# Modelling of the Water Absorption Kinetics and Determination of the Water Diffusion Coefficient in the Pith ofRaffia vinifera of Bandjoun, Cameroon

Authors: E. Tiaya Mbou; E. Njeugna; A. Kemajou; N. R. Tagne Sikame; D. Ndapeu
Journal: Advances in Materials Science and Engineering
 (2017)

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## Abstract

The present work focuses on the study of the water absorption phenomenon through the pith ofRaffia vinifera along the stem. The water absorption kinetics was studied experimentally by the gravimetric method with the discontinuous control of the sampling mass at temperature of 30°C. The samples of 70 mm × 8 mm × 4 mm were taken from twelve sampling zones of the stem ofRaffia vinifera. The result shows that the percentage of water absorption of the pith ofRaffia vinifera increases from the periphery to the center in the radial position and from the base to the leaves in the longitudinal position. Fick’s second law was adopted for the study of the water diffusion. Eleven models were tested for the modelling of the water absorption kinetics and the model of Sikame Tagne (2014) is the optimal model. The diffusion coefficients of two stages were determined by the solution of the Fick equation in the twelve sampling zones described by Sikame Tagne et al. (2014). The diffusion coefficients decreased from the center to the periphery in the radial position and from the base to the leaves in the longitudinal position.


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## Body

## 1. Introduction

The raffia is a plant which is found in the tropical zones particularly in Madagascar and which is generally grown in the swampy zone [1, 2]. In the world, there are about twenty (20) species among which theRaffia viniferais found in the West, the Northwest, the South, the Center and the East Regions of Cameroon. This raffia is constituted with a feather grass bearing stick with stout rachis and large petiole [2, 3].Raffia is generally used in the realization of arts, crafts, decoration, braces, clothes, baskets, hats, beds, and food (sap/wine) [1, 4]. Raffia is also used in the West Region of Cameroon as a building material specially as ceiling material, but this use is still local. Our main objective is to extend the use of raffia as ceiling material over the world particularly in our country. The pith of raffia will be combined with another material, such as polymer and other vegetal material, to produce a new composite building material suitable for ceiling applications. The physical behavior of our pith ofRaffia vinifera has to be known because the realization of the composite material will use the matrix, which is liquid and will be absorbed by our pith of raffia. It is important in our study to know the water diffusion coefficient and the percentage of the water absorption that will help us to predict the behavior of the composite material.Several works have been carried out on the raffia such as the study of the thermal properties on trunk ofRaffia hookeri which is used as ceiling material [5]. Some studies are also carried out on the study and the use of raffia and other vegetal products as insulation material in Cameroon [6]. Other works include the use of raffia in the reinforcement of concrete [7]. The use of raffia in the textile industry is equally growing. Studies are also carried out on the study of the microstructure and the physical properties of fibers coming from the leaves of raffia; the drying kinetics of these fibers is carried in view of its use as roofing elements [8, 9]. Mechanical studies have been carried out on the long-term creep behavior of theRaffia vinifera stem in compression and flexion tests [10–13]. A study on the long-term mechanical behavior and the mechanical properties of fibers from theRaffia vinifera has been done [4]. Studies have been equally done on the traction and compression of composite cement matrix reinforcement by the raffia fiber [14].In the same vein and in the view of improving the scientific knowledge so as to optimize the use ofRaffia vinifera, this study focuses on the study of the water absorption phenomenon in theRaffia vinifera pith. The main objective of this work is to evaluate the percentage of the water absorbed, do the modelling of the water absorption kinetics, and evaluate the diffusion coefficient of water through the pith ofRaffia vinifera.In this article, we analyzed the distribution of water absorption rate of our material through the stem of raffia respectively on the sampling zones described by Sikame Tagne et al. [4]. The modelling of the water absorption kinetics is also taken in place. The diffusion coefficient of water through the material was evaluated in the sampling zones and followed by the conclusion.


## 2. Methods

### 2.1. Materials

The samples used in this work come from the stem ofRaffia vinifera originating from Bandjoun in the West Region of Cameroon particularly the Koung-Khi division. In the stem, prismatic samples are cut particularly on the parallelepiped shape with 70 mm × 8 mm × 4 mm ±1mm. The largest dimension of the sample is along the length of the stem. The samples were taken from the twelve sampling zones described by Sikame Tagne et al. [4] and presented in Figure 1. In the stem, we distinguished four parts (Figure 1(a)) from the bottom to the top named P1/4 (near the root), P2/4, P3/4, and P4/4 (near the leaves). In the radial position, we distinguished three zones (Figure 1(b)) called center, half-radius, and periphery (near the bark) zones.Figure 1
Localization of the sampling zone on the stem of raffia. (a) Longitudinal position and (b) radial position corresponding to each longitudinal position [1, 4, 15].
(a) (b)A microwave of Bosch mark was used as oven for the drying of the sample at a constant temperature until the mass is stabilized. Digital scales of ADAM mark with a maximum weight of 750 g and to the milligram accuracy level are used for the various weighing. A numerical slide caliper with the hundredth of the millimeter accuracy is used for the measurement of the sample dimensions. The distilled water at the ambient temperature of30°C±1°C is used for the immersion of the samples as in the case of the absorption of other types of wood such asAfra, Ojamlesh, andRoosi [1, 16, 17, 28].


### 2.2. Methods

The section of the stem of raffia has an elliptic form. The large diameter of the raffia stem is about 62 mm on the base, 40 mm on the middle, and 34 mm on the top. The small diameter of the raffia stem is about 51 mm on the base, 36 mm on the middle, and 29 mm on the top. After subdivision of the stem in four parts named P1/4, P2/4, P3/4, and P4/4, we removed the bark of each part. According to each longitudinal position, the remaining diameter was divided into three parts to obtain primary samples following radial position (center, half-radius, and periphery zones). On this primary samples, we extracted the final samples which have a parallelepiped shape with the following dimensions 70 mm × 8 mm × 4 mm. The samples were extracted in the twelve sampling zones on the stem.The obtained samples are now conditioned in the microwave used as the oven at the constant temperature of 90°C until stabilization of the mass. This gravimetric analysis method is done in order to eliminate residual water found in the samples. The samples were then introduced in plastic bags to avoid reabsorption of moisture from the air during the samples cooling period. Conditioned samples are weighed and labeled before being introduced into the distilled water. After the drying phase, we introduced our samples in the distilled water at the laboratory room temperature estimated at30°C±1°C. The samples are maintained under water by nuts made of rustproof steel. During the regular time intervals, samples were removed from the water; surface water of the sample was then eliminated with a dry fabric base on cotton and the samples were weighted and reintroduced into the distilled water. This operation was done in very short time so as to enable us to neglect the time that was spent out of the distilled water. At the beginning of the test, immersion time is taken at five minutes. After one hour of test, we changed it to ten minutes. We prolonged the weight time until we arrive at weight one time per day, one time after two days, three days, and eventually one time per week. The process of weighting is repeated until the constant mass, that is, until samples reach water saturation level [16, 29–32].In each sampling zone, twenty (20) samples were tested for a total of two hundred and forty (240) samples ofRaffia vinifera along the stem. These samples were extracted from two dry mature stems ofRaffia vinifera for each zone. These samples were mixed and tested in the group. The test was stopped when the mass no longer varies.The statistical analysis of experimental data was done in the Matlab 2009b software environment which enabled drawing experimental curves and doing the modelling with different models as found in Table4. The best model was the one which has an average correlation coefficient near to the unit, the square root of mean error average near zero, and the sum of square error average near zero. These two last statistical parameters are defined by the following [1, 26]: (1)RMSE=∑i=1nmci-mei2n,SSE=∑i=1nmci-mei,where mci, mei, and n are the theoretical mass, the predicted mass, and the number of observation, respectively.


