^{1}

^{2}

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^{3}

^{4}

^{1}

The study is focused on the phenomenon of diffusion of water through the shells of two coconut species (coconut nucifera) of Cameroun. The kinetics absorption of water was studied experimentally by the gravimetric method with discontinuous control of the mass of the samples at the temperature of 23℃. The mature coconut shells were cleaned mechanically, cut in a spherical shape and placed in a drying oven with 105 ℃ for 4 hours before being plunged in distilled water at 23 ℃. This study made it possible not only to determine the rate of water absorbed, but also to model the water kinetic absorption of the shells. Of the two models tested (Peleg and Page), the Page model predicted very well the experimental data. The Fick law made it possible to evaluate the effective diffusivity coefficients at the initial and final phases of absorption. The effective diffusivity coefficient was given from the Arrhenius equation.

The by-products of the coconut occupy an important place in the current research. This is justified by the dash of valorization of green energies and the fight against pollution. The shell of coconut (CS) is the subject of the research and several research tasks are devoted to it. Certain works are interested in the production of the activated carbon from CS. Other works deal with the use of the CS like charges in the composites [

Coconut shells used in this study come from the areas of the South, the Littoral and the South-west of Cameroun. Two varieties of Shells are concerned, and are characterized by the form of their nut: a lengthened form (species 1) and a round form (species 2). Coconuts shells were separated from nuts and remained at the laboratory in approximate ambient moisture of 60% and at a temperature varying between 20˚C and 23˚C, for two months. The separation of coconuts shells from nuts was carried out mechanically and CSs was cleaned and cut in the form of the shape of a sphere in the southernmost direction (

Nomenclature | |||
---|---|---|---|

w (%) | Water content | MR_{pre,i } | Predicted moisture ratio |

m_{0 } | Initial dry mass (g) | MR_{exp,i } | Experimental moisture ratio |

m_{(t) } | Massat instant t | D_{eff} | effective diffusivity(m²/s) |

m_{eq } | Mass at equilibrium (g) | D_{0}^{ } | Pre-exponential factor of the Arrhenius equation (m²/s) |

MR | Moisture ratio | R_{i} | Interior radius (mm) |

a, b, c, k, h, n | Models constants | R_{e} | Exterior radius (mm) |

RMSE | Root Mean Square Errors | t | Absorption time |

SSE | Mean of the squares errors | r^{2} | Coefficient of determination |

N | Number of Observations | R | Constant of perfect gas (kJ/mol∙K). |

MR | Moisture ratio | T | Drying temperature (˚C) |

P | Number of constants | E | Standard deviation |

The samples intended for the tests were dried in a standard drying oven of mark memmert model UN 160 to 105˚C for 4 hours. Under the above drying conditions after 4 hours, the mass of the samples of CS do not vary any longer. This was proven experimentally by [_{0}. For each sample, one notes the moment to which it was immerged, the sample was weighted aftertime duration in water. Making sure that the hygroscopy water was delicately removed and by minimizing the removing time from water. We note m(t) the wet mass of the sample at the instance t. We repeat this experiment until the mass of the sample does not vary anymore and it was notes the equilibrium mass m_{eq}.

The rate of absorption of water W compared to the dry matter of the samples is calculated starting from the drying mass m_{0} and of the equilibrium mass m_{eq} according to the formula (2). The instantaneous humidity content noted M(T) compared to the dry matter is calculated according to the formula (1). The ratio of the instantaneous rate of absorption which is the equivalent without dimension of the instantaneous water content is calculated according to the formula (3).

The software Matlab 2009 and Excel 2007 was used for the identification of the parameters of the various models. The effectiveness of a model is evaluated starting from the statistical criteria such as: the root mean square errors (RMSE) and the coefficient of correlation r^{2}. In fact a model is better if r^{2} tends towards 1 for a lower value and with RMSE which tends towards 0 for a higher value. These parameters are expressed by the Equations ((4) and (5)) respectively.

where

The Equation (5) is that of Ficks which governs the diffusion of mass through the vegetation products [

N˚ | Names of Models | Models | References |
---|---|---|---|

1 | Peleg | [ | |

2 | Page | [ |

where D_{eff} is the effective coefficient of diffusion in m^{2}/s and M is the rate of absorption. The analytical solution of the Equation (5), for a radial diffusion in a hollow sphere (

where MR is the moisture ratio of the water content of all the sample at the moment t and n is a positive entirety.

