# A Mechanistic Model of Mass Transfer in the Extraction of Bioactive Compounds from Intact Sorghum Pericarp

^{1}

^{2}

^{*}

## Abstract

**:**

_{ep}), mass transfer coefficient from the pericarp surface to the solvent (k

_{c}), and distribution coefficient (H). The model simulates the extraction performance, including the yield and the concentrations of bioactive compound in the extract and inside the pericarp at various positions and times. A sensitivity analysis of the changes in each involved parameter provided sufficient information for increasing the performance of the model. A validation test that compared the results of the simulation with those of established analytical solutions showed that the model has high accuracy. Hence, the model can be applied in quantitative evaluations to improve productivity in the pharmaceutical industry.

## 1. Introduction

## 2. Methods

#### 2.1. Model Development

_{P}is the mass flux of the phytochemical compound or

**P**compound (g/min. cm

^{2}), C

_{P}is the concentration of

**P**compound in a solid (g/cm

^{3}), D

_{eP}is the effective diffusivity (cm

^{2}/min), and x is the position (cm). The mass transfer from the surface of the pericarp to the bulk of the solvent is described while using film theory, as shown in Equation (2).

_{c}is the mass transfer coefficient (cm/min), ${C}_{pf}$is the concentration of P in the solid surface (g/cm

^{3}), and ${C}^{*}{}_{Pf}$ is the concentration of P in the solvent (g/cm

^{3}solvent), which is in equilibrium with the concentration of P in the solid surface.

_{eP}), mass transfer coefficient from the surface of the solid to the bulk of the liquid (k

_{c}), and coefficient distribution (H). The system of extraction was conducted in a stirred tank extractor, in which the grains (N

_{b}particles) are immersed in a solvent of volume V. The details of the derivation of the model are presented in this section.

- The phytochemical compounds of sorghum (phenolic compounds, mainly condensed tannin or proanthocyanidins) are located in the pericarp of a sorghum grain [33].
- The sorghum grain is assumed to have a spherical shape with an average radius of R.
- The geometry of an intact pericarp is assumed to be a slab of thickness L that covers the spherical sorghum grain, since the pericarp’s thickness is very thin when compared to the size of the pericarp.
**P**compound diffuses from the inside part of the pericarp to the surface of the pericarp, and the rate of diffusion is assumed to follow Fick’s diffusion law with an effective diffusivity of Dep.**P**compound is transferred from the surface of the pericarp to the bulk of the liquid.

- The pericarp is a homogenous medium; the position of the pericarp is between x = 0 and x = L.
- The effective diffusivity of P (${D}_{eP}$) is assumed to be constant.
- The solid–liquid equilibrium of P follows the coefficient distribution model with a distribution coefficient value of $H$ (Equation (2)).
- The solvent’s penetration to the pericarp is relatively fast when compared with the diffusion process, so it does not control the overall process of mass transfer of P [12].

_{eP}by assuming that the rate of solvent penetration and the dissolution of P to the solvent are much faster than diffusion processes [35]. The whole process is transient (unsteady state) (Equation (10)). The mass balance equation is applied to the volume element, as shown in Figure 1. The volume element in the solid is of thickness ∆x and it has the cross-section area S. The mass balance of P in the volume element is as follows.

_{b}grains of red sorghum are immersed is described as in Equation (11) or Equation (12).

#### 2.2. Model Solution and Simulation

#### 2.3. Validation

#### 2.3.1. Pretreatment of Red Sorghum Grains

_{equivalent}/cm

^{3}pericarp. The grains were cleaned and dried before extraction. The grains were dried while using a sun drier for 10 h over two successive days.

#### 2.3.2. Extraction Process

#### 2.3.3. Proanthocyanidins Quantification in Aqueous Extract

## 3. Results and Discussion

#### 3.1. Simulation Results

_{equivalent}/cm

^{3}pericarp. The values of the other parameters were D

_{eP}= 0.0000025 cm

^{2}/min. [39], H = 0.5, and k

_{c}= 0.0004 cm/min. Table 1 lists the values of the applied parameters. Figure 2, Figure 3 and Figure 4 show the simulation results.

#### 3.2. Analysis of the Sensitivity of the Model to Fluctuations in Parameters

_{p}

_{0}) of 0.1 g P

_{equivalent}/cm

^{3}; an effective diffusivity (D

_{eP}) of 0.0000025 cm

^{2}/min.; a mass transfer coefficient (k

_{c}) of 0.0004 cm/minute, and a slab thickness (L) of 0.03 cm.

_{P}

_{0}, 0.5 C

_{P}

_{0}, 1 C

_{P}

_{0}, and 2 C

_{P}

_{0}). The results show that the P concentration in the extract was very sensitive to C

_{P}

_{0}. In contrast, C

_{P}

_{0}did not significantly affect the yield. Under real conditions, this value depends on the variety of the plant, the location of the crop, and the age of the harvested grain. Figure 5b,c illustrate the sensitivity of the model in predicting the P concentration in extract as the impact of the effective diffusivity value (D

_{eP}) and the mass transfer coefficient (k

_{c}), respectively, while Figure 6b,c illustrate the sensitivity of the model in predicting the yield as the impact of the effective diffusivity value (D

_{eP}) and the mass transfer coefficient (k

_{c}), respectively. It can be seen that, as the D

_{eP}value and k

_{c}value increase, the P concentration in the extract and the yield increase. Similar impacts were observed for k

_{c}. Figure 5d and Figure 6d illustrate the sensitivity of the model to the thickness of the pericarp. The occurrence of these phenomena is conceivable, since, the thinner the slab, the faster the diffusion.

