# Macroscopic Geometrical Modelling of Oil Palm Mesocarp Fibers of Three Varieties of Palm Nut

E. Njeugna, P.W.M. Huisken, D. Ndapeu, N.R.T. Sikame, J. Y. Dréan

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E. Njeugna, P.W.M. Huisken, D. Ndapeu, N.R.T. Sikame, J. Y. Dréan. Macroscopic Geometrical Modelling of Oil Palm Mesocarp Fibers of Three Varieties of Palm Nut. Mechanics, Materials Science & Engineering Journal, 2016, 10.13140/RG.2.1.2275.9921. [hal-01367494](https://hal.science/hal-01367494)

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# Macroscopic Geometrical Modelling of Oil Palm Mesocarp Fibers of Three Varieties of Palm Nut

E. Njeugna¹, ², P. W. M. Huisken¹, ², a, D. Ndapeu², N. R. T. Sikame², J. Y. Dréan³

1. Laboratory of Mechanics and Production (LMP), Douala, Cameroon;

2. Laboratory of Mechanics and Appropriate Materials (LAMMA), Douala, Cameroon;

3. Laboratory of Physics and Mechanics Textile (LPMT), Mulhouse, France;

a. huiskenwilliam@gmail.com

<figure><img src="image_2.png" /></figure>

DOI 10.13140/RG.2.1.2275.9921

**Keywords:** Oil Palm Mesocarp fibers, modeling, geometry, cross section, profile.

**ABSTRACT.** This work is part of a process of characterization of plant fibers. The macroscopic geometric parameter of Oil Palm Mesocarp Fibers (length, diameter) was measured. A mathematical model of the evolution of the cross-section is provided in order to facilitate a digital reconstruction of the geometry of these fibers. In this context, we manually isolated with great care many fibers of several oil palm varieties *Dura*, *Tenera* and *Pissifera* distinguishing for the last two varieties two extraction position. Five different partitions of fibers have been studied. The lengths of these fibers were measured and the transverse dimensions of each of the fibers were taken at five equally spaced discrete and different sections. For each section, we made two measurements at 90 ° in the front plane and the profile view. The mathematical model of the evolution of the profile were determined in each plan and the evolution of the cross section model was described for each of the five partitions on the basic assumption that this cross section is elliptical according to SEM images and flattening rate of the cross section we calculated.

**Introduction.** The use of vegetable fibers as reinforcement in composite materials to replace synthetic fibers is growing rapidly. Many researches are interested in it in order to optimize their physical and mechanical properties [1-5]. Many fibers (sisal, coir, raffia, jute ...) are studied in the literature [6]. The fibers from palm oil tree (EFBF<fnref n="4" />, OPMF<fnref n="5" />, OPTF<fnref n="6" />) also have a particular interest [2, 3, 5, 7, 8].

It is clear from the various work that knowing the geometry of the fiber (section and length) is indispensable for the analysis of the results of a physicochemical characterization test (water absorption kinetic of fiber) [9], mechanical characterization (determination of Young's modulus of the fiber) [7, 10, 11] and during predictive calculations of composite properties [12, 13] (application of composites homogenization theories). All Authors are unanimous on the versatility of the geometry of the cross section and its non-uniformity along the vegetable fibers. It is shown [14-18] that the mechanical properties of sisal, jute, hemp, bamboo and coconut fiber may be influenced by the dimensions and aspect of their cross-section. Most of the works in the literature consider an approximation of vegetable fibers to a straight beam of constant circular cross section [6, 19] (in this case, the diameter measurement is made on a singular section); on the other hand, consider that the fiber is a straight beam of elliptic section and constant (measuring the major axis and the minor axis of a singular cross section) on which several measurements can be made and the average of these measurements considered as mean cross section of the fiber. Some author [11] propose a method for assessing the cross section of kenaf fibers by image analysis but this study is limited to a single section

<footnote n="4">⁴ Empty Fruit Bunch Fibers</footnote>

<footnote n="5">⁵ Oil Palm Mesocarp Fibers</footnote>

<footnote n="6">⁶ Oil Palm Trunk Fibers</footnote>
by considering it to be constant. Other authors [20] propose a characterization approach considering
the variability of the cross section along the hemp fibers and its non-circularity.

These various techniques have some inaccuracies including:

(i) Some are an average of local measures that do not take into consideration all the variations and features of the fiber profile;

(ii) Others exclude non circularity of the cross section of plant fibers.

In this context, the physicochemical and mechanical properties such as Young's modulus, tensile stress limits, the kinetics of water absorption and drying kinetics will be determined with some imprecision introducing a measurement uncertainty.

In the particular case of Oil Palm Mesocarp Fiber (OPMF), very few works are devoted to their study.
Okafor and Owolarafe et al [12, 21, 22] indicate that there are three main types of palm fruit (Tenera,
Pissifera and Dura) depending on the size of the shell and the oil content. They consider as [23-24]
that the cross section of OPMF is circular and constant along the fiber.

