Analytical and numerical investigation of water diffusion through Raffia vinifera pith

ABSTRACT This work aims to develop analytical and numerical approaches for predicting the water diffusion through raffia vinifera pith using Fick’s law. A representative elementary volume consisting of a fiber surrounded by a layer of natural matrix is considered. Assuming that the physical properties of fiber and matrix are constant, the resulting equations are solved analytically by the separation of variables method and numerically by finite difference method. A constant water concentration is imposed at the external surface of material while the continuity of the moisture and flux is applied at the fiber/matrix interface. To test the computational results, physical properties of the two layers are set equal. The experimental solutions in this case are used as the reference results, and the present computational method results are compared. Results demonstrate that the maximum value of the relative error is less than 6.5% for numerical and 11% for analytical methods. The case of two layers with different physical properties is examined. Results indicate that the heterogeneity of raffia vinifera pith reduces considerably the water absorption in the fiber. This study can be applied to composite materials where the polymers are combined to natural fibers to achieve high mechanical properties of materials.


Introduction
Raffia vinifera (Figure 1a) is a fast-growing palm tree of the Arecacea family, found in swampy areas and along rivers (Obahiagbon 2009). The stem of this palm (petiole), commonly called raffia bamboo, is made of a fragile cork or pith, surrounded by a hard and smooth shell (Nimjieu et al. 2022) ( Figure 1b). Cork consists of about 35.8% to 54.5% fibers (Sikame et al. 2020) aligned lengthwise in a natural matrix (Figure 1c). To date, raffia vinifera bamboo is very little studied among contemporary building materials and is paradoxically widely used for the construction of precarious housing, small bridges, furniture, and granaries for storing agricultural products (Sikame et al. 2013).
Composite materials are increasingly used in industries such as automotive and aerospace industries where light and high resistance materials are required. The most commonly used fibers that meet these requirements are carbon and glass fibers (Neto et al. 2021). However, synthetics or man-made fibers increase economic, ecological and social cost of materials. Thus, renewable, eco-friendly, biodegradable, and low cost of natural materials like raffia vinifera bamboo and its derivatives products can be used as reinforcement in composite materials usable in many sectors (Ismail et al. 2019;Oliveira et al. 2020;Naveen et al. 2019;Nayak et al. 2020;Safri et al. 2018;Sultan et al. 2009;Tido et al. 2022). The low cost of the composite materials should be associated with its hydrophilicity since natural fibers adjust naturally to environment in which they are placed. This means that composite materials absorb or loose water depending on how wet or dry is the environmental medium. This process, called water diffusion, is important for studying how moisture can affect raffia vinifera properties, predicting its mechanical behavior and guaranteeing it sustainability.
Several works on thermal (Foadieng et al. 2017;Nimjieu et al. 2022), mechanical (Foadieng et al. 2012;Foadieng, Talla, and Fogue 2019;Talla et al. 2004Talla et al. , 2007Talla et al. , 2010Talla, Foudjet, and Fogue 2005), and physicochemical (Foadieng et al. 2014) aspects exist to advance the understanding of the behavior, and properties of raffia vinifera bamboo as well as its derivatives products (Abu, Kofi Yalley, and Adogla 2016;Elenga et al. 2006Elenga et al. , 2009Elenga et al. , 2013Ohimain, Inyan, and Osai 2015;Rober and Shedrack 2016). Most of these studies considered raffia vinifera bamboo at a macroscopic scale or a single component of material. For the purpose of having light raffia vinifera bamboo material, attention was paid on raffia vinifera fibers from stem for determining its Young module (Njeugna et al. 2012;Sikame et al. 2017, Sikame et al. 2018) and its diffusion coefficient (Sikame et al. 2013(Sikame et al. , 2014. Beside these studies, a theoretical investigation of the volume variations of a single raffia vinifera fiber and the water absorption kinetic of fiber was carried out (Nimjieu et al. 2021). Although this important study on the water diffusion through raffia vinifera fibers, the effect of the natural matrix surrounding the fibers is still to be consider.
As shown in Figure 1c, raffia vinifera pith is made up of a set of unidirectional fibers randomly distributed in transverse plane and surrounded by a natural matrix. The physical properties of the fibers and those of the matrices are quite different. Accordingly, raffia vinifera pith cannot be treated as a homogeneous material as in the literature (Mbou et al. 2017), but as a composite material. The purpose of this work is to develop analytical and numerical methods to predicting the concentration of the water through raffia vinifera pith in considering raffia vinifera pith as two layered material. The numerical method consists of approximating the spatial and temporal derivatives by finite differences while the analytical method consists of finding the solution of the problem as a product of two independent functions of time and space. In the implementation of these methods, it is assumed at constant temperature in the pith, a perfect contact between the fiber and matrix. In Section 2, raffia vinifera pith is first modeled as a multilayer material. Secondly, water concentration is determined using numerical and analytical approaches, respectively. The mass and water absorption ratio are finally deduced. The obtained results are presented and discussed in section 3. Conclusion and perspectives are presented in section 4.

