One of the successful methods to valorize green energies and fight against pollution is to transform agricultural biomass into raw materials. Cameroon is the third African country in oil palm production and the industrial and craftsmen transformation co-produce several residues among which we have palm kernel shell (PKS). The PKS has been regarded as waste from oil palm processing . It has been shown in Nigeria, first African producer of oil palm that approximately 15 to 18 tonnes of fresh fruit bunches are produced per hectare per year and PKS comprises about 64% of the bunch mass . In Cameroon, the sub-product PKS is either burned to supply energy at palm mills machines or left in piles to the air. The process of burning PKS releases significant volatiles and particulates which pose pollution concerns and generates greenhouse effect and his natural degradation is too slow. Also, PKS is pouring on road to improve vehicular traction along plantation farms where there are no tarred roads. Researches works have been carried out on PKS, among which the determination of physical and chemical characteristics of PKS , which are essential parameters in utilization, development of processing methods, and design of equipment, is related to their transformation . Other researchers deal with the use of both charred and non-charred PKS for filtering effluent from sedimentation tanks of water treatment plant , and also as concrete reinforcement agent in substitution to conventional aggregates such as granite . The use of PKS in the composites materials implies water diffusion phenomenon. Therefore, we are interested in the kinetics of water absorption in the shell. In the present study, objectives are: Determination of the rate of water absorption and moisture diffusivity in the PKS using hydration data in the Becker’s model suitable for arbitrary shapes . Palm kernels, even on the same cob, vary in their shape from oval to round enabling the use of standard relations of regular shape body for moisture diffusion.
Palm kernel shells used in this investigation were obtained from the production areas of the west and south-west region of Cameroon. One variety of shell (DURA) is concerned, and it is characterized by the considerable volume of the shell. PKS aggregate, Figure 1 below obtained was made through mechanical method. The initial moisture content of selected sizes of PKS was within the interval of [6.18%; 11.74%] respectively Tombel PKS and Bafang PKS and was determined by oven drying method of the whole kernels at 105˚C for 6 h  and expressed as percentage dry basis (db). 20 samples were produced for this study viz. 10 per region.
Figure 1. PKS aggregate sample (source: authors).
3.1. Experimental Measurements
1) Granulometric analysis
After measuring the mass of PKS aggregate, we pour it into a set of sieves arranged in descending order of fineness. Thereafter it is taken to the automatic sieves shaker D407 for 15 min to achieve complete classification. At the end of sieving operation, according to the pre-selected sieves we have recorded three different sizes respectively in (mm): 12.5 ≤ Sizes < 16; Sizes ≥ 16 and Sizes < 12.5. This last size was not significant for us to carry experimentation.
2) Gravimetric method
Samples were dried in a standard drying oven of mark Memmert model UN 160 at 105˚C for 6 hours, until the mass of the samples doesn’t vary any more . PKS aggregate samples were conditioned in test bowl and then the test bowl was placed in an ambient temperature (23˚C) water bath for soaking experiments as described by . During soaking experiments, the samples were removed at specific variable intervals of time and the soaked samples were quickly blotted with the paper towels (3 - 5 times) to remove hygroscopic surface moisture and weighed. The moisture gain was calculated from the variation in weight of the sample to an accuracy of 0.01 g using electronic precision scale. The saturation mass is reached when the mass of the PKS remains constant.
By using Excel 2007 and Matlab R2014 b software with 95% rate of confidence, the solution of differential diffusion equation for arbitrary shapes provides by Becker was applied. The final mass is obtained when the mass of sample becomes constant. The duration of immersion is estimated about 28 days.
3) Percentage of water absorbed
The rate of absorption of water W compared to the dry matter of the samples is calculated starting from the drying mass M0 and the equilibrium mass Meq according to Equation (1).
The instantaneous humidity content M(t) compared to the dry matter is computed by Equation (2).
where M0 and Mt are the mass at respectively initial time and the actual time.
