# Modelling of Catechin Extraction from Red Grape Solids under Conditions That Simulate Red Wine Fermentation

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

^{2}) for all test scenarios exceeded 0.94, indicating a good fit to the experimental data. Sensitivity analysis for the extraction rate and internal cross-validation showed the model to be robust, with a small standard error in cross-validation (SECV) of 0.11 and a high residual predictive deviation (RPD) of 17.68. The model that was developed is well suited to digital technologies where low computational overheads are desirable, and industrial application scenarios are presented for future implementation of the work.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental

#### 2.1.1. Simulated Catechin Extraction—Experimental Design

#### 2.1.2. Preparation of Grape Solids

^{−1}, with total soluble solids of 14.3° Baumé, pH 3.6, and titratable acidity (to pH 8.2) of 3.9 g L

^{−1}.

#### 2.1.3. Solid–Liquid Extractions

#### 2.1.4. Catechin Quantification

#### 2.2. Extraction Model Development

^{0}denotes the extractable amount of catechin in grape skin and seed at the beginning, and P(t) denotes the amount at time t. The total amount of catechin in grape skin and seed is hence P

^{0}$+$ P(t). The amount of catechin extracted in the liquid is denoted by C(t). The instantaneous change in quantity over time is assumed proportional to the quantity currently present, with the proportionality provided by the extraction rate constant k [22], which is independent of time but varies for different values of temperature [23] and solvent concentrations (ethanol) [24]. Due to the conversion of sugar into ethanol during alcoholic fermentation [15], concentrations of these two components are directly connected. With these notations and assumptions, the symbolic Equation (1) can be formulated through the following first-order system of ordinary differential equations (ODEs) with initial values P

^{0}and C

^{0}at t = 0.

^{0}+ P

^{0}:

^{∞}(= C

^{0}+ P

^{0}), this solution can be rewritten in the form of

^{0}to zero with rate k, and Equation (7) shows that the extracted catechin amount, C(t), increases from C

^{0}to the maximal value C

^{∞}accordingly with the degradation of P(t). There are three free parameters (k, C

^{0}and C

^{∞}) for fitting the curve C(t) with measured data. Adding Equations (2) and (3) shows that the sum P(t) + C(t) = P

^{0}+ C

^{0}= constant = C

^{∞}. This demonstrates that the law of mass conservation is fulfilled. In an ideal experiment, C

^{0}is zero and practically it is close to zero, such that C

^{∞}is equal to P

^{0}. Usually, only the concentration of a substance in solution is measured rather than its total mass. The catechin concentration $\tilde{C}\left(t\right)$ in the solution is hence provided by

_{l}is the fixed volume (1.5 L in these experiments). The loss by periodically removing samples is neglected as this volume (10 mL) was notably lower than the total volume overall.

^{0}(≈ C

^{∞}) that can be extracted depends on temperature and ethanol concentration [25,26]. As sugar is fermented into alcohol, the concentrations of these two components are dependent on each other in a real wine fermentation, but considering that the objective of this study was to develop a model for future control system application, only one of these needs to be included in the model. Since sugar/glucose measurements are already routinely collected by wineries for monitoring fermentation progress, the rate constant k was investigated in dependence of temperature (T) and glucose concentration (G). Equation (7) corresponds to equations stated for flavonoid extraction derived from grape skin by Boulton et al. [15] and total phenolic content by Zanoni et al. [19]. However, unlike those studies where k was taken to be a fixed value, k was assumed here (from work by Setford et al. [12]) to vary based on temperature T and glucose concentration G, which defines a function k of these two variables. This function was then approximated by a Taylor polynomial of second order and coefficients were computed by fitting. This will yield different mathematical solutions of function C(t) for different values of k. Dimensionless temperature and glucose concentrations (Table 1 and Table 2) were used to preserve dimensional consistency as shown by respective Equations (9) and (10):

_{0}and G

_{0}being defined as the centre point of the system at the medium temperature condition (12.2 °C).

_{1}to c

_{6}can be computed by non-linear regression:

^{∞}, the maximal extractable catechin concentration, is also dependent on the temperature and glucose concentration [25], hence a Taylor polynomial of second order was again used, with parameters likewise determined by non-linear regression:

#### 2.3. Model Fitting and Statistical Analysis

^{2}) and root mean square error (RMSE) were calculated:

^{2}.

#### 2.4. Control System Design

^{2}

_{cal}), the standard error in cross validation (SECV), and the residual predictive deviation (RPD) were calculated. The RPD is calculated as the ratio of the standard deviation of the reference (experimental) values to the SECV. A robust model is indicated by a relatively high RPD value compared to that of the SECV [35].

