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 [8]. 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 [8]. 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 M_eq according to Equation (1).

w ( % ) = M e q − M 0 M 0 ∗ 100 (1)

The instantaneous humidity content M(t) compared to the dry matter is computed by Equation (2).

M ( t ) = M t − M 0 M 0 (2)

where M₀ and M_t are the mass at respectively initial time and the actual time.

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).

M R ( t ) = M t − M 0 M e q − M 0 (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

v s = 4 π r 3 3 s ⇒ v s = r 3 [ 4 π r 2 s ] (4)

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 ϕ . Substituting ϕ , Equation (4) gets reduced to

v s = r 3 ϕ ⇒ v s = d 6 ϕ (5)

where d is the equivalent diameter which can be expressed by a relationship as d = ( 6 v / π ) 1 / 3 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,

v s = [ 6 v π ] 1 / 3 ϕ 6 ⇒ v s = ϕ v 1 / 3 4.836 (6)

Similar expression has also been reported by [7]. The sphericity of sample was therefore computed from Equation (7) as:

ϕ = ( L B T ) 1 / 3 L (7)

where ϕ is sphericity; L the length, mm; B the breadth, mm; and T thickness, mm [7].

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).

v = ∑ m / ρ n (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).

∂ c ∂ t = D e f f ( ∂ 2 c ∂ x 2 + ∂ 2 c ∂ y 2 + ∂ 2 c ∂ z 2 ) (9)

where, D_eff 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 [7].

c − c s c 0 − c s = 1 − 2 π X + B X 2 (10)

Subject to the following initial and boundary conditions

c = c 0 at t = 0 ,

c = c s at s = 0 and t > 0 ,

where s is a general coordinate with origin at bounding surface, c the average concentration, c_s 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:

M R = M t − M e q M 0 − M e q = 1 − 2 π X + B X 2 (11)

where MR is moisture ratio. M_t, M₀ and M_eq 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:

1 − M R = ( 2 / π ) X − B X 2 (12)

where,

X = ( s / v ) D t (13)

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

M t − M 0 = α t − μ t (14)

where,

α = ( 2 / π ) ( M e q − M 0 ) ( s / v ) D (15)

and

μ = B ( M e q − M 0 ) ( s / v ) 2 D (16)

from Equation (15),

D = [ α π / { 2 ( M e q − M 0 ) ( s / v ) } ] 2 (17)

In Equation (17), M_eq 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 [7]. A plot of ( M t − M 0 ) with t 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 M_eq can be estimated by the method provided by [7] from absorption data. Since the quantities s/v, D , M_eq and M₀ do not vary for a particular sample of PKS in Equation (17), Equation (14) can be then rewritten as:

M t − M 0 = ( β − ω ) ( M e q − M 0 ) (18)

where,

β = ( 2 / π ) ( s / v ) D t ; ω = B ( s / v ) 2 ( D t ) (19)

In Equation (18) we can easily see that if the diffusion time t is kept constant, then the quantity ( M t − M 0 ) would be a linear function of M₀ with a slope of ( β − ω ) with ω > β . On extending the plot to abscissa axis when ( M t − M 0 ) = 0, the intercept can be designated as moisture content M_eq.

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.

Figure 2 presents the curves of experimental points of the variation of moisture gain ( M t − M 0 ) 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.

The surface moisture content (M_eq) 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, M_eq was 0.092 g/g and 0.082 g/g at ambient temperature and For Bafang PKS, M_eq 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.

Sphericity of PKS aggregate was computed from [7] 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 [9], 80% by [10] 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.

To determine the different moisture diffusivity, we used Becker’s Model [7] 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.