In this study, we want to make a contribution on the evaluation of the shape of the cross section of fiber from the mesocarp of the three varieties of palm nuts and its evolution along the mean line. We will develop a method for measuring the cross section and detecting the profile of the OPMF. We will determine with precision the real geometry of the profile of OPMF and propose a mathematical model of evolution of the cross section of fiber from the three varieties of palm nuts extracted at different positions.

**Materials and Methods.**

**Materials. Oil Palm Mesocarp Fiber (OPMF).** The fibers (OPMF) objects of study are from a palm tree farm in Nkongsamba, a city of Mungo division in Littoral region in Cameroon.

We worked on the three varieties of palm nuts as described in the literature [12, 21, 22] and illustrated
in Figure 1: (Pissifera, Tenera and Dura).

- *Dura* (wild variety having a thick shell);

- *Tenera* (hybrid variety having a small shell);

- *Pissifera* (variety without shell).

<figure><img src="image_3.png" /><figcaption>Fig. 1. Different types of palm fruit: (a) Dura, (b) Tenera, (c) Pissifera [22].</figcaption></figure>

These fibers were extracted manually after having been placed dry nut in pure water for 20 days. They were later rinsed thoroughly in water heated at 60 °C. They were then dried in open air and stored in plastic bags. To check the influence of the position of the fiber on the morphology, we had each nut partitioned into two extraction zones for *Tenera* and *Pissifera* varieties: the peripheral fibers (external position) and those near the shell (internal position) as shown in Figure 2. For the variety *Dura*, the
size of the shell did not allow us to distinguish the two areas. So we distinguished 5 different partitions of fibers. We worked with 54 fibers per partition thus a total of 270 fibers.

<figure><img src="image_4.png" /><figcaption>Fig. 2. Schematic illustration of internal and peripheral fibers of palm fruit.</figcaption></figure>

**Collecting images and measurement equipment.** We used a Celestron LCD Digital Microscope with the lens corresponding to a magnification of 40x to collect images. The measurement of each section was then carried out by using a COOLING Tech USB 8LEDS microscopic camera previously calibrated on its operating software. The lengths of fibers were measured using a digital slide caliper having 0.01mm of accuracy. We processed the data collected using Matlab R2009b.

**Methods.**

**Data collection techniques.** We measured the transverse dimensions of the fibers on the five sections of the face (a₁, a₂, a₃, a₄ and a₅) and profile (b₁, b₂, b₃, b₄ and b₅) along the fiber as illustrated in Figure 3 below. We were inspired by the technique stated by Ilczyszyn et al. [20]. Section 1 is positioned at a distance of 3 to 5 mm from the fiber tip and section 5 is defined by the beginning of dislocation of the fiber.

<figure><img src="image_5.png" /><figcaption>Fig. 3. Illustration of the measuring points of transverse dimensions.</figcaption></figure>
The measurement process is performed in several steps. The first step is to immobilize the fibers and marking measuring points. Six fibers are positioned on the front plan, strained and then bonded by adhesive tape on wooden plates of rectangular shape, dimension 800x300 mm covered by graph paper. On each fiber we placed five equidistant points on the working length of the fiber.

<figure><img src="image_6.png" /><figcaption>Fig. 4. Schematic illustration of the fixation of fibers.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. Photo of a bonded fiber divided into five sections.</figcaption></figure>

The second step consists of images taken from the vicinity of each section labeled with the LCD Digital Microscope Celestron. A light source is used to illuminate the fibers to have a clear image on the microscope screen. A magnification of 40x is used to film the sections of fibers.

<figure><img src="image_8.png" /><figcaption>Fig. 6. Diagram showing the principle (a) and image editing and filming equipment for the fibers (b).</figcaption></figure>

After all marked sections are filmed, the images are stored in Microscope memory and then transferred and stored in a computer. The measurements of each section are made from the Cooling Tech driven software. At this stage we obtain the values of **a1, a2, a3, a4** and **a5** with 0.001mm accuracy.
<figure><img src="image_9.png" /><figcaption>Fig. 7. Microscopic camera Cooling USB 8LEDS (a) and illustration of the measured image.</figcaption></figure>

The fibers are then carefully peeled off and turned over 90 ° to the position on the profile plane. The previous steps are repeated to permit measurement of the second transverse dimension on the same sections. The values of **b1**, **b2**, **b3**, **b4** and **b5** with 0.001mm accuracy were thus obtained. At the end, the length of each fiber is measured by a digital caliper. This is the distance between the two extreme points (point 1 and 5).