#### 2.2.1. Mass Diffusion Theory through Solid

The mass transfer equation of solid result from the second Fick law given by(2)∂C∂t=div⁡DgradC→,where C is the molar concentration (mol/m3), D is the diffusion coefficient (m2/s), and t is the time (s).To simplify the resolution of (2), it is supposed that the diffusion coefficient is independent of the space direction [33, 34]. This hypothesis enables us to write (2) in the following form:(3)∂C∂t=D∂2C∂x2+∂2C∂y2+∂2C∂z2.It is assumed that the sample is the plane shape and it is assumed to have a single long direction, thez direction in Figure 2(b), so that the diffusion equation can be solved in one dimension across the shortest dimension of the sample. In our case, the x direction is the one on which diffusion has taken place. Figure 2 presents the samples obtained for the test (Figure 2(a)) and the plan which presents the dimensions of the sample (Figures 2(b) and 2(c)). These hypotheses enabled the reduction of (3) which then gives [29, 33, 34] (4)∂C∂t=D∂2C∂x2.Figure 2
Illustration of the samples used for the test. (a) Picture of the sample, (b) axis orientation of the sample, and (c) extraction zone of the samples in the radial position.
(a) (b) (c)By taking into account the boundary’s conditions, we have(5)C=C0,0<x<L,t=t0,C=C1,where L is the thickness of the sample.The solution of (4) can be given in the following form:(6)C-C0C1-C0=1-4π∑n=0∞-1n2n+1exp⁡-D2n+12π2t4L2cos⁡2n+1πx2L.If we callMt the total mass of water which diffuses in the material at t instant and M∞, the same quantity at the infinite time, that is to say, when the saturation of the material is reached, then [23, 25, 29, 35] (6) can be written in the following form:(7)MtM∞=1-∑n=0∞82n+12π2exp⁡-Deff2n+12π2t4L2.


#### 2.2.2. Percentage of Water Absorption

The percentage of water absorption of our sample (WA) is given by the following relation, which was also used for the determination of the water absorption by [15, 16, 36](8)WA=mf-mimi∗100,where mf and mi are, respectively, the final and initial mass of the sample.


#### 2.2.3. Study of the Absorption Kinetics

The ratio of the water absorption calledg(t) is defined as follows:(9)gt=MtM∞=mt-m0m∞-m0.The similarity between (7) and (9) enables us to write(10)gt=MtM∞=mt-m0m∞-m0=1-∑n=0∞82n+12π2exp⁡-Deff2n+12π2t4L2,where m0, mt, and m∞ are, respectively, the anhydrous mass (at the initial instant), at current moment and at the infinite time (t=∞), that is, at saturation.


## 2.1. Materials

The samples used in this work come from the stem ofRaffia vinifera originating from Bandjoun in the West Region of Cameroon particularly the Koung-Khi division. In the stem, prismatic samples are cut particularly on the parallelepiped shape with 70 mm × 8 mm × 4 mm ±1mm. The largest dimension of the sample is along the length of the stem. The samples were taken from the twelve sampling zones described by Sikame Tagne et al. [4] and presented in Figure 1. In the stem, we distinguished four parts (Figure 1(a)) from the bottom to the top named P1/4 (near the root), P2/4, P3/4, and P4/4 (near the leaves). In the radial position, we distinguished three zones (Figure 1(b)) called center, half-radius, and periphery (near the bark) zones.Figure 1
Localization of the sampling zone on the stem of raffia. (a) Longitudinal position and (b) radial position corresponding to each longitudinal position [1, 4, 15].
(a) (b)A microwave of Bosch mark was used as oven for the drying of the sample at a constant temperature until the mass is stabilized. Digital scales of ADAM mark with a maximum weight of 750 g and to the milligram accuracy level are used for the various weighing. A numerical slide caliper with the hundredth of the millimeter accuracy is used for the measurement of the sample dimensions. The distilled water at the ambient temperature of30°C±1°C is used for the immersion of the samples as in the case of the absorption of other types of wood such asAfra, Ojamlesh, andRoosi [1, 16, 17, 28].


## 2.2. Methods

The section of the stem of raffia has an elliptic form. The large diameter of the raffia stem is about 62 mm on the base, 40 mm on the middle, and 34 mm on the top. The small diameter of the raffia stem is about 51 mm on the base, 36 mm on the middle, and 29 mm on the top. After subdivision of the stem in four parts named P1/4, P2/4, P3/4, and P4/4, we removed the bark of each part. According to each longitudinal position, the remaining diameter was divided into three parts to obtain primary samples following radial position (center, half-radius, and periphery zones). On this primary samples, we extracted the final samples which have a parallelepiped shape with the following dimensions 70 mm × 8 mm × 4 mm. The samples were extracted in the twelve sampling zones on the stem.The obtained samples are now conditioned in the microwave used as the oven at the constant temperature of 90°C until stabilization of the mass. This gravimetric analysis method is done in order to eliminate residual water found in the samples. The samples were then introduced in plastic bags to avoid reabsorption of moisture from the air during the samples cooling period. Conditioned samples are weighed and labeled before being introduced into the distilled water. After the drying phase, we introduced our samples in the distilled water at the laboratory room temperature estimated at30°C±1°C. The samples are maintained under water by nuts made of rustproof steel. During the regular time intervals, samples were removed from the water; surface water of the sample was then eliminated with a dry fabric base on cotton and the samples were weighted and reintroduced into the distilled water. This operation was done in very short time so as to enable us to neglect the time that was spent out of the distilled water. At the beginning of the test, immersion time is taken at five minutes. After one hour of test, we changed it to ten minutes. We prolonged the weight time until we arrive at weight one time per day, one time after two days, three days, and eventually one time per week. The process of weighting is repeated until the constant mass, that is, until samples reach water saturation level [16, 29–32].In each sampling zone, twenty (20) samples were tested for a total of two hundred and forty (240) samples ofRaffia vinifera along the stem. These samples were extracted from two dry mature stems ofRaffia vinifera for each zone. These samples were mixed and tested in the group. The test was stopped when the mass no longer varies.The statistical analysis of experimental data was done in the Matlab 2009b software environment which enabled drawing experimental curves and doing the modelling with different models as found in Table4. The best model was the one which has an average correlation coefficient near to the unit, the square root of mean error average near zero, and the sum of square error average near zero. These two last statistical parameters are defined by the following [1, 26]: (1)RMSE=∑i=1nmci-mei2n,SSE=∑i=1nmci-mei,where mci, mei, and n are the theoretical mass, the predicted mass, and the number of observation, respectively.