## 3. Results and Discussion

#### 3.1. Experimental Catechin Extrction and Model Performance

^{2}and RMSE values obtained from the experimental data as described in Equations (13) and (14). Small RSME values and R

^{2}> 0.94 were observed in all cases (Table 3).

^{2}of 0.98 and a small RSME for these conditions indicate an appropriate model fit. The extraction curve for simulated juice at high temperature showed similar behaviour, but likewise had a favourable R

^{2}of 0.94 and a small RSME, demonstrating a satisfactory fit overall.

^{2}of 0.98 in more than 91% of the cases), but also that the more complex models converged to the first-order model after a short time. According to Simonin, first- and second-order rate laws are both valid to describe reaction kinetics; however, second order is unable to describe steep extraction rates in short times [39] as seen in the current experiments. Previous models used in other wine-related extraction studies, however, do not account for any dependencies of the extraction rate.

_{1}to c

_{6}describing the extraction rate k, and d

_{1}to d

_{6,}describing the maximum extracted catechin $\tilde{C}$∞.

_{undiluted}/V

_{l}= 8.79 must be included in the calculation for ${\tilde{\mathrm{C}}}_{\mathrm{corr}}^{\infty}\left(\tilde{T},\tilde{G}\right)$ to estimate the maximum extractability of catechin that could be observed in undiluted industrial red wine fermentation:

^{2}values ranged between 0.93 and 0.96, suggesting a high degree of model robustness. Table S4 containing the re-calculated R

^{2}values can be found in Supplementary Materials. Additionally, an internal cross-validation method was used to test and validate the model, as explained in Section 2.4. The data were divided into ten subsets, one of which was reserved for validation while the remaining nine subsets were utilised for training. This process was repeated multiple times, with each subset serving as the validation set. The R

^{2}

_{cal}, the SECV, and the RPD were calculated. The R

^{2}

_{cal}obtained from this validation was 0.93, which is considered to be a good value. The high RPD of 17.68 compared to the small SECV of 0.11 indicate a robust model.

#### 3.2. Industrial Application: Future Implementation of Models for Process Control

## 4. Conclusions

^{2}(>0.94) and low RMSE values in all cases. The robustness of the model was evaluated using four metrics. R

^{2}

_{cal}obtained from this validation was 0.93, indicating good performance. The standard error in cross-validation (SECV) was calculated to be 0.11, and the residual predictive deviation (RPD) was calculated to be 17.68, which suggests a robust model. In addition, a sensitivity analysis on extraction rate k determined that even at a +50% adjustment to the fitted values, R

^{2}was maintained between 0.93 and 0.96, suggesting a high degree of model robustness.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

C | Catechin amount (mg) |

C^{0} | Initial catechin amount (mg) |

$\tilde{C}$ | Catechin concentration (mg/L) |

${\tilde{C}}_{}^{\infty}$ | Maximal extractable catechin concentration (mg/L) |

${\tilde{C}}_{corr}^{\infty}$ | Maximal extractable catechin concentration in undiluted industrial red wine fermentation (mg/L) |

${C}_{pred,i}$ | Catechin concentration predicted by model (mg/L) |

${C}_{obs,i}$ | Catechin concentration observed experimentally (mg/L) |

${\overline{C}}_{obs,i}$ | Mean of the catechin concentration observed experimentally (mg/L) |

c_{1}-c_{6} | Constants describing catechin extraction rate |

d_{1}-d_{6} | Constants describing maximum extracted catechin |

G | Glucose concentration (g/L) |

G_{0} | Glucose concentration at centre point of the system (g/L) |

${\tilde{G}}_{}^{}$ | Dimensionless glucose |

k | Catechin extraction rate (1/h) |

N | Number of replicates |

P(t) | Extractable amount of catechin in grape pomace at time t (mg) |

R^{2} | Coefficient of determination |

RMSE | Root mean square error |

t | Time (h) |

T | Temperature (°C) |

T_{0} | Temperature at centre point of the system (°C) |

${\tilde{T}}_{}^{}$ | Dimensionless temperature |

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**Figure 1.**Catechin extraction in simulated juice (266 g/L glucose) and simulated wine (14% v/v ethanol) at 4.4 °C, showing experimental values (symbol) and fitted models (solid line).

**Figure 2.**Catechin extraction in simulated mid-fermentation (133 g/L glucose, 7% v/v ethanol) at 12.2 °C, showing experimental values (symbol) and fitted model (solid line).

**Figure 3.**Catechin extraction in simulated juice (266 g/L glucose) and simulated wine (14% v/v ethanol) at 23.1 °C, showing experimental values (symbol) and fitted models (solid line).

**Figure 4.**Catechin extraction rate constant k (1/h) depending on dimensionless glucose ($\tilde{G}$) and temperature ($\tilde{T}$) values, showing experimental data (•) and model response surface.