<figure><img src="image_10.png" /><figcaption>Fig. 8. Illustration of measurement according to the front plan ($a_i$) and the profile plane ($b_i$).</figcaption></figure>

**Theoretical considerations.** We have been working under the assumption that the cross section is elliptical (it admits two planes of symmetry: the front plan and the profile plane) with regard to the observations made in the scanning electron microscope, an example is shown in Figure 9. The shape and proportions of the lumen have been neglected in this work. This hypothesis has also been stated by Ilczyszyn et al. [20] in the context of the characterization of hemp fiber.

<figure><img src="image_11.png" /><figcaption>Fig. 9. SEM image of the cross section of OPMF (LPMT Mulhouse).</figcaption></figure>

Under these conditions, the calculation of the cross section is done by equation (1) [21].
$$S = \frac{\pi}{4} a \cdot b \qquad (1)$$

With S the fiber section in mm², a and b transverse dimensions in mm (major and minor axis of the ellipse).

Considering that the cross section is variable along the fiber, we set that:

$$s(x) = \frac{\pi}{4} a(x) \cdot b(x) \qquad (2)$$

$$\text{With } a(x) = f(x) \text{ and } b(x) = g(x) \qquad (3)$$

We are thus supposed to search for the respective functions f(x) and g(x) that best fit the discrete experimental data aᵢ and bᵢ respectively.

Moreover we will focus on the statistical distribution of aᵢ and bᵢ to get an idea about the law of distribution of these experimental data.

For this purpose we will focus on the determination of the mean μ, standard deviation σ, and shape parameters of the distribution [25] that are the skewness (asymmetry coefficient) γ₁ and the kurtosis (flattening coefficient) γ₂ of aᵢ and bᵢ.

The standard deviation σ is calculated using equation (4).

$$\sigma^2 = \frac{1}{N} \sum (X_i - \bar{X})^2 \qquad (4)$$

The skewness coefficient γ₁ is calculated by relation (5).

$$\gamma_1 = \mathbb{E} \left[ \left( \frac{x - \mu}{\sigma} \right)^3 \right] \qquad (5)$$

The flattening γ₂ also called normalized Kurtosis coefficient is calculated by equation (6).

$$\gamma_2 = \mathbb{E} \left[ \left( \frac{x - \mu}{\sigma} \right)^4 \right] - 3 \qquad (6)$$

For this, recall that the probability density function of the normal distribution or Gauss-Laplace law is given by equation (7).

$$q(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} \qquad (7)$$

where μ – is the mean and σ² the variance.

## Results and Discussion.
Statistical Distribution of Geometric Parameters of OPMF.

**Determination of statistical parameters.** Considering the number of surveys, we conducted a statistical study to investigate the distribution of measurements around the mean values. For each sample we determined statistical distribution parameters that are the mean value and standard deviation. To find out whether the trend is centered or decentered (left or right of the mean), we calculated the skewness. We also calculated the kurtosis to deduce the spreading of data to conclude whether the distribution [25] is normal (mesokurtic) or not (leptokurtic or platikurtic). Table 1 summarizes these parameters for each variety of fibers.

