given to the investigation of coupled heat transfer by conduction, natural convection and radiation through a structure with two air cells deep in direction oy submitted to an incident heat flux G. We examined the effect of thermal conductivity k_{s}, the incident solar flux and the number of cells in a hollow concrete block.

Based on the findings of the present work, the following conclusions are drawn:

The number of holes has a significant impact on the value of the overall heat flux through the structure studied, the heat transfer through the system could be considerably reduced by using a structure with two cells. This is increasing thermal resistance of the envelope and hence, reducing transmission loads.

The incident solar flux alters significantly the thermal fields through the studied structure. Indeed, the studied system has decreased the temperature by about 42°C between the outside and the inside of the roof of the building, which helps to ensure the reduction of thermal loads.

It has also been found that the effect of conductivity is important on the structure of the flow and the temperature field, Thus the heat transfer through the hollow block strongly depends on the thermal conductivity of the solid partitions k_{s}, This corresponds to an increase of about 64% passing from k_{s} = 0.5 W/m.K to k_{s} = 1.0 W/m.K.

These results can contribute to the future building envelope design in terms of insulation capacity and fill the gaps for research in the area of insulation systems.

structure height (m)

cavity height (m)

view factor between finite surfaces

finite area (m^{2})

_{i}

horizontal partition thickness (m)

_{j}

vertical partition thickness (m)

incident solar flux (W.m^{−2})

gravity (m.s^{−2})

radiosity (W.m^{−2})

thermal conductivity (W.m^{−1}.K^{−1})

structure depth (m)

cavity width (m)

_{r}

radiation to conduction number, σ_{s}ΔT)

_{k}

thermal conductivity ratio, _{f}/k_{s}

dimensionless pressure_{0}_{0}(_{f}/^{2}

pressure (Pa)

average heat flux, _{s}xQ_{a}xΔT)/H

_{a}

dimensionless average heat flux

_{r, k}

net radiative heat flux at surface k (W.m^{−2})

_{r, k}

dimensionless net radiative heat flux at surface k,

incident radiative heat flux (W.m^{−2})

position on the cavity surface

dimensionless position associated with r

Prandtl number, _{f} /α_{f}

Rayleigh number, ^{4}/(ν_{f} α_{f})

temperature (K)

dimensionless velocity components in x and y directions, _{f}/H)

dimensionless Cartesian coordinates in x and y directions,

thermal expansion coefficient (K^{−1})

thermal diffusivity (m^{2}.s^{−1})

cavity surface emissivity

fluid density, (kg.m^{−3})

Stephan-Boltzman constant (W.m^{−2}.K^{−4})

dimensionless coordinate normal to a cavity surface

kinematics viscosity (m^{2}.s^{−1})

dimensionless temperature, _{i})/

dimensionless streamline function

fluid

inside

outside

solid