# 3-D Modeling of Dehydration Kinetics and Shrinkage of Ellipsoidal Fermented Amazonian Cocoa Beans

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## Abstract

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## 1. Introduction

## 2. Morphological and Physical Characterization of Ellipsoidal Cocoa Beans

## 3. Mathematical Models

- the influence of the ellipsoidal shape on dehydration kinetics in the simplest case of a uniform initial water distribution, i.e., ${\varphi}_{0}\left(\tilde{\mathbf{x}}\right)=constant$ and in the absence of shrinkage, i.e., $\alpha \left(\varphi \right)=0$;
- the influence of a nonuniform initial water distribution, i.e., ${\varphi}_{0}\left(\tilde{\mathbf{x}}\right)={\varphi}_{0}^{core}$ for $\tilde{\mathbf{x}}\in {\tilde{V}}^{core}$ and ${\varphi}_{0}\left(\tilde{\mathbf{x}}\right)={\varphi}_{0}^{shell}$ for $\tilde{\mathbf{x}}\in {\tilde{V}}^{shell}$, on dehydration rates, in the absence of shrinkage, $\alpha \left(\varphi \right)=0$;
- the influence, on dehydration rates, of a constant shrinkage factor $\alpha \left(\varphi \right)={\alpha}_{0}$.

^{®}v. 3.5. (COMSOL AB, Stockholm, Sweden). The convection-diffusion package (conservative form) was coupled with ALE (Arbitrary Lagrangian Eulerian) moving mesh, allowing re-meshing during the time evolution of the physical domain. Free displacement induced by boundary velocity conditions was set. Lagrangian quadratic elements were chosen. The linear solver adopted is UMFPACK, with relative tolerance ${10}^{-3}$ and absolute tolerance ${10}^{-6}$. The number of finite elements was $4\times {10}^{5}$–$6\times {10}^{5}$ with a non-uniform mesh. Smaller elements (minimum element size 0.01 dimensionless unit) have been located close to the moving boundary and to the core/shell boundary in order to accurately compute concentration gradients controlling the velocity of the moving front and the water transport at the core/shell interface. Figure 3 shows the computational domain (1/4 of the ellipsoidal bean) and the FEM mesh adopted.

#### 3.1. Uniform Initial Water Distribution (No Shrinkage)

#### 3.2. Non-Uniform Initial Water Distribution (No Shrinkage)

#### 3.3. The Influence of Shrinkage on Dehydration Curves

## 4. Analysis of Dehydration Kinetics and Shrinkage of Ellipsoidal Cocoa Beans

#### 4.1. Estimate of Water Effective Diffusivity D

#### 4.2. Influence of Air Velocity and Non-Uniformity of the Initial Water Distribution

#### 4.3. Temporal Evolution of the Moisture Content and Thickness of Core and Shell

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Scaled representation of an ellipsoidal cocoa bean (core and shell). a, b and c are the three principal axes and $\lambda $ is the shell thickness.

**Figure 3.**Representation of the computational domain (1/4 of the ellipsoidal bean) and the FEM (finite elements method) mesh adopted.

**Figure 4.**(

**A**) Moisture ratio $(X-{X}_{eq})/({X}_{0}-{X}_{eq})$ vs. dimensionless time $\tau =tD/{L}_{z}^{2}$ for $B{i}_{m}=1,2,3,5,9$ and a uniform initial water distribution ${\varphi}_{0}^{core}={\varphi}_{0}^{shell}={\varphi}_{0}=0.5$. (

**B**) Log-normal plot of dehydration curves reported in Figure (

**A**). Arrows indicate increasing values of $B{i}_{m}$. Dashed black lines show the asymptotic theoretical exponential behaviour, Equations (17) and (18).

**Figure 5.**Dimensionless dehydration rate ${J}_{d}$ vs. moisture ratio ${X}_{r}=(X-{X}_{eq})/({X}_{0}-{X}_{eq})$ for $B{i}_{m}=1,2,3,5,9$ for a uniform initial water distribution, ${X}_{0}^{core}/{X}_{0}={V}_{0}^{core}/{V}_{0}$. Arrow indicates increasing values of $B{i}_{m}$. Dashed lines indicate the theoretical long-drying time-scales linear behaviour, Equation (20).