<table><caption>Table 1. Summary table of statistical distribution parameters</caption><thead><tr><th rowspan="2"></th><th rowspan="2">Statistical Parameters</th><th colspan="5">Front Plan</th><th colspan="5">profile Plan</th></tr><tr><th>a1</th><th>a2</th><th>a3</th><th>a4</th><th>a5</th><th>b1</th><th>b2</th><th>b3</th><th>b4</th><th>b5</th></tr></thead><tbody><tr><td rowspan="4">μ σ γ₁ γ₂</td><td>OPMF-Pe<sup>7</sup></td><td>0,154 0,158 0,173 0,198 0,232</td><td>0,035 0,028 0,039 0,047 0,052</td><td>1,167 0,589 0,696 0,237 -0,063</td><td>1,559 0,466 -0,074 -0,875 -1,036</td><td>0,142 0,151 0,164 0,186 0,218</td><td>0,046 0,038 0,032 0,037 0,041</td><td>0,797 1,407 0,449 0,115 -0,521</td><td>0,080 1,596 0,056 -0,664 -0,508</td><td></td></tr><tr><td>OPMF-Pi<sup>8</sup></td><td>0,171 0,183 0,206 0,237 0,288</td><td>0,031 0,031 0,033 0,038 0,044</td><td>0,100 0,588 0,362 0,022 -0,187</td><td>-0,776 0,148 -0,544 -0,391 -0,708</td><td>0,170 0,177 0,197 0,220 0,270</td><td>0,035 0,029 0,031 0,031 0,035</td><td>0,087 0,509 0,421 0,105 -0,271</td><td>-0,979 0,132 0,713 -0,148 0,266</td><td></td></tr><tr><td>OPMF-Te<sup>9</sup></td><td>0,193 0,201 0,221 0,247 0,297</td><td>0,030 0,035 0,037 0,039 0,051</td><td>0,248 0,300 0,279 0,150 0,235</td><td>0,126 -0,664 -0,810 -0,501 -0,338</td><td>0,186 0,195 0,209 0,232 0,274</td><td>0,035 0,034 0,035 0,038 0,043</td><td>-0,045 -0,245 0,094 0,291 0,599</td><td>-0,565 -0,614 -0,418 0,016 2,297</td><td></td></tr><tr><td>OPMF-μ</td><td>0,231 0,245 0,263 0,296 0,343</td><td>0,043 0,045 0,048 0,059 0,054</td><td>0,044 0,460 0,926 0,590 0,331</td><td>-0,766 0,164 1,388 -0,126 -0,300</td><td>0,216 0,230 0,251 0,276 0,314</td><td>0,045 0,042 0,050 0,060 0,057</td><td>-0,275 0,430 0,804 0,348 0,122</td><td>-0,960 0,323 0,497 -0,478 -0,510</td><td></td></tr><tr><td rowspan="4">μ σ γ₁ γ₂</td><td>OPMF-Pe<sup>7</sup></td><td>0,208 0,218 0,247 0,275 0,312</td><td>0,047 0,049 0,050 0,049 0,051</td><td>0,448 0,359 0,542 0,240 0,042</td><td>-0,527 -0,595 -0,122 -0,478 -0,431</td><td>0,202 0,215 0,234 0,260 0,306</td><td>0,042 0,051 0,054 0,047 0,048</td><td>0,289 0,491 0,553 0,291 -0,326</td><td>-0,350 -0,330 0,092 -0,028 0,118</td><td></td></tr><tr><td>OPMF-Pi<sup>8</sup></td><td>0,047 0,049 0,050 0,049 0,051</td><td>0,448 0,359 0,542 0,240 0,042</td><td>0,448 0,359 0,542 0,240 0,042</td><td>-0,527 -0,595 -0,122 -0,478 -0,431</td><td>0,042 0,051 0,054 0,047 0,048</td><td>0,042 0,051 0,054 0,047 0,048</td><td>0,289 0,491 0,553 0,291 -0,326</td><td>-0,350 -0,330 0,092 -0,028 0,118</td><td></td></tr><tr><td>OPMF-Te<sup>9</sup></td><td>0,208 0,218 0,247 0,275 0,312</td><td>0,047 0,049 0,050 0,049 0,051</td><td>0,448 0,359 0,542 0,240 0,042</td><td>-0,527 -0,595 -0,122 -0,478 -0,431</td><td>0,202 0,215 0,234 0,260 0,306</td><td>0,042 0,051 0,054 0,047 0,048</td><td>0,289 0,491 0,553 0,291 -0,326</td><td>-0,350 -0,330 0,092 -0,028 0,118</td><td></td></tr><tr><td>OPMF-μ</td><td>0,208 0,218 0,247 0,275 0,312</td><td>0,047 0,049 0,050 0,049 0,051</td><td>0,448 0,359 0,542 0,240 0,042</td><td>-0,527 -0,595 -0,122 -0,478 -0,431</td><td>0,202 0,215 0,234 0,260 0,306</td><td>0,042 0,051 0,054 0,047 0,048</td><td>0,289 0,491 0,553 0,291 -0,326</td><td>-0,350 -0,330 0,092 -0,028 0,118</td><td></td></tr></tbody></table>

The mean values and standard deviations in Table 1 show the existing disparity between the measures when going from one fiber to another partition. This demonstrates the interest of modeling distinguishing several different partitions (each partition is unique in its kind).

The asymmetry coefficient values oscillate around zero; we can say that the distribution is centered in all partitions.

<footnote n="7">⁷ OPMF-Pe: Oil Palm Mesocarp Fiber of the Peripheral *Pissifera*</footnote>

<footnote n="8">⁸ OPMF-Pi: Oil Palm Mesocarp Fiber of the Internal *Pissifera*</footnote>

<footnote n="9">⁹ OPMF-Te: Oil Palm Mesocarp Fiber of the Peripheral *Tenera*</footnote>

<footnote n="10">¹⁰ OPMF-Ti: Oil Palm Mesocarp Fiber of the Internal *Tenera*</footnote>

<footnote n="11">¹¹ OPMF-Du: Oil Palm Mesocarp Fiber of *Dura*</footnote>
Kurtosis also oscillates in almost all cases around zero; under these conditions, it can be said that the distribution of the geometric parameters of OPMF follows a normal distribution. So this distribution is mésokurtic.

**Distribution according to the Normal Law or Gauss-Laplace Law.** Considering the foregoing conclusions about the nature of the distribution, we have built a density probability curve of Normal distribution for transverse dimensions of each partition of fiber defined by equation (7).

We determined the number of class using equation (8):

$$n = 1 + 3.3\text{Log } N \quad (8)$$

where n – is the number of classes;

N – the number of samples.

Having worked with 54 fibers per partition, equation (8) allows us to get 7 classes for each partition.

