MC Ndukwu et al. /Future Technology November 2023| Volume 02 | Issue 04 | Pages 33-44 33 Review Energy in buildings: A review of models on hygrothermal transfer through the porous materials for building envelope Macmanus Chinenye Ndukwu1*, Merlin Simo-Tagne2, Ifiok Edem Ekop3, Matthew. I. Ibeh4, Maureen .A. Allen4, Fidelis. I. Abam4, Lyes Bennamoun5, Razika Kharchi6 1Department of Agricultural and Bioresources Engineering, Michael Okpara University of Agriculture, P.M.B. 7267, Umuahia, Nigeria 2LERMaB, ENSTIB, 27 rue Philippe Séguin, PO Box 1041, F-88051 Epinal, France 3Department of Building, University of Uyo, Akwa Ibom State, Nigeria 4Department of Mechanical Engineering, Michael Okpara University of Agriculture, P.M.B. 7267, Umuahia, Nigeria 5Department of Mechanical Engineering, University of New Brunswick, New Brunswick, Canada 6Centre de Développement des Energies Renouvelables, CDER, B.P. 62, Route de l’Observatoire, 16340 Bouzaréah, Alger, Algérie A R T I C L E I N F O Article history: Received 20 February 2023 Received in revised form 19 March 2023 Accepted 24 March 2023 Keywords: Porous materials, Building walls, Modelling, Moisture absorption, Green Building *Corresponding author Email address: ndukwumcu@mouau.edu.ng DOI: 10.55670/fpll.futech.2.4.4 A B S T R A C T The hygrothermal transfer is very important for the design of a building envelope for thermal comfort, economic and energy analysis of the building envelope. The lack of reference materials on models of moisture and temperature behavior in the building, including wooden walls, is a challenge. This paper reviewed the hygrothermal transfer models for building walls. Energy and mass conservation equations with boundary and input conditions were presented in this paper for concrete, bricks, and wooden walls. The review showed the presence of mainly physical-based models, while there is a dearth of data-based models. The influence of the type of wall, orientation, thickness, the density of the material, and climatic variations on the temperature and moisture evolutions within the building materials influenced the model mechanisms. Future research gaps should include shrinkage influence on hygroscopic materials like wood due to their behavior under ambient conditions. Data-based models should be explored too. 1. Introduction Heat and moisture transport has been studied simultaneously in building envelope. The relative humidity of the air indoors can affect the micro-climate of the building envelope and, by extension, the energy consumption. While there is a clamor for energy-efficient buildings [1], this should be achieved in consonance with the material that has high moisture buffering capacity to avoid moisture damage within the building. Some researchers have estimated that building alone consumes about 36 -70 % of global energy, generating close to 50 % of global greenhouse gas emissions [1-3]. This energy is expended right from the operational phase of a building, material extraction, production, construction, transportation, and the end of the life span of the building [1- 9]. Therefore emphasis now is to cut down energy utilization in a building to reduce greenhouse generation on the environment, acidification of the environment, depletion of the ozone layer, global warming, abiotic resources reduction, eutrophication, etc [1,10,11]. Countries are adopting a different green strategy in building, hoping to cut down energy consumption by 20- 42 % by 2050 with a 35 % reduction in greenhouse emissions [12, 13]. While some have considered the entire structural envelope, others have looked at the materials for construction or various components of the structures ranging from the walls, the floor, or the roof envelope [9, 14, 15]. The adoption of hygroscopic materials throws up the issue of moisture adsorption for these hygroscopic materials. Condensations inside the building envelope are most common due to variations in temperature and humidity indoors and outdoors the building. The Future Technology Open Access Journal https://doi.org/10.55670/fpll.futech.2.4.4 November 2023| Volume 02 | Issue 04 | Pages 33-44 Journal homepage: https://fupubco.com/futech ISSN 2832-0379 mailto:ndukwumcu@mouau.edu.ng https://doi.org/10.55670/fpll.futech.2.4.4 https://fupubco.com/futech https://fupubco.com/ MC Ndukwu et al. /Future Technology November 2023| Volume 02 | Issue 04 | Pages 33-44 34 presence of moisture within the building envelope will cause thermal discomfort and can corrode the metallic structures within the building and also make the indoors moldy [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. Understanding the physics of this physical process is important to be able to predict them. 