Mathematical formulation of the problem
The mass transfer through a solid is described according to Fick's second law (Carslaw and Jaeger 1959;Crank 1995) by the equation where C (in mol=m 3 ) is the concentration of the diffusing molecule and D (in m=s) the diffusion coefficient. Assimilating the raffia cork to a composite cylinder of radiusR m , the diffusion of water through the cork is described by Equation 1 in cylindrical coordinates (r; θ; z) as (Hahn and Özisik 2012;Özisik 1993;Sikame et al. 2013Sikame et al. , 2014: As illustrated in Figure 1c, raffia cork is made up of a set of unidirectional fibers in the longitudinal direction and randomly distributed in a natural matrix in the radial plane. Therefore, it is clear that the physical properties of the fiber and the matrix are quite different. To simplify the problem, a representative elementary volume consisting of a cylindrical fiber of radius R f surrounded by a matrix layer of radiusR m is considered ( Figure 2). For infinite raffia vinifera pith in cylindrically symmetry, Equation 2 for each layer i (i ¼ f ; mÞ reads where C i and D i represent, respectively, the water concentration and the diffusion coefficient in the medium i. The index i ¼ f denotes the fiber and i ¼ m the matrix. A similar decomposition was made by (Carr and March 2018;Castiglione et al. 2003;Kaoui, Lauricella, and Pontrelli 2018;LaBolle, Quastel, and Fogg 1998). Assuming that the physical properties of each layer of the raffia vinifera pith (fiber and matrix) are constant, the raffia vinifera pith is considered as homogeneous two layer cylinders ( Figure 2).

Initial and boundary conditions
At the initial time, it is assumed a uniform water distribution throughout the raffia vinifera pith and the initial condition can be written as Due to the cylindrical symmetry of material, the flux vanishes at the center of the cylinder (Hahn and Özisik 2012;Özisik 1993) and the boundary condition is given as At the interface fiber/matrix, the continuity condition used in multilayer problems imposes for the flux and water concentration that (Bakhtiyor et al. 2020;Carr and Pontrelli 2018;Carr 2018;Carr and March 2018;Carr and Turner 2016;Dejam, Hassanzadeh, and Chen 2016;Hickson, Barry, and Mercer 2009;Kaoui, Lauricella, and Pontrelli 2018;LaBolle, Quastel, and Fogg 1998;Monte 2000;Rodrigo and Worthy 2016) and where σ f is a coefficient that takes into account the contact between the fiber and the matrix. A constant water concentration, C e ; imposed at the outer surface of the material allows writing the boundary condition

Numerical approach
The equations are discretized according to an implicit Euler scheme and implemented in the MatlabR2015a software. The radius is divided into n intervals of length Δr ¼ 0:0001 mm and a time step Δt ¼ 0:1s is chosen. Let us denote by n 1 the number of nodes corresponding to the fiber/matrix interface. The approximation of C i r; t ð Þ at a point r j and at time t k is denoted as C k i;j with j ¼ 1; 2; . . . ; n and k ¼ 1; 2; . . . ; j ¼ 0 corresponds to the center of the cylinder. The index j ¼ n 1 at the interface fiber/matrix �and at the external surface j ¼ n. With this notation, the spatial and temporal derivatives are approximated by finite differences as (Silva 2007) and Using these approximations for j ¼ 0; 1; 2; . . . ; n, yields to the matrix equation in the form where A ½ � and B ½ � are square matrices of size n 1 and n À n 1 , respectively. Also, C f � � and C m ½ � are vectors of length n 1 and n À n 1 , respectively.