For each partition, we plotted the Gaussian curve using the equation (7). Figure 10 is an example of a Gaussian curve to the transverse dimension of the internal a₃ *Pissifera* variety. The shape of this curve is generalized for other partitions.

<figure><img src="image_12.png" /><figcaption>Fig. 10. Gauss distribution curve of transverse dimensions a3 (section 3) of the internal variety *Pissifera*.</figcaption></figure>

**Distribution parameters of front plan.** Here we study the evolution of the transverse dimensions on the front plan. The distribution of Figure 11 shows how these data are changing along the fiber in each extraction zone.
<figure><img src="image_13.png" /><figcaption>Fig. 11. Distribution of dimensions of cross-sections of OPMF in front plan.</figcaption></figure>

From the distribution of figure 11, it appears that the external fibers have smaller transverse dimensions than the internal fibers whatever the variety. This distribution on the front plan shows that the variety *Dura* is intermediate between *Pissifera* variety that has the lowest transverse dimensions and *Tenera* variety that has the largest dimensions. It is also noted that for all varieties, the profile evolves increasingly from position 1 to position 5 (from the tip of the fiber to the root) as one might have imagined by visual inspection.

**Distribution parameters of profile plan.** We also conducted the study of the evolution of the transverse dimensions of the profile plan. The distribution of Figure 12 shows the distribution of these parameters.

<figure><img src="image_14.png" /><figcaption>Fig. 12. Distribution of dimensions of cross-sections of OPMF in profile plan.</figcaption></figure>

This distribution reinforces the thesis that the peripheral fibers have lower transverse dimensions than the internal fibers whatever the variety. In the profile plan too, the *Dura* range is intermediate between *Pissifera* variety that has the lowest transverse dimensions and *Tenera* variety that has the largest dimensions. Just as on the front plan, profile dimensions here are evolving increasingly from the head of the fiber (section 1) to the base (section 5).

**Aspect Ratio (Specific length) of the fibers.** The specific length is an important parameter in the knowledge of plant fibers. From This parameter it can be concluded if the fiber is short, long or belongs to the family of particles. It describes the ratio of the length to the diameter by equation (9).
$$ Specific\ length = l = \frac{L}{d} \qquad (9) $$

where L – the length of the fiber in mm;

d – is the diameter of the fiber in mm.

Given the fact that the OPMF section varies along the fiber and is non-circular and taking into consideration that:

$$ a_i < a_{i+1} ; b_i < b_{i+1} \text{ and } a_i > b_i \text{ with } i \in [1; 5], $$

We can write the equations (10) and (11) below:

$$ l_{min} = \frac{L}{b_5} \qquad (10) $$

$$ l_{max} = \frac{L}{a_1} \qquad (11) $$

We summarized in Table 2 the average values of aspect ratio for each partition of fibers.

<table><caption>Table 2. Table of average specific lengths (aspect ratio)</caption><thead><tr><th>Type of Fiber</th><th>a<sub>1m</sub><br>(mm)</th><th>b<sub>5m</sub><br>(mm)</th><th>L<sub>m</sub><br>(mm)</th><th>l<sub>min</sub></th><th>l<sub>max</sub></th></tr></thead><tbody><tr><td>OPMF-Pe</td><td>0,147</td><td>0,218</td><td>29,12</td><td><strong>133,578</strong></td><td><strong>198,095</strong></td></tr><tr><td>OPMF-Pi</td><td>0,171</td><td>0,302</td><td>29,23</td><td><strong>96,788</strong></td><td><strong>170,936</strong></td></tr><tr><td>OPMF-Te</td><td>0,194</td><td>0,242</td><td>24,65</td><td><strong>101,859</strong></td><td><strong>127,062</strong></td></tr><tr><td>OPMF-Ti</td><td>0,277</td><td>0,341</td><td>26,99</td><td><strong>79,149</strong></td><td><strong>97,437</strong></td></tr><tr><td>OPMF-Du</td><td>0,212</td><td>0,306</td><td>18,11</td><td><strong>59,183</strong></td><td><strong>85,424</strong></td></tr></tbody></table>

According to the classification proposed in the literature, it is concluded in relation to the values in the table 2 that OPMF belong to the class of short fibers (50<L/D<1000). We also note that whatever the variety, internal fibers are far shorter than peripheral fibers. *Tenera* fibers are intermediate between *Dura* (the shortest) and *Pissifera* (the longer).

**Flattening rate of the fibers.** The flattening rate also called circularity rate is an important parameter in describing the shape of the cross section of a fiber or particle. It is the relationship between two diameters measured along the same section as described by equation (12).

$$ \tau = \frac{d}{D} \qquad (12) $$

With τ with the flattening rate, d and D are respectively the small and large diameter value of the same cross section in mm.
As part of our work, we calculate the average flattening rate of each cross section by equation (13).

$$ \tau_{im} = \frac{b_{im}}{a_{im}} \qquad (13) $$

With $\tau_{im}$, $b_{im}$ and $a_{im}$ respectively the average values of the flattening rate, the small diameter and large diameter section number $i$.

