Numerical computation of heat transfer, moisture transport and thermal comfort through walls of buildings made of concrete material in the city of Douala, Cameroon: An ab initio investigation

Abstract

The buildings of the city of Douala in Cameroon have been experiencing degradation for several decades due to the climate characterized by high humidity and oppressive heat. As a result, large grayish or black stains can be observed on these buildings. We sometimes witness the subsidence of the slab of the balconies, the cracking of the walls and the collapse of the buildings worn by the humidity. These damages are generally caused by infiltration and capillary rise. In addition, it has been demonstrated that people living in damp buildings are at risk of illnesses such as asthma and lung infections. Therefore, the novelty of this work is threefold: (i) it proposes for the very first time a numerical study of the transport of humidity and heat through the porous walls of buildings constructed with concrete material, the main construction material in the city of Douala; (ii) It was determined what level of indoor thermal comfort was appropriate for sleeping inside a real three-dimensional G+1 complex residential building constructed with concrete blocks; and (iii) Using the geographical coordinates, and time data, the sun radiation's direction of incidence was assessed throughout the simulation. The computation was performed using Comsol Multiphysics 6.0 software. The distributions of temperature, relative humidity as well as moisture level were presented at various periods. It appears from the results that face to this high humidity, the concrete material retains a large quantity of water for a considerable periods of time, which weakens the steel reinforcement of concrete which is corroded by rust. The computation of thermal comfort in the 3D building showed that the various rooms of the building were not comfortable during the night since temperature inside the building increased progressively due to diffusion of heat. In addition, the numerical solutions indicated that the energy stored within the walls diffused from the external walls to the internal walls during the night. It was also demonstrated that the walls of the building were warmer than the windows, doors and the roof at the computational times, which simply revealed a greater storage capacity of heat in the concrete blocks material. The findings highlighted that the temperature decreased rapidly in a thickness of 0.06 m of the concrete block during the nine days and this decrease was attenuated in the second part of the thickness of the concrete block (0.14 m).

Nomenclature

$d_{v,i}$

Mass flux vector of vapour water, kg/(m²s)

$P_{sat}$

Saturation pressure of vapour, Pa

$\phi$

Relative humidity

${\phi }_0$

Initial relative humidity

$h_e$

Coefficient of heat transfer, external, W/m².K

$h_i$

Coefficient of heat transfer, internal, W/m².K

${\beta }_e$

Coefficient of moisture transfer, external, s/m

${\beta }_i$

Moisture transfer coefficient, internal, s/m

$T$

Temperature, Kelvin

$T_0$

Temperature, Kelvin

$T_i$

Temperature, internal, Kelvin

$T_e$

Temperature, external, Kelvin

$T_{ref}$

Reference temperature, Kelvin

$M_w$,

Molar water mass, kg/mol

$d_{w,i}$

Mass flux vector of liquid water, kg/(m²s)

$K_l$

Permeability coefficient of liquid, kg/m.s.Pa

${\rho }_w$

Water's density, kg/m³

$R_w$

Water's gas constant, J/(kg.K)