### 2.2.1. Mass Diffusion Theory through Solid

The mass transfer equation of solid result from the second Fick law given by(2)∂C∂t=div⁡DgradC→,where C is the molar concentration (mol/m3), D is the diffusion coefficient (m2/s), and t is the time (s).To simplify the resolution of (2), it is supposed that the diffusion coefficient is independent of the space direction [33, 34]. This hypothesis enables us to write (2) in the following form:(3)∂C∂t=D∂2C∂x2+∂2C∂y2+∂2C∂z2.It is assumed that the sample is the plane shape and it is assumed to have a single long direction, thez direction in Figure 2(b), so that the diffusion equation can be solved in one dimension across the shortest dimension of the sample. In our case, the x direction is the one on which diffusion has taken place. Figure 2 presents the samples obtained for the test (Figure 2(a)) and the plan which presents the dimensions of the sample (Figures 2(b) and 2(c)). These hypotheses enabled the reduction of (3) which then gives [29, 33, 34] (4)∂C∂t=D∂2C∂x2.Figure 2
Illustration of the samples used for the test. (a) Picture of the sample, (b) axis orientation of the sample, and (c) extraction zone of the samples in the radial position.
(a) (b) (c)By taking into account the boundary’s conditions, we have(5)C=C0,0<x<L,t=t0,C=C1,where L is the thickness of the sample.The solution of (4) can be given in the following form:(6)C-C0C1-C0=1-4π∑n=0∞-1n2n+1exp⁡-D2n+12π2t4L2cos⁡2n+1πx2L.If we callMt the total mass of water which diffuses in the material at t instant and M∞, the same quantity at the infinite time, that is to say, when the saturation of the material is reached, then [23, 25, 29, 35] (6) can be written in the following form:(7)MtM∞=1-∑n=0∞82n+12π2exp⁡-Deff2n+12π2t4L2.


### 2.2.2. Percentage of Water Absorption

The percentage of water absorption of our sample (WA) is given by the following relation, which was also used for the determination of the water absorption by [15, 16, 36](8)WA=mf-mimi∗100,where mf and mi are, respectively, the final and initial mass of the sample.


### 2.2.3. Study of the Absorption Kinetics

The ratio of the water absorption calledg(t) is defined as follows:(9)gt=MtM∞=mt-m0m∞-m0.The similarity between (7) and (9) enables us to write(10)gt=MtM∞=mt-m0m∞-m0=1-∑n=0∞82n+12π2exp⁡-Deff2n+12π2t4L2,where m0, mt, and m∞ are, respectively, the anhydrous mass (at the initial instant), at current moment and at the infinite time (t=∞), that is, at saturation.


## 2.2.1. Mass Diffusion Theory through Solid

The mass transfer equation of solid result from the second Fick law given by(2)∂C∂t=div⁡DgradC→,where C is the molar concentration (mol/m3), D is the diffusion coefficient (m2/s), and t is the time (s).To simplify the resolution of (2), it is supposed that the diffusion coefficient is independent of the space direction [33, 34]. This hypothesis enables us to write (2) in the following form:(3)∂C∂t=D∂2C∂x2+∂2C∂y2+∂2C∂z2.It is assumed that the sample is the plane shape and it is assumed to have a single long direction, thez direction in Figure 2(b), so that the diffusion equation can be solved in one dimension across the shortest dimension of the sample. In our case, the x direction is the one on which diffusion has taken place. Figure 2 presents the samples obtained for the test (Figure 2(a)) and the plan which presents the dimensions of the sample (Figures 2(b) and 2(c)). These hypotheses enabled the reduction of (3) which then gives [29, 33, 34] (4)∂C∂t=D∂2C∂x2.Figure 2
Illustration of the samples used for the test. (a) Picture of the sample, (b) axis orientation of the sample, and (c) extraction zone of the samples in the radial position.
(a) (b) (c)By taking into account the boundary’s conditions, we have(5)C=C0,0<x<L,t=t0,C=C1,where L is the thickness of the sample.The solution of (4) can be given in the following form:(6)C-C0C1-C0=1-4π∑n=0∞-1n2n+1exp⁡-D2n+12π2t4L2cos⁡2n+1πx2L.If we callMt the total mass of water which diffuses in the material at t instant and M∞, the same quantity at the infinite time, that is to say, when the saturation of the material is reached, then [23, 25, 29, 35] (6) can be written in the following form:(7)MtM∞=1-∑n=0∞82n+12π2exp⁡-Deff2n+12π2t4L2.


## 2.2.2. Percentage of Water Absorption

The percentage of water absorption of our sample (WA) is given by the following relation, which was also used for the determination of the water absorption by [15, 16, 36](8)WA=mf-mimi∗100,where mf and mi are, respectively, the final and initial mass of the sample.


## 2.2.3. Study of the Absorption Kinetics

The ratio of the water absorption calledg(t) is defined as follows:(9)gt=MtM∞=mt-m0m∞-m0.The similarity between (7) and (9) enables us to write(10)gt=MtM∞=mt-m0m∞-m0=1-∑n=0∞82n+12π2exp⁡-Deff2n+12π2t4L2,where m0, mt, and m∞ are, respectively, the anhydrous mass (at the initial instant), at current moment and at the infinite time (t=∞), that is, at saturation.


## 3. Results and Discussions

### 3.1. Determination of the Percentage of Water Absorption

The percentage of water absorption of our samples for the twelve sampling zones is done by (8). The synthesis result is presented in Table 1.Table 1
Average rate of water absorption and standard deviation of theRaffia vinifera pith.
Center Half-radius Periphery Part 1/4 1141 . 509 ± 156.063 847 . 525 ± 95.568 392 . 044 ± 36.618 Part 2/4 1256 . 696 ± 87.415 1156 . 705 ± 79.810 565 . 967 ± 100.128 Part 3/4 1331 . 956 ± 49.144 1207 . 133 ± 107.053 612 . 198 ± 132.146 Part 4/4 1308 . 677 ± 53.766 1137 . 932 ± 61.979 705 . 018 ± 113.600It is observed that the average percentage of water absorption stands between392.044%±36.618% and 1331.956%±49.144% during the immersion period estimated at 45 days.A comparative study of the percentage of the water absorption of theRaffia vinifera pith and other vegetal products has been done, in Table 2. This study indicates that theRaffia vinifera pith absorbs more water than the fibers taken in the same position of the stem [4]. This is probably due to the fact that the samples of pith ofRaffia vinifera are the natural composites materials comprising fibers and a spongy part that work as a natural binder between the fibers and which in addition absorbs water.Table 2
Comparison of the percentage of water absorption of the pith ofRaffia vinifera with other natural materials.
Types Percentage of water absorbed (%) Absorption duration Temperature (°C) References Afra wood 102 25 Days 25 [16] Ojamlesh Wood 54 Roosi Wood 120  Hemp 62 13 Days 27–67 [17] Okra 64 Betel nut 38  Fibers ofRaffia vinifera 303–662 25 Days 23 [4]  Pith of the Raffia vinifera 392–1331 45 Days 30 ± 1 Studied caseThe average of the percentage of water absorption along the stem is presented on Figure3. Globally the observation of Figure 3 enables us to see that the percentage of water absorption increases from the periphery to the center on the radial position and from the base to the leaves on the longitudinal position along the stem ofRaffia vinifera.Figure 3
Distribution of the water absorption rate percentage of the pith ofRaffia vinifera.