3.2. Analytical Approach
In the literature several analytical models are proposed to describe the water diffusion kinetic of seeds and PKS can be ranged. Moisture ratio which is the equivalent without dimension of the instantaneous water content is given by Equation (3).
The volume-surface area ratio (v/s) was calculated by dividing the equivalent volume of sphere by surface area of crushed palm kernel shell as
where r is equivalent radius. The factor (4πr2/s) is the ratio of equivalent surface area of a sphere of equal volume to the surface area of the crushed palm kernel shell, and can be denoted as sphericity . Substituting , Equation (4) gets reduced to
where d is the equivalent diameter which can be expressed by a relationship as from Equation (4). Substituting d, in Equation (5) and further simplifying, we get Equation (6) which describes the relationship between volume and surface area explicitly,
Similar expression has also been reported by . The sphericity of sample was therefore computed from Equation (7) as:
where is sphericity; L the length, mm; B the breadth, mm; and T thickness, mm .
Spatial dimensions, viz. length, breadth and thickness of 150 units, were measured using digital vernier caliper with 0.05 mm accuracy. These are the dimensions along the longest axis, across the axis perpendicular to the longest in the horizontal direction, and across the third axis perpendicular to both the first and second axis. Sphericity was averaged from 150 crushed palm kernels shells. The mean volume of sample was obtained through Equation (8).
where m is the mass of sample, ρ the solid density of palm kernel shell and n the number of sample.
Moisture diffusion of a liquid in a solid is primarily caused by concentration gradient. This gradient tends to move the water molecules to equalize concentration. The diffusion coefficient is defined by Fick’s second law as Equation (9).
where, Deff is the diffusivity coefficient; c is the concentration of diffusing substance at a point in solid; x, y and z are Cartesian coordinates of the point under consideration; and t is the diffusion time. It has been demonstrated that Equation (9) can be integrated approximately through Equation (10) for diffusion in solids of arbitrary shape to correlate the moisture gain during soaking in water .
Subject to the following initial and boundary conditions
at and ,
where s is a general coordinate with origin at bounding surface, c the average concentration, cs the concentration at the bounding surface and c0 the initial concentration and t the diffusion time. After replacing the concentration term (c) by the moisture content (m) the equation can be rewritten as:
where MR is moisture ratio. Mt, M0 and Meq are average moisture contents at any given soaking duration, initial moisture content and moisture content at the equilibrium respectively. Equation (11) can be arranged by Equation (12) and Equation (13) as:
in which (s/v) is surface to volume ratio, D moisture diffusivity and t diffusion time. B is a parameter dependent of solid shape. Combining Equations (11)-(13), we get
from Equation (15),
In Equation (17), Meq and M0 are constants for a particular PKS sample and the ratio of volume-to-surface area (v/s) may be taken as constant irrespective of moisture content . A plot of with would give a polynomial curve of coefficients (α) and (µ), Equation (14). Moisture diffusivity, D, can be determined at ambient temperature using (α) values in Equation (17). The value of moisture content Meq can be estimated by the method provided by  from absorption data. Since the quantities s/v, , Meq and M0 do not vary for a particular sample of PKS in Equation (17), Equation (14) can be then rewritten as:
In Equation (18) we can easily see that if the diffusion time t is kept constant, then the quantity would be a linear function of M0 with a slope of with . On extending the plot to abscissa axis when = 0, the intercept can be designated as moisture content Meq.
4. Results and Discussions
4.1. Water Absorbed
After evaluating the percentage of water gain by PKS aggregate for the different sizes using Equation (1), the summary of the results for the various samples is illustrated in Table 1.
We note that the percentage of water absorption of PKS aggregate of the two regions is approximately the same. Table 2 represents some materials with the comparison of their water absorbed percentage. It is observed that PKS water absorption is close to coconut nucifera absorption.
Table 1. Percentage of water absorbed of PKS.
Table 2. Comparison of water absorbed of some kernels.