In the case that the interior radius r = R_{i} and the exterior radius r = R_{e} are maintained with concentrations C_{1} and C_{2} respectively such as C_{1} = C_{2}, the solution of Carslaw and Jaeger is given by the expression of the equation while limiting at the first term of this series we obtain Equation (7) given by;

The Neperien logarithm of the Equation (7) by taking in account the two phases of absorption, gives the Equation (8) which makes it possible to determine the coefficients of diffusion D_{eff}_{1} and D_{eff}_{2} at the initial and final phases respectively. Where

To determine the effective coefficient of diffusion, it is enough to trace the linear straight regression line of _{eff} according to Equation (9).

The instantaneous rates of absorption made it possible to plot the curves of the rate of absorption as a function of time.

To evaluate the kinetics of absorption of these Shells, starting from the experimental data, we calculate the moisture ratio of the instantaneous rate of absorption noted MR from Equation (3). The curve of MR for each species, presents two phases: an initial phase having a very great slope as of the first instance that the CS is in contact with water; and a final stage characterized by a very weak slope which is asymptotic with MR = 1. The

durations of absorption of the two species are almost identical and equilibrium is reached at the end of 35 days of immersion.

To model this kinetics of absorption, we tested the models of Peleg, and Page.

Equation (10) present the expression of the Peleg model:

where k_{1} is the parameter which characterizes the kinetics of absorption as it can show in Equations (11) and (12).

N˚ | % of absorption (d.b) | |
---|---|---|

Species 1 | Species 2 | |

1 | 15.95 | 18.07 |

2 | 16.39 | 19.64 |

3 | 18.72 | 18.81 |

4 | 18.53 | 21.00 |

5 | 18.89 | 17.79 |

6 | 16.22 | 19.45 |

7 | 16.55 | 18.68 |

8 | 18.31 | 19.36 |

9 | 15.73 | 19.57 |

10 | 16.92 | 22.39 |

11 | 18.98 | 21.73 |

12 | 16.99 | 22.09 |

13 | 18.14 | 17.42 |

14 | 16.17 | 24.08 |

15 | 17.42 | 23.21 |

16 | 17.22 | 23.58 |

17 | 18.32 | 20.53 |

18 | 16.32 | 20.23 |

19 | 17.62 | 21.43 |

20 | 17.02 | 19.43 |

A | 17.32 | 20.42 |

S | 1.06 | 1.95 |

and

k_{2} characterizes the rate of absorption indeed at equilibrium, when

The advantage of this model is that it can be put in the form of Equation (14):

The exploitation of the Equation (14) makes it possible to obtain the values of the parameters k_{1} and k_{2} of Peleg, by linear regression. These statistical constants and these parameters are presented in _{1} and k_{2} are a bite different from one species to another since the confidence intervals overlap.

N˚ | Species 1 | Species 2 | ||||||
---|---|---|---|---|---|---|---|---|

R² | RSME | k_{1} (min∙%^{−1}d.b) | k_{2} (%^{−1}d∙b) | R² | RMSE | k_{1} (min∙%^{−1}d∙b) | k_{2} (%^{−1}d∙b) | |

1 | 0.9995 | 0.96 | 19.380 | 0.0631 | 0.9995 | 0.76 | 13.06 | 0.056 |

2 | 0.9996 | 0.97 | 13.240 | 0.0542 | 0.9993 | 0.84 | 14.92 | 0.057 |

3 | 0.9997 | 0.96 | 13.000 | 0.0608 | 0.9993 | 0.76 | 13.68 | 0.052 |

4 | 0.9987 | 0.97 | 18.590 | 0.0555 | 0.9992 | 0.98 | 16.04 | 0.064 |

5 | 0.9991 | 0.98 | 17.340 | 0.0540 | 0.9993 | 0.57 | 19.8 | 0.051 |

6 | 0.9998 | 0.99 | 13.100 | 0.0619 | 0.9997 | 0.79 | 18.27 | 0.052 |

7 | 0.9997 | 0.98 | 15.330 | 0.0639 | 0.9977 | 0.84 | 14.85 | 0.046 |

8 | 0.9996 | 0.97 | 16.850 | 0.0716 | 0.9995 | 0.72 | 14.79 | 0.042 |

9 | 0.9996 | 0.98 | 18.590 | 0.0555 | 0.9981 | 0.79 | 18.1 | 0.055 |

10 | 0.9998 | 0.99 | 17.340 | 0.0540 | 0.9975 | 0.81 | 16.73 | 0.047 |

11 | 0.9983 | 0.99 | 15.330 | 0.0639 | 0.9986 | 0.58 | 15.57 | 0.049 |

12 | 0.9987 | 0.99 | 16.850 | 0.0716 | 0.9989 | 0.99 | 16.71 | 0.053 |

13 | 0.9968 | 0.96 | 18.720 | 0.0545 | 0.9983 | 0.76 | 17.47 | 0.046 |

14 | 0.9991 | 0.97 | 18.590 | 0.0555 | 0.9984 | 0.70 | 16.32 | 0.043 |

15 | 0.9991 | 0.98 | 16.589 | 0.0600 | 0.9988 | 0.80 | 16.16 | 0.05 |

16 | 0.9996 | 0.99 | 13.240 | 0.0542 | 0.9993 | 0.83 | 14.92 | 0.057 |

17 | 0.9991 | 0.98 | 17.350 | 0.0540 | 0.9993 | 0.58 | 19.8 | 0.051 |

18 | 0.9996 | 0.98 | 18.590 | 0.0555 | 0.9981 | 0.59 | 18.1 | 0.055 |

19 | 0.9991 | 0.99 | 18.590 | 0.0555 | 0.9984 | 0.63 | 16.32 | 0.044 |

20 | 0.9981 | 0.97 | 19.380 | 0.0631 | 0.9995 | 0.74 | 15.623 | 0.056 |

A | 0.9991 | 0.98 | 16.799 | 0.0591 | 0.9988 | 0.76 | 16.362 | 0.051 |

S | 0.000 | 0.01 | 2.198 | 0.005 | 0.000 | 0.12 | 1.823 | 0.006 |

The Page model is governed by the Equation (15).

where a and n are the parameters which are obtained by a nonlinear regression using the software Matlab (2009). The statistical parameters and the values of the parameters of this model are given in

The Page model predicts very well the kinetics absorption of the CS according to the values of the statistical parameters such as: r² and RMSE.

To determine the effective coefficients of diffusion, one takes into account the two phases of absorption kinetics. The Fick law expressed by Equation (7) was adopted for each phase. Equations ((16) and (17)) translate Fick law for the initial and final phases of the water absorption in this study.

Species 1 | Species 2 | ||||||||
---|---|---|---|---|---|---|---|---|---|

N˚ | R² | RMSE | a | n | N˚ | R² | RMSE | a | n |

1 | 0.9944 | 0.0230 | 0.0692 | 0.3853 | 1 | 0.9695 | 0.0551 | 0.0983 | 0.3950 |

2 | 0.9938 | 0.0225 | 0.1087 | 0.3427 | 2 | 0.9952 | 0.0228 | 0.0724 | 0.4366 |

3 | 0.9962 | 0.0179 | 0.0995 | 0.3311 | 3 | 0.9935 | 0.0247 | 0.0664 | 0.3682 |

4 | 0.9953 | 0.0197 | 0.0996 | 0.3648 | 4 | 0.9830 | 0.0475 | 0.1139 | 0.3409 |

5 | 0.9967 | 0.0174 | 0.0900 | 0.4053 | 5 | 0.9828 | 0.0421 | 0.1684 | 0.2813 |

6 | 0.9979 | 0.0145 | 0.0884 | 0.3996 | 6 | 0.9876 | 0.1252 | 0.2373 | 0.2492 |

7 | 0.9969 | 0.0177 | 0.1158 | 0.3673 | 7 | 0.9871 | 0.1568 | 0.1498 | 0.2630 |

8 | 0.9969 | 0.0165 | 0.0571 | 0.3982 | 8 | 0.9916 | 0.0272 | 0.0982 | 0.3600 |

9 | 0.9851 | 0.0378 | 0.0571 | 0.3982 | 9 | 0.9676 | 0.0593 | 0.0578 | 0.4799 |

10 | 0.9811 | 0.0388 | 0.0689 | 0.3904 | 10 | 0.9917 | 0.0284 | 0.0780 | 0.3798 |

11 | 0.9908 | 0.0295 | 0.0700 | 0.4204 | 11 | 0.9614 | 0.0632 | 0.0912 | 0.4008 |

12 | 0.9766 | 0.0491 | 0.1554 | 0.3363 | 12 | 0.9937 | 0.0251 | 0.0709 | 0.3902 |

13 | 0.9944 | 0.0230 | 0.0692 | 0.3853 | 13 | 0.9796 | 0.0426 | 0.1176 | 0.3546 |

14 | 0.9938 | 0.0225 | 0.1087 | 0.3427 | 14 | 0.9847 | 0.0381 | 0.1012 | 0.3674 |

15 | 0.9962 | 0.0179 | 0.0995 | 0.3311 | 15 | 0.9927 | 0.0622 | 0.1160 | 0.3758 |

16 | 0.9953 | 0.0197 | 0.0996 | 0.3648 | 16 | 0.9839 | 0.0615 | 0.1149 | 0.3500 |

17 | 0.9962 | 0.0179 | 0.0995 | 0.3311 | 17 | 0.9737 | 0.0583 | 0.1468 | 0.3364 |

18 | 0.9969 | 0.0177 | 0.1158 | 0.3673 | 18 | 0.9943 | 0.0504 | 0.1410 | 0.3065 |

19 | 0.9944 | 0.0230 | 0.0692 | 0.3853 | 19 | 0.9957 | 0.0214 | 0.0664 | 0.4092 |

20 | 0.9969 | 0.0165 | 0.0571 | 0.3982 | 20 | 0.9917 | 0.0251 | 0.0770 | 0.3120 |

A | 0.9926 | 0.0242 | 0.0911 | 0.3727 | A | 0.9851 | 0.0518 | 0.1092 | 0.3578 |

S | 0.0062 | 0.0097 | 0.0256 | 0.0291 | S | 0.0101 | 0.0343 | 0.0435 | 0.0568 |

where D_{eff}_{1} and D_{eff}_{2} indicate the coefficients of effective diffusivity at initial and final phases respectively.

_{eff}_{1} at the initial phase. The same reasoning is used to determine the coefficients of diffusion of the final phase of the two species.

Species 1 | Species 2 | ||||
---|---|---|---|---|---|

N˚ | Deff_{1} (m²/min) | Deff_{2} (m²/min) | N˚ | Deff_{1} (m²/min) | Deff_{2} (m²/min) |