### 3.2. Kinetics of Water Absorption

Equation (9) allows us to plot the curve of water absorption ratio according to the time and the obtained curve is presented by Figure 4. This curve presents the results of water absorption test of the pith ofRaffia vinifera taken from the center of part 4/4.Figure 4
Average of the water absorption kinetics of the pith ofRaffia vinifera from the center and at the base of the stem (1/4).The obtained curves on the twelve sampling zones have the same profile as that presented in Figure4.An observation of the curve presented in Figure4 shows that, during the first 5000 minutes, approximately 50% of the sample masses reached saturation level.The curve presented in Figure4 is similar to the curve obtained by other authors who have studied water absorption kinetics in natural materials [4, 16, 26, 27, 34, 37]. It was noticed that the water absorption kinetics of the pith ofRaffia vinifera takes place in two phases: the first phase rapidly takes place and enables reaching approximately 50% of the mass at saturation. This is done in the first 5000 minutes and the second phase that is relatively slow and is represented by the second part of the curve. This brings out two diffusion coefficients which indicate the velocity of the water diffusion in the pith ofRaffia vinifera samples.


### 3.3. Modelling of the Water Absorption Kinetics

The water absorption kinetics in the pith ofRaffia vinifera is done by using the mathematical models found in the literature. In this fact, we found eleven models which are in Table 3.Table 3
Utilized models summary for the modelling of the absorption kinetics.
Number Model’s name Model’s equation References 1 Pilosof g t = a + b ∗ t c + t [18] 2 Mohsenin g t = a ∗ 1 - exp ⁡ ( - k ∗ t ) + c + d ∗ t [16, 19] 3 Singh g t = a + b ∗ t c ∗ t + 1 [20] 4 Peleg g t = a + t b ∗ t + c [21, 22] 5 García-Pascual g t = 1 - exp ⁡ - t a b [23] 6 Gowen g t = a - b ∗ exp ⁡ - k ∗ t + b [24] 7 Czel and Czigany g t = a ∗ t m [25] 8 Vega-Gálvez g t = a ∗ exp ⁡ - b 1 + t α [26] 9 Gornicki g t = a ∗ b - exp ⁡ - k ∗ t [27] 10 Gornicki g t = a + b ∗ 1 - 1 1 + b ∗ c ∗ t [27] 11 Sikame Tagne g t = c - a ∗ exp ⁡ - k ∗ t - b ∗ exp ⁡ ( - m ∗ t ) [4]Table 4
Distribution of the average parameters models of the pith ofRaffia vinifera at the base of the stem (1/4).
Model’s name Radial position a b c k   ( m i n - 1 ) m   ( m i n - 1 ) R 2 RMSE SSE Peleg Center 0.239 ± 0.034 9312.150 ± 2044.966 1.176 ± 0.090   0.972 0.039 0.135 Half-radius 0.245 ± 0.033 9  256.250 ± 2  090.859 1.183 ± 0.109   0.971 0.040 0.143 Periphery 0.216 ± 0.030 4  763.550 ± 1  040.602 1.252 ± 0.071   0.966 0.047 0.194  Pilosof Center 0.230 ± 0.042 0.817 ± 0.045 6492.5 ± 549.047   0.968 0.042 0.152 Half-radius 0.234 ± 0.041 0.810 ± 0.049 6  315.350 ± 879.532   0.967 0.043 0.160 Periphery 0.215 ± 0.029 0.802 ± 0.044 3  762.600 ± 923.578   0.966 0.046 0.193  Singh Center 0.240 ± 0.033 0.157 ± 0.321 6861.700 ± 4310.801   0.973 0.054 0.135 Half-radius 0.245 ± 0.033 0.161 ± 0.331 6  806.700 ± 4  230.220   0.971 0.058 0.143 Periphery 0.217 ± 0.031 0.168 ± 0.344 2  885.300 ± 1  656.826   0.966 0.046 0.193  Witrowa Center 0.228 ± 0.057 0.858 ± 0.068 1.563 × 10 - 4 ± 5.405 × 10 - 5   0.974 0.039 0.139 Half-radius 0.245 ± 0.033 0.854 ± 0.078 1.652 × 10 - 4 ± 6.949 × 10 - 5   0.960 0.0432 0.101 Periphery 0.216 ± 0.030 0.801 ± 0.044 3.489 × 10 - 4 ± 1.023 × 10 - 4   0.966 0.047 0.194  Gowen Center 0.262 ± 0.036 0.972 ± 0.019  1.910 × 10 - 4 ± 2.567 × 10 - 5  0.963 0.046 0.184 Half-radius 0.268 ± 0.038 0.974 ± 0.022  1.076 × 10 - 4 ± 2.358 × 10 - 5  0.961 0.067 0.198 Periphery 0.251 ± 0.036 0.932 ± 0.022  1.856 × 10 - 4 ± 3.828 × 10 - 5  0.942 0.060 0.324  Wesolowski Center 0.710 ± 0.053 1.374 ± 0.080  1.045 × 10 - 4 ± 2.504 × 10 - 5  0.963 0.067 0.184 Half-radius 0.706 ± 0.058 1.386  ± 0.087  1.465 × 10 −4 ± 1.662 × 10 - 5  0.961 0.067 0.198 Periphery 0.681 ± 0.037 1.371 ± 0.070  1.827 × 10 −4 ± 6.106 × 10 - 5  0.942 0.060 0.323  Mohsenin Center 0.571 ± 0.110 3.025 × 10 - 6 ± 2.261 × 10 - 6 0.265 ± 0.080 1.629 × 10 −4 ± 5.107 × 10 - 5  0.963 0.043 0.164 Half-radius 0.568 ± 0.080 2.429 × 10 −5 ± 6.555 × 10 - 5 0.255 ± 0.036 1.638 × 10 −4 ± 5.019 × 10 - 5  0.967 0.044 0.175 Periphery 0.557 ± 0.046 3.894 × 10 −6 ± 1.356 × 10 - 6 0.222 ± 0.036 3.505 × 10 −4 ± 1.351 × 10 - 4  0.958 0.074 0.241  Weibull Center 6207.600 ± 2386.766 0.379 ± 0.042    0.939 0.058 0.293 Half-radius 5  938.200 ± 2  202.506 0.373 ± 0.038    0.938 0.058 0.293 Periphery 3  756.650 ± 933.131 0.388 ± 0.029    0.978 0.037 0.131  Czel and Czigany Center 0.085 ± 0.026    0.231 ± 0.028 0.973 0.039 0.135 Half-radius 0.089 ± 0.030    0.226 ± 0.028 0.978 0.057 0.129 Periphery 0.105 ± 0.023    0.214 ± 0.020 0.978 0.037 0.131  Vega-Gálvez Center 7.871 ± 2.119 5.471 ± 0.727    0.958 0.048 0.198 Half-radius 7.734 ± 2.295 5.402 ± 0.791    0.958 0.047 0.182 Periphery 6.899 ± 1.268 5.028 ± 0.493    0.986 0.029 0.085  Sikame Tagne Center 0.241 ± 0.043 0.686 ± 0.062 0.985 ± 0.079 3.662 × 10 −2 ± 2.291 × 10 - 2 8.682 × 10 −5 ± 1.918 × 10 - 5 0.992 0.022 0.041 Half-radius 0.252 ± 0.042 0.679 ± 0.075 1.001 ± 0.016 3.917 × 10 −2 ± 2.024 × 10 - 2 8.563 × 10 −5 ± 1.445 × 10 - 5 0.992 0.022 0.040 Periphery 0.298 ± 0.047 0.600 ± 0.051 0.991 ± 0.013 1.035 × 10 −2 ± 1.078 × 10 - 2 1.053 × 10 −4 ± 2.123 × 10 - 5 0.993 0.021 0.037The experimental data of the pith ofRaffia vinifera of the center from the base of the stem and the eleven models found in the literature are represented in the Figure 5.Figure 5
Experimental curve of the pith ofRaffia viniferaand curves of models.Table4 provides a summary of the parameters of the various models previously cited concerning samples of the pith coming from part 1/4 of theRaffia vinifera stem.The variations presented by some models such as those by Singh and Kulshrestha (1987) and Pilosof et al. (1987) can be interpreted by the fact that these models are not ideal to describe the kinetics of water absorption of the pithof Raffia vinifera.According to Table4, it can be seen that the models used in this work have the coefficients of correlation higher than 0.93. In the case of this work the model of Weibull has the lowest correlation coefficient. The better model is the model of Sikame Tagne et al. which has the correlation coefficient close to the unit. In this particular work, the correlation coefficient is 0.99 for the Sikame Tagne et al. model.The model of Sikame Tagne et al. (2014) is the one which has better statistical parameters for the sampling zones. This model is used for the modelling of the absorption kinetics in the twelve sampling zones of the stem ofRaffia vinifera. The average parameters of this model are provided in Table 5, the coefficient of linear correlation, the square root of the mean error, and the squared sum of errors of theRaffia vinifera stem.Table 5
Average parameters of the model of Sikame Tagne [4] on the twelve sampling zones of the stem of Raffia vinifera.
Longitudinal position Radial Position a b c k   m i n - 1 m   ( m i n - 1 ) R 2 RMSE SSE 1/4 Center 0.241 ± 0.043 0.688 ± 0.062 0.985 ± 0.079 3.662 × 10 - 2  ± 2.291 × 10 - 2 8.689 × 10 - 5  ± 1.918 × 10 - 5 0.992 0.022 0.042 Half-radius 0.252 ± 0.042 0.679 ± 0.075 1.002 ± 0.016 3.917 × 10 - 2 ± 2.024 × 10 - 2 8.563 × 10 - 5  ± 1.445 × 10 - 5 0.999 0.022 0.040 Periphery 0.298 ± 0.048 0.599 ± 0.051 0.9913 ± 0.013 1.035 × 10 - 2  ± 1.078 × 10 - 2 1.053 × 10 - 4  ± 2.123 × 10 - 5 0.993 0.021 0.037  2/4 Center 0.2003 ± 0.0219 0.721 ± 0.033 1.026 ± 0.031 1.120 × 10 - 2  ± 1.280 × 10 - 3 6.530 × 10 - 5      ± 1.550 × 10 - 5 0.992 0.022 0.039 Half-radius 0.2133 ± 0.0446 0.713 ± 0.049 1.019 ± 0.029 2.840 × 10 - 2  ± 3.221 × 10 - 3 7.390 × 10 - 5      ± 2.520 × 10 - 5 0.990 0.027 0.043 Periphery 0.2975 ± 0.0469 0.599 ± 0.051 0.991 ± 0.013 1.041 × 10 - 2  ± 1.080 × 10 - 3 1.050 × 10 - 4      ± 2.120 × 10 - 5 0.993 0.021 0.037  3/4 Center 0.1877 ± 0.0410 0.711 ± 0.055 1.029 ± 0.024 8.220 × 10 - 3  ± 1.520 × 10 - 3 5.571 × 10 - 5     ± 1.280 × 10 - 5 0.994 0.021 0.056 Half-radius 0.2009 ± 0.0627 0.694 ± 0.076 1.032 ± 0.017 1.370 × 10 - 2  ± 2.820 × 10 - 3 5.530 × 10 - 5      ± 1.130 × 10 - 5 0.993 0.022 0.055 Periphery 0.2319 ± 0.0267 0.64 ± 0.075 1.022 ± 0.011 1.410 × 10 - 2  ± 1.730 × 10 - 3 5.870 × 10 - 5      ± 1.040 × 10 - 5 0.990 0.024 0.066  4/4 Center 0.2492 ± 0.0334 0.640 ± 0.092 1.021 ± 0.042 4.260 × 10 - 2  ± 5.600 × 10 - 3 5.210 × 10 - 5  ± 2.520 × 10 - 5 0.988 0.028 0.056 Half-radius 0.2409 ± 0.0470 0.664 ± 0.090 1.031 ± 0.051 1.740 × 10 - 2 ± 2.701 × 10 - 3 4.820 × 10 - 5  ± 1.790 × 10 - 5 0.992 0.025 0.046 Periphery 0.2421 ± 0.0548 0.671 ± 0.092 1.023 ± 0.034 9.700 × 10 - 3 ± 1.500 × 10 - 3 5.681 × 10 - 5  ± 2.333 × 10 - 5 0.995 0.021 0.039It can be noticed from Table5 that the correlation coefficient is always more than 0.98 and remains higher than the obtained values for the other models in Table 4. Then, we can say that the model of Sikame Tagne [4] is the one which enables the modelling of the experimental data and is illustrated by Figure 6.Figure 6
Experimental points and model of Sikame Tagne [4] of the pith ofRaffia vinifera from the center of the base.It can be concluded that the mathematical equation which enables the description of the water absorption kinetics of the pith ofRaffia vinifera from the stem is given by (11)gt=c-a∗exp⁡-k∗t-b∗exp⁡-m∗t,where a, b, and c are constants and k and m are the diffusion parameters expressed per minutes.