Analytical approach
To simplify the calculations, the following transformations are adopted and With the news variables, Equation 3 to Equation 8 become and U m R m ; t ð Þ ¼ 0: Using the separation of variables method (Delouei, Kayhani, and Norouzi 2012;Hahn and Özisik 2012;Kaoui, Lauricella, and Pontrelli 2018;Lubin 2017;Monte 2000;Özisik 1993;Nimjieu et al. 2022;Tamene, Bougriou, and Bessaïh 2006), the solution of Equation 15 is written as where J 0 and Y 0 represent the Bessel functions of first and second kind, respectively (Epelde 2015;Watson 1922;Zakharov 2009). a i;, and b i;, are the integration constants and λ i;, the eigen values of the characteristics equation.

Determination of the coefficients a i and b i
At the center of the fiber (r ¼ 0), the water concentration must be finite. This imposes thatb f ¼ 0. The continuity equation on the water concentration at the interface, Equation 18, yields to Also, the continuity of the diffusion flux at the interface and the surface condition lead to the following system of equations For the non-trivial solutions of Equation (23), we pose β ¼ b m =a m andγ ¼ a f =a m and the third equation of Equation 23 yields: Inserting Equation 24 into the first equation of Equation 23 yields to Replacing β and γ by their expression, taking into account Equation 22 and introducing the variableZ ¼ λ m R m , the second equation of Equation 23 leads to the following equation and To determine a f ;, , we use the initial condition and the orthogonality of Bessel functions (Hahn and Özisik 2012;Özisik 1993;Tamene, Bougriou, and Bessaïh 2006;Watson 1922). It comes that where α , ¼ λ f ;,; R f . The other coefficients are determined by writing a f ;, ¼ γ , a m;, and b m;, ¼ β , a m;, .
The final solutions are therefore given by: and where

Numerical approach
As in (Carr and Pontrelli 2018) and (Kaoui, Lauricella, and Pontrelli 2018), the mass of the pith at a time t is defined by Knowing the local concentration at each point and time in the pith, the mass of the pith can be calculated numerically by the Gaussian quadrature method (Michel and Antoine 2014;Quarteroni, Sacco, and Saleri 2004). Denoting by M k the mass of the cork at time t k ¼ kΔt, it comes that where r j ¼ j:Δr and n denotes the total number of quadrature points. The coefficients A i verify the relation

Analytical approach
The total mass of the raffia vinifera pith is the sum of the masses of all the constituents. Let M f t ð Þ, M m t ð Þ and M t ð Þ be the mass of the fiber, the matrix and the total mass of the raffia vinifera pith, respectively. Replacing Equation 30 and Equation 31, respectively, into Equation 33 and applying the properties of Bessel functions (Epelde 2015;Hahn and Özisik 2012;Lerner 2015;Özisik 1993;Nimjieu et al. 2022;Watson 1922;Zakharov 2009), the following results are obtained and where M f ;0 and M m;0 denote the fiber and matrix mass at initial time (t ¼ 0), respectively; M f ;1 and M m;1 denote the fiber and matrix mass at saturation (t ¼ 1), respectively. The water absorption ratio noted as g t ð Þ is defined by (Mbou et al. 2017;Sikame et al. 2014) g According to Equation 38, Equation 39 yields where V f , g f t ð Þ and g m t ð Þ denote the volume fraction of fibers, the water absorption ratio of the fiber and the water absorption ratio of the matrix, respectively. Equation 40 shows that the physical properties of raffia vinifera pith can be deduced from the properties of raffia vinifera fiber and raffia vinifera matrix using the mixture rule (Berthelot 2010; Matthews et al. 2000). This law was already used to determine the mechanical properties and the density of raffia vinifera matrix (Sikame et al. 2020). Thus, in writing , the diffusion coefficient of raffia vinifera matrix is calculated and introduced into the previous equations.

