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The thermal performance of three roofing models: tile, corrugated and earth terrace is numerically analyzed. The mathematical equations which govern the three roofing models are established by the electrical method of analogies. These equations are discretized by an implicit finite difference method and solved by the Gauss-Seidel algorithm. We analyze the influences of geometric parameters (Xlo, Xlarg, α and Ep) on the evolution of the temperatures of the different environments of our three roof models. In particular, we have shown that the effectiveness of a roof in reducing the temperature inside a room is linked to its physical properties. The results obtained that for the same geometric parameters, the earth roof terrace and the earth tile roof compared to the corrugated metal roof improve thermal comfort by lowering the interior temperature of 5ºC and 4.6ºC.

Burkina Faso, a country with low energy resources, is characterized by a hot and dry climate that favours the transfer of heat inside the habitat, hence the thermal discomfort. The rate of sweating exceeds 80% during the hot periods of the year. To improve thermal comfort, we proceed by active air treatment methods, such as air conditioners. These methods generate excessive energy consumption. Faced with this, it is imperative to develop other methods of ventilation by natural convection. To do this, several possibilities can be considered, notably by taking into account in the design of the roof envelope. Thus, the search for thermal comfort through the improvement of the housing envelope is a very important area of research and has undergone great development in recent years. Several numerical and experimental studies have been carried out on the search for thermal comfort by improving the roof envelope. Among these studies, [

The tile roof has been used in Burkina Faso for about ten years. It is used in modern constructions, of size 0.3 m × 0.15 m. The interior consists of a 0.005 - 0.010 m thick wood ceiling that reduces heat transfer by conduction into the habitat as shown in

To simulate the thermal behavior of our three roof models, we divide the length (L) of the roof into several dummy sections according to the direction (X). To write the thermal balance of each slice of the length (L), we proceed by the method of analogies that exists between thermal and electrical transfers. Hypothetically, we assume that there is no mass transfer in the three roof models, and the air movement is one-dimensional (X). Also, we assume that the physical properties of air and roofing materials are constant (

material | Density (ρ) (kg/m^{3}) | Calorific capacity (Cp) (J·kg^{−1}·K^{−1}) | Thermal conductivities (λ) (W/m^{−1}·K^{−1}) | Emissivity ε |
---|---|---|---|---|

Clay tile | 780 | 1800 | 0.8 | 0.6 |

Compacted earth | 1500 | 1700 | 0.658 | 0.834 |

Aluminium | 2750 | 936 | 204 | 0.09 |

ceiling | 500 | 3000 | 0.15 | - |

law allows us to write the thermal balance equations according to the direction (X). The thermal balance equations are applied to the three roof models studied. To avoid the repetition of the heat balance equations in the text, we present the equations of the tile roof, which are also valid for the corrugated roof and the terrace roof. Indeed, these are the three modes of transfer which are convection, conduction and radiation.

Roof in Tiles

The external face

M c e ⋅ C p c e S ∂ T c e ∂ t = λ c e E p c e ( T c e − T c i ) + h c e ( T a m b − T c e ) + h r v c ( T v c − T c e ) + h r s o l ( T s o l − T c e ) (1)

The inside face

M c e C p c e S ∂ T c i ∂ t = λ c e E p c e ( T c e − T c i ) + h c i ( T a − T c i ) (2)

Air in the roof

M a C p a S ∂ T a ∂ t = h c i ( T c i − T a ) + h c p ( T p e − T a ) (3)

Upper side of ceiling

M p C p p S ∂ T p ∂ t = λ p E p p ( T p e − T p i ) + h c p ( T a − T p e ) (4)

The discretization of Equations (1) to (4) is expressed in the form:

External wall of the roof

M c e ⋅ C p c e S ⋅ T c e t + 1 − T c e t Δ t = λ c e E p c e ( T c e t − T c i t ) + h c e ( T a m b t − T b e t ) + h r v ( T v c t − T c e t ) + h r s o ( T s o t − T c e t )

T c e t + 1 = ( 1 + λ c e ⋅ S ⋅ Δ t E p c e ⋅ M c e ⋅ C p c e − h c e ⋅ S ⋅ Δ t M c e ⋅ C p c e − h r v ⋅ S ⋅ Δ t M c e ⋅ C p c e − h r s o ⋅ S ⋅ Δ t M c e ⋅ C p b e ) T c e t − λ c e ⋅ S ⋅ Δ t E p c e ⋅ M c e ⋅ C p c e T b i t + h c e ⋅ S ⋅ Δ t M b e ⋅ C p c e T a m b t + h r v ⋅ S ⋅ Δ t M c e ⋅ C p c e T v c t + h r s o ⋅ S ⋅ Δ t M c e ⋅ C p c e T s o t (5)

Inside the roof face

M c e ⋅ C p c e S ⋅ T c i t + 1 − T c i t Δ t = λ c e E p c e ( T c e t − T c i t ) + h c i ( T a t − T c i t )

T c i t + 1 = λ b e ⋅ S ⋅ Δ t E p c e ⋅ M c e ⋅ C p c e T b e t + ( 1 − λ c e ⋅ S ⋅ Δ t E p b e ⋅ M c e ⋅ C p c e − h c i ⋅ S ⋅ Δ t M c e ⋅ C p c e ) T c i t + h c i ⋅ S ⋅ Δ t M c e ⋅ C p c e T a t (6)

Air in the roof

M a ⋅ C p a S ⋅ T a t + 1 − T a t Δ t = h c i ( T c i t − T a t ) + h c p ( T p , s u p t − T a t )