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]. This will determine the magnitude of humidity, heating, and cooling required [18, 19]. Materials like wood, metal, steel, concrete, pozzolan, glass, bricks, and other multi-layered composites have been adopted for building walls [20-25]. 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 [27-30]. In several cases, walls can be made of more than one material 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 physics of heat and moisture transfer of various materials in response to variation in weather parameters, which includes solar radiation, temperature, moisture, and relative humidity. These models are developed and present as single or multidimensional cases in the literature [31-47]. These models are resolved using finite elements, finite control volume etc, in a steady and non-steady numerical scheme. They are majorly predictive models to determine the temperature, relative humidity, and moisture condition of the inside of the building as a variation to the ambient conditions [32-34, 36-43]. The various studies used the thermo-physical properties of these materials, the nature of airflow, the dimension of moisture transfer, and the meteorological data of each area to develop the simulation codes to make their predictions with good result. The vapor 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 hysteresis is yet well established, some authors included it in their modeling approach, while others neglected it. Simo-Tagne et al. [9] stated that studies like this would help in the selection of materials to satisfy the desired heat load at minimum dissipation of energy with less environmental impact. Some researchers have tried to review these models, though they presented only computer-based model tools, like UMIDUS, Wuf, MATCH, DELPHIN etc [46, 47]. Prior to that, the Canadian mortgage and housing cooperation in 2003 reviewed and showed that about forty five hygrothermal transfer models were in existence, which increased to about 57 models by 2008 [47]. The use of a dynamic coupled co- simulation approach for forecasting the hygrothermal behavior of building envelopes was discussed by Ferroukhi et al. [46]. However, with the development of new material and the creation of different kinds of boundary conditions and different interactions with the environment considered, new numerical models on hygrothermal transfer are presented and validated with experimental data. Busser et al. [78] towed this line in presenting the recent trends in the experimental validation of hygrothermal transfer models, but the specific equations were lacking in their presentations. Prior knowledge of energy behavior is important to design a building for optimum comfort, taking into consideration all the heat loads. Moisture and temperature changes in the building are coupled together to study this moisture and temperature gradient. This requires the development of modeling tools for optimal condition prediction for different kinds of walls. Due to the effect of greenhouse gas emissions generated from the production of non–bio–based walls, interest is shifting to environmentally friendly walls. Wooden walls are now of interest, especially in Africa, when the cost of non-biobased walls is also considered. This review is an updated review of hygrothermal transfers with the addition of research work on wooden walls lacking in other previous reviews. Therefore these will bring up to date various reviews conducted with the same theme. The review will single out each model and discuss its pros and cons. 2. Methodology This review involves searching and requesting available open literature on hygrothermal behavior for different walls. Emphasis was on the literature on models that have not been reviewed, although where the models are based on existing models, the old models were presented, and the new models can be discussed under them. Subsequently, the models were