**Figure 6.**Dimensionless dehydration rate ${J}_{d}$ vs. moisture ratio ${X}_{r}=(X-{X}_{eq})/({X}_{0}-{X}_{eq})$ for $B{i}_{m}=1,3,5,9$ for a uniform initial water distribution, ${\beta}_{0}^{core}={\beta}_{0}^{shell}=1$ (continuous lines) and for two non-uniform initial water distributions (dashed lines), ${\beta}_{0}^{core}=0.9,0.8$. Arrows indicate decreasing values of ${\beta}_{0}$, i.e., increasing non-uniformity.

**Figure 7.**Dimensionless dehydration rate ${J}_{d}$ vs. moisture ratio ${X}_{r}=(X-{X}_{eq})/({X}_{0}-{X}_{eq})$ for $B{i}_{m}=1,3,5,9$ and for a uniform initial water distribution, ${\beta}_{0}^{core}={\beta}_{0}^{shell}=1$. Continuous lines: no-shrinkage. dashed lines: shrinkage factor $\alpha \left(\varphi \right)={\alpha}_{0}=0.1,0.3,0.5$. Arrows indicate increasing values of ${\alpha}_{0}$.

**Figure 8.**Dimensionless dehydration rate ${J}_{d}$ vs. moisture ratio ${X}_{r}=(X-{X}_{eq})/({X}_{0}-{X}_{eq})$ for $B{i}_{m}=1,3,5,9$ and for a uniform initial water distribution, ${\beta}_{0}^{core}={\beta}_{0}^{shell}=1$. Shrinkage factor ${\alpha}_{0}=0.5$, ${V}_{eq}/{V}_{0}\simeq 0.85$. Arrow indicates increasing values of $B{i}_{m}$. Dashed lines indicate the theoretical long-drying time-scales linear behaviour, Equation (23).

**Figure 9.**(

**A**) Dehydration curves $X/{X}_{0}$ vs. t (h) and (

**B**) dehydration-rate curves ${j}_{d}$ (h${}^{-1}$) vs. ${X}_{r}$ for $T=30$${}^{\circ}$C, 40${}^{\circ}$C, 50 ${}^{\circ}$C, 60 ${}^{\circ}$C and $v=0.6$ m/s. Cocoa bean from data set (1). Comparison between experimental data (points) and model predictions (continuous lines) with water diffusivities D and Biot numbers $B{i}_{m}$ reported in Table 5. Arrows indicate increasing values of T. Dashed line indicates the asymptotic rescaled moisture content ${X}_{eq}/{X}_{0}=0.055$.

**Figure 10.**Diffusion coefficient D (m${}^{2}$/s) vs. dehydration temperature T (${}^{\circ}$C). Continuous line represents the Arrhenius behaviour $D={D}_{0}exp[-E/(RT\left)\right]$ with ${D}_{0}=5.77\times {10}^{-5}$ m${}^{2}$/s and $E/R=4037.78$ (K).

**Figure 11.**Influence of air velocity $v=0.3,0.6,1$ m/s on dehydration kinetics at $T=60$${}^{\circ}$C. Comparison between experimental data and model predictions with $D=3.12\times {10}^{-10}$ m${}^{2}$/s and $B{i}_{m}=10.27,13.61,16.93$ for $v=0.3,0.6,1.0$ m/s, respectively. Arrows indicate increasing values of v and $B{i}_{m}$. (

**A**) Rescaled moisture content $X/{X}_{0}$ vs. time (h). (

**B**) Dehydration rate ${j}_{d}$ (h${}^{-1}$) vs. moisture ratio ${X}_{r}$.

**Figure 12.**Dehydration kinetics at $T=60$${}^{\circ}$C and $v=0.6$ m/s for two different types of cocoa beans (data set (1) and data set (2)), characterized by two different initial water distributions and final shrinkage, see Table 4. Comparison between experimental data and model predictions with $D=3.12\times {10}^{-10}$ m${}^{2}$/s and $B{i}_{m}=13.61$ for data set (1) and $B{i}_{m}=12.87$ for data set (2). (