$g$

Constant of gravity, m/s²

$w_w$

Moisture content of water, kg/m³

$t$

time, s

${\delta }_v$

Vapour's diffusion coefficient, kg/m.s.Pa

$D_{\phi }$

Capillary transport coefficient, kg/ms

${\rho }_s$

Solid's material density at the dry state, kg/m³

$c_{p,s}$

Solid's material specific heat capacity, J/kg.K

$c_{p,w}$

Water's specific heat capacity, J/kg.K

$\lambda$

Thermal conductivity, W/m/K

$c_{p,v}$

Specific heat capacity of vapour, J/kg.K

$h_v$

Enthalpy of evaporation, J/kg

$w_{sat}$

Saturation moisture content, kg/m³

${\mu }_{dry}$

Dynamic viscosity of dry vapour, Pa.s

p

Diffusion coefficient

${\phi }_i$

Internal relative humidity

${\phi }_e$

External relative humidity

$k_{dry}$

Dry material's thermal conductivity, W/m/K

$k_{moist}$

Moist material's thermal conductivity, W/m/K

1. Introduction

The accumulation of humidity in porous materials of the building causes not only the reduction of the envelope's thermal efficiency but moreover the metal component's deterioration and the degradation of the structure. The excessive presence of humidity in the premises is the result of humidity and heat's propagation through the structure's walls. Douala, a coastal city located in Cameroon, is characterized by relative humidity rates ranging between 55 and 96 % and temperatures ranging from 23 to 32 °C. The combination of high relative humidity rate and temperatures constitutes a real danger for buildings which are deteriorated (the walls turn black, the vegetable moss covers the walls, the metal reinforcement of the concrete is corroded by rust). A full comprehension of the phenomena of the transmission of moisture and heat taking into account the climate of the city of Douala would be an advantage for the explanation of the aforementioned damages. An extensive research has been conducted on the transmission of heat and moisture. Indeed, moisture and heat propagation were computed in exposed building elements [1]. They found an excellent fit between experimental and theoretical data. Srinivasan and Wijeysundera studied the transmission of moisture and heat in 2–5 mm damp cork surfaces in the presence of temperature fluctuations [2]. In comparison with the dry state, it was justified that the perceived thermal conductivity rose by almost 60 % for a soaked cork. Lü modeled the transmission of moisture and heat inside structures using thermodynamic laws [3]. He took into account building envelopes and indoor air and the model was validated using real test houses. Isgor and Razaqpur modeled heat transfer and the transmission of moisture and compared the solutions with experimental data [4]. Charoenvai et al. used an industrial software for simulating the moisture and heat transmission [5]. It was demonstrated that the moisture's impact on the perceived thermal efficiency increased with the absorption of water. Mendes and Philippi developed an approach to forecast the transmission of moisture and heat [6]. They determined that the significant fluctuations of the computed values for the moisture content and temperature profiles could result on some discontinuities of the moisture content. A mathematical approach for the simulation of the spread of moisture propagation in hygroscopic construction materials was formulated [7]. A joint computation of moisture and heat's spread in some porous media was reported [8]. A simulation carried out on a enclosed painting-related microclimate demonstrated that the phenomena were perfectly forecast. A mathematical approach to simulate the moisture and heat propagation in construction materials was investigated [9]. The investigators concluded that the method only required some transport properties, which were easy to determine. Slanina and Silarova investigated the moisture transmission via vapour retarders with perforations [10]. The authors finally proposed a model to describe the flux of water vapour diffusion via vapour retarders with perforations. Desta et al. reported experiments on the transmission of moisture and heat through the walls of a building taking into consideration atmosphere's conditions [11]. The quantity of absorbed wetness was exactly correlated with the interior decor's vapour permeability. A study of moisture and heat transmission in a space with radiator panels was carried out [12]. The findings showed that the consumed energy could be drastically decreased as well as improving thermal comfort. Traoré et al. studied the thermal characteristics of a timber panel [13]. The investigators noticed that when vapour converted into water, a sharp drop in the density was observed. The impact of wetness content on the transmission of moisture and heat was analyzed [14]. Computational simulations of the propagation of moisture and heat were developed in stacked exterior walls [15]. Zaknoune et al. estimated coefficients of the transmission of moisture in porous media [16]. They proved that two factors characterizing the liquid phase transfer parameters may be estimated concurrently. The process of uniformizing the moisture and heat simultaneously was developed [17]. The influence of the transmission of moisture on the energy-related effectivemess was carried out [18]. The findings indicated that if moisture transmission pathways are not taken into account, the energy of cooling and heating may be overestimated. Litavcova et al. simulated the diffusion of moisture into buildings [19]. The computations took into account the materials characteristics. A computational model for simulating moisture propagation was examined [20]. The heat transfer and moisture transmission were evaluated in concrete blocks [21]. Min et al. presented some indicators to study the moisture and heat transmission in concrete materials [22]. A numerical investigation of moisture and heat transmission was described [23]. The impact of ouside surroundings was evaluated [24]. The solutions demonstrated enhancements to the yearly energy balances. The simulation of moisture and heat transmission within porous media was developed [25,27]. Min et al. investigated how damage caused by load affected the moisture transfer [26]. A model presenting the transfer of moisture and heat in construction materials by considering the quantity of water in capillaries was developed [28]. Lee et al. proposed a review of thermal characteristics and energy economy of adapted concretes in tropical environments [36]. The authors concluded that concrete's thermal characteristics could be used for minimizing the energy consumed by the building. The thermal efficiency of hybrid concrete was investigated [37]. Kueh et al. performed the measurement and simulation of mechanical and acoustic characteristics of some construction materials [38]. The land surface temperature of a city of Bangladesh was estimated by Uddin and Swapnil [39]. From this literature review, it is worth noticing that no research work coupling the transmission of moisture and heat was carried out within the Douala's town characterized by high relative humidity rate and temperatures.