We present in Table 3 the mean values of the flattening rate of five sections on which we performed the measurements for each partition of fibers.

<table><caption>Table 3. Table of flattening rates per section</caption><thead><tr><th>N° of section</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr><tr><th>τ/section</th><th>τ<sub>1</sub></th><th>τ<sub>2</sub></th><th>τ<sub>3</sub></th><th>τ<sub>4</sub></th><th>τ<sub>5</sub></th></tr></thead><tbody><tr><th>OPMF-Pe</th><td>0,922</td><td>0,955</td><td>0,947</td><td>0,939</td><td>0,939</td></tr><tr><th>OPMF-Pi</th><td>0,994</td><td>0,967</td><td>0,956</td><td>0,928</td><td>0,937</td></tr><tr><th>OPMF-Te</th><td>0,963</td><td>0,970</td><td>0,945</td><td>0,939</td><td>0,922</td></tr><tr><th>OPMF-Ti</th><td>0,935</td><td>0,938</td><td>0,954</td><td>0,932</td><td>0,915</td></tr><tr><th>OPMF-Du</th><td>0,966</td><td>0,982</td><td>0,943</td><td>0,938</td><td>0,936</td></tr><tr><th>min value</th><td><strong>0,922</strong></td><td><strong>0,938</strong></td><td><strong>0,943</strong></td><td><strong>0,928</strong></td><td><strong>0,915</strong></td></tr><tr><th>max value</th><td><strong>0,994</strong></td><td><strong>0,982</strong></td><td><strong>0,956</strong></td><td><strong>0,939</strong></td><td><strong>0,939</strong></td></tr><tr><th>Standard Deviation</th><td><strong>0,022</strong></td><td><strong>0,012</strong></td><td><strong>0,004</strong></td><td><strong>0,004</strong></td><td><strong>0,009</strong></td></tr></tbody></table>

In view of the values obtained, we can say that the OPMF section is not circular ($\tau_i \neq 1 \forall i$). This section, however, may be likened to an ellipse.

## Geometric Modeling.

From the experimental data we have collected, we plotted curves materializing the profile of each of the fibers (Figure 13). We have subsequently shown in a graph the overall trend of the average profile of each partition of the fibers on the front plan (a) and the profile plane (b).

We note that OPMF have a wider front plan than the profile plan. Moreover, the two plans seem to evolve in the same manner along the fiber. Generally, the internal fibers have a larger section than those of the periphery.

Given the absence of data on the geometric model of this type of fiber, we performed a test of convergence with several mathematical functions. We got the best correlations (in the field of definition of fiber lengths) for the second-order polynomial model and the exponential model with two elements.

**Modeling with a second order polynomial function.** Figure 14 below shows the case of modeling with a second order polynomial function to the fibers of the periphery of *Tenera* partition (*Te*) on the front plan (a) and side view (b).
<figure><img src="image_15.png" /><figcaption>Fig. 13. Geometry of front (a) and side view (b) of OPMF.</figcaption></figure>

Under these conditions, we considered:

$$a(x) = f(x) = \alpha_{0}x^{2} + \alpha_{1}x + \alpha_{2}, \qquad (14)$$

$$b(x) = g(x) = \beta_{0}x^{2} + \beta_{1}x + \beta_{2} \qquad (15)$$

where $\alpha_0, \alpha_1, \alpha_2, \beta_0, \beta_1$ and $\beta_2$ – are constants.

Table 4 summarizes the parameters of the polynomial model for the different partitions of the fibers on the front plan (a) and the profile view (b).

