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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 

 

 

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Open Access Journal 

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November 2023| Volume 02 | Issue 04 | Pages 33-44 

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ISSN 2832-0379 

mailto:ndukwumcu@mouau.edu.ng
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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. The authors adhere to 
publication requirements that the submitted work is original 
and has not been published elsewhere. 

Data availability statement 
Data sharing is not applicable to this article as no datasets 

were generated or analyzed during the current study. 

Conflict of interest 

The authors declare no potential conflict of interest. 

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