The buildings of the city of Douala in Cameroon have been experiencing degradation for several decades due to the climate characterized by high humidity and oppressive heat. As a result, large grayish or black stains can be observed on these buildings. We sometimes witness the subsidence of the slab of the balconies, the cracking of the walls and the collapse of the buildings worn by the humidity. These damages are generally caused by infiltration and capillary rise. In addition, it has been demonstrated that people living in damp buildings are at risk of illnesses such as asthma and lung infections. Therefore, three major points can justify the significancy of the present work: (i) it proposes a numerical study of the transport of humidity and heat through the porous walls of buildings constructed with concrete material, the main construction material in the city of Douala; (ii) the temperature distribution inside a real three-dimensional G+1 complex residential building constructed with concrete blocks was carried out; and (iii) using the geographical coordinates, and time data, the sun radiation's direction of incidence was assessed throughout the simulation. The remainder of the document is organized like this: Section 2 depicts the methodology. In section 3, numerical simulations are conducted. Lastly, some closing thoughts are expressed.

2. Methodology

2.1. Heat and moisture propagation's mathematical approach

2.1.1. Moisture transport

The processes describing the spread of moisture in porous media can be divided into three steps, namely.

• -The diffusion of vapour generated by the gradients in the pressure of vapour. It was modeled by the Fick's law [29]. It reads:

$$\begin{array}{cc}{d}_{v,i}=-\left(\delta P_{sat}\frac{d\phi }{d{x}_i}+\delta \phi \frac{d{P}_{sat}}{dT}\frac{dT}{d{x}_i}\right)& \forall i=\left\{1,2,3\right\}\end{array}$$ (1)

• -The capillary suction defined by the Darcy's law and generated by gradients in suction pressure [29]. The Darcy's law in this mechanism reads:

$$d_{w,i}=-K_l\left[{\rho }_w{R}_w\left(\frac{T}{\phi }\frac{d\phi }{d{x}_i}+\ln \left(\phi \right)\frac{dT}{d{x}_i}\right)+{\rho }_{w}g\overrightarrow{e_z}\right]\forall i=\left\{1,2,3\right\}$$ (2)

• -The transfer of vapour by transport of air generated by the gradients in the pressure of air.

By considering the conservation of mass for moisture content and by using (1), (2), the final equation defining the moisture transport reads [29]:

$$\frac{d{w}_w}{d\phi }\frac{d\phi }{dt}=\frac{d}{d{x}_i}\left(\left[{\delta }_v{P}_{sat}+D_{\phi }\right]\frac{d\phi }{d{x}_i}+\left[\left({\delta }_v\frac{d{P}_{sat}}{dT}+D_{\phi }\frac{1}{T}\ln \left(\phi \right)\right)\frac{dT}{d{x}_i}+\frac{D_{\phi }}{R_w}\frac{g}{T}\overrightarrow{e_z}\right]\phi \right)$$ (3)