**A**) Rescaled moisture content $X/{X}_{0}$ vs. time (h). (

**B**) Dehydration rate ${j}_{d}$ (h${}^{-1}$) vs. moisture ratio ${X}_{r}$.

**Figure 13.**Moisture content in the core ${X}^{core}$ and in the shell ${X}^{shell}$ vs. the total moisture content $X={X}^{core}+{X}^{shell}$ (kg/kg db) for cocoa beans of data set (2), $T=60$${}^{\circ}$C and $v=0.6$ m/s. Comparison between experimental data (vertical bars) and model predictions (continuous and dashed lines) with $D=3.12\times {10}^{-10}$ m${}^{2}$/s, $B{i}_{m}=12.87$. Continuous lines indicate model predictions with a constant shrinkage factor ${\alpha}_{0}=0.39$. Dashed lines indicate model predictions without shrinkage ${\alpha}_{0}=0$.

**Figure 14.**Rescaled lengths of the three principal axes and shell thickness vs. total moisture content $X={X}^{core}+{X}^{shell}$ (kg/kg db) for cocoa beans of data set (2), $T=60$${}^{\circ}$C, $v=0.6$ m/s. Comparison between experimental data and model predictions with $D=3.12\times {10}^{-10}$ m${}^{2}$/s, $B{i}_{m}=12.87$, ${\alpha}_{0}=0.39$.

**Figure 15.**Evolution of the water volume fraction $\varphi (\tilde{\mathbf{x}},t)$ inside the shrinking bean (shell and core) for decreasing values of the rescaled total moisture content $X\left(t\right)/{X}_{0}$, $T=60$${}^{\circ}$C, $v=0.6$ m/s, $D=3.12\times {10}^{-10}$ m${}^{2}$/s, $B{i}_{m}=12.87$, ${\alpha}_{0}=0.39$. (

**A**) $X\left(t\right)/{X}_{0}=1$ (initial time instant); (

**B**) $X\left(t\right)/{X}_{0}=0.7$; (

**C**) $X\left(t\right)/{X}_{0}=0.3$; (

**D**) $X\left(t\right)/{X}_{0}=0.1$.

Data | ${\mathit{a}}_{0}$ | ${\mathit{b}}_{0}$ | ${\mathit{c}}_{0}$ | ${\mathit{\lambda}}_{0}$ | ${\mathit{X}}_{0}$ | ${\mathit{X}}_{0}^{\mathbf{core}}$ | ${\mathit{X}}_{0}^{\mathbf{core}}/{\mathit{X}}_{0}$ | ${\mathit{V}}_{0}^{\mathbf{core}}/{\mathit{V}}_{0}$ |
---|---|---|---|---|---|---|---|---|

Set | (mm) | (mm) | (mm) | (mm) | (kg/kg db) | (kg/kg db) | (-) | (-) |

(1) | 24.7 ± 0.72 | 12.73 ± 0.5 | 8.94 ± 0.86 | 0.45 ± 0.09 | 0.82 ± 0.09 | 0.6 ± 0.08 | ≃ 0.73 | ≃ 0.8 |

(2) | 24.28 ± 1.57 | 13.34 ± 0.5 | 8.52 ± 0.35 | 0.58 ± 0.09 | ≃ 0.88 | ≃ 0.49 | ≃ 0.56 | ≃ 0.72 |

${\mathbf{Bi}}_{\mathit{m}}=1$ | ${\mathbf{Bi}}_{\mathit{m}}=2$ | ${\mathbf{Bi}}_{\mathit{m}}=3$ | ${\mathbf{Bi}}_{\mathit{m}}=5$ | ${\mathbf{Bi}}_{\mathit{m}}=7$ | ${\mathbf{Bi}}_{\mathit{m}}=9$ | |
---|---|---|---|---|---|---|

${\gamma}_{0}$ | 1.35252 | 1.79058 | 2.06105 | 2.38064 | 2.56061 | 2.67429 |

**Table 3.**Values of ${\tilde{\gamma}}_{0}$ for ${R}_{eq}/{L}_{r}=0.7$, ${\alpha}_{0}=0.5$, ${V}_{eq}/{V}_{0}=0.85$ and different values of $B{i}_{m}$.

