INTRODUCTION
Various methods have been used in the determination of diffusivities by solving Fick’s second law of diffusion (Eq. 1).
where, X is moisture content (kg water/kg dry matter) and D_{eff} is effective diffusivity (m^{2} sec^{1}).
The effective diffusivity is an overall mass transport property of water in
the drying material which includes liquid diffusion, vapour diffusion, hydrodynamic
flow and other possible mass transfer mechanisms (Karathanos
et al., 1990). By using appropriate initial and boundary conditions,
together with some reasonable assumptions, the analytical solution can be derived
for some standard geometry such as slab, cylinder and sphere (Crank,
1975). Equation 2 shows an example solution derived for
spherical object.
where, MR is moisture ratio (dimensionless) and R is radius in (m).
Many researchers predicted the effective diffusivity values by taking only the first term of Eq. 2 through regression analysis. However, this method predicts a single value of diffusivity for the entire process, which could result in not representing the kinetics of the entire drying process, because diffusivity should vary with time and moisture content throughout the drying cycle.
Karathanos et al. (1990) reported the use of
the slopes method to predict the variation of diffusivity with moisture content.
This was done by calculating the slope of the drying curve (dMR/dt)_{exp}
and the theoretical curve (dMR/dF_{0})_{theo} at a given moisture
ratio and the effective diffusivity is hence estimated from Eq.
3 and 4.
where, F_{ο} is fourier number.
However, the theoretical equation used in this model is similar to Eq.
2, which assumed constant diffusivity in the first place. Nevertheless,
the use of this method is reported in some published literatures (Luangmalawat
et al., 2008; Tuwapanichayanan et al.,
2008; Chemkhi and Zagrouba, 2005).
The objective of this study was to determine a variable diffusivity model by using the onedimensional Fick’s second law of diffusion in the drying of cocoa beans.
MATERIALS AND METHODS
Cocoa beans: Fresh cocoa beans were obtained from Jengka, Pahang and fermented using wooden boxes for 5 days. The fermenting mass weighed about 25 kg based on the fresh beans weight using box dimension measuring 30.5×30.5×30.5 cm. The beans were turned every 48 h to ensure uniformity during fermentation. The conditions of the beans were carefully monitored such that the temperature developed in the proper manner (range within 30 to 50°C from the start to the end) and that no mould growth was observed.
Drying: A hybrid heat pump dryer (Fig. 1) was used in all the drying trials. Two product chambers measuring 101×32×33 cm each were connected to the hot and cold air supplied from the heat pump. The cold temperature setting was adjusted by using a temperature controller and the corresponding hot temperature would depend on the amount of heat evolved from the condenser. Drying was conducted at cold temperature setting of 5°C with heater at 60°C (HHPD1) and 10°C without heater (HHPD2). These combinations were able to generate hot air at 53.3 and 39.3°C for HHPD1 and HHPD2, respectively. About 700 g of fermented beans were placed thinly on meshed surface and air flow parallel to the bean layer during drying in each chamber.
Moisture content: The beans used in each experiment were weighed prior to mixing during drying by using an analytical balance. The moisture content of the beans was determined with reference to the bonedry weight of the beans using Eq. 5.
where, the subscripts W_{i} and W_{bd} refer to the intial and bonedry weight, respectively. The Equilibrium Moisture Contents (EMC) were determined by prolonging the drying process until no further change in weight was observed for the beans in each treatment.
Method of slopes: The method of slopes was carried out by calculating
the slope of the drying curve (dMR/dt)_{exp} and the theoretical curve
(dMR/dF_{0})_{theo} at a given moisture ratio (Karathanos
et al., 1990).

Fig. 1: 
The hybrid heat pump dryer 
The slopes of the experimental drying curves were
determined using experimental data while the slopes of the theoretical curves
were calculated by differentiating the general solution with respect to the
Fourier number (F_{ο}). Calculation was carried out using Microsoft Excel (MS Office, version 2003,
USA). Equation 3 and 4 are referred.
Constant diffusivity model: The one dimensional Fick’s second law diffusional model for spherical geometry was discretized according to the radius (i) and time (t) intervals. The model was solved using explicit finite difference method with 0.01 h time step and grid with 20 constant subdivisions on the radius.
Boundary conditions:
The effective diffusivity was determined through a curve optimization process to the experimental data by minimizing the Sum Squared of the Residuals (SSR) using the SOLVER tool in Microsoft Excel spreadsheet (MS Office, version 2003, USA).
Variable diffusivity model: The one dimensional Fick’s second law
diffusional model taking into account the variation of moisture diffusivity
with local moisture content (X) was used. The model was discretized and solved
using explicit finite difference method with 0.01 h time step and grid with
20 constant subdivisions on the radius. Similar boundary conditions were used
Eq. 79.
The following empirical equations were defined for the variable diffusivity model.
Coefficients a and b were generated through a curve optimization process to the experimental data by minimizing the Sum Squared of the Residuals (SSR) using the SOLVER tool in Microsoft Excel spreadsheet (MS Office, version 2003, USA).
Fitting criteria: The mean relative error (E) between the experimental and predicted data was obtained to select the bestfit solution. Percentage of Evalue of less than 5 indicates an excellent fit.
The coefficient of determination (R^{2}) was also determined to compare the goodness of fit.
RESULTS AND DISCUSSION
Comparison between the predicted and experimental data can be shown from Fig.
2a and b. Table 1 shows the results
of the coefficient of determination and E (%) values obtained from the modeling.
Results showed that the constant diffusivity model obtained the lowest R^{2}
and the highest E (%) values which indicate poor fitting. This shows that the
model with constant diffusivity assumption is not adequate in describing the
mass transfer process inside the cocoa beans. Good fitting was observed in the
entire variable diffusivity model with E (%) values ranging from 3.0 to 11.1
and 4.9 to 8.2 in HHPD1 and HHPD2, respectively. The corresponding coefficient
of determination was found ranging from 0.9947 to 0.9995 and 0.9935 to 0.9987,
respectively. Excellent fitting was observed when using the quadratic model
with E (%) values of less than 5.