2.1.2. Heat transport

By considering the equation of enthalpy in the conservative form, the heat transport can be modeled by the following [29]:

$${\rho }_s\left(c_{p,s}+\frac{w_w{c}_{p,w}}{{\rho }_s}\right)\frac{dT}{dt}=\frac{d}{d{x}_i}\left(\lambda \frac{dT}{d{x}_i}-\left[c_{p,w}{d}_{w,i}+c_{p,v}{d}_{v,i}\right]T-h_v{d}_{v,i}\right)$$ (4)

2.2. Study site description

The site of study is the city of Douala with the geographical coordinates (Latitude: 4.051056 North; longitude: 9.767869 East). Its elevation is 41.029. It has the largest port of the Central Africa [30]. The average maximum temperature is 32 °C during the year (28 °C in July and 34 °C in February). It rains 2918 mm over the year, with a minimum of 54 mm in December and a maximum of 494 mm in July [31]. The location of the city of Douala in Africa is shown in Fig. 1 [32]. The high humidity in the city of Douala leads to infiltration and capillary rise which generate black stains on the buildings walls (see Fig. 2) [33].

Fig. 1.

Location of the city of Douala in Africa [32].

Fig. 2.

Black stains caused by humidity on a building wall in city of Douala [33].

2.3. Comsol Multiphysics

Comsol Multiphysics 6.0 is a simulation software based on advanced numerical methods [34] and which was used to solve Eq (1), (2), (3), (4) on a 0.2 m long and 0.01 m wide chunk of two-dimensional concrete material. The description of the building and the properties of the concrete material used in the computations are given in Table 1, Table 2 respectively. Fig. 3 presents the distribution plan of the real building (ground floor (Fig. 3a) and upper floor (Fig. 3b)). Fig. 4 shows the computational domain of the G+1 residential building performed in Comsol Multiphysics 6.0 (Fig. 4a) and the various heat transfers (Fig. 4b).

Table 1.

Description of the building.

Level	Part	Length (m)	Width (m)	Area (m2)
Ground floor	Stay	5,90	5,60	33,04
	Living room	6,00	5,33	31,98
	Verandas	6,43	1,86	11, 96
	Dining room	5,90	3,67	21,65
	Kitchen	4,85	4,45	21,58
	Visitor's room	4,84	4,00	19,36
	Toilet in the visitor's room	1,75	1,60	2,80
	Toilet	1,81	1,50	2,72
	Closet	5,29	3,41	18,03
	Stairwell	4,84	3,14	15,20
Total 1				178,32
Upper floor	Master bedroom	5.65	5.01	28.86
	Dressing room for the master bedroom	4.26	3.28	13.99
	Balcony	6.25	1.95	12.29
	Bedroom I	4.95	4.67	21.12
	Bedroom II	7.61	3.91	29.79
	Bedroom III	5.65	5.35	30.25
	Bedroom IV	5,23	4,20	2,97
	Toilet for bedroom I	2.20	1.97	4.33
	Toilet for bedroom II	2.7	2.20	5.94
	Toilet for bedroom III	1.85	1.70	3.155
	Toilet for bedroom IV	2,00	2,00	4,00
	Corridor	7.61	1.28	9.78
Total 2				166,48
Total				344,79

Table 2.

Some properties of the concrete material implemented in Comsol Multiphysics.

Property	Expression
Thermal conductivity	$k\left(\phi \right)=k_{dry}+k_{moist}{w}_c\left(\phi \right)/{\rho }_w$
Density	212 kg/m3
Heat capacity at constant pressure	1000 J/kg.K
Moisture content	$w_c\left(\phi \right)=w_{sat}\left({k_1/\left(1+{\left(a_{1}h\left(\phi \right)\right)}^{n_1}\right)}^{1-1/n_1}+\left({k_2/\left(1+{\left(a_{2}h\left(\phi \right)\right)}^{n_2}\right)}^{1-1/n_2}\right)\right)$ $k_1=0.41$; $k_2=0.59$; $a_1=0.006$; $a_2=0.012$; $n_1=2.4883$; $n_2=2.3898$
Vapour permeability	$M_w/\left(RT{D}_{vap}\left(\phi \right)\right)$

Fig. 3.

Distribution plan of the real building: ground floor (a) and upper floor (3b).

Fig. 4.