T a t + 1 = h c i ⋅ S ⋅ Δ t M a ⋅ C p a T p i t + ( 1 − h c i ⋅ S ⋅ Δ t M a ⋅ C p a − h c p ⋅ S ⋅ Δ t M a ⋅ C p a ) T a t + h c p ⋅ S ⋅ Δ t M a ⋅ C p a T p e t (7)

Upper side of ceiling

M p ⋅ C p p S ⋅ T p e t + 1 − T p i t Δ t = λ p E p p ( T p e t − T p i t ) + h c p ( T a t − T p e t )

T p e t + 1 = ( 1 + λ p ⋅ S ⋅ Δ t M p ⋅ C p p ⋅ E p p − h c p ⋅ S ⋅ Δ t M p ⋅ C p p ) T p e t − λ p ⋅ S ⋅ Δ t M p ⋅ C p p ⋅ E p p T p i t + h c p ⋅ S ⋅ Δ t M p ⋅ C p p T a t (8)

The Equations (1)-(4) are discretized (5)-(8) using an explicit method of finite differences. This method transforms the transfer equations into a system of algebraic equations which are solved by the Gauss-Seidel method. The coefficients of heat transfer by natural convection and radiation depend on the temperatures of the different environments which are unknown [

We present the numerical results of our calculations for the three roof models, in the form of temperature curves in the different roof environments. Thus,

environments of the roof terrace during our typical day. We observe an increase in the temperature curves of the different environments from 10 h until reaching the maximum values at 13 h. After 13 h the temperature starts to drop. At the external wall of this roof, we have a temperature of 311.7 K at 10 h which evolves and reaches its maximum value at 13 h or 314.6 K, then it drops to 306.4 K at 18 h. For the inner wall, the temperatures are 310.7 K at 10:00, 313.6 K at 13:00 and 305.4 K at 18:00. These temperature differences are due to the thermal conductivity of the earth material. By comparing the temperature curves of the Figures 2-4, we find that the roof terrace presents the lowest temperatures followed by the roof tiles.

➢ Influence of calculation parameters on temperature distributions:

➢ To find the optimal parameters of our three roof models,we vary the following parameters:length(Xlo),width(Xlarg),inclination(α)and thickness of materials(Ep).

The curves of the Figures 5-10 show the influence of the geometric parameters (Xlo, Xlarg, α and Ep) on the evolution of the temperatures of the different environments of our three roof models.

The curves in

At the level of the roof terrace

By comparing the curves of the Figures 5-10, we find that the curves of

For the following parameters: Xlo = 2.35 m; Xlarg = 1.75 m; α = 35˚; Ep = 0.3 m, we obtain the following results:

By varying the parameters:Xlo = 3.15 m; Xlarg =2.2 m; α = 20˚; Ep = 0.2 m,we obtain the following results.

The obvious interest of the roof on the reduction of the internal temperature in a room, we have proceeded to the development of mathematical models based on the electric analogy that can predict the temperature of the different roof environments. These models integrate both meteorological data, geometric and physical parameters of the materials of our roof models. Through this study, we have provided an analysis on the thermal behavior of three roof models used in the Burkinabè habitat. A good use of the numerical method allows the thermal characterization of roofing materials.

At the end of this study, we show that the main causes of thermal discomfort in the habitat remain certain roofing materials with high thermal conductivity such as sheet metal, as well as the geometric shape. Thus, the corrugated roof shows a maximum temperature of 318.8 K for a thermal conductivity of λ Aluminum = 204 W / K − 1 ⋅ m − 1 . The terracotta tile roof shows a maximum temperature of 313.6 K for a thermal conductivity of λ terracotta = 0.8 W / K − 1 ⋅ m − 1 . As for the compressed earth roof deck, we observe a maximum temperature of 313.4 K for a thermal conductivity of λ earth = 0.658 W / K − 1 ⋅ m − 1 . By way of comparison the roof terrace offers better indoor temperature of the habitat.

Whatever the variations in the geometric parameters of the three roof models, materials with low thermal conductivity are those that generate the lowest temperatures within the habitat. The results show that the compressed earth roof terrace shows a satisfactory thermal behavior. However, the temperatures of the different media of each roof decrease as geometric parameters increase. The effectiveness of a roof to reduce the interior temperature of a room is related to its geometric shape and physical properties. This digital study results in an endogenous solution on heat transfer of three roof models in hot and dry climates. Our results also show that the choice of roof of a habitat has a significant impact on thermal comfort. It is always considered as the main element of the thermal regulation of the heat exchange between the interior and the exterior. This leads us to say that the roof terrace is the most suitable for the hot and dry climate that is that of Burkina. As a result, this work contributes to the awareness and promotion of local materials.

The authors declare no conflicts of interest regarding the publication of this paper.

Koumbem, W.N.D., Ouédraogo, I., Ilboudo, W.D.A. and Kieno, P.F. (2021) Numerical Study of the Thermal Performance of Three Roof Models in Hot and Dry Climates. Modeling and Numerical Simulation of Material Science, 11, 35-46. https://doi.org/10.4236/mnsms.2021.112003

Cp: Heat capacity (J·kg^{−1}·K^{−1})

Ep: Thickness (m)

hr: Coefficient of heat transfer by natural convection between a wall and the fluid that circulates in its vicinity. (W·m^{−2}·K)

hc: Coefficient of heat transfer by radiation between two walls (W·m^{−2}·K)

M: Mass (kg)

T: Temperature (K)

t: time (s)

grec symbols

λ: Thermal conductivity (W·m^{−2}·K)

Δt: Time step (s)

Indice

ce: Exterior face of roof tile

ci: Roof tile inside face

se: Exterior face of roof terrace

si: Interior face of roof terrace

vc: celestial vault

sol: sol

amb: Ambiance

a: Confined air between roof and false ceiling

pi: inside of ceiling

pe: outside of ceiling