separated into those developed and validated with concrete and bricks and those validated with wood. After reading and deducing the mechanisms governing the behavior of the hygrothermal transfer and important equations and results, this is presented and discussed. Figure 1 gives the flow chart of the review methodology. Figure 1. Schematics of the review process Energy and moisture transfer in buildings: A review of hygrothermal transfer models through the porous building walls Literature sourcing Models for concrete and brick walls Models for woods Discussion Conclusion MC Ndukwu et al. /Future Technology November 2023| Volume 02 | Issue 04 | Pages 33-44 35 3. Models for concrete and brick walls Several modeling and simulation investigations exist 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 were validated experimentally. The various models are as follows. 3.1 Xingguo models Xingguo et al. [20] modeled 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 oK and 11.4 % for indoor and outdoor conditions. 𝜕𝑊 𝜕𝑡 = 𝑊𝑠 𝜉𝜌𝑚 (𝐷𝑣𝑅𝑣𝑇𝑚𝜌𝑎 + 𝜉𝜌𝑚 𝑊𝑠 𝐷𝑤) 𝜕2𝑊 𝜕𝑥2 + ᶲ 𝜕𝑊𝑠 𝜕𝑇 𝜕𝑇 𝜕𝑡 (1) The resultant boundary conditions for the above equation were given as follows. −𝐷𝑣𝑅𝑣𝑇𝑚𝜌𝑎 𝜕𝑊 𝜕𝑥 = ℎ𝑚(𝑊∞ −𝑊𝑠𝑢𝑟𝑓) (2) For heat transport, the following governing equation was used. (𝜌𝑚𝐶𝑣𝑚) 𝜕𝑇 𝜕𝑡 = 𝐾 𝜕2𝑇 𝜕𝑥2 + ℎ𝑓𝑔𝐷𝑣𝑅𝑣𝑇𝑚𝜌𝑎 𝜕2𝑊 𝜕𝑥2 (3) The resultant boundary conditions for the above equation were given as follows. −𝐾 𝜕𝑇 𝜕𝑥 = ℎ𝑚(𝑇∞ − 𝑇𝑠𝑢𝑟𝑓) + 𝑄𝑟𝑎𝑑 + ℎ𝑓𝑔𝑚𝑠 (4) 3.2 Lelievre model Lelievre et al. [17] combined two sub-models of Pederson and a phenomenological model from Mualem II to develop a numerical simulation model for multilayer hemp concrete. Hemp concrete is usually coated with plasters of different levels of permeability inside and outside, and the thickness is non-homogenous. 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. 𝜌𝑠(𝐶𝑝,𝑠 + 𝑤𝐶𝑝,𝑙) 𝜕𝑇 𝜕𝑡 = −∇ × (−𝜆∇𝑇) + −∇ × (𝐷𝑣 𝜑 ∇𝜑 + 𝐷𝑣 𝑇∇𝑇) × (𝑙𝑣 + (𝐶𝑝,𝑠 − 𝐶𝑝,𝑙)(𝑇 − 𝑇𝑟𝑒𝑓)) (5) 𝜌𝑠𝜃 𝜕𝜑 𝜕𝑡 = −∇ × (−(𝐷𝑙 𝜑 + 𝐷𝑣 𝜑 ∇𝑇)∇𝜑 − 𝐷𝑣 𝑇∇𝑇) (6) The sorption capacity of equation 6 was deduced using two hysteresis models in sorption and desorption phases from Pederson (equations 7 and 8) and Mualem (equations 9 and 10) as follows. 𝜃𝑎𝑑,ℎ𝑦𝑠 = 𝐵(𝑤−𝑤𝑎𝑑) 𝐴𝜃𝑑𝑒𝑠+(𝑤−𝑤𝑑𝑒𝑠) 𝐴𝜃𝑎𝑑 (𝑤𝑑𝑒𝑠−𝑤𝑎𝑑) 𝐴 (7) 𝜃𝑎𝑑,ℎ𝑦𝑠 = (𝑤−𝑤𝑎𝑑) 𝐴𝜃𝑑𝑒𝑠+ 𝐶(𝑤−𝑤𝑑𝑒𝑠) 𝐴𝜃𝑎𝑑 (𝑤𝑑𝑒𝑠−𝑤𝑎𝑑) 𝐴 (8) Using Mualem models, the following sorption equations were deduced. 𝑤𝑑𝑒𝑠,ℎ𝑦𝑠(𝜑) = 𝑤𝑗 − 𝑝𝑑 𝑤𝑠 (𝑤𝑠 −𝑤𝑎𝑑(𝜑)) (𝑤𝑎𝑑(𝜑𝑗) − 𝑤𝑎𝑑(𝜑)) (9) 𝑤𝑎𝑑,ℎ𝑦𝑠(𝜑) = 𝑤𝑗 − 𝑤𝑗−𝑤𝑖 (𝑤𝑎𝑑(𝜑𝑗)−𝑤𝑎𝑑(𝜑)) (𝑤𝑎𝑑(𝜑𝑗) − 𝑤𝑎𝑑(𝜑)) (10) 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. 3.3 Djongyang model Djongyang et al. [40] 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 conditions with variations in latitude for the three cities of Cameroun. In solving the numerical equations, they considered the periodic solution approach and validated their model with two works of Menghao et al. [48, 49]. The model developed was one dimensional in which liquid water and air and water vapor 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 [50] as follows. 𝜕𝑢(𝑥,𝑡) 𝜕𝑥 = 𝑎𝑚 𝜕2𝑢(𝑥,𝑡) 𝜕𝑥2 + 𝑎𝑚𝛿 𝜕2𝑇(𝑥,𝑡) 𝜕𝑥2 (11) 𝜕𝑇(𝑥,𝑡) 𝜕𝑡 = 𝛼 𝜕2𝑇(𝑥,𝑡) 𝜕𝑥2 + 휀𝛽 𝜕𝑢(𝑥,𝑡) 𝜕𝑡 , 0 < 𝑥 < 𝑙 (12) Te boundary conditions for the two equations above were defined as follows. −𝑘𝑞𝑜𝑢𝑡 𝜕𝑇(𝑥,𝑡) 𝜕𝑥 | x=0 + ℎ𝑜𝑢𝑡(𝑇𝑥=0 − 𝑇𝑜𝑢𝑡) + 𝜆𝑜𝑢𝑡(1 − 휀𝑜𝑢𝑡)(𝑢𝑥=0 − 𝑢𝑜𝑢𝑡) = 0 (13) 𝑘𝑚𝑜𝑢𝑡 𝜕𝑢(𝑥,𝑡) 𝜕𝑥 | x=0 + 𝑘𝑚𝑜𝑢𝑡𝛿𝑜𝑢𝑡 𝜕𝑇(𝑥,𝑡) 𝜕𝑥 | x=0 + 𝛼𝑜𝑢𝑡(𝑢𝑥=0 − 𝑢𝑜𝑢𝑡) = 0 (14) −𝑘𝑞𝑖𝑛 𝜕𝑇(𝑥,𝑡) 𝜕𝑥 | x=0 + ℎ𝑖𝑛(𝑇𝑥=0 − 𝑇𝑖𝑛) + 𝜆𝑖𝑛(1 − 휀𝑖𝑛)(𝑢𝑥=0 − 𝑢𝑖𝑛) = 0 (15) 𝑘𝑚𝑖𝑛 𝜕𝑢(𝑥,𝑡) 𝜕𝑥 | x=0 + 𝑘𝑚𝑖𝑛𝛿𝑖𝑛 𝜕𝑇(𝑥,𝑡) 𝜕𝑥 | x=0 + 𝛼𝑖𝑛(𝑢𝑥=0 − 𝑢𝑖𝑛) = 0 (16) 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 represent the mass balance with the components of moisture gradient, temperature gradient and the convective flux exchanged MC Ndukwu et al. /Future Technology November 2023| Volume 02 | Issue 04 | Pages 33-44 36 between the ambient and the surface of the materials. Djongyang et al. [40], 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 𝜕2𝑢𝑛(𝑥) 𝜕𝑥2 − 𝑖𝑛𝑤 𝑎𝑚 𝑢𝑛 + 𝛿 𝜕2𝑇𝑛(𝑥) 𝜕𝑥2 = 0 (17) 𝜕2𝑇𝑛(𝑥) 𝜕𝑥2 − 𝑖𝑛𝑤 𝛼 𝑇𝑛 + 휀𝛽 𝑖𝑛𝑤 𝛼 𝑢𝑛 = 0 (18) Using a numerical approach, they solved equations 17 and 18 to generate the temperature and moisture profile of the material under variable external conditions. Validation of the results using thermo-physical properties of earth bricks gave good results. They showed that latitudes affect hygrothermal transfer. Menghao et al. [49] also used the same one-dimensional Lukoiv equation presented in equations 11 and 12 in the modeling 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. 