<figure><img src="image_16.png" /><figcaption>Fig. 14. Geometric modeling by the polynomial function of the mean fibers of the periphery of Tenera partition.</figcaption></figure>
<table><caption>Table 4. Parameters of the polynomial mathematical model of the geometry of OPMF</caption><thead><tr><th rowspan="2">Type of fibers</th><th colspan="3">Model coefficients</th><th rowspan="2">R<sup>2</sup></th><th rowspan="2">RSME</th></tr><tr><th>α<sub>0</sub></th><th>α<sub>1</sub></th><th>α<sub>2</sub></th></tr></thead><tbody><tr><td rowspan="5">Front Plan</td><td>OPMF-Pe-a</td><td>9,43 e-5</td><td>-5,49 e-5</td><td>0,1540</td><td>0,9999</td><td>4,47 e-4</td></tr><tr><td>OPMF-Pi-a</td><td>1,15 e-4</td><td>5,79 e-4</td><td>0,1717</td><td>0,9987</td><td>2,38 e-3</td></tr><tr><td>OPMF-Te-a</td><td>1,69 e-4</td><td>-5,10 e-5</td><td>0,1939</td><td>0,9967</td><td>3,36 e-3</td></tr><tr><td>OPMF-Ti-a</td><td>1,27 e-4</td><td>6,46 e-4</td><td>0,2322</td><td>0,9985</td><td>2,45 e-3</td></tr><tr><td>OPMF-Du-a</td><td>1,85 e-4</td><td>2,51 e-3</td><td>0,2066</td><td>0,9968</td><td>3,38 e-3</td></tr><tr><td rowspan="5">Profile view</td><td>OPMF-Pe-b</td><td>7,41 e-5</td><td>4,10 e-4</td><td>0,1427</td><td>0,9989</td><td>1,38 e-3</td></tr><tr><td>OPMF-Pi-b</td><td>1,19 e-4</td><td>-1,54 e-4</td><td>0,1709</td><td>0,9944</td><td>4,25 e-3</td></tr><tr><td>OPMF-Te-b</td><td>1,41 e-4</td><td>-2,09 e-5</td><td>0,1873</td><td>0,9959</td><td>3,17 e-3</td></tr><tr><td>OPMF-Ti-b</td><td>8,16 e-5</td><td>1,39 e-3</td><td>0,2164</td><td>0,9991</td><td>1,68 e-3</td></tr><tr><td>OPMF-Du-b</td><td>2,54 e-4</td><td>9,81 e-4</td><td>0,2032</td><td>0,9968</td><td>3,29 e-3</td></tr></tbody></table>

**Modeling by an exponential function with two elements.** Figure 15 below shows the case of the modeling by an exponential function with two elements for the fibers of the periphery of the partition *Tenera* (Te) on the front plan (a) and side view (b).

<figure><img src="image_17.png" /><figcaption>Fig. 15. Geometric modeling by the exponential function with two elements of the average Tenera peripheric fibers.</figcaption></figure>

Under these conditions we have,

$$a(x) = f(x) = \alpha'_{0}e^{\gamma_{0}x} + \alpha'_{1}e^{\gamma_{1}x} \qquad (16)$$

$$b(x) = g(x) = \beta'_{0}e^{\delta_{0}x} + \beta'_{1}e^{\delta_{1}x} \qquad (17)$$
where $\alpha'_o, \alpha'_1, \beta'_o, \beta'_1, \gamma_0, \gamma_1, \delta_0$ and $\delta_1$ – are constants.

Table 5 summarizes the parameters of the exponential model with two elements for different fiber partitions on the front plan (a) and the profile view (b).

<table><caption>Table 5. Parameters of exponential mathematical model with two elements of the geometry of OPMF</caption><thead><tr><th rowspan="2"></th><th rowspan="2">Type of fibers</th><th colspan="4">Model coefficients</th><th rowspan="2">R<sup>2</sup></th><th rowspan="2">RSME</th></tr><tr><th>α'<sub>o</sub></th><th>α'<sub>1</sub></th><th>γ<sub>0</sub></th><th>γ<sub>1</sub></th></tr></thead><tbody><tr><td rowspan="5">Front plan</td><td>OPMF-Pe-a</td><td>0,0481</td><td>0,1058</td><td>-0,0634</td><td>0,0258</td><td>0,9999</td><td>0,0002</td></tr><tr><td>OPMF-Pi-a</td><td>0,1393</td><td>0,0314</td><td>-0,0025</td><td>0,0553</td><td>0,9997</td><td>0,0016</td></tr><tr><td>OPMF-Te-a</td><td>0,1327</td><td>0,0605</td><td>-0,0184</td><td>0,0508</td><td>0,9985</td><td>0,0032</td></tr><tr><td>OPMF-Ti-a</td><td>0,2017</td><td>0,0295</td><td>-0,0014</td><td>0,0599</td><td>0,9997</td><td>0,0017</td></tr><tr><td>OPMF-Du-a</td><td>0,0137</td><td>0,1943</td><td>-0,6626</td><td>0,0259</td><td>0,9992</td><td>0,0025</td></tr><tr><td rowspan="5">Profile view</td><td></td><th>β'<sub>o</sub></th><th>β'<sub>1</sub></th><th>δ<sub>0</sub></th><th>δ<sub>1</sub></th><td></td><td></td></tr><tr><td>OPMF-Pe-b</td><td>0,1172</td><td>0,0249</td><td>-0,0033</td><td>0,0515</td><td>0,9999</td><td>0,0006</td></tr><tr><td>OPMF-Pi-b</td><td>0,1638</td><td>0,0053</td><td>0,0039</td><td>0,0954</td><td>0,9981</td><td>0,0035</td></tr><tr><td>OPMF-Te-b</td><td>0,1816</td><td>0,0043</td><td>0,0041</td><td>0,1145</td><td>0,9999</td><td>0,0001</td></tr><tr><td>OPMF-Ti-b</td><td>0,0611</td><td>0,1553</td><td>-0,0352</td><td>0,0231</td><td>0,9992</td><td>0,0022</td></tr><tr><td></td><td>OPMF-Du-b</td><td>0,1825</td><td>0,0198</td><td>0,0013</td><td>0,0987</td><td>0,9994</td><td>0,0019</td></tr></tbody></table>

According to the correlation coefficients of the models tested (polynomial and exponential), we can say that these two models can describe with some accuracy the geometry of OPMF whatever the variety and the position of the latter. However, the exponential model with two elements appears to be more appropriate for its stability (the exponential function is monotonically increasing). So we have chosen the exponential model with two elements in the rest of the study.