### 3.4. Determination of the Effective Diffusion Coefficient

The determination of the water diffusion coefficient through the pith ofRaffia vinifera is done by using Fick’s dual stage law. It was used in the prediction and interpretation of mass transfer through natural fibers [4, 36]. This method is based on Fick’s law and introduces two effective diffusion coefficients D1eff and D2eff and two water saturation levels corresponding to the two diffusion coefficients. The use of this method on water absorption kinetics for the pith ofRaffia vinifera is done by (12) which described two phases of diffusion, that is, an initial and a final stage. The sum of the two saturation levels of the two phases gave the total absorption rate of the pith ofRaffia vinifera. The effective diffusion coefficient will be determined by using (12):(12)Mt=M∞11-∑0+∞82n+12π2exp⁡-D1eff2n+12π24e2t+M∞21-∑0+∞82n+12π2exp⁡-D2eff2n+12π24e2t,(13)M∞=M∞1+M∞2.The parameters of this equation are determined in the software environment Matlab 2009b by using experimental data of water absorption Mt depending on the time. D1eff and D2eff are the respective values of the diffusion coefficient, respectively, at the initial and final phases. M∞1 and M∞2 are the respective water saturation levels at the initial and final stages.Experimental data and the curve corresponding to the Fick dual stage formula are presented in Figure7. An observation of this figure shows that the dual stage saturation method that experimented with Fick’s law equation result is the better approach of experimental data.Figure 7
Curves of the water absorption rate and the two-phase diffusion model in part 3/4 in the half-radius zone.This work has been done on twelve sampling zones and permits us to obtain the distribution of the effective diffusion coefficient of water in the pith ofRaffia vinifera. Those distributions are presented in Figures 8 and 9.Figure 8
Distribution of the effective diffusion coefficient in the pith ofRaffia vinifera in twelve sampling zones during the initial stage.Figure 9
Distribution of the effective diffusion coefficient in the pith ofRaffia vinifera in twelve sampling zone during the final stage.Table6 provides the effective diffusion coefficient, the water absorption rate, and the correlation coefficient in the twelve sampling zones.Table 6
Effective diffusion coefficient of water of the pith ofRaffia vinifera during the initial and final stage.
Longitudinal position Radial position D 1 e f f  m 2 / s × 10 - 9 D 2 e f f  m 2 / s × 10 - 11 M ∞ R 2 Part 1/4 Center 8.746 14.00 1148.800 0.988 Half-radius 5.337 13.55 790.455 0.987 Periphery 4.808 16.16 367.719 0.985  Part 2/4 Center 2.415 10.48 1304.794 0.985 Half-radius 2.260 11.83 1163.629 0.986 Periphery 2.242 14.78 585.027 0.986  Part 3/4 Center 1.493 8.130 1477.300 0.990 Half-radius 1.163 8.234 1219.823 0.991 Periphery 1.966 7.792 557.978 0.991  Part 4/4 Center 1.063 8.438 1414.600 0.990 Half-radius 1.150 7.668 1149.307 0.989 Periphery 1.149 9.159 646.913 0.988The analysis of Table6 enables us to see that the correlation coefficient of the pith ofRaffia vinifera resulting from the modelling of the dual stage of the second law of Fick is higher than 0.98. This table also shows that the effective diffusion coefficient in the pith ofRaffia vinifera increase from 1.063×10-9 to 8.746×10-9m2/s in the initial stage and from 7.668×10-11 to 1.615×10-10m2/s in the second or final stage. Globally, the effective diffusion coefficient decreases from the periphery to the center in the radial position and from part 1/4 to part 4/4 in the longitudinal position in the first part or initial stage. In the second stage, the effective diffusion coefficient has the same trend such as the initial stage. The values of the effective diffusion coefficient obtained in this work are in the same greatness order of certain vegetal species such as the sisal fiber, the jute fiber, the flax fiber, the hemp fiber [36], and the raffia fiber [4].