𝐶𝑃𝜌 𝜕𝑇 𝜕𝑡 = 𝑘 𝜕2𝑇 𝜕𝑥2 + 𝐶𝑚𝜌(𝜖ℎ𝑙𝑣 + 𝛾) 𝜕𝑚 𝜕𝑡 (19) 𝐶𝑚𝜌 𝜕𝑚 𝜕𝑡 = 𝐷𝑚 𝜕2𝑚 𝜕𝑥2 + 𝐷𝑚𝛿 𝜕2𝑇 𝜕𝑥2 (20) The latent heat of vaporization will be integrated as part of the energy balance at the building materials boundary, which will be affected by mass diffusion due to deviations in temperature and moisture content [40, 51]. The boundary equations were given as follows. 𝑘 𝜕𝑇(𝑥,𝑡) 𝜕𝑥 | x=0 = 𝛼1(𝑇 (0, 𝑡) − 𝑇1) + 𝛽1ℎ𝑙𝑣(1 − 휀)(𝑚(0, 𝑡) − 𝑚1) (21) −𝑘 𝜕𝑇(𝑥,𝑡) 𝜕𝑥 | x=0 = 𝛼2(𝑇 (𝑙, 𝑡) − 𝑇2) + 𝛽2ℎ𝑙𝑣(1 − 휀)(𝑚(𝑙, 𝑡) − 𝑚2) (22) 𝐷𝑚 𝜕𝑇(𝑥,𝑡) 𝜕𝑥 | x=0 + 𝑘𝑚𝛿 𝜕𝑇(𝑥,𝑡) 𝜕𝑥 | x=0 = 𝛽1(𝑚(0, 𝑡) − 𝑚1) (23) −𝐷𝑚 𝜕𝑇(𝑥,𝑡) 𝜕𝑥 | x=0 − 𝑘𝑚𝛿 𝜕𝑇(𝑥,𝑡) 𝜕𝑥 | x=0 = 𝛽2(𝑚(𝑙, 𝑡) − 𝑚2) (24) 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 setup in Figure 2. The porous material is a multilayer material whereby that one side is permeable and exposed to outdoor humid conditions while the other side of the wall, which is impermeable, is subjected to cold temperatures. Validations of the numerical simulation model, although showed the same trend as the experimental data looking at Figure 3 plotted by the authors it shows that there is still temperature variations which implies 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. Figure 2. Experimental set of the testing rig for model evaluation [40, 51] Figure 3. Comparison of the numerically simulated temperature profiles with the experimental result after quasi- steady state [40] 3.4 Simo-Tagne model Simo-Tagne et al. [9] presented a numerical model for predicting the hygrothermal transfer for concrete walls 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 vapor phases in equation 25-27, respectively, as follows. 𝜕(𝛼𝑆𝜌𝑙) 𝜕𝑡 + 𝛻 → . 𝐽𝑙 → = −𝐾𝑙 (25) 𝜕(𝑋𝑏𝜌𝑆) 𝜕𝑡 + 𝛻 → . 𝐽𝑎𝑠 → = −𝐾𝑎𝑠 (26) The contribution of water vapor to the mass balance equations is given by: 𝜕(𝛼(1−𝑆)𝜌𝑔𝐶𝑔) 𝜕𝑡 + 𝛻 → . (𝜌𝑔𝑉𝑔 → + 𝐽𝑔) → = 𝐾𝑎𝑠 + 𝐾𝑙 (27) 𝜌𝑔𝑉𝑔 → is the flux characteristic of the movement of the vapour phase. The solution of equation 27 gave equation 28 as follows. 𝜕(𝜌𝑆𝐻) 𝜕𝑡 + 𝛻 → . (𝜌𝑔𝑉𝑔 → + 𝐽𝑙 → + 𝐽𝑎𝑠 → + 𝐽𝑔 → ) = 0 (28) MC Ndukwu et al. /Future Technology November 2023| Volume 02 | Issue 04 | Pages 33-44 37 The heat energy balance was given by equation 29 as follows. [𝜌𝐶𝑝 𝜕𝑇 𝜕𝑡 + �⃗� 𝐽 𝑇] 𝑠 + [𝜌𝐶𝑝 𝜕𝑇 𝜕𝑡 + �⃗� . �⃗� 𝑇] 𝑔 + [𝜌𝐶𝑝 𝜕𝑇 𝜕𝑡 + �⃗� . �⃗� 𝑇] 𝑙 + 𝐾𝑙𝐿 + 𝐾𝑎𝑠(𝐿 + 𝐸𝑏) = 0 (29) The boundary conditions are given in equations 30 and 31 as follows. [−ρs(Eb + L)DH ∂H ∂x − (λ + αtρs(Eb + L)DH) ∂T ∂x ]| x=0 = hc,ext(Taext − T(0, t)) + ρwLhm,ext(Xeq(Taext, HRaext) − H(0, t))+Ggloi (30) [−𝐷𝐻 𝜕𝐻 𝜕𝑥 − 𝛼𝑡𝐷𝐻 𝜕𝑇 𝜕𝑥 ]| 𝑥=0 = ℎ𝑚,𝑒𝑥𝑡(𝑋𝑒𝑞(𝑇𝑎𝑒𝑥𝑡, 𝐻𝑅𝑎𝑒𝑥𝑡) − 𝐻(0, 𝑡)) (31) 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. 3.5 Philip and De-Vries model Philip and De-Vries proposed a model applied to study the heat and mass transfer through a porous medium using a building wall based on the thermodynamics of irreversible processes [78]. This model has been the bases for some other models, which will also be discussed in this section. This model took into consideration the effect of temperature and humidity gradients, and also, the altitude of the location was taken into account. The medium is assumed to be stable and homogenous. Kevin’s law was applied at every point of the medium, and the hysteresis effect between adsorption and desorption phenomena is neglected. This model is given as follows. { 𝜕𝑋 𝜕𝑡 + 𝑎 𝜕𝑋 𝜕𝑡 + 𝑐 𝜕𝑇 𝜕𝑡 = 𝑑𝑖𝑣 (𝐷𝑞𝑔𝑟𝑎𝑑 → 𝑋 + 𝐷𝑇𝑔𝑟𝑎𝑑 → 𝑇) − 𝜕𝐾 𝜕𝑧 𝜌𝐶𝑝 𝜕𝑇 𝜕𝑡 = 𝑑𝑖𝑣 (𝜆𝑔𝑟𝑎𝑑 → 𝑇) + 𝜌𝑙𝐿𝑑𝑖𝑣 (𝐷𝑞𝑣𝑔𝑟𝑎𝑑 → 𝑋 + 𝐷𝑇𝑣𝑔𝑟𝑎𝑑 → 𝑇 − 𝑎 𝜕𝑋 𝜕𝑡 − 𝑐 𝜕𝑇 𝜕𝑡 ) + 𝜌𝑙𝐶𝑙 (𝐷𝑞𝑙𝑔𝑟𝑎𝑑 → 𝑋 + 𝐷𝑇𝑙𝑔𝑟𝑎𝑑 → 𝑇 − 𝐾)𝑔𝑟𝑎𝑑 → 𝑇 + 𝜌𝑙𝐶𝑣 (𝐷𝑞𝑣𝑔𝑟𝑎𝑑 → 𝑋 + 𝐷𝑇𝑣𝑔𝑟𝑎𝑑 → 𝑇) 𝑔𝑟𝑎𝑑 → 𝑇 (32) Where 𝐾 = 𝑘𝑙𝑔 𝑔𝑙 , 𝑦 = − 𝑃𝑐 𝜌𝑙𝑔 , 𝐷𝑞𝑙 = 𝐾 ( 𝜕𝑦 𝜕𝑥 ) 𝑇 (33) 𝐷𝑇𝑙 = 𝐾 ( 𝜕𝑦 𝜕𝑇 ) 𝑥 , 𝐷𝑞𝑣 = 𝑓𝐷 ( 𝑀𝑣 𝑅𝑇 ) 2 𝑔𝑃𝑣𝑠 𝜌𝑙 ( 𝜕𝑦 𝜕𝑥 ) 𝑇 𝑒𝑥𝑝 ( 𝑀𝑣𝑔𝑦 𝑅𝑇 ) , 𝐷𝑞 = 𝐷𝑞𝑙 + 𝐷𝑞𝑣 (34) 𝐷𝑇𝑣 = 𝑓𝐷 𝑃 𝑃−𝑃𝑣 ( 𝑀𝑣 𝑅𝑇 ) 2 𝐿𝑃𝑣𝑠 𝑇𝜌𝑙 𝑒𝑥𝑝 ( 𝑀𝑣𝑔𝑦 𝑅𝑇 ) + 𝑓𝐷 𝑃 𝑃−𝑃𝑣 ( 𝑀𝑣 𝑅𝑇 ) 2 𝑔𝑃𝑣𝑠 𝜌𝑙 (( 𝜕𝑦 𝜕𝑇 ) 𝑥 − 𝑦 𝑇 )𝐻𝑅 (35) 𝐷𝑇 = 𝐷𝑇𝑙 + 𝐷𝑇𝑣 , 𝑎 = (𝐽−𝑞𝑙)𝐷𝑞𝑣 𝑓𝐷 𝑃 𝑃−𝑃𝑣 − 𝑃𝑣𝑀𝑣 𝜌𝑙𝑅𝑇 , 𝑐 = (𝐽−𝑞𝑙)𝐷𝑇𝑣 𝑓𝐷 𝑃 𝑃−𝑃𝑣 − (𝐽−𝑞𝑙)𝑃𝑣 𝜌𝑙𝑅𝑇 (36) Neglecting the heat transfer due to the mass transfer and the local temporal variation of the condensed water content in the vapor state, the simplified model of Philip and De-Vries is obtained, and it is well described by the experimental data obtained by Larbi in 1990 [78]. { 𝜕𝑋 𝜕𝑡 = 𝑑𝑖𝑣 (𝐷𝑞𝑣𝑔𝑟𝑎𝑑 → 𝑋 + 𝐷𝑇𝑣𝑔𝑟𝑎𝑑 → 𝑇) − 𝜕𝐾 𝜕𝑧 𝜌𝐶𝑝 𝜕𝑇 𝜕𝑡 = 𝑑𝑖𝑣 (𝜆𝑔𝑟𝑎𝑑 → 𝑇) + 𝜌𝑙𝐿𝑑𝑖𝑣 (𝐷𝑞𝑣𝑔𝑟𝑎𝑑 → 𝑋 + 𝐷𝑇𝑣𝑔𝑟𝑎𝑑 → 𝑇) (37) The model of Umidus [79] assumed that the humidity is transferred through the wall only in vapor and liquid forms. The liquid form is transferred by capillary, and the vapor form is transferred due to the partial pressure of the vapor. Thus, the model in one dimension is as follows: { 𝜕𝑋 𝜕𝑡 = 𝜕 𝜕𝑥 (𝐷𝑇 𝜕𝑇 𝜕𝑥 ) + 𝜕 𝜕𝑥 (𝐷𝐻 𝜕𝑋 𝜕𝑥 ) 𝜌𝑜(𝐶𝑝𝑜 + 𝐶𝑝𝑙 𝜌𝑙 𝜌𝑜 𝑋) 𝜕𝑇 𝜕𝑡 = 𝜕 𝜕𝑥 (𝜆 𝜕𝑇 𝜕𝑥 ) + 𝐿𝑣𝜌𝑙 ( 𝜕 𝜕𝑥 (𝐷𝑇𝑣 𝜕𝑇 𝜕𝑥 ) + 𝜕 𝜕𝑥 (𝐷𝐻𝑣 𝜕𝑋 𝜕𝑥 ) ) (38) The boundary equations are given by: Humidity: −𝜌𝑙(𝐷𝑇 𝜕𝑇 𝜕𝑥 + 𝐷𝐻 𝜕𝑋 𝜕𝑥 )| 𝑥=0,𝑒 = ℎ𝑀,𝑒(𝜌𝑣𝑒,𝑎,𝑒 − 𝜌𝑣𝑒,𝑠,𝑒) (39) −𝜌𝑙 (𝐷𝑇 𝜕𝑇 𝜕𝑥 +𝐷𝐻 𝜕𝑋 𝜕𝑥 )| 𝑥=𝐿,𝑖 = ℎ𝑀,𝑖(𝜌𝑣𝑒,𝑎,𝑖 − 𝜌𝑣𝑒,𝑎,𝑖) (40) Temperature: −𝜆 𝜕𝑇 𝜕𝑥 − 𝐿𝑣𝜌𝑙 (𝐷𝑇𝑣 𝜕𝑇 𝜕𝑥 + 𝐷𝐻𝑣 𝜕𝑋 𝜕𝑥 )| 𝑥=0,𝑒 = ℎ𝑇,𝑒(𝑇𝑎,𝑒 − 𝑇𝑠,𝑒) + 𝐿𝑣ℎ𝑀,𝑒(𝜌𝑣𝑒,𝑎,𝑒 − 𝜌𝑣𝑒,𝑠,𝑒) (41) −𝜆 𝜕𝑇 𝜕𝑥 − 𝐿𝑣𝜌𝑙 (𝐷𝑇𝑣 𝜕𝑇 𝜕𝑥 + 𝐷𝐻𝑣 𝜕𝑋 𝜕𝑥 )| 𝑥=𝐿,𝑖 = ℎ𝑇,𝑖(𝑇𝑠,𝑖 − 𝑇𝑎,𝑖) + 𝐿𝑣ℎ𝑀,𝑖(𝜌𝑣𝑒,𝑎,𝑖 − 𝜌𝑣𝑒,𝑎,𝑖) (42) Using this model to study the heat mass transfer through the wall building with and without a perfect contact on a double- layer wall, Le [80] shows that the assumption with a perfect contact can produce some errors during the determination of the physical parameters influenced by the humidity. When the contact is real (not perfect), the model shows that the energy consumption is reduced by 10%. Using the experimental data taken from the literature, Le [80] validated this model. Using the same assumptions by Philip and De- Vries, the model of Duforestel is given by Simo-Tagne [78] as follows. { 𝑎𝑇 𝜕𝑝𝑣 𝜕𝑡 − 𝑎𝑇𝑝𝑣 𝜌𝑣𝑇 2 (ℎ𝑚 + 𝐿) 𝜕𝑇 𝜕𝑡 = 𝑑𝑖𝑣 (( 𝑝𝑣 𝑝𝑡 + 𝐾𝑛 + 𝜌𝑙𝐾𝑙𝜌𝑣𝑇 𝑝𝑣 )𝑔𝑟𝑎𝑑 → 𝑝𝑣 − 𝜌𝑙𝐾𝑙 𝐿 𝑇 𝑔𝑟𝑎𝑑 → 𝑇) (−ℎ𝑚𝑎𝑇 + ℎ𝑚 𝐿 𝜌𝑣𝑇 ) 𝜕𝑝𝑣 𝜕𝑡 + (𝐶′+ 𝑎𝑇𝑝𝑣 𝜌𝑣𝑇 2 ℎ𝑚(ℎ𝑚 + 𝐿)) 𝜕𝑇 𝜕𝑡 = 𝑑𝑖𝑣 ( 𝐿 ( 𝑃 𝑃𝑡 + 𝐾𝑛) 𝑔𝑟𝑎𝑑 → 𝑝𝑣 +𝜆𝑔𝑟𝑎𝑑 → 𝑇 ) (43) This model is easy to use than the one of Philip and De-Vries and can be applied to non-hygroscopic material of construction, but the fact it did not take into account the gradient of moisture content is often the reason for the difference between experimental data and numerical data. Based on the same assumptions of the model of Philip and De- Vries, Luikov presented his model with other parameters. This model gives satisfaction when the partial pressure of the gas phases is uniform during the heat mass transfer. Luikov’s model is given by Simo-Tagne [78]. MC Ndukwu et al. /Future Technology November 2023| Volume 02 | Issue 04 | Pages 33-44 38 { 𝜕𝑋 𝜕𝑡 = 𝑑𝑖𝑣 [𝑎𝑚 (𝑔𝑟𝑎𝑑 → 𝑋 + 𝛿𝑔𝑟𝑎𝑑 → 𝑇)] 𝜌𝐶𝑝 𝜕𝑇 𝜕𝑡 = 𝑑𝑖𝑣 (𝜆𝑔𝑟𝑎𝑑 → 𝑇) + 𝑒𝐿 𝜕𝑋 𝜕𝑡 (44) The difficulty faced in using this model is the difficulty in obtaining the conversion factor. Though it is mostly obtained using the inverse method, which is cumbersome. When the wall is constructed by the mortar slab, Saadani et al. [81] show that Luikov and Philip and De-Vries models can give similar values of temperature evolutions with some differences in the evolutions of moisture content, using the time steps between 0.1 and 1 s. Nitcheu et al. [82] used Luikov’s model with satisfaction to estimate the numerical heat mass transfer through the wall building constructed in earth bricks stabilized with thatch fibers. They used the finite differences method and implicit scheme of Crank–Nicolson to generate the numerical results, but the thermophysical parameters of the medium were considered constant during the process. 3.6 Whitaker model In 1977, Whitaker presented a model based on the morphology and the average thermophysical values of the building wall. The mass transfer of water in gas and liquid phases were separately written. This model is very difficult to use, but Thaï used it with satisfaction in 2006 [78]. This model is given by: In 1992, Nicolas proposed a model that took into account the capillary and hysteresis effects with the temperature gradient as a motor term. The Kevin’s law has been modified, and the pressures of water in saturation state, respectively in porous medium and in the free medium, have been differently written. The model obtained is given by: { 𝜌𝐶𝑝 ( 𝜕𝑇 𝜕𝑡 + 𝑉 → . 