Material properties
The properties of cellulosic materials in general vary from one sampling area to another. For raffia, one distinguishes 12 sampling zones along the stem: 4 zones in the longitudinal direction and 3 zones in the radial direction as illustrated in the Figure 3 (Sikame et al. 2013(Sikame et al. , 2014. This study is limited to a sample located at the center of the stem base (PL1/4-R3). Results for other sampling zones can be obtained in the same way. The diffusion coefficient of raffia vinifera fiber is taken in the work done by Sikame et al. (2014). This diffusion coefficient is D f ¼ 1:94 � 10 À 10 m 2 =s. The diffusion coefficient of raffia vinifera pith, considered as homogeneous material, is taken in the work done by Mbou et al. (2017). This diffusion coefficient is D ¼ 8:746 � 10 À 9 m 2 =s.

Water concentration in the pith
Since raffia cork is a natural composite, a perfect contact between the fiber and matrix is assumed and σ f ¼ 1 throughout the work. The case where the fiber and the matrix have the same properties is first considered. Under this consideration, raffia vinifea pith is viewed as a homogeneous material. The physical properties are those determined experimentally by Mbou et al. (2017). Therefore, Figure 4 is plotted for D f ¼ D m ¼ 8:746 � 10 À 9 m 2 =s. Figure 4 gives the evolution of water concentration in the cork as function of time.
In Figure 4, it is observed that the local water concentration in the cork grows with time and tends to a limit value for time greater than 50,000 seconds. This asymptotic behavior reflects the saturation of the pith observed in (Carr and Pontrelli 2018;Mbou et al. 2017;Nimjieu et al. 2021;Nouri et al. 2021;Sikame et al. 2014). A deviation between analytical and numerical results is observed. This deviation would be due either to the truncations made in the approximation of the derivatives in the numerical method or to the approximation of z , , solution of Equation 26. In solving Equation 26, the successive bisection method was used. This method gives only real solutions. However, Equation 26 may have complex solutions that are not included in the computation. Another reason of discrepancy may be the sensitivity of the Bessel functions on the truncation and/or rounding errors.

Mass of cork
Considering the properties D f ¼ D m ¼ 8:746 � 10 À 9 m 2 =s, the profile of the mass of cork is represented in Figure 5. It is observed that the relative mass of raffia vinifera pith decreases with time. From t > 4 � 10 4 s, the mass of raffia vinifera pith tends to zero. This characterizes the saturation of the pith. Similar curves were obtained by (Carr and Pontrelli 2018;Kaoui, Lauricella, and Pontrelli 2018). Qualitatively, a good agreement between the analytical and numerical solutions is observed in Figure 5. The quantitative difference would be due either to the truncations made in the approximation of the derivatives in the numerical method or to the approximation of the values of z , :

Absorption ratio
The knowledge of the mass of the pith at a given time allows determining the absorption kinetics of the raffia pith given by its absorption ratio. It is observed in Figure 6 that, for D f ¼ D m ¼ 8:746 � 10 À 9 m 2 =s, the water absorption ratio of raffia vinifera pith increases rapidly with time and then stabilizes from t > 3 � 10 4 seconds. This means that, for t > 3 � 10 4 seconds, the raffia vinifera pith becomes saturated and does not absorb water in a significant way. Like the mass, the small deviation observed between the analytical and numerical solutions would be related to the sensitivity of Bessel function to decimal approximations. These solutions (analytical and numerical) are compared to the experimental results obtained by Mbou et al. (2017) and a good correlation is observed. The numerical results agree better to the experimental results than the analytical results. Considering the experimental results as the reference, the relative errors are calculated and tabulated in Table 1. It can be seen from Table 1 at any time that the maximum value of the relative error is less than 6.5% for numerical and 11% for analytical methods. Although results presented are somewhat different, the methods develop in this work are correct.

Influence of diffusion coefficient on solutions
In this section, the physical properties of the fiber and the matrix are different and the raffia vinifera pith is viewed as a heterogeneous material consisting of a fiber surrounded by a natural matrix. The diffusion coefficient of raffia vinifera fiber were determined experimentally by Sikame et al. (2014) and is D f ¼ 1:94 � 10 À 10 m 2 =s while the diffusion coefficient of the matrix is calculated using the mixture rule and is D m ¼ 1:578 � 10 À 8 m 2 =s. The results obtained are compared with the experimental results and with the results obtained in the case where raffia vinifera pith is considered as homogenous material.
It can be seen from Figure 7 that for D f �D m , the water absorption ratio does not changed. This result is explained by the fact that the water absorption ratio does not depend on the spatial coordinate. Also, the diffusion coefficient of the raffia vinifera pith remains unchanged in applying the rule of mixture. However, Figure 8 shows that, the water concentration in the raffia vinifera pith increases from the center to the periphery with a peak occurring at the fiber/matrix interface in the heterogeneous case. This peak highlights the heterogeneous character of raffia vinifera pith. Moreover, when raffia vinifera pith is considered as heterogeneous material, the local water concentration in the fiber is lower compared to the case where raffia vinifera pith is considered as homogeneous material. The gap between these values is considerable and demonstrates that, studying the raffia vinifera pith as homogeneous material causes enormous errors. According to the heterogeneous case, it is observed in Figure 8 that the water concentration in the matrix increases rapidly than the water concentration in the fiber. This result implies that, for the same sample zone, the matrix absorbs more water than fiber and reflects the fact that the matrix is very porous than the fiber.

Conclusion
The objective of this work was to developed analytical and numerical approaches to solve the problem of water diffusion through raffia vinifera pith. The model was based on Fick's second law and a representative elementary volume consisting of a fiber surrounded by a layer of natural matrix was considered. The physical properties of each constituent were kept constant. The diffusion equations were solved numerically by finite difference method and analytically by separation of variables method. The analytical and numerical resolutions gave almost the same results despite a slight difference observed in some values. This slight difference could be due to the sensitivity of the Bessel functions to the decimal approximations made in the calculation of the z , values or to the decimal approximations made in the calculation of the derivatives in the finite difference method. To evaluate the efficiency of each method developed in this work, a particular case where all the constituents of the material have the same physical properties was studied. In comparing the computational results with the experimental literature results a good correlation was observed. Thus methods developed in this work are correct. To evaluate the effect of the heterogeneity of material on the solutions, the case where the fiber and matrix have the different physical properties was examined. The results showed a peak occurring at the fiber/ matrix interface of the water concentration profile. In addition, a considerable gap between the water concentrations in the fiber of composite material was observed. These results demonstrated that, study the raffia vinifera as homogeneous material causes enormous errors. The present study can be applied to hybrid composite materials where polymers are combined to natural fibers to achieve high mechanical properties of materials. All the methods implemented in this work are approximated methods and so have some limitations. Other advanced method can be used to improve the quality of this work. In addition, restricting the raffia vinifera pith at only two-layer material represents a limit of this work. The authors will consider more sophisticated and realistic representations of the fiber/matrix geometry in future work and the dependence of the diffusion coefficient with the radius of the bamboo.

Highlights
• The present paper is a continuation work on raffia vinifera pith and deals the modeling of water diffusion through raffia vinifera pith; • Raffia vinifera pith is modeled in this work as heterogeneous material; • A numerical investigation using finite differences method is done; • An analytical investigation using separation of variable is done; • A good correlation between analytical and numerical results is observed; • A good correlation between theoretical result and experimental results in literature is observed; • The result obtained would help to predict moisture transfer through composite materials with natural fibers reinforcement.

Disclosure statement
No potential conflict of interest was reported by the authors.

Funding
The author(s) reported there is no funding associated with the work featured in this article.

Ethics declarations
I confirm that all the research meets ethical guidelines and adheres to the legal requirements of the study country.