Computational domain of the G+1 residential building performed in Comsol Multiphysics 6.0 (a) and various heat transfers (b).

2.3.1. Description of the building

The G+1 building consists of a house with a single slab, with an interior communication. It has a ground floor and an upper floor. Its measurements are 19.75 m long by 12.45 m wide, i.e. an 245.60 m² area. The structure is of the beam-column type, with 15 $\times$ 20$\times 40$ agglomerates filling. The foundation is made of 20 $\times$ 20$\times 40$ packed agglomerates. The roof frame has four slopes of 13 %, required for good evacuation of precipitation water, made of local wood, bubinga species. All openings are glazed. The roof is made of repainted sheets 2 m long. For the architectural design, the facades of the building are oriented according to the compass rose (North, South, East and West). The vertical covering is made with a cement mortar coating, dosed at 400 kg/m³. On the other hand, the horizontal covering is almost uniform in all the rooms, in ceramic stoneware tiles.

The ground floor is composed of.

• -A stay;
• -A living room;
• -Verandas;
• -A dining room (SAM);
• -A kitchen;
• -A visitor's room;
• -A toilet in the visitor's room;
• -A toilet;
• -A closet;
• -A stairwell.

The upper floor consisted of.

• -A master bedroom;
• -A dressing room for the master bedroom;
• -A balcony;
• -A bedroom I;
• -A bedroom II;
• -A bedroom III;
• -A bedroom IV;
• -A toilet for bedroom I;
• -A toilet for bedroom II;
• -A toilet for bedroom III;
• -A toilet for bedroom IV;
• -A balcony;
• -A corridor (which serves as communication between the different rooms).

An overview of the building components is provided in Table 1.

The parameters values utilized in this work are presented in Table 3. The boundary conditions implemented for the 2D geometry in this paper are shown in Fig. 5, Fig. 6. Indeed, a convective heat flux on the internal and external surfaces was considered. On both the interior and outside surfaces, the coefficients of convective heat transmission were determined to be: $h_i=h_e=10\mathrm{W}/\left(\mathrm{m}²·\mathrm{K}\right)$. In addition, the interior and exterior relative humidity rates were respectively defined as: ${\phi }_i=0.7$ and ${\phi }_e=0.8$. The upper and lower parts of the concrete blocks were supposed thermally insulated. Fig. 7 (a) and (b) show the mesh of the 3D and 2D computational domains. The 2D mesh consists of 1248 triangular elements.

Table 3.

Parameters values used in this work.

${\rho }_w$	1000 kg/m³
${\phi }_0$	0.95
$h_e$	10 W/(m2·K)
$h_i$	10 W/(m2·K)
${\beta }_e$	7.38E-12 s/m
${\beta }_i$	2E-7 s/m
$T_0$	293.15 K
$T_i$	293.15 K
$T_e$	305.15 K
$T_{ref}$	293.15 K
$M_w$	0.018 kg/mol
$R_v$	461.91 J/(kg·K)
$w_{sat}$	871 kg/m³
${\mu }_{dry}$	5.6
$p$	0.2
${\phi }_i$	0.7
${\phi }_e$	0.8
$k_{dry}$	0.06 W/(m·K)
$k_{moist}$	0.56 W/(m·K)

Fig. 5.

Boundary conditions implemented for heat transport in Comsol Multiphysics.

Fig. 6.

Boundary conditions implemented for moisture transport in Comsol Multiphysics.

Fig. 7.

2D (b) and 3D (a) meshes of the computational domains.

The initial conditions are given by Ref. [35]:

$$T\left(t=0\right)=T_0$$ (5)

$$\phi \left(t=0\right)={\phi }_0$$ (6)

3. Numerical results and discussion

The findings of moisture and temperature profiles through the concrete material under the climate of Douala during nine days are presented. In Fig. 8, it can be observed that the temperature decreases rapidly in a thickness of 0.06 m of the concrete block during the nine days and this decrease is attenuated in the second part of the thickness of the concrete block (0.14 m).

Fig. 8.