𝑔𝑟𝑎𝑑 → 𝑇) = 𝑑𝑖𝑣 (𝜆𝑔𝑟𝑎𝑑 → 𝑇) −((𝐶𝑙 − 𝐶𝑣)𝑇 − 𝐿𝑜) 𝑅𝑇 𝑘𝑀𝑣 𝑙𝑛 ( 𝑝𝑣 𝑝𝑣𝑠 ) 𝜕(𝑏𝑎𝜌𝑎) 𝜕𝑡 + 𝑑𝑖𝑣 (𝑏𝑎𝜌𝑎𝜐 → 𝑎) = 0 𝜕(𝑏𝑣𝜌𝑣) 𝜕𝑡 + 𝑑𝑖𝑣 (𝜌𝑣𝑏𝑣𝜈 → 𝑣) + 𝜕(𝑏𝑙𝜌𝑙) 𝜕𝑡 + 𝑑𝑖𝑣 (𝜌𝑙𝑏𝑙𝜈 → 𝑙) = 0 𝜕(𝑏𝑙𝜌𝑙) 𝜕𝑡 + 𝑑𝑖𝑣 (𝑏𝑙𝜌𝑙𝜈 → 𝑙) = 𝑅𝑇 𝑘𝑀𝑣 𝑙𝑛 ( 𝑝𝑣 𝑝𝑣𝑠 ) (46) With : 𝑏𝑎𝜈 → 𝑎 = −(𝐾𝑔 + 𝑝𝑣 𝑝𝑎 𝐾𝑎𝑣)𝑔𝑟𝑎𝑑 → 𝑝𝑎 − (𝐾𝑔 − 𝐾𝑎𝑣)𝑔𝑟𝑎𝑑 → 𝑝𝑣 (47) 𝑏𝑣𝜈 → 𝑣 = −(𝐾𝑔 −𝐾𝑎𝑣)𝑔𝑟𝑎𝑑 → 𝑝𝑎 − (𝐾𝑔 + 𝑝𝑎 𝑝𝑣 𝐾𝑎𝑣)𝑔𝑟𝑎𝑑 → 𝑝𝑣 (48) 𝑏𝑙𝜈 → 𝑙 = −𝐾𝑙𝑔𝑟𝑎𝑑 → (𝑝𝑙 + 𝜌𝑙𝑅𝑇 𝑀𝑣 ℎ(𝑏𝑙) + 𝜌𝑙𝑔) (49) 𝑝𝑎 = 𝜌𝑎𝑅𝑇 𝑀𝑎 , 𝑝𝑣 = 𝜌𝑣𝑅𝑇 𝑀𝑣 , 𝑝𝑙 = 𝑝𝑎 + 𝑝𝑣 + 𝑝𝑣 𝑑ℎ(𝑏𝑙) 𝑑𝑏𝑙 (50) Using his model, Nicolas studied the influence of the variation of the total pressure of the gas on the phenomenon of imbibition of cement. He obtained that the variation of the moisture content is influenced by the variation of the value of the total pressure of gas. The high pressure of the gas decreases the rate of transfer of humidity in the medium. This model permits the study of the effects of the hysteresis phenomenon during the heat mass transfer through the humid building wall [78]. 4. Models for wooden walls The impact of wood as a building material for walls has been explored greatly in literature. An investigation has been carried out on the effects of wood on thermal comfort in terms of relative humidity, temperature, background noise levels, and CO2 concentrations [53-59]. 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 room conditions and thermal comfort [60]. Therefore, the type of wood, thickness, moisture isotherm etc. has been factored in the predictive models to describe the hygrothermal behavior of wood for indoor and outdoor conditions. 4.1 Luikov models Due to the flexibility of Luikov models, it has formed the basis for modeling the coupled heat and mass transfer for porous material independent of its hygroscopic nature. The model accounts for all the bonding water simplistically without restricting the water transfer mechanism [61]. Therefore, these models can be applied for one, two, or three- dimensional coupled heats and mass transfer in woods. Younis et al. [61] used this model to predict 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 behavior of the wooden slab. Comparison of this model with analytical and experimental data showed closer agreement with the analytical model rather than experimental data. 4.2 Osayintola model Osayintola et al. [26] used a combined Knudsen and Fickian diffusion and neglected thermal diffusion in the modeling of heat and mass transfer evolution in spruce wood. The energy and mass conservation equations were defined in equations 51-54 as follows: 𝜌ℓ 𝜕𝜀ℓ 𝜕𝑡 + �̇� = 0 (51) { 𝜌𝑤 𝜕𝜀𝑤 𝜕𝑡 + 𝜌𝑤𝛻 → < 𝑉𝑤 → > +< 𝑚 ∗ >= 0 𝜕 𝜕𝑡 (휀𝑔 < 𝜌𝑣 > 𝑔) + 𝛻 → (< 𝜌𝑎 > 𝑔< 𝑉𝑔 > → ) = 𝛻 → [< 𝜌𝑔 > 𝑔 𝐷𝑒𝑓𝑓. 𝛻 → ( <𝜌𝑣> 𝑔 <𝜌𝑔>𝑔 )] < 𝜌 > 𝐶𝑃 𝜕<𝑇> 𝜕𝑡 + [𝜌𝑤𝐶𝑃𝑤 < 𝑉 → 𝑤 > +< 𝜌𝑔 > 𝑔< 𝐶𝑃 > 𝑔< 𝑉𝑔 → >]𝛻 → < 𝑇 > +𝛥ℎ𝑣 < 𝑚 ∗ >= 𝛻 → (𝜆𝑒𝑓𝑓𝛻 → < 𝑇 >) (45) MC Ndukwu et al. /Future Technology November 2023| Volume 02 | Issue 04 | Pages 33-44 39 𝜕(𝜌𝑣𝜀𝑔) 𝜕𝑡 − �̇� = 𝜕 𝜕𝑥 (𝐷𝑒𝑓𝑓 𝜕𝜌𝑣 𝜕𝑥 ) (52) 𝜌𝐶𝑝𝑒𝑓𝑓 𝜕𝑇 𝜕𝑡 + �̇�ℎ𝑓𝑔 = 𝜕 𝜕𝑥 (𝑘𝑒𝑓𝑓 𝜕𝜌𝑣 𝜕𝑥 ) (53) �̇� = −𝜌0 𝜕𝑢 𝜕𝑡 (54) 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 2nd 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, they did not show this with their experimental data. Rather they only fitted the sorption data with moisture content, thermal conductivity, and water vapor permeability with good results. 4.3 Simonson model Simonson et al. [69] presented a numerical simulation model to predict the moisture transfer (indoor climate) between the indoor air and the structural materials for wooden buildings 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: 𝑞𝑀 = −𝑘𝑑(𝑢, 𝑇)∇𝑝𝑣 − 𝜌0𝐷𝑤(𝑢, 𝑇)∇𝑢 + 𝑣𝑎𝜌𝑣 + 𝐾𝜌𝑤𝑔 (55) Equation 55 is adopted from IEA ECBCS Annex 24 ‘Heat, Air and Moisture Transfer in Insulated Envelope Parts presented in detail in Hens [62]. 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 [64- 69]. 4.4 Talukdar model Talukdar et al. [70] 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 an analytical approach to set the moisture penetration depth through the woods using equation 56 and 57. For a semi-porous material, the analytical equation developed for vapor transport is given in equation 30 as follows. (𝜌𝑣−𝜌𝑣𝑖) (𝜌∞−𝜌𝑣𝑖) = 𝑒𝑟𝑓𝑐 ( 𝑥 √𝛼𝑚𝑒𝑓𝑓 𝑡2 ) − [𝑒𝑥𝑝 ( 𝐷𝑚𝑥 𝐷𝑒𝑓𝑓 + 𝑘 ℎ𝑚 2 𝛼𝑚𝑒𝑓𝑓 𝑡 𝐷𝑒𝑓𝑓 2 )] × [𝑒𝑟𝑓𝑐 ( 𝑥 √𝛼𝑚𝑒𝑓𝑓 𝑡2 + ℎ𝑚√𝛼𝑚𝑒𝑓𝑓 𝑡 𝐷𝑒𝑓𝑓 )] (56) (𝜌𝑣𝛿𝑚−𝜌𝑣𝑖) (𝜌∞−𝜌𝑣𝑖) = 0.01 (57) The obtained results show that increasing the air velocity increases the temperature, relative humidity, and moisture accumulation within the plywood. 4.5 Watt model Watt et al. [71] proposed a two-dimension model for a light timber wall with an air barrier with changes in air tightness in a Swedish. The model was used to predict the moisture accumulation and mold 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. 