4.2. Kinetic of Moisture Diffusion
Figure 2 presents the curves of experimental points of the variation of moisture gain according to the root square of time of PKS for different sizes. The experimental study of the phenomenon of water diffusion in PKS aggregate has two phases, namely a rapid (initial) absorption phase and a slow (final) absorption phase. This last phase has a low absorption rate of about 7% to 9% of the total absorption compared to the first phase, which is in a range of 91% to 93% of the total absorption of the samples, which express the approach of saturation. For this, the application of Equation (14) will be done over a period of 150 minutes.
Bafang PKS has slow rate for bigger particles compared to others and also has low absorption compared to Tombel PKS. This could be because of the microstructure of Bafang PKS with poor porosity and weak void index according to their geography zone of production.
4.3. Evaluation of Moisture Content Surface Meq
The surface moisture content (Meq) was evaluated to determine the moisture diffusivity using Equation (16). The experimental data of moisture gain in PKS for a soaking time of 150 min were plotted against diverse initial moisture contents of PKS ranging from 0.045 to 0.07 g/g (db) for Tombel and 0.098 to 0.139 g/g (db) for Bafang at ambient water soaking temperatures as shown in Figure 3.
For Tombel PKS, Meq was 0.092 g/g and 0.082 g/g at ambient temperature and For Bafang PKS, Meq was 0.18 g/g and 0.143 g/g respectively for the two different sizes. Coefficients of soaking curves were determined from Figure 2, surface moisture content from Figure 3 and they are reported in Table 3.
Figure 2. Relation between the moisture gain and square root of the soaking time. (a) Tombel PKS of size 12.5 ≤ S < 16 (mm); (b) Tombel PKS of size S ≥ 16 (mm); (c) Bafang PKS of size 12.5 ≤ S < 16 (mm); (d) Bafang PKS of size S ≥ 16 (mm).
Figure 3. Extrapolation of (Mt − M0) as a function of M0 to obtain the surface moisture content.
Table 3. Surface moisture content, coefficients of soaking curves of PKS and pertinent statistics parameters.
4.4. Sphericity of Palm Kernel Shell Aggregate
Sphericity of PKS aggregate was computed from  equation. Average sphericity for 150 broken palm kernels shells has been found to be 0.477 (47.7%). Sphericity of non-broken Palm kernel shells of 70% has also been reported by , 80% by  and 78% by Mijinyawa and Omokhoje in 2005 in the determination of some physical properties of palm kernel, presented in the Proceedings of the Nigerian Institution of Agricultural Engineering.
4.5. Moisture Diffusivity
To determine the different moisture diffusivity, we used Becker’s Model  expressed by Equations (10)-(17). Experimental absorption data recorded allow us to determine moisture contents at any given soaking duration and moisture content at the equilibrium respectively to be plotted. Results are reported in Table 4.
By looking at the different values presented in Table 5, it is observed that, moisture diffusivity in the case study approach Okra fibre and rice grain. The little difference between diffusivity coefficients of PKS for the two regions is due to their initial moisture content.
Palm kernel shell aggregate of Cameroon dried in oven to obtain initial moisture content within the interval of [6.18%; 11.74%] was studied with an objective to assess the effective coefficient of diffusion through the phenomenon of water absorption by the shells. Samples were immersed in the distilled water at the ambient temperature of 23˚C and total moisture equilibrium balance of samples is reached after a period of approximately 28 days in water. The rate of water
Table 4. Moisture diffusivity of palm kernel shell.
Table 5. Comparison of the effective coefficients of diffusion during absorption at constant temperature (23˚C).
absorption percentage is found to be 18.20% and 18.58% respectively for Bafang and Tombel PKS aggregate. Experimental moisture gain following Becker’s equation with the square root of soaking time of 150 min has been plotted. Moisture diffusivity of 5.19 × 10−8 and 7.90 × 10−9 m2/s was respectively determined for Tombel and Bafang PKS.
The knowledge of moisture diffusion phenomenon through PKS allows mastering the behaviour of composite in their elaboration process, above all total liquid soil that can influence cohesion energies within composite material.
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