**Evolutionary model of the fiber section.** The fiber cross section is assumed to be elliptical and the profile is described by an exponential model with two elements, the equations (1) and (2) are used to write the model of the section.

$$S(x) = \frac{\pi}{4}(\alpha'_{o}e^{\gamma_{0}x} + \alpha'_{1}e^{\gamma_{1}x})(\beta'_{o}e^{\delta_{0}x} + \beta'_{1}e^{\delta_{1}x}) \quad (18)$$

To simplify the calculations, we pose

$$A = \alpha'_{0}\beta'_{0}; B = \alpha'_{0}\beta'_{1}; C = \alpha'_{1}\beta'_{0}; D = \alpha'_{1}\beta'_{1} \text{ with } A, B, C \text{ et } D \in \mathbb{R}$$

$$K_0 = \gamma_0 + \delta_0, K_1 = \gamma_0 + \delta_1, K_2 = \gamma_1 + \delta_0, K_3 = \gamma_1 + \delta_1 \text{ with } K_0, K_1, K_2 \text{ et } K_3 \in \mathbb{R}$$

The cross section $S(x)$ of the fiber can be written:

$$S(x) = \frac{\pi}{4} [A e^{K_0 x} + B e^{K_1 x} + C e^{K_2 x} + D e^{K_3 x}] \quad (19)$$
The parameters of the section model for all five partitions are given in Table 6 below.

<table><caption>Table 6. Mathematical model parameters of the cross section of OPMF.</caption><thead><tr><th rowspan="2">Type of fibers</th><th colspan="8">Model coefficients</th></tr><tr><th>A</th><th>B</th><th>C</th><th>D</th><th>K<sub>0</sub></th><th>K<sub>1</sub></th><th>K<sub>2</sub></th><th>K<sub>3</sub></th></tr></thead><tbody><tr><td>OPMF-Pe</td><td>0,005637</td><td>0,001197</td><td>0,012399</td><td>0,002634</td><td>-0,0667</td><td>-0,0119</td><td>0,0225</td><td>0,0773</td></tr><tr><td>OPMF-Pi</td><td>0,022817</td><td>0,000738</td><td>0,005143</td><td>0,000166</td><td>0,0014</td><td>0,0929</td><td>0,0592</td><td>0,1507</td></tr><tr><td>OPMF-Te</td><td>0,024098</td><td>0,000570</td><td>0,010987</td><td>0,000260</td><td>-0,0143</td><td>0,0961</td><td>0,0549</td><td>0,1653</td></tr><tr><td>OPMF-Ti</td><td>0,012324</td><td>0,031324</td><td>0,001802</td><td>0,004581</td><td>-0,0366</td><td>0,0217</td><td>0,0247</td><td>0,0830</td></tr><tr><td>OPMF-Du</td><td>0,002500</td><td>0,000271</td><td>0,035459</td><td>0,003847</td><td>-0,6613</td><td>-0,5639</td><td>0,0272</td><td>0,1246</td></tr></tbody></table>

Worth noting that the presentation of the results in Table 6 is based on average values. In practice, we researched the coefficients [(A, B, C, D); (K₀, K₁, K₂, K₃)] for each of the 270 fibers; i.e. 54 fibers per partition. Thus the evolution of the section of each fiber should help in the counting of a single fiber tensile test by implementing the variation of the strain energy.

The knowledge of the mean models of the section is also of some interest for the implementation of certain calculation theories [12-13] and the digital multi-scale modeling of composite that would be reinforced by OPMF.

**Summary.** Our interest in this study was focused on the geometrical modeling of five fibers partitions from three varieties of palm nuts (Tenera, Pissifera and Dura) cultivated in the Mungo division in Cameroon. It appears that whatever the varieties, the OPMF are short fibers having a flattened cross-section and assimilated to a variable ellipse along the fiber. The peripheral fibers (on the pulp) are thinner and slightly shorter than those near the shell. We have proposed a geometrical model of diameter evolution as a function of the length of both the front plan than profile plan in exponential form with two elements. We have also written the model equation of the cross section in exponential form with four elements. A statistical study on the fiber diameters concluded that each of these fibers is unique although data follow a normal distribution whose parameters have been defined.

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