4.6 Simo-Tagne model Simo-Tagne et al. [9] numerical model presented in equations 25-31 above was also used to predict the hygrothermal transfer for different wooden walls for outdoor conditions in sub-Saharan Africa. In this case, the model took into account the bound water presents in the wood with the integration of the flow pattern in the boundary conditions. Validation of the model was with Norway Spruce wood with a low mean relative error. 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. [72] also presented a numerical simulation model for building walls in Nancy, France. The same approach and solutions in Simo-Tagne et al. [72] were 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 canceling the effect of ambient conditions. Simo-Tagne et al. [74, 75] present a novel model of heat mass transfer through wooden material. The model-based initially on the description of each type of water (bound water, vapor, and free water); equations obtained are given by: { 𝜕𝑊 𝜕𝑡 = 𝜕 𝜕𝑥 (𝐷𝐻𝐻 𝜕𝑊 𝜕𝑥 + 𝐷𝐻𝑇 𝜕𝑇 𝜕𝑥 ) 𝜌𝐶 𝜕𝑇 𝜕𝑡 = 𝜕 𝜕𝑥 (𝐷𝑇𝐻 𝜕𝑊 𝜕𝑥 + (𝜆 + 𝐷𝑇𝑇) 𝜕𝑇 𝜕𝑥 ) (58) With: 𝐷𝐻𝐻 = 𝐷𝐻 − 𝜌𝑙𝑘 𝜌𝑠 ( 𝑘𝑟 𝜇 ) 𝑙 𝜕𝑃𝑐 𝜕𝑊 + 𝜌𝑔𝐷𝑔 𝜌𝑠(1−𝐶𝑔) 𝜕𝐶𝑔 𝜕𝑊 (59) 𝐷𝐻𝑇 = 𝐷𝑇 − 𝜌𝑙𝑘 𝜌𝑠 ( 𝑘𝑟 𝜇 ) 𝑙 𝜕𝑃𝑐 𝜕𝑇 + 𝜌𝑔𝐷𝑔 𝜌𝑠(1−𝐶𝑔) 𝜕𝐶𝑔 𝜕𝑇 (60) 𝐷𝑇𝑇 = (𝐸+𝐿)𝜌𝑔𝐷𝑔 1−𝐶𝑔 𝜕𝐶𝑔 𝜕𝑇 − 𝐸𝜌𝑙𝑘 ( 𝑘𝑟 𝜇 ) 𝑙 𝜕𝑃𝑐 𝜕𝑇 (61) 𝐷𝑇𝐻 = (𝐸+𝐿)𝜌𝑔𝐷𝑔 1−𝐶𝑔 𝜕𝐶𝑔 𝜕𝑊 − 𝐸𝜌𝑙𝑘 ( 𝑘𝑟 𝜇 ) 𝑙 𝜕𝑃𝑐 𝜕𝑊 (62) 𝐷𝑇 = 𝐸𝑏𝐻𝑅 𝑅𝑇2 𝜕𝑋𝑒 𝜕𝐻𝑅 𝐷𝐻 (63) The boundary equations are given by: If x=0, thus: ∂W ∂x = 0 ; ∂T ∂x = 0 (64) If x=±e/2, thus: −𝐷𝐻𝐻 𝜕𝑊 𝜕𝑥 = ℎ𝑚(𝑊 − 𝑋𝑒) (65) −(𝜆 + 𝐷𝑇𝑇) 𝜕𝑇 𝜕𝑥 = ℎ𝑐(𝑇 − 𝑇𝑎𝑖𝑟) + 𝜌𝑠𝐿𝐷𝐻𝐻 𝜕𝑊 𝜕𝑥 (66) MC Ndukwu et al. /Future Technology November 2023| Volume 02 | Issue 04 | Pages 33-44 40 Using the real variations in thermophysical parameters presented in equations 47-55, Simo-Tagne et al. [75] showed that the model is flexible and will allow taking 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 have to be experimentally determined for each wood. Generally, to have an 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 is defined [75-77]. These values and equations mainly differentiated most of the presented equations in this review from each other. 4.7 Berger model Berger et al. [73] proposed a numerical model using Scharfetter-Gummel scheme 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 follows: { 𝜕 𝜕𝑡 (𝑤𝑣 +𝑤𝑙) = −𝛻. (𝐽𝑐,𝑣 + 𝐽𝑐,𝑙) 𝜕 𝜕𝑡 (𝑤𝑣 +𝑤𝑑𝑎) = −𝛻. (𝐽𝑐,𝑣 + 𝐽𝑐,𝑑𝑎) + 𝑙𝑐,𝑣 (67) The energy balance was given as follows: (𝑐𝑜𝜌𝑜 + 𝐶𝑣𝑤𝑣 + 𝐶𝑙𝑤𝑙 + 𝐶𝑑𝑎𝑤𝑑𝑎) 𝜕𝑇 𝜕𝑡 = −𝛻. 𝐽𝑞 − 𝑟𝑣𝑙𝑙𝑐,𝑣 − 𝛻. (𝐶𝑣𝑇)𝐽𝑐,𝑣 − 𝛻. (𝐶𝑙𝑇)𝐽𝑐,𝑙 − 𝛻. (𝐶𝑑𝑎𝑇)𝐽𝑐,𝑑𝑎 (68) Where the volumetric vapor source lc,vis given by: 𝑙𝑐,𝑣 = 𝛱(1−𝜎)𝑃𝑣 𝑅𝑣𝑇 2 𝜕𝑇 𝜕𝑡 + −𝛻. 𝐽𝑐,𝑣 (69) The heat flux was expressed as: 𝐽𝑞 = −𝜆𝑞𝛻𝑇 + (𝐶𝑣𝑤𝑣 + 𝐶𝑑𝑎𝑤𝑑𝑎) 𝑇 𝛱(1−𝜎) 𝑉 (70) V is the vapor velocity taken equal to the air velocity and given by: 𝑉 = − 𝑘𝑣𝑑𝑎 𝜇 𝛻𝑃 (71) The flux of water vapor was given by: 𝐽𝑐,𝑣 = −𝑘𝑣𝛻𝑃𝑣 + 𝑃𝑣 𝑅𝑣𝑇 𝑉 (72) The flux of dry air was given by: 𝐽𝑐,𝑑𝑎 = 𝑤𝑣+𝑤𝑑𝑎 𝛱(1−𝜎) 𝑉 − 𝐽𝑐,𝑣 (73) The flux of liquid water was given by: 𝐽𝑐,𝑙 = −𝑘𝑚𝛻𝑃𝑣 + 𝑃𝑣 𝑅𝑣𝑇 𝑉 − 𝐽𝑐,𝑣 (74) Applying this model to the wood fiber using the constant properties (without influences of humidity and temperature), Berger et al. [73] used a program translated with MatlabTM to obtain the curves that defined well the experimental data presented. Based on Whitaker’s model, Perré and Turner [83] presented a model applied to the wooden wall byRafidiarison [84]. The medium is supposed to be partially saturated, and mass transfers are applied in each phase. This model is given by Simo-Tagne [78]. { 𝜌𝑙 ∂𝜀𝑙 ∂𝑡 + ∇ → (𝜌𝑙𝑢𝑙 → ) = −𝑚 ∗ ∂𝜌𝑣 𝑔 ∂𝑡 + ∇ → (𝜌𝑣 𝑔 𝑢𝑔 → ) = 𝑚 ∗ +𝑚 ∗ 𝑏 ∂𝜌𝑏 ∂𝑡 + ∇ → (𝜌𝑣 𝑔 𝑢 → 𝑣) = 𝑚 ∗ +𝑚𝑏 ∗ ∂𝜌𝑏 ∂𝑡 + ∇ → (𝜌𝑏𝑢𝑏 → ) = −𝑚𝑏 ∗ (75) { 𝜌𝑣 𝑔 𝑢𝑣 → = 𝜌𝑣 𝑔 𝑢𝑔 → − 𝜌𝑔𝐷𝑒𝑓𝑓∇ → ( 𝜌𝑣 𝜌𝑔 ) 𝜌𝑏𝑢𝑏 → = −𝜌𝑐𝐷𝑏∇ → ( 𝜌𝑏 𝜌𝑐 ) 𝑢𝑔 → = − 𝐾𝑔 𝜇𝑔 ∇ → (𝑝𝑔) 𝑢𝑙 → = − 𝐾𝑙 𝜇𝑙 ∇ → (𝑝𝑙) (76) 𝑝𝑙 = 𝑝𝑔 − 𝑝𝑐 (77) 𝜌𝐶𝑝 𝜕𝑇 𝜕𝑡 + 𝛥ℎ𝑣 (𝑚 ∗ +𝑚𝑏 ∗ ) + ℎ𝑠𝑚 ∗ 𝑏 − 𝜌𝑏𝑢𝑏 . → 𝛻 → (ℎ𝑠) + [(𝜌𝑙𝑢𝑙 → + 𝜌𝑏𝑢𝑏 → ) 𝐶𝑝𝑙 + ∑ (𝜌𝑖 𝑔 𝑢𝑖 → 𝐶𝑝𝑖)𝑖=𝑎,𝑣 ] . 𝛻 → 𝑇 = 𝛻 → (𝜆𝑒𝑓𝑓𝛻 → 𝑇) (78) Using experimental data given by Rafidiarison [84] and Rafidiarison et al. [85], the literature shows that this model gives satisfaction to predicting relative humidity, temperature, and moisture content of the wooden building walls in a temperate climate. 