_{P}

_{0}) and a high slab thickness (L), by increasing the effective diffusivity value (D

_{eP}), for example, via ultrasound, or by increasing the mass transfer coefficient (k

_{c}) via agitation.

#### 3.3. Validation of the Model Using Analytical Results

#### 3.3.1. Validation of the Model under the Conditions of a Very Thick Slab and Very Slow Diffusivity

_{P}= C

_{P}

_{0}for t = 0, all x and C

_{P}= 0 for x = L all t.

_{p}is the concentration of

**P**compound, x is the distance from the inside of the pericarp, and C

_{pf}

_{0}is the concentration at x = 0.

^{−8}cm

^{2}/min., a mass transfer coefficient of 4·10

^{−4}cm/min., an extraction time of 400 min., and a relatively large liquid volume of 50,000 mL (at an almost constant extract concentration). This shows the consistency of the simulation that was applied in this case.

#### 3.3.2. Validation of the Model under the Condition of a Constant Concentration in the Liquid Phase

_{P}= C

_{P}

_{0}for t = 0, all x; C

_{P}= 0 $\mathrm{at}x=L;all\mathrm{t};\frac{\partial {C}_{p}}{\partial x}$ = 0 at x = 0, all t and C

_{p}= 0 for t = ∞, all x.

^{−6}cm

^{2}/min., a mass transfer coefficient of 4·10

^{−4}cm/min., an extraction time of 4000 min., a radius of sphere of 4 cm, and a relatively large liquid volume of 500,000 mL. From Figure 8, it can be observed that the numerical solution matches well with the analytical solutions.

#### 3.4. Validation of the Model Using Experimental Results

_{c}). An experimental extraction was performed while using 178 grains of red sorghum with a radius of 0.19 cm, a pericarp thickness of 0.01 cm, an initial concentration of 0.1263 g P

_{equivalent}/cm

^{3}pericarp, and a free solvent volume of 45 mL. A series of P concentration values in aqueous extract measured in 30-min. intervals over 150 min. were used to test the accuracy of the proposed model and to determine the values of the parameters of the simulated model. From the results of the simulation, the values of some of the parameters of the model have been determined and they are shown in Table 2. The parameter’s values were determined by minimizing the sum of the squares of errors (SSE) to the experimental data. The simulation results were compared with the experimental data (Figure 9) to check the accuracy of the proposed model.

_{c}, and it supports the results of previous studies that found that a higher temperature improves extraction performance [24,48,49]. The profile of the extraction process as a function of time and temperature is also precisely illustrated through this model, as shown in Figure 9.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The profile of the phytochemical concentration inside the pericarp as a function of: (

**a**) position; and (

**b**) extraction time.

**Figure 3.**The profile of the phytochemical concentration inside the pericarp as a function of: (

**a**) position; and (

**b**) extraction time.

**Figure 4.**The profile of: (

**a**) P concentration in aqueous extract; and, (

**b**) the yield of extraction as a function of extraction time.

**Figure 5.**The effect of the: (

**a**) initial P concentration in the pericarp; (

**b**) effective diffusivity; (

**c**) mass transfer coefficient; and, (

**d**) slab thickness on the P concentration in the extract.

**Figure 6.**The effect of the: (

**a**) initial P concentration in the pericarp; (

**b**) effective diffusivity; (

**c**) mass transfer coefficient; and, (

**d**) slab thickness on the yield of the extraction.

**Figure 7.**The comparison of the analytical solution and the numerical solution under the conditions of a very thick slab and very slow diffusivity.

**Figure 8.**The comparison of the analytical solution and the numerical solution under the conditions of a constant phytochemical concentration in the solvent and very slow diffusion.

Parameter | Value | Unit |
---|---|---|

D_{eP} | 0.0000025 | cm^{2}/min |

R | 0.3 | cm |

L | 0.03 | cm |

H | 0.5 | - |

V | 500 | mL |

π | 3.14159 | - |

N_{b} | 20 | grain |

dx | 0.003 | cm |

k_{c} | 0.0004 | cm/min |

C_{P}_{0} | 0.1 | g P_{equivalent}/cm^{3} _{pericarp} |

C_{Pf}_{0} | 0 | g P_{equivalent}/cm^{3} _{solvent} |

Temperature (°C) | D_{e}_{P} × 10^{7} (cm^{2}/min) | H × 10^{4} | k_{c} × 10^{2} (cm/min) | SSE × 10^{10} |
---|---|---|---|---|

50 | 0.926 | 4.18 | 3.8 | 1.65 |

60 | 1.11 | 6.33 | 2.53 | 0.264 |

70 | 167 | 11 | 2 | 2.4 |

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**MDPI and ACS Style**

Susanti, D.Y.; Sediawan, W.B.; Fahrurrozi, M.; Hidayat, M.
A Mechanistic Model of Mass Transfer in the Extraction of Bioactive Compounds from Intact Sorghum Pericarp. *Processes* **2019**, *7*, 837.
https://doi.org/10.3390/pr7110837

**AMA Style**

Susanti DY, Sediawan WB, Fahrurrozi M, Hidayat M.
A Mechanistic Model of Mass Transfer in the Extraction of Bioactive Compounds from Intact Sorghum Pericarp. *Processes*. 2019; 7(11):837.
https://doi.org/10.3390/pr7110837

**Chicago/Turabian Style**

Susanti, Devi Yuni, Wahyudi Budi Sediawan, Mohammad Fahrurrozi, and Muslikhin Hidayat.
2019. "A Mechanistic Model of Mass Transfer in the Extraction of Bioactive Compounds from Intact Sorghum Pericarp" *Processes* 7, no. 11: 837.
https://doi.org/10.3390/pr7110837