## 3.1. Determination of the Percentage of Water Absorption

The percentage of water absorption of our samples for the twelve sampling zones is done by (8). The synthesis result is presented in Table 1.Table 1
Average rate of water absorption and standard deviation of theRaffia vinifera pith.
Center Half-radius Periphery Part 1/4 1141 . 509 ± 156.063 847 . 525 ± 95.568 392 . 044 ± 36.618 Part 2/4 1256 . 696 ± 87.415 1156 . 705 ± 79.810 565 . 967 ± 100.128 Part 3/4 1331 . 956 ± 49.144 1207 . 133 ± 107.053 612 . 198 ± 132.146 Part 4/4 1308 . 677 ± 53.766 1137 . 932 ± 61.979 705 . 018 ± 113.600It is observed that the average percentage of water absorption stands between392.044%±36.618% and 1331.956%±49.144% during the immersion period estimated at 45 days.A comparative study of the percentage of the water absorption of theRaffia vinifera pith and other vegetal products has been done, in Table 2. This study indicates that theRaffia vinifera pith absorbs more water than the fibers taken in the same position of the stem [4]. This is probably due to the fact that the samples of pith ofRaffia vinifera are the natural composites materials comprising fibers and a spongy part that work as a natural binder between the fibers and which in addition absorbs water.Table 2
Comparison of the percentage of water absorption of the pith ofRaffia vinifera with other natural materials.
Types Percentage of water absorbed (%) Absorption duration Temperature (°C) References Afra wood 102 25 Days 25 [16] Ojamlesh Wood 54 Roosi Wood 120  Hemp 62 13 Days 27–67 [17] Okra 64 Betel nut 38  Fibers ofRaffia vinifera 303–662 25 Days 23 [4]  Pith of the Raffia vinifera 392–1331 45 Days 30 ± 1 Studied caseThe average of the percentage of water absorption along the stem is presented on Figure3. Globally the observation of Figure 3 enables us to see that the percentage of water absorption increases from the periphery to the center on the radial position and from the base to the leaves on the longitudinal position along the stem ofRaffia vinifera.Figure 3
Distribution of the water absorption rate percentage of the pith ofRaffia vinifera.


## 3.2. Kinetics of Water Absorption

Equation (9) allows us to plot the curve of water absorption ratio according to the time and the obtained curve is presented by Figure 4. This curve presents the results of water absorption test of the pith ofRaffia vinifera taken from the center of part 4/4.Figure 4
Average of the water absorption kinetics of the pith ofRaffia vinifera from the center and at the base of the stem (1/4).The obtained curves on the twelve sampling zones have the same profile as that presented in Figure4.An observation of the curve presented in Figure4 shows that, during the first 5000 minutes, approximately 50% of the sample masses reached saturation level.The curve presented in Figure4 is similar to the curve obtained by other authors who have studied water absorption kinetics in natural materials [4, 16, 26, 27, 34, 37]. It was noticed that the water absorption kinetics of the pith ofRaffia vinifera takes place in two phases: the first phase rapidly takes place and enables reaching approximately 50% of the mass at saturation. This is done in the first 5000 minutes and the second phase that is relatively slow and is represented by the second part of the curve. This brings out two diffusion coefficients which indicate the velocity of the water diffusion in the pith ofRaffia vinifera samples.