Temperature profile within the concrete material in Douala's environment during nine days.

In Fig. 9, the relative humidity distribution is shown. It is observed that it increases in a thickness of 0.06 m while the temperature decreases. It stabilizes between the thickness of 0.06 m and 0.12 m of the concrete block (95 % of relative humidity) and decreases between 0.18 m and 0.2 m. This simply means that the concrete block retains a high humidity in its core during a long period a time. This weakens the steel reinforcement of concrete which is corroded by rust.

Fig. 9.

Relative humidity profile within the concrete material in Douala's environment during nine days.

The moisture content is illustrated in Fig. 10. The profile of relative humidity corresponds quite well in the same period of time.

Fig. 10.

Moisture content profile within the concrete material in Douala's environment during nine days.

The three-dimensional findings are also depicted. Indeed, the temperature profile for the G+1 building were depicted in Fig. 11, Fig. 12 at times 24h, and 28h correspondingly. It can be seen that the external walls of the building have cooled between 12 a.m. and 4 a.m. because the energy stored in the walls diffused from the external walls to the internal walls. It can be also noticed that the walls of the building were warmer than the windows, doors and the roof at the aforementionned times. This reflected a greater storage capacity of heat in the concrete blocks material.

Fig. 11.

Temperature contours at t = 24h (12 a.m.).

Fig. 12.

Temperature contours at t = 28h (4 a.m.).

Fig. 13, Fig. 14 presented the temperature contours for a XY slice of the ground and the first floors of the building at t = 28h (4 a.m.) respectively. These solutions demonstrated that the temperature increased progressively inside the building due to the diffusion of temperature from the environment to the inner part of the building. These solutions of thermal comfort showed that at these particular times of the night, the various rooms of the building were not comfortable. The outcomes were fairly consistent with those in Ref. [35].

Fig. 13.

Temperature contours for a XY slice of the groundfloor of the 3D building at t = 28h (4 a.m.).

Fig. 14.

Temperature contours for a XY slice of the first floor of the 3D building at t = 28h (4 a.m.).

4. Conclusion

The present investigation examined the simulation of moisture and heat transmission through the porous walls of buildings constructed with concrete material in the city of Douala was investigated for the very first time using Comsol Multiphysics 6.0 software. The final thoughts that were sketched were as follows.

• -The energy stored in the walls diffused from the external walls to the internal walls during the night;
• -The walls of the building were warmer than the windows, doors and the roof at the computational times. This behavior reflected a greater storage capacity of heat in the concrete blocks material;
• -The various rooms of the building were not comfortable during the night;
• -The temperature increased progressively inside the building at time t = 28 h due to the diffusion of temperature from the environment to the inner surface of the structure;
• -The building's outside walls cooled between 12 a.m. and 4 a.m. because the energy stored in the walls diffused from the external walls to the internal walls;
• -The temperature decreased rapidly in a thickness of 0.06 m of the concrete block during the nine days and this decrease was attenuated in the second part of the thickness of the concrete block (0.14 m);
• -The humidity increased in a thickness of 0.06 m while the temperature decreased. It stabilized between the thickness of 0.06 m and 0.12 m of the concrete block (95 % of relative humidity) and decreased between 0.18 m and 0.2 m. This simply meant that the concrete block retained a high humidity during a long period a time;
• -The profiles of relative humidity and moisture agreed fairly well during the same time period.

In the present work, only the concrete block material was considered. Indeed, the most used construction materials in the city of Douala are concrete block and earth brick materials. Therefore, the simulation of the transmission of heat and humidity through porous walls of buildings constructed with each of the aformentioned materials in the city of Douala was left as future works.

CRediT authorship contribution statement

Andre Yves Moyou: Writing – review & editing, Writing – original draft, Validation, Software, Formal analysis, Conceptualization. Abdou Njifenjou: Writing – review & editing, Visualization, Supervision. Pascalin Tiam Kapen: Writing – review & editing, Writing – original draft, Validation, Supervision, Software, Methodology, Formal analysis, Conceptualization. Didier Fokwa: Visualization, Supervision.

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.