5. Discussions The comprehensive bibliography of old and more recent hygrothermal transfer models for various building walls is reviewed in this paper. The hygrothermal transfer is very important for the design of a building envelope 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 wooden building walls. Some of the models are easier to use than others, though this doesn’t make them give better results because, in most cases, the validation with the experimental results has more errors. For example, the model of Lukoiv poses a lot of challenges because the conversion factor of water from a liquid state to a vapor state is located between 0 and 1 due to the fact, they require the conversion factor to be obtained through a cumbersome process of the inverse method. Again, Duforestel model is easier to use than the one of Philip and De-Vries and can be applied to non-hygroscopic material in building, but the fact it did not take into account the gradient of moisture content is often the reason for the difference between experimental data and numerical data. The accuracy or otherwise of this model depends on the establishment of the right boundary conditions. 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 literature review shows that most models were linked to Luikov equations for heat and mass transfer, and solutions were mostly by finite difference methods and implicit scheme of Crank–Nicolson to generate the numerical results. The thermophysical parameters of the medium were always considered constant during the process, but it will be good to consider swelling and shrinkage over time in the model as the material is influenced by environmental conditions. For example, in a high moisture environment, certain wood absorbs moisture and expands if used as a wall, while some shrink during the winter period. According to Ndukwu et al. [86], materials like wood, concrete or bricks MC Ndukwu et al. /Future Technology November 2023| Volume 02 | Issue 04 | Pages 33-44 41 used as a base material in wall construction if subjected to outdoor condition, undergoes continuous drying under ambient conditions. These materials, because they are hygroscopic, desorb or absorb moisture unless an equilibrium condition is maintained between the indoor and outdoor air, or the wall material surface is impermeable [87]. The subjection of these materials to external heat load from solar radiation and other high-temperature-producing sources like industrial activities results in continued moisture modification. Thus, moisture desorption or adsorption of most materials is a continuous process for the lifetime of the walls. The repercussion is the continuous modification of the thermophysical properties of this material. These require consideration in modeling the hygrothermal transfer for building walls. Therefore, as suggested by Ndukwu et al. [86], long-term behavioral models that will incorporate the continuous moisture loss or gain from building materials during structural application require investigation. The model should consider the effect of seasonal variations (winter, spring, summer, and autumn) and long-term meteorological data. 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. Imaging models using software like COMSOL multi-physics, CFD etc. are scarce in the review 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 materials like wood, as they respond to ambient variations. 6. Conclusion The applications of various materials in building structures have been studied extensively. The study presents different models applied for predicting hygrothermal transfer for various building walls. Energy and mass conservation equations were applied with different boundary conditions, and thermophysical properties were presented in this paper for concrete, bricks, and wooden walls. Luikov models formed the basis of most models for porous materials. The parameters considered in most models were the nature of the materials of the wall, building orientation, variation in climate, the thickness of the wall, temperature, moisture changes, and the density of the material. 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 behavior under environmental conditions. Ethical issue The authors are aware of and comply with best practices in publication ethics, specifically with regard to authorship (avoidance of guest authorship), dual submission, manipulation of figures, competing interests, and compliance with policies on research ethics. 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