Fig. 2: 
Comparison between the experimental and the predicted data,
(a) HHPD1 (53.3°C) and (b) HHPD2 (39.3°C) 
Table 1: 
Comparison of R^{2} and Evalues (%) 

The variation of the effective diffusivity with moisture content for HHPD1
and HHPD2 are as shown in Fig. 3a and b.
The effective diffusivity values are in the order of 10^{10} to 10^{11}
which is within the range as those reported in studies (Zogzas
et al., 1996).
It can be seen that the exponential model tend to predict a higher effective
diffusivity as compared to the other models most of the time. Meanwhile, the
quadratic model and slopes method predicted a sudden increase in effective diffusivity
towards the final stage of drying. Comparison between the various models and
the slopes method showed big difference between the predicted diffusivity values.

Fig. 3: 
Predicted variable diffusivity models, (a) HHPD1 (53.3°C)
and (b) HHPD2 (39.3°C) 

Fig. 4: 
Predicted moisture ratio at t = 9 h along bean radius during
drying, (a) HHPD1 (53.3°C) and (b) HHPD2 (39.3°C) 
It must be noted that the basis of the slopes method is based on the constant
diffusivity model of the Fick’s law general solution. The current applied models could be more realistic and reasonable as compared
to the slopes methods as it was predicted using the partial differential equation
of the Fick’s diffusional model taking into account the variation of diffusivity
with local moisture content.
The drop of effective diffusivity with moisture content is apparent as cocoa
bean consists of two main layers namely the external testa and the internal
cotyledon. The testa will dry faster as compared to the inner part and this
increases the resistance to moisture diffusion as moisture content reduces.
Furthermore, the testa is covered by a thin film of slimy mucilage resulted
from the fermentation process and this will add an additional layer of resistance
to moisture diffusion as drying progresses. However, the quadratic model predicted
a slight increase in effective diffusivity towards the later stage of drying.
Such prediction may not reflect the actual mechanism, since it was purely a
curve fitting analyses. Nevertheless, this could imply that some structural
changes have occurred inside the beans during drying. This was observed in the
drying of fresh blueberries where big changes in intercellular space created
more opened structure and contribute to the higher increase of effective diffusivity
(Shi et al., 2008). Similar observation can be
observed in cocoa beans as the cotyledon starts to form crevices as moisture
content is reduced.
The variation of moisture ratios inside the cocoa beans during drying are as
shown in Fig. 4 and 5. It can be seen that
the constant diffusivity assumption predicted a much lower profiles as compared
to those predicted by the variable diffusivity models after 9 and 24 h of drying.

Fig. 5: 
Predicted moisture ratio at t = 24 h along bean radius during
drying, (a) HHPD1 (53.3°C) and (b) HHPD2 (39.3°C) 
In the first 9 h of drying, the quadratic model predicted higher moisture ratios
in the middle section of the beans in both treatments (Fig. 4a,
b). The quadratic model predicted about 99 and 24% higher
in HHPD1 and HHPD2, respectively, compared to the constant diffusivity model
based on average moisture ratio at this specific hour of drying. The drop in
moisture ratio is slower from the center and gradually increases towards the
surface of the beans.
Towards the end of drying, after 24 h in HHPD1 (Fig. 5a), the linear model shows higher moisture ratios profile followed by the exponential and the quadratic models. The linear model predicted a much higher average moisture profile at about 42% higher than the constant diffusivity model. In HHPD2 (Fig. 5b), the quadratic model predicted the highest average moisture ratio (4% higher) as compared to the constant diffusivity model. Modelling showed that there is no specific trend among the models either in over or under predicting the moisture ratios throughout the drying process and it solely depends on the variation of effective diffusivity at the specific drying time.
CONCLUSION
Modelling showed good fitting in the entire variable diffusivity model between the experimental and predicted data. The quadratic model showed excellent fitting with E (%) values of less than 5. The effective diffusivity values are in the order of 10^{10} to 10^{11} which is within the range reported in literatures. The quadratic model predicted possible structural changes within the beans which could be due to the formation of crevices within the cotyledon as moisture content is reduced. Modelling showed that the prediction of the moisture contents throughout the drying process solely depends on the variation of effective diffusivity at the specific drying time.
ACKNOWLEDGMENTS
The study was made possible through the Research Assistantship Grant from the School of Chemical and Environmental Engineering, University of Nottingham, Malaysia Campus and also the support given by the Ministry of Science, Technology and Innovation through the eScience research Grant (050212SF0014).