1 | 6.49E−09 | 1.30E−10 | 1 | 6.49E−09 | 1.46E−10 |

2 | 8.11E−09 | 1.46E−10 | 2 | 8.11E−09 | 1.30E−10 |

3 | 8.11E−09 | 9.74E−11 | 3 | 4.87E−09 | 1.46E−10 |

4 | 8.11E−09 | 9.74E−11 | 4 | 6.49E−09 | 1.30E−10 |

5 | 4.87E−09 | 1.62E−10 | 5 | 6.49E−09 | 1.62E−14 |

6 | 4.87E−09 | 9.74E−11 | 6 | 4.87E−09 | 1.30E−10 |

7 | 4.87E−09 | 1.30E−10 | 7 | 6.49E−09 | 1.30E−10 |

8 | 4.87E−09 | 1.14E−10 | 8 | 4.87E−09 | 1.14E−10 |

9 | 4.87E−09 | 1.30E−10 | 9 | 6.49E−09 | 1.46E−10 |

10 | 4.87E−09 | 9.74E−11 | 10 | 6.49E−09 | 1.46E−10 |

11 | 6.49E−09 | 1.46E−10 | 11 | 6.49E−09 | 1.14E−10 |

12 | 6.49E−09 | 9.74E−11 | 12 | 8.11E−09 | 9.74E−11 |

13 | 6.49E−09 | 1.30E−10 | 13 | 8.11E−09 | 1.46E−10 |

14 | 8.11E−09 | 9.74E−11 | 14 | 8.11E−09 | 1.46E−10 |

15 | 8.11E−09 | 9.74E−11 | 15 | 8.11E−09 | 9.74E−11 |

16 | 8.11E−09 | 9.74E−11 | 16 | 4.87E−09 | 1.46E−10 |

17 | 6.49E−09 | 9.74E−11 | 17 | 4.87E−09 | 9.74E−11 |

18 | 6.49E−09 | 1.30E−10 | 18 | 4.87E−09 | 1.14E−10 |

19 | 8.11E−09 | 9.74E−11 | 19 | 4.87E−09 | 1.14E−10 |

20 | 8.11E−09 | 9.74E−11 | 20 | 4.87E−09 | 1.14E−10 |

A | 6.65E−09 | 1.14E−10 | A | 6.25E−09 | 1.25E−10 |

S | 1.38E−09 | 2.13E−11 | S | 1.31E−09 | 1.90E−11 |

Products | Temp | D_{eff} (m²/s) | References | |
---|---|---|---|---|

Initiale phase | Final phase | |||

CS espèce 1 | 23˚C | 1.10 ± 0.23 × 10^{−10 } | 1.90 ± 0.35 × 10^{−12} | Study case |

CS espèce 2 | 23˚C | 1.04 ± 0.21 × 10^{−10} | 2.08 ± 0.31 × 10^{−12} | |

wood afra | 23˚C | 1.38 × 10^{−3} | [ | |

wood ojamlesh | 3.71 × 10^{−4} | |||

wood roosi | 4.88 × 10^{−4} | |||

grain of Amaranth | 23˚C | [10^{−11} - 10^{−12}] | [ | |

Grain of rice | 23˚C | 7 × 10^{−10} | [ | |

Chickpeas | 23˚C | [1.85 × 10^{−10} - 1.98 × 10^{−10} | [ | |

Beans | 23˚C | [2.56 × 10^{−9} - 8.18 × 10^{−11}] | ||

Wheatkernel | 25˚C | 2.8 × 10^{−12} | [ | |

chestnuts | 40˚C | 0.98 ± 0.037 × 10^{−8} | 0.96 ± 1.85 × 10^{−8 } | [ |

Food paste | 23˚C | 5.69 × 10^{−11} | 4.20 × 10^{−11} | [ |

fiber of hemp | 5.29 × 10^{−12} | 5.80 × 10^{−13} | [ | |

jute fiber | 2.33 × 10^{−12} | 2.30 × 10^{−13} | ||

flax fiber | 23˚C | 2.11 × 10^{−12} | 2.11 × 10^{−13} | |

fiber of sisal | 4.00 × 10^{−12} | 4.38 × 10^{−13} | ||

fiber of Okra | 23˚C | 5.40 × 10^{−10} | [ | |

nut fiber of bétel | 80 × 10^{−10} |

The shells of two coconut species were separated from nuts and remained at the laboratory in approximate ambient moisture of 60% and at a temperature varying between 20˚C and 23˚C, for two months. Sample was obtained by cutting shells in portion of hollow sphere. Samples were dried in oven and immersed in the distilled water with an aim of studying their absorption kinetics. It was noted that at a temperature of 23˚C ± 1˚C, equilibrium content of balance is reached after a period of approximately 35 days in water. The rate of absorption of species 1 is 17.32% ± l.06% and that of species 2 is 20.42% ± 1.95%. The absorption kinetics of CSs presents two phases: an initial phase with great absorption kinetics in the first 28 minutes; and a final phase for the rest of time. It appeared clearly that the absorption kinetics of the two species is nearly identical. Of the 2 models tested (Peleg and Page), the Page models model very well the experimental data with a coefficient of correlation r^{2} > 0.98. The effective coefficients of diffusion obtained starting from the law of Fick: in the initial phase, they are (1.10 ± 0.23) × 10^{−10} m²/s and (1.04 ± 0.21) × 10^{−10} m²/s for species 1 and 2 respectively; in the final phase, they are (1.90 ± 0.35) × 10^{−12} m²/s and (2.08 ± 0.31) × 10^{−12} m²/s for species 1 and 2 respectively.

Thanks are due to the Head of the Civil Engineering Department of Fotso Victor University Institute of Technology for his help in various stages of the experimental work.

Dieunedort Ndapeu,Ebénezer Njeugna,Nicodème Rodrigue Sikame,Sophie Brogly Bistac,Jean Yves Drean,Médard Fogue, (2016) Experimental Study of the Water Absorption Kinetics of the Coconut Shells (Nucifera) of Cameroun. Materials Sciences and Applications,07,159-170. doi: 10.4236/msa.2016.73016