${\mathbf{Bi}}_{\mathit{m}}=1$ | ${\mathbf{Bi}}_{\mathit{m}}=2$ | ${\mathbf{Bi}}_{\mathit{m}}=3$ | ${\mathbf{Bi}}_{\mathit{m}}=5$ | ${\mathbf{Bi}}_{\mathit{m}}=7$ | ${\mathbf{Bi}}_{\mathit{m}}=9$ | |
---|---|---|---|---|---|---|

${\tilde{\gamma}}_{0}$ | 1.29016 | 1.71879 | 1.98915 | 2.31671 | 2.50135 | 2.62754 |

**Table 4.**Review of physical, geometrical and shrinkage parameters adopted in the moving-boundary model for the analysis of data set (1) and (2).

Data Set | ${\mathit{\varphi}}_{0}$ | ${\mathit{R}}_{\mathbf{eq}}$ (mm) | ${\mathit{\beta}}_{0}^{\mathbf{core}}$ | ${\mathit{\beta}}_{0}^{\mathbf{shell}}$ | ${\mathit{V}}_{\mathbf{eq}}/{\mathit{V}}_{0}$ | ${\mathit{\varphi}}_{\mathbf{eq}}$ | ${\mathit{\alpha}}_{0}$ |
---|---|---|---|---|---|---|---|

Equation (2) | Equation (19) | Equation (4) | Equation (4) | Equations (5) & (6) | Equation (8) | Equation (22) | |

(1) | 0.57 | 6.23 | 0.912 | 1.35 | 0.82 | 0.038 | 0.33 |

(2) | 0.59 | 6.13 | 0.78 | 1.53 | 0.78 | 0.042 | 0.39 |

**Table 5.**Estimated water diffusivities ${D}_{min}$, D and Biot number $B{i}_{m}$ for cocoa beans of data set (1) for different drying temperatures T and $v=0.6$ m/s.

$\mathit{T}=30{\phantom{\rule{0.166667em}{0ex}}}^{\circ}$C | $\mathit{T}=40{\phantom{\rule{0.166667em}{0ex}}}^{\circ}$C | $\mathit{T}=50{\phantom{\rule{0.166667em}{0ex}}}^{\circ}$C | $\mathit{T}=60{\phantom{\rule{0.166667em}{0ex}}}^{\circ}$C | |
---|---|---|---|---|

${D}_{min}$ (m${}^{2}$/s) | $(7.65\pm 0.1)\times {10}^{-11}$ | $(1.34\pm 0.03)\times {10}^{-10}$ | $(1.62\pm 0.05)\times {10}^{-10}$ | $(2.48\pm 0.03)\times {10}^{-10}$ |

D (m${}^{2}$/s) | $(9.11\pm 0.1)\times {10}^{-11}$ | $(1.54\pm 0.03)\times {10}^{-10}$ | $(2.09\pm 0.05)\times {10}^{-10}$ | $(3.12\pm 0.03)\times {10}^{-10}$ |

$B{i}_{m}$ | $(17.93\pm 0.1)$ | $(16.24\pm 0.1)$ | $(14.83\pm 0.1)$ | $(13.61\pm 0.12)$ |

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**MDPI and ACS Style**

Adrover, A.; Brasiello, A.
3-D Modeling of Dehydration Kinetics and Shrinkage of Ellipsoidal Fermented Amazonian Cocoa Beans. *Processes* **2020**, *8*, 150.
https://doi.org/10.3390/pr8020150

**AMA Style**

Adrover A, Brasiello A.
3-D Modeling of Dehydration Kinetics and Shrinkage of Ellipsoidal Fermented Amazonian Cocoa Beans. *Processes*. 2020; 8(2):150.
https://doi.org/10.3390/pr8020150

**Chicago/Turabian Style**

Adrover, Alessandra, and Antonio Brasiello.
2020. "3-D Modeling of Dehydration Kinetics and Shrinkage of Ellipsoidal Fermented Amazonian Cocoa Beans" *Processes* 8, no. 2: 150.
https://doi.org/10.3390/pr8020150