## 3.3. Modelling of the Water Absorption Kinetics

The water absorption kinetics in the pith ofRaffia vinifera is done by using the mathematical models found in the literature. In this fact, we found eleven models which are in Table 3.Table 3
Utilized models summary for the modelling of the absorption kinetics.
Number Model’s name Model’s equation References 1 Pilosof g t = a + b ∗ t c + t [18] 2 Mohsenin g t = a ∗ 1 - exp ⁡ ( - k ∗ t ) + c + d ∗ t [16, 19] 3 Singh g t = a + b ∗ t c ∗ t + 1 [20] 4 Peleg g t = a + t b ∗ t + c [21, 22] 5 García-Pascual g t = 1 - exp ⁡ - t a b [23] 6 Gowen g t = a - b ∗ exp ⁡ - k ∗ t + b [24] 7 Czel and Czigany g t = a ∗ t m [25] 8 Vega-Gálvez g t = a ∗ exp ⁡ - b 1 + t α [26] 9 Gornicki g t = a ∗ b - exp ⁡ - k ∗ t [27] 10 Gornicki g t = a + b ∗ 1 - 1 1 + b ∗ c ∗ t [27] 11 Sikame Tagne g t = c - a ∗ exp ⁡ - k ∗ t - b ∗ exp ⁡ ( - m ∗ t ) [4]Table 4
Distribution of the average parameters models of the pith ofRaffia vinifera at the base of the stem (1/4).
Model’s name Radial position a b c k   ( m i n - 1 ) m   ( m i n - 1 ) R 2 RMSE SSE Peleg Center 0.239 ± 0.034 9312.150 ± 2044.966 1.176 ± 0.090   0.972 0.039 0.135 Half-radius 0.245 ± 0.033 9  256.250 ± 2  090.859 1.183 ± 0.109   0.971 0.040 0.143 Periphery 0.216 ± 0.030 4  763.550 ± 1  040.602 1.252 ± 0.071   0.966 0.047 0.194  Pilosof Center 0.230 ± 0.042 0.817 ± 0.045 6492.5 ± 549.047   0.968 0.042 0.152 Half-radius 0.234 ± 0.041 0.810 ± 0.049 6  315.350 ± 879.532   0.967 0.043 0.160 Periphery 0.215 ± 0.029 0.802 ± 0.044 3  762.600 ± 923.578   0.966 0.046 0.193  Singh Center 0.240 ± 0.033 0.157 ± 0.321 6861.700 ± 4310.801   0.973 0.054 0.135 Half-radius 0.245 ± 0.033 0.161 ± 0.331 6  806.700 ± 4  230.220   0.971 0.058 0.143 Periphery 0.217 ± 0.031 0.168 ± 0.344 2  885.300 ± 1  656.826   0.966 0.046 0.193  Witrowa Center 0.228 ± 0.057 0.858 ± 0.068 1.563 × 10 - 4 ± 5.405 × 10 - 5   0.974 0.039 0.139 Half-radius 0.245 ± 0.033 0.854 ± 0.078 1.652 × 10 - 4 ± 6.949 × 10 - 5   0.960 0.0432 0.101 Periphery 0.216 ± 0.030 0.801 ± 0.044 3.489 × 10 - 4 ± 1.023 × 10 - 4   0.966 0.047 0.194  Gowen Center 0.262 ± 0.036 0.972 ± 0.019  1.910 × 10 - 4 ± 2.567 × 10 - 5  0.963 0.046 0.184 Half-radius 0.268 ± 0.038 0.974 ± 0.022  1.076 × 10 - 4 ± 2.358 × 10 - 5  0.961 0.067 0.198 Periphery 0.251 ± 0.036 0.932 ± 0.022  1.856 × 10 - 4 ± 3.828 × 10 - 5  0.942 0.060 0.324  Wesolowski Center 0.710 ± 0.053 1.374 ± 0.080  1.045 × 10 - 4 ± 2.504 × 10 - 5  0.963 0.067 0.184 Half-radius 0.706 ± 0.058 1.386  ± 0.087  1.465 × 10 −4 ± 1.662 × 10 - 5  0.961 0.067 0.198 Periphery 0.681 ± 0.037 1.371 ± 0.070  1.827 × 10 −4 ± 6.106 × 10 - 5  0.942 0.060 0.323  Mohsenin Center 0.571 ± 0.110 3.025 × 10 - 6 ± 2.261 × 10 - 6 0.265 ± 0.080 1.629 × 10 −4 ± 5.107 × 10 - 5  0.963 0.043 0.164 Half-radius 0.568 ± 0.080 2.429 × 10 −5 ± 6.555 × 10 - 5 0.255 ± 0.036 1.638 × 10 −4 ± 5.019 × 10 - 5  0.967 0.044 0.175 Periphery 0.557 ± 0.046 3.894 × 10 −6 ± 1.356 × 10 - 6 0.222 ± 0.036 3.505 × 10 −4 ± 1.351 × 10 - 4  0.958 0.074 0.241  Weibull Center 6207.600 ± 2386.766 0.379 ± 0.042    0.939 0.058 0.293 Half-radius 5  938.200 ± 2  202.506 0.373 ± 0.038    0.938 0.058 0.293 Periphery 3  756.650 ± 933.131 0.388 ± 0.029    0.978 0.037 0.131  Czel and Czigany Center 0.085 ± 0.026    0.231 ± 0.028 0.973 0.039 0.135 Half-radius 0.089 ± 0.030    0.226 ± 0.028 0.978 0.057 0.129 Periphery 0.105 ± 0.023    0.214 ± 0.020 0.978 0.037 0.131  Vega-Gálvez Center 7.871 ± 2.119 5.471 ± 0.727    0.958 0.048 0.198 Half-radius 7.734 ± 2.295 5.402 ± 0.791    0.958 0.047 0.182 Periphery 6.899 ± 1.268 5.028 ± 0.493    0.986 0.029 0.085  Sikame Tagne Center 0.241 ± 0.043 0.686 ± 0.062 0.985 ± 0.079 3.662 × 10 −2 ± 2.291 × 10 - 2 8.682 × 10 −5 ± 1.918 × 10 - 5 0.992 0.022 0.041 Half-radius 0.252 ± 0.042 0.679 ± 0.075 1.001 ± 0.016 3.917 × 10 −2 ± 2.024 × 10 - 2 8.563 × 10 −5 ± 1.445 × 10 - 5 0.992 0.022 0.040 Periphery 0.298 ± 0.047 0.600 ± 0.051 0.991 ± 0.013 1.035 × 10 −2 ± 1.078 × 10 - 2 1.053 × 10 −4 ± 2.123 × 10 - 5 0.993 0.021 0.037The experimental data of the pith ofRaffia vinifera of the center from the base of the stem and the eleven models found in the literature are represented in the Figure 5.Figure 5
Experimental curve of the pith ofRaffia viniferaand curves of models.Table4 provides a summary of the parameters of the various models previously cited concerning samples of the pith coming from part 1/4 of theRaffia vinifera stem.The variations presented by some models such as those by Singh and Kulshrestha (1987) and Pilosof et al. (1987) can be interpreted by the fact that these models are not ideal to describe the kinetics of water absorption of the pithof Raffia vinifera.According to Table4, it can be seen that the models used in this work have the coefficients of correlation higher than 0.93. In the case of this work the model of Weibull has the lowest correlation coefficient. The better model is the model of Sikame Tagne et al. which has the correlation coefficient close to the unit. In this particular work, the correlation coefficient is 0.99 for the Sikame Tagne et al. model.The model of Sikame Tagne et al. (2014) is the one which has better statistical parameters for the sampling zones. This model is used for the modelling of the absorption kinetics in the twelve sampling zones of the stem ofRaffia vinifera. The average parameters of this model are provided in Table 5, the coefficient of linear correlation, the square root of the mean error, and the squared sum of errors of theRaffia vinifera stem.Table 5
Average parameters of the model of Sikame Tagne [4] on the twelve sampling zones of the stem of Raffia vinifera.
Longitudinal position Radial Position a b c k   m i n - 1 m   ( m i n - 1 ) R 2 RMSE SSE 1/4 Center 0.241 ± 0.043 0.688 ± 0.062 0.985 ± 0.079 3.662 × 10 - 2  ± 2.291 × 10 - 2 8.689 × 10 - 5  ± 1.918 × 10 - 5 0.992 0.022 0.042 Half-radius 0.252 ± 0.042 0.679 ± 0.075 1.002 ± 0.016 3.917 × 10 - 2 ± 2.024 × 10 - 2 8.563 × 10 - 5  ± 1.445 × 10 - 5 0.999 0.022 0.040 Periphery 0.298 ± 0.048 0.599 ± 0.051 0.9913 ± 0.013 1.035 × 10 - 2  ± 1.078 × 10 - 2 1.053 × 10 - 4  ± 2.123 × 10 - 5 0.993 0.021 0.037  2/4 Center 0.2003 ± 0.0219 0.721 ± 0.033 1.026 ± 0.031 1.120 × 10 - 2  ± 1.280 × 10 - 3 6.530 × 10 - 5      ± 1.550 × 10 - 5 0.992 0.022 0.039 Half-radius 0.2133 ± 0.0446 0.713 ± 0.049 1.019 ± 0.029 2.840 × 10 - 2  ± 3.221 × 10 - 3 7.390 × 10 - 5      ± 2.520 × 10 - 5 0.990 0.027 0.043 Periphery 0.2975 ± 0.0469 0.599 ± 0.051 0.991 ± 0.013 1.041 × 10 - 2  ± 1.080 × 10 - 3 1.050 × 10 - 4      ± 2.120 × 10 - 5 0.993 0.021 0.037  3/4 Center 0.1877 ± 0.0410 0.711 ± 0.055 1.029 ± 0.024 8.220 × 10 - 3  ± 1.520 × 10 - 3 5.571 × 10 - 5     ± 1.280 × 10 - 5 0.994 0.021 0.056 Half-radius 0.2009 ± 0.0627 0.694 ± 0.076 1.032 ± 0.017 1.370 × 10 - 2  ± 2.820 × 10 - 3 5.530 × 10 - 5      ± 1.130 × 10 - 5 0.993 0.022 0.055 Periphery 0.2319 ± 0.0267 0.64 ± 0.075 1.022 ± 0.011 1.410 × 10 - 2  ± 1.730 × 10 - 3 5.870 × 10 - 5      ± 1.040 × 10 - 5 0.990 0.024 0.066  4/4 Center 0.2492 ± 0.0334 0.640 ± 0.092 1.021 ± 0.042 4.260 × 10 - 2  ± 5.600 × 10 - 3 5.210 × 10 - 5  ± 2.520 × 10 - 5 0.988 0.028 0.056 Half-radius 0.2409 ± 0.0470 0.664 ± 0.090 1.031 ± 0.051 1.740 × 10 - 2 ± 2.701 × 10 - 3 4.820 × 10 - 5  ± 1.790 × 10 - 5 0.992 0.025 0.046 Periphery 0.2421 ± 0.0548 0.671 ± 0.092 1.023 ± 0.034 9.700 × 10 - 3 ± 1.500 × 10 - 3 5.681 × 10 - 5  ± 2.333 × 10 - 5 0.995 0.021 0.039It can be noticed from Table5 that the correlation coefficient is always more than 0.98 and remains higher than the obtained values for the other models in Table 4. Then, we can say that the model of Sikame Tagne [4] is the one which enables the modelling of the experimental data and is illustrated by Figure 6.Figure 6
Experimental points and model of Sikame Tagne [4] of the pith ofRaffia vinifera from the center of the base.It can be concluded that the mathematical equation which enables the description of the water absorption kinetics of the pith ofRaffia vinifera from the stem is given by (11)gt=c-a∗exp⁡-k∗t-b∗exp⁡-m∗t,where a, b, and c are constants and k and m are the diffusion parameters expressed per minutes.


