# Model-Based Real Time Operation of the Freeze-Drying Process

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## Abstract

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## 1. Introduction

- It shows how the identification problem (i.e., characterization of product and equipment) can be simplified by solving a (decoupled) sequence of parameter identification problems, this is, by considering food matrix and freeze-drying chamber as two separate subsystems, which are described by physics-based operational models developed to satisfy identifiability.
- It provides a robust, digital tool that supports decision-making in real time for complex food structuring processes (such freeze-drying). This has potential for significant reductions in cost and processing times, waste generation (e.g., reducing batch rejection) and energy demand (e.g., off-line and in-line strategies can be used to minimize energy use during processing) [10,28].

## 2. Lyophilization Plant and Mathematical Model

#### 2.1. Lyophilization Pilot Plant and Experimental Setup Description

#### 2.2. A Physical Model for the Freeze-Drying Process

- Freezing. The product is frozen in controlled conditions to avoid possible damage, by crystal growth, to the food or biological material. Our model does not include this stage. The product temperature is assumed to be uniform at the end of the freezing stage, and it is used as the initial temperature for the primary drying.
- Primary drying. In this stage, ice is removed from the product by sublimation. Pressure conditions are kept below the triple point and the product is heated from the bottom. An excessive temperature increase in this stage will cause product to collapse so it must be kept below a given value, see Section 2.2.4 for details.
- Secondary drying. The aim is to remove water bound to the solid matrix by desorption. This stage allows for reaching low moisture contents. The food product is more stable during secondary drying so its temperature may be increased to accelerate the process.

- The frozen region has uniform heat and mass transfer properties.
- The interface between the frozen and dried layers (sublimation front) is continuous and has infinitesimal thickness.
- Vapor and ice are at equilibrium at the interface.
- The matrix pore structure is permeable to the vapor flux and it is not deformable.

#### 2.2.1. The Primary Drying

#### 2.2.2. The Secondary Drying

#### 2.2.3. The Condenser Model

#### 2.2.4. The Desorption Model

## 3. Strategies for Model Parameter Identification

- A set of model parameters to be estimated. Usually, these are parameters that are difficult to measure or cannot be found in the literature.
- Experimental measurements to compare against model results. Measured variables are usually state variables (e.g., temperature, concentration, pressure) or combinations of state variables. The quantities to be measured are known as the observables. Changes on the unknown model parameters should have an effect on the observables, otherwise such parameters cannot be estimated.
- A measure of the distance between model predictions and experimental data (cost function).
- An optimization algorithm to find the parameter values that minimize the cost function.

- Estimation of the unknown parameters involved in the product model, using the indirect measurements of ${P}_{ch}^{v}$ instead of the condenser model. On the one hand, this reduces the number of parameters to be estimated together since the unknown condenser parameter is not taken into account. On the other hand, possible modeling errors on the condenser model are avoided.
- Estimation of the unknown parameters involved in the condenser model. As it is sketched in Figure 2, this estimation requires the product model and the parameters estimated in the previous step to compute the front velocity. Note that, if a procedure to measure w, such as an on-line sensor of front velocity, were available, product model could be disregarded in this step.

- Product freezing using −50 ${}^{\xb0}\mathrm{C}$ as a set point (cooling rate of 0,6 ${}^{\xb0}\mathrm{C}\text{}\mathrm{min}{-}^{1}$).
- Primary drying varying from freezing temperature to −10, 0 or 20 ${}^{\xb0}\mathrm{C}$ depending on the experiment.
- Secondary drying using 25 ${}^{\xb0}\mathrm{C}$ as the set point.
- Total chamber pressure was kept at 20 or 60 $\mathