Energy in buildings: A review of models on hygrothermal transfer through the porous materials for building envelop

The hygrothermal transfer is very important for the design of a building envelope for thermal comfort and economic and energy analysis of the building envelope. The applications of various materials in building envelope have been studied extensively. The study presents several models for the hygrothermal transfer for various building walls. Several energy and mass conservation equations with different boundary conditions and input considerations were presented in this paper for concrete, bricks and wooden walls. The effect of hysteresis was ignored in developing most model equations, while few considered flow pattern of fluid through the wall surfaces. Due to the flexibility of Luikov models, it formed the basis for modelling the coupled heat and mass transfer for porous material independent of hygroscopic nature with different boundary conditions defined according to the geometry and orientations. The influence of type of wall, orientation, thickness, the density of the material and climatic variations on the temperature and moisture evolutions within the building materials was more pronounced. Literature, presenting imaging models using imagery software like COMSOL multi-physics, CFD etc. were scarce considering that microscopic imagery is now deployed to measure the heat and moisture evolution in materials. Future models should include shrinkage or expansion influence on the fibrous material like wood due to their behaviour under environmental condition.


Introduction
All over the globe, structures such as housing, storage cabin for agricultural products (barns, concrete silos, mud rhombus etc.), animal houses etc. utilizes much energy for their life span. As the global population grows, more demand is placed on energy supply. Strains are imposed on both renewable and non-renewable energy resources. Therefore, energy efficiency within the building envelope has been advocated all over the world due to the large energy demand of the building and construction sector and the overall greenhouse gas emitted through the process to the environment [1]. Some researchers have estimated that building alone consumes about 36 -40 % of global energy, generating close to 50 % of global greenhouse gas emission [1,2]. However, in some countries, the IEA report has revealed that energy utilization in building alone consume over 70 % of the total available energy in those countries releasing a large amount of greenhouse gas into the environment [3]. Energy demands in a building can be categorized into energy utilization in the constructions phase and energy utilization in the operational phase [1]. Researchers have noted that the operational phase of a building consumes 75-90 % of the energy in a building while the rest comes from material extraction, production, construction, transportation and the end of the life span of the building [1,4,5]. However, in the operational phase, the aspect of building that consumes a lot of energy is the cooling [6,7] of building structures in tropical environment and heating of building in temperate zones. The global requirement for energy in cooling houses rose to about 11000GW in 2016 from 4000 GW in 19 90 [8,9].
The demand for energy efficiency in building structure has become more imperative because of the need to tame the global production of greenhouse gases produced during energy generation caused by building activities. There is the need to reduce globally the environmental impact caused directly or indirectly by building, through the development of a green strategy to enhance energy efficiency utilization on buildings. Although most researchers in this area have looked at the energy generation impact on transportation, manufacturing, construction and operational phases of the building on greenhouse generation on the environment, others have included, the acidification of the environment, depletion of the ozone layer, global warming, abiotic resources reduction, eutrophication etc. [1,[10][11]. Therefore emphasis is targeted on cutting down energy utilization in a building. Countries are adopting a green strategy in building, hoping to cut down energy consumption by 20-42 % by 2050 with a 35 % reduction in greenhouse emission [12,13]. Different strategies have been adopted by researchers in this area. While some has considered the entire structural envelop, others have looked at the materials for construction or various components of the structures ranging from the walls, the floor or the roof envelop [9, 14, and 15].
Another challenge faced by building developers besides energy utilization is the issue of moisture adsorption within the building envelope for hygroscopic materials. Condensations inside the building envelop is most common due to variation of temperature and humidity of the indoor and the outside of the building. The presence of moisture within the building envelope will cause thermal discomfort and can corrode the metallic structures within the building and also make the indoor mouldy [16]. This can lead to structural degradation and failure. Therefore the study of heat and mass transfer between the porous materials interface is done to elucidate their thermal performance and strength. The discontinuous moisture profile of two porous materials at the interface because of their hygroscopic characteristics has been used to predict the temperature and moisture gradients. Therefore, it is obvious that the nature of the materials used in building construction affects the temperature and moisture variations within the building as a function of the ambient weather changes and by extension the energy required for cooling or heating satisfaction [9,17]. Therefore it can determine the magnitude of humidity, heating and cooling required [18,19].
Materials like wood, metal, steel, concrete, pozzolan, glass, bricks and other multilayered composites have been adopted for building walls [20][21][22][23][24][25]. Osayintola et al [26] and Lelievre et al [17] classified building materials as classical and hygroscopic building materials. Lelievre et al [17] further sub-classified the building materials as bio-based and non-bio based materials. The authors stated that bio-based materials like hemp concrete have potential low carbon emission, good thermo-hydric properties and moisture buffering properties [27][28][29][30]. In several cases walls can be made of more than one materials layer, considering building insulations and cement plasters in some walls. Insulated walls (Figure 1) will regulate the heat and moisture transfer within the building envelope. Therefore experimental and numerical studies have been carried out to study the characteristics of heat and moisture transfer of various materials in response to variation in weather parameters which includes, solar radiation, temperature, moisture and relative humidity. Therefore studies on the hygrothermal transfer through different building walls have been presented in the literature [31-43, 16, 44-52]. Most of the studies developed predictive models to determine the temperature, relative humidity and moisture condition of the inside of the building as a variation to the ambient conditions [32][33][34][36][37][38][39][40][41][42][43]. The various studies used the thermo-physical properties of these materials to develop the simulation models to make their predictions with good result. The vapour adsorption and desorption isotherm is controlled by the main adsorption isotherm with the occurrence of hysteresis in the sorption curves [17]. Although the nature of the hysteresis is yet well established, some authors included it in their modelling approach while others neglected it. Simo-Tagne et al. [9] stated that studies like this will help in the selection of materials to satisfy the desired heat load at minimum dissipation of energy with less environmental impact. Therefore this research is focused on the presentation and review of various studies and models developed for different building walls made from different materials used in wall construction. This will help to elucidate different model approach adopted and help to improve building design to conserve energy and achieve satisfactory living comfort at minimum operating conditions.

Hygrothermal transfer models for concrete and brick walls
There exists several modelling and simulation investigation with a different approach for concrete and bricks based walls in literature. To reduce energy consumption or improve hygrothermal transfer in buildings, researchers have developed different kinds of wall composite. Walls can be single, double or multilayer. A multilayer wall is a wall made up of more than one porous material. Concrete (bio-based and non-bio-based), bricks, cement plasters etc can be used to develop a multilayer wall. These walls have also been numerically simulated to obtain different varying conditions for indoor and outdoor conditions and the results of the models validated experimentally. Xingguo et al [20] modelled a multi-layered porous wall made of cement mortar, red bricks and cement plaster in the southern Chinese city of Hunan. The model developed was a modified one dimensional transient hygrothermal model with temperature and humidity as the driving potential. Two key transport equations (1 and 3) for mass and heat transport respectively were solved using the finite element with established boundary conditions and the thermophysical properties of these materials. The maximum temperature and humidity difference obtained by the researchers were 1.87 o K and 11.4 % for indoor and outdoor condition.
The resultant boundary conditions for the above equation were given as follows: Where ᶲ is the relative humidity, T is temperature ξ is the slope of moisture equilibrium curve, Ws is the saturation moisture content of air, t is time, W is Air humidity ratio, Dv is the vapour diffusion coefficient of the material, Rv is the water vapour characteristic constant, ρa is the density of air, ρm is the density of the material, Tm is the temperature of the material, Dw is the liquid diffusivity coefficient of the material, Cm is the volumetric moisture capacity, hm is the mass transfer coefficient, x is location.
For the heat transport, the following governing equation was used The resultant boundary conditions for the above equation were given as follows Where Qrad is direct solar heat gain, ms is the rate of mass transfer at the surface of the wall, K is the thermal conductivity.
Lelievre et al [17] combined two sub-models of Pederson and phenomenological model developed Mualem to develop a numerical simulation model for multilayer hemp concrete. Hemp concrete is usually coated with plasters of different level of permeability inside and outside and the thickness is non-homogenous [17]. The model developed which accounted for phase change and hysteresis depended on the temperature and moisture transfer as a function of the hygrothermal properties of the hemp. They gave the energy and moisture conservation equations as follows Where ρs is the apparent density of the material, Dv is the Vapor diffusion coefficients, w is the moisture and is the sorption capacity lv is the heat of vaporization at the reference temperature Tref.
The sorption capacity of equations 6 was deduced using two hysteresis models in sorption and desorption phases from Pederson (equation 7 and 8) and Mualem (equations 9 and 10) as follows Where wdes and wad are the adsorption and desorption moisture isotherms for a particular relative humidity. A B and C are coefficients determined empirically. Using Mualem models, the following sorption equations were deduced Where φi and φj is the minimum and maximum values of the relative humidity while wi and wj are the minimum and maximum values of moisture content respectively in the adsorption-desorption cycle. Pd is a function that depends on pore water blockage against air entry, while ws is the saturated moisture content.
Validation of the above models showed that using sorption isotherm from Mualem gave a good agreement between the experimental and predicted results while the model of Pederson was off the mark.
Djongyang et al [42] presented a hygrothermal transfer model for porous building components which they validated with earth bricks wall. They considered a plane geometrical shape and the influence of inter-tropical condition with variations in latitude for the three cities of Cameroon. In solving the numerical equations, they considered the periodic solution approach and validated their model with two works of Menghao et al [45,50]. The model developed was one dimensional in which liquid water and air and water vapour as a single binary gas mixture was considered. The conservation equations for mass and heat transport were taken from the equation of Luikov which has been used by other researchers [52] as follows The boundary conditions for the two equations above were defined as follows Where T is temperature, t is time am is moisture diffusion coefficient (m2/s), h is convective heat coefficient (W/m 2 K), km is moisture conductivity (kg/m s), kq is thermal conductivity (W/kg K), I is the thickness (m), u is moisture rate (kg/kg), x is the distance (m), is thermal diffusivity (W m2/J), is thermogradient coefficient (K -1 ), is the phase change coefficient, is the heat of phase change (J/kg) Equations 13 and 15 represent the heat balance with the three components of the quantity of heat exchanged for outdoor and indoor, the convective heat transfer and evaporative flux while equations 14 and 16 represents the mass balance with the components of moisture gradient, temperature gradient and the convective flux exchanged between the ambient and the surface of the materials. Djongyang et al [42] considering the length of the day and the declination of the sun converted equations 11 and 12 to oscillatory (periodic) equations using the Fourier series method as presented in equations 17 and 18 which they used in their validation adopting a periodic approach Where w is frequency over a whole day cycle. Using a numerical approach they solved equations 17 and 18 to generate the temperature and moisture profile of the material under the variable external conditions. Validation of the results using thermo-physical properties of earth bricks gave good results. They showed that latitudes affect the hygrothermal transfer.
Menghao et al [45] also used the same one-dimensional Lukoiv equation presented in equations 11 and 12 in the modelling of a hygrothermal transfer for a fibrous slab but they added the effect of adsorption and desorption heat which is also one of the source or sink terms in coupled heat and mass transfer equations. They assumed a localized thermodynamic equilibrium between the fluid and the porous matrix in presenting the equations as follows.
The latent heat of vaporization will be integrated as part of the energy balance at the building materials boundary which will be affected by the mass diffusion due to deviation in temperature and moisture content [42,53]. The boundary equations were given as follows Where Cm is the specific moisture, Cp is the specific heat, Dm is the conductivity coefficient of moisture content, h is the latent heat, hlv is the heat of phase change, k is the thermal conductivity, l is the thickness of the specimen, m is the moisture potential, T is temperature, w is the moisture content, α is the convective heat transfer coefficient, Β is the convective moisture transfer coefficient, γ is the Heat of adsorption or desorption, ε is the Ratio of vapour diffusion coefficient to total moisture diffusion coefficient, δ is the Thermo-gradient coefficient, φ is the Transformation function In the resolution of the equations using Laplace transformations, dimensionless terms were introduced to equations 19-24 to change it to dimensionless form. Validation of the equations was done using the experimental set up in Figure 2. The porous material is a multilayer material whereby that one side is permeable and exposed to outdoor humid condition while the other side of the wall which is impermeable were subjected to cold temperature. Figure 2: the experimental set of the testing rig for model evaluation [45,54].
Validations of the numerical simulation model although showed the same trend with the experimental data but looking at Figure 3 plotted by the authors it shows that there is still temperature variations which imply that there is still some omitted parameters like hysteresis which is yet to be understood by researchers because of non-homogeneous nature of some porous materials, however, moisture profile was more accurately predicted as shown in Figure 4.  Simo-Tagne et al [9] presented a numerical model for predicting the hygrothermal transfer for concrete wall for outdoor conditions in sub-Saharan Africa. The model took into account all kinds of bound and integrated the type of flow in the boundary conditions which is not common in other established models. They also stated that all water type occurring in the material is modified during the moisture transfer, therefore they presented the mass conservation equations for liquid water, bound water and vapour phases in equation 25-27 respectively as follows Where as J → Kl and Kas the flux vector of the bound water phase and the volume mass flowrate to the conversion of bound water to unbound water are respectively. The Contribution of water vapour to the mass balance equations is given by: The above equations were discretized with finite differences using the Crank-Nicolson scheme and the solution is with the Gauss-Seidel relaxation iteration method using Fortran 90 language. Validation of the model was with hemp concrete with a low mean relative error is shown in Figure 5 below.

Hygrothermal transfer models for wooden walls
The impact of wood as a building material for walls has been explored greatly in literature. The investigation has been carried out on the effects of woods on thermal comfort in terms of, relative humidity, temperature, background noise levels and CO2 concentrations [55][56][57][58][59][60][61]. Therefore wood is energy efficient with lower CO2 concentrations. Wood as a hygroscopic material absorbs moisture from the environment which has a direct and indirect impact on the room conditions and thermal comfort [62]. Therefore the type of wood, thickness, moisture isotherm etc. has been factored in the predictive models to describe the hygrothermal behaviour of wood for indoor and outdoor conditions. Due to the flexibility of Luikov models, it has formed the basis for modelling the coupled heat and mass transfer for porous material independent of hygroscopic nature. The model accounts for all the bonding water simplistically without restricting the water transfer mechanism [63]. Therefore these models can be applied for one, two or three dimensional coupled heats and mass transfer in woods. Younsi et al [63] used this model for predicting the temperature and moisture transfer in a wooden slab. After defining the entire dimensionless variables, they used the finite element method which operates in Matlab to solve the partial differential equations in three dimensions. They concluded that different dimensionless numbers (Luikov, Kossovitch, Posnov and Biot numbers) in the coupled heat and mass transfer equations affected the overall heat and mass transfer behaviour of wooden slab. Comparison of this model with analytical and experimental data showed closer agreement with the analytical model rather than experimental data. Osayintola et al. [26] used a combined Knudsen and Fickian diffusion and neglected thermal diffusion in the modelling heat and mass transfer evolution in spruce wood. The energy and mass conservation equations were defined in equations 32-35 as follows: Figure 5. Comparison of experimental and simulated data (moisture content (a) and temperature (b)) through the wall thickness in concrete, and variations of the mean relative error (c) [9].
Where Cpeff is the specific heat capacity at constant pressure (J/(kg K)), Deff is the effective vapour diffusion coefficient (m 2 /s), hfg is the latent heat of vaporization/sorption (J/kg), ε is the volume fraction, ρ is the density (kg/m3), ρ0 dry density of the plywood specimen (kg/m 3 ), m˙ phase change rate per unit volume (kg/(m 3 s)) or mass flow rate (kg/s), u is the mass of moisture adsorbed per kg of dry spruce plywood (kg/kg),x distance from the top of plywood specimen (m), t time (s), T temperature ( o C) The above equations were discretely solved and a stable solution was provided with a relaxed, Gauss-Seidel iteration method. Discretization was by finite difference method with 2 nd order accuracy for the implicit scheme. Although the authors suggested that the presented diffusion model can be used to compare the experimental results and set benchmarks for similar materials but they did not show this with their experimental data. Rather they only fitted the sorption data with moisture content, thermal conductivity and water vapour permeability with good results. Figure 6 showed the fitting of the sorption data with a moisture content which is very good Figure 6: fitting of the Sorption isotherm for spruce plywood with moisture content data [26].
Simonson et al [64] presented a numerical simulation model to predict the moisture transfer (indoor climate) between the indoor air and the structural materials for wooden building in Belgium, Germany, Finland, and Italy. The model was developed to predict the temperature and relative humidity and applied to measure the comfort thresholds of the occupants in the building. The analysis of the model showed that the permeable interior layer is more satisfactory than when the interior layer is made with a water-resistant layer. The model is presented following the Ficks law of diffusion as follows = − ( , )∇ − 0 ( , )∇ + + (36) Where Pv partial pressure of water vapour (Pa), ρo dry density of the porous material (kg/m3), ρv density of water vapour (kg/m 3 ), ρw density of liquid water (kg/m3), T temperature (C), u moisture content (kg/kg), va velocity of air (m/s), Dw liquid moisture diffusivity (m 2 /s), g acceleration of gravity (m/s 2 ), K moisture permeability (s), kd vapour permeability (kg/s m Pa) predicted Equation 36 is adopted from IEA ECBCS Annex 24 'Heat, Air and Moisture Transfer in Insulated Envelope Parts presented in detail in Hens [65]. The model took into consideration all the energy components of the moisture transfer process for adsorption, desorption, condensation, evaporation, freezing and thawing. The conservation equations were solved simultaneously to predict the variable indoor conditions under different experimental data obtained in the field [66][67][68][69][70][71].
Talukdar et al [72] used a numerical model to set a benchmark for 1-D transient heat and moisture transfer models of spruce wood as a building material. Although the partial differential equations presented in equations 25-28 was used to set the governing energy conservation equations they used analytical approach to set the moisture penetration depth through the woods using equation 37 Figure 7 showed the result of the model and the effect of moisture penetration, at a step change of relative humidity and varying airflow rates. The obtained results show that increasing the air velocity increases the temperature, relative humidity and moisture accumulation within the plywood.
Watt et al [73] proposed a two-dimension model for a light timber wall with an air barrier with changes in air tightness in a Swedish environment shown in figure 8. The model was used to predict the moisture accumulation and mould growth within the building envelop. The magnitude of moisture accumulation is higher behind the outdoor air-tight layer of the simulated wall with the interior of the wall unsealed when compared to the sealed inside.
Simo-Tagne et al [9] numerical model presented in equations 25-31 above was also used to predict the hygrothermal transfer for different wooden wall for outdoor conditions in sub-Saharan Africa. In this case the model took into account the bound water present in wood with the integration of the flow patter in the boundary conditions. Validation of the model was with Norway Spruce wood with a low mean relative error is shown in Figure 9 below.
Further analysis of the model showed that less dense wood with increased thickness provided better thermal comfort. Therefore the nature of the wood and climatic factors is an important consideration that affects the hygrothermal transfer in woods.
Simo-Tagne et al [74] also presented a numerical simulation model for building walls in Nancy France. The same approach and solutions in Simo-Tagne et al [74] was adopted but they integrated the Dufour and Soret effect. The driving potential for the heat and mass transfer was temperature and moisture gradient. The model showed the negligible influence of the wooden structure while thickness is important in cancelling the effect of ambient conditions.   Berger et al. [75] proposed a numerical model using Scharfetter-Gummel sheme combined with a two-step Runge-Kutta approach. Three phases were distinguished: water vapor, liquid water and dry air. The moisture mass balance was given as follow: The energy balance was given as follow: The heat flux was expressed as: V is the vapour velocity taken equals to the air velocity and given by: The flux of water vapour was given by: The flux of dry air was given by: The flux of liquid water was given by: is the global moisture permeability, Π is the porosity of the material, Indexes v, l and da denote respectively water vapour, water liquid and dry air, P is the pressure, w is the volumetric concentration, R is the specie gas constant, k is the permeability, σ is the saturation rate, T is the temperature, C is the specific heat, is the thermal conductivity of the material, r vl is the latent heat of vaporization, t is the time, c o is the specific heat, ρ o is the dry-basis specific mass and ∇ is the gradient symbol.
Applying this model to the wood fiber using the constant properties (without influences of humidity and temperature), Berger et al. [75] used a programme translated with Matlab TM to obtained the curves that defined well the experimental data presented in Figure 10.
The thermal and hydric diffusivities (DHH, DHT, DTH and DTT) were given by capillary pressure (P c ); air permeability (k, k r ); diffusivity of the vapor in air drying (D g ); concentration of the vapour (C g ); densities of drying drying (ρ g ), free water (ρ l ) and dry wood (ρ s ); dynamical viscosity of the air drying (μ); mass diffusivity of wood material (D H ); heat of vaporization (E, E b ); heat of sorption (L); the sorption isotherm (Xe) and relative humidity of drying air (HR); with R the gas perfect constant; e is the thickness, hc and hm are the heat and mass convective transfer coefficients respectively.
Using the real variations in thermophysical parameters presented in equations 47-55, Simo-Tagne et al. [77] showed that the model is flexible and will allow to take into account all properties of the wood types and the movement of all types of water (bound water, free water, vapor of water). However they suggested that to adopt the model, the hydric diffusivities has to be experimentally determined for each wood.
Generally, to have accurate model for all the heat and mass transfer model equations, the sorption and desorption isotherm for all the materials have to be accurately determined and also the accurate boundary condition defined [77][78][79]. This values and equations mainly differentiated most of the presented equations in this reviewed with each other.

Conclusions
The hygrothermal transfer equations for various building walls are reviewed in this paper. The hygrothermal transfer is very important for the design of a building envelop for thermal comfort and economic and energy analysis of the building. Several numbers of energy and mass conservation equations with different boundary conditions and input considerations have been presented in this paper for soil-based and the wooden building wall. Most of the research ignored the effect of hysteresis in their models while very few considered the flow pattern of fluid through the wall surfaces. A review of the literature shows that most of the models were based on Luikov equations for heat and mass transfer and solution were mostly by finite difference methods. Validations of the models showed the influence of wall, thickness, the density of the material and climatic variations on the temperature and moisture evolutions within the building materials. The choice of building material in the building envelop has shown to influence the energy utilization in the building. However, literature, presenting imaging models using software like COMSOL multi-physics, CFD etc. is scarce bearing in mind that microscopic imagery is now deployed to measure the heat and moisture evolution in materials. Future models should include shrinkage or expansion influence especially in fibrous material like wood as they respond to ambient variations.