## 3.4. Determination of the Effective Diffusion Coefficient

The determination of the water diffusion coefficient through the pith ofRaffia vinifera is done by using Fick’s dual stage law. It was used in the prediction and interpretation of mass transfer through natural fibers [4, 36]. This method is based on Fick’s law and introduces two effective diffusion coefficients D1eff and D2eff and two water saturation levels corresponding to the two diffusion coefficients. The use of this method on water absorption kinetics for the pith ofRaffia vinifera is done by (12) which described two phases of diffusion, that is, an initial and a final stage. The sum of the two saturation levels of the two phases gave the total absorption rate of the pith ofRaffia vinifera. The effective diffusion coefficient will be determined by using (12):(12)Mt=M∞11-∑0+∞82n+12π2exp⁡-D1eff2n+12π24e2t+M∞21-∑0+∞82n+12π2exp⁡-D2eff2n+12π24e2t,(13)M∞=M∞1+M∞2.The parameters of this equation are determined in the software environment Matlab 2009b by using experimental data of water absorption Mt depending on the time. D1eff and D2eff are the respective values of the diffusion coefficient, respectively, at the initial and final phases. M∞1 and M∞2 are the respective water saturation levels at the initial and final stages.Experimental data and the curve corresponding to the Fick dual stage formula are presented in Figure7. An observation of this figure shows that the dual stage saturation method that experimented with Fick’s law equation result is the better approach of experimental data.Figure 7
Curves of the water absorption rate and the two-phase diffusion model in part 3/4 in the half-radius zone.This work has been done on twelve sampling zones and permits us to obtain the distribution of the effective diffusion coefficient of water in the pith ofRaffia vinifera. Those distributions are presented in Figures 8 and 9.Figure 8
Distribution of the effective diffusion coefficient in the pith ofRaffia vinifera in twelve sampling zones during the initial stage.Figure 9
Distribution of the effective diffusion coefficient in the pith ofRaffia vinifera in twelve sampling zone during the final stage.Table6 provides the effective diffusion coefficient, the water absorption rate, and the correlation coefficient in the twelve sampling zones.Table 6
Effective diffusion coefficient of water of the pith ofRaffia vinifera during the initial and final stage.
Longitudinal position Radial position D 1 e f f  m 2 / s × 10 - 9 D 2 e f f  m 2 / s × 10 - 11 M ∞ R 2 Part 1/4 Center 8.746 14.00 1148.800 0.988 Half-radius 5.337 13.55 790.455 0.987 Periphery 4.808 16.16 367.719 0.985  Part 2/4 Center 2.415 10.48 1304.794 0.985 Half-radius 2.260 11.83 1163.629 0.986 Periphery 2.242 14.78 585.027 0.986  Part 3/4 Center 1.493 8.130 1477.300 0.990 Half-radius 1.163 8.234 1219.823 0.991 Periphery 1.966 7.792 557.978 0.991  Part 4/4 Center 1.063 8.438 1414.600 0.990 Half-radius 1.150 7.668 1149.307 0.989 Periphery 1.149 9.159 646.913 0.988The analysis of Table6 enables us to see that the correlation coefficient of the pith ofRaffia vinifera resulting from the modelling of the dual stage of the second law of Fick is higher than 0.98. This table also shows that the effective diffusion coefficient in the pith ofRaffia vinifera increase from 1.063×10-9 to 8.746×10-9m2/s in the initial stage and from 7.668×10-11 to 1.615×10-10m2/s in the second or final stage. Globally, the effective diffusion coefficient decreases from the periphery to the center in the radial position and from part 1/4 to part 4/4 in the longitudinal position in the first part or initial stage. In the second stage, the effective diffusion coefficient has the same trend such as the initial stage. The values of the effective diffusion coefficient obtained in this work are in the same greatness order of certain vegetal species such as the sisal fiber, the jute fiber, the flax fiber, the hemp fiber [36], and the raffia fiber [4].


## 4. Conclusion

This work whose objective is the study of water absorption phenomenon through the pith ofRaffia vinifera consisted of the determination of the rate of water absorption. The modelling of the water absorption kinetics and the determination of the effective diffusion coefficient in the twelve sampling zones of the pith ofRaffia vinifera were also the objective of this work. The procedure consisted of the immersion of the sample in the distilled water at the constant and room temperature of30°C±1°C and weighing the samples in the regular time intervals. This water absorption rate has values between 392.044%±36.618% and 1331.956%±49.144%. We have noticed that the percentage of water absorption of the pith ofRaffia vinifera increases from the periphery to the center in the radial position and from the base (part 1/4) to the leaves (part 4/4) in the longitudinal position. The modelling of the water absorption kinetics was done by eleven models and the model of Sikame Tagne et al. (2014) gives better statistical parameters with the experimental data. The determination of the diffusion coefficient is done by Fick’s second law in the dual stage. The values of the diffusion coefficient increase from 1.063×10-9 to 8.746×10-9m2/s in the initial stage and from 7.668×10-11 to 1.615×10-10m2/s. Globally, the diffusion coefficient decreases from the periphery to the center in the radial position and from part 1/4 to part 4/4 in the longitudinal position. This is done in the initial stage and final stage.



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*Source: 1953087-2017-02-13.xml*