**Figure 5.**Maximum extracted catechin concentration $\tilde{C}$∞ (mg/L) depending on dimensionless glucose ($\tilde{G}$) and temperature ($\tilde{T}$) values, showing experimental data (•) and model response surface.

**Figure 6.**Model response surface for predicted maximum extracted catechin concentration corrected for dilution ${\tilde{C}}_{corr}^{\infty}\left(\tilde{T},\tilde{G}\right)$ (mg/L) depending on dimensionless glucose ($\tilde{G}$) and temperature ($\tilde{T}$) values.

**Figure 7.**Process instrumentation diagram (PID) of a typical red wine fermentor, showing glucose and temperature measurements influencing the utilisation of cooling system via a process-control system.

**Table 1.**Summary of the temperature conditions used, with corresponding dimensionless temperature ($\tilde{T}$) values based on Equation (8).

$\mathit{T}$(°C) | $\tilde{\mathit{T}}$ | |
---|---|---|

T(low) | 4.4 | −0.639 |

T(med) | 12.2 | 0 |

T(high) | 23.1 | 0.893 |

**Table 2.**Summary of glucose (g/L) conditions used, with corresponding dimensionless glucose ($\tilde{G}$) values based on Equation (9).

$\mathit{G}$(g/L) | $\tilde{\mathit{G}}$ | |
---|---|---|

G(low) | 0 | −1 |

G(med) | 133 | 0 |

G(high) | 266 | 1 |

**Table 3.**Summary of trial conditions (simulated juice (sim-juice), mid-ferment (sim-mid-ferment), and wine (sim-wine)) and model parameters.

Trial Conditions | Model Parameters | Model Fit | ||||||
---|---|---|---|---|---|---|---|---|

Temp. (°C) | Glucose (g/L) | Ethanol (% v/v) | $\tilde{\mathit{C}}$_{0} (mg/L)
| $\tilde{\mathit{C}}$_{∞} (mg/L)
| k (1/h) | RMSE | R^{2} | |

Sim-Juice | Low (4.4) | 266 | 0 | 0.452 | 4.856 | 0.152 | 0.268 | 0.971 |

Sim-Wine | Low (4.4) | 0 | 14 | 0.867 | 9.457 | 0.109 | 0.480 | 0.976 |

Sim-Mid-Ferment | Med (12.2) | 133 | 7 | 0.889 | 8.603 | 0.116 | 0.407 | 0.979 |

Sim-Juice | High (23.1) | 266 | 0 | 1.473 | 9.898 | 0.118 | 0.737 | 0.944 |

Sim-Wine | High (23.1) | 0 | 14 | 0.059 | 11.190 | 0.595 | 0.451 | 0.984 |

**Table 4.**Fitted values for the rate constants c

_{1}, c

_{2}, c

_{3}, c

_{4}, c

_{5}and c

_{6}describing the extraction rate from Equation (11).

Constant | Fitted Value (h^{−1}) |
---|---|

c_{1} | 0.116 |

c_{2} | 0.090 |

c_{3} | −0.087 |

c_{4} | 0.225 |

c_{5} | −0.021 |

c_{6} | −0.170 |

**Table 5.**Fitted values for the rate constants d

_{1}, d

_{2}, d

_{3}, d

_{4}, d

_{5}and d

_{6}describing the maximum extracted catechin from Equation (12).

Constant | Fitted Value (mg/L) |
---|---|

d_{1} | 8.603 |

d_{2} | 2.072 |

d_{3} | −1.610 |

d_{4} | 0.544 |

d_{5} | −0.344 |

d_{6} | 1.079 |

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## Share and Cite

**MDPI and ACS Style**

Unterkofler, J.; Jeffery, D.W.; Setford, P.C.; Macintyre, J.; Muhlack, R.A.
Modelling of Catechin Extraction from Red Grape Solids under Conditions That Simulate Red Wine Fermentation. *Fermentation* **2023**, *9*, 394.
https://doi.org/10.3390/fermentation9040394

**AMA Style**

Unterkofler J, Jeffery DW, Setford PC, Macintyre J, Muhlack RA.
Modelling of Catechin Extraction from Red Grape Solids under Conditions That Simulate Red Wine Fermentation. *Fermentation*. 2023; 9(4):394.
https://doi.org/10.3390/fermentation9040394

**Chicago/Turabian Style**

Unterkofler, Judith, David W. Jeffery, Patrick C. Setford, Jean Macintyre, and Richard A. Muhlack.
2023. "Modelling of Catechin Extraction from Red Grape Solids under Conditions That Simulate Red Wine Fermentation" *Fermentation* 9, no. 4: 394.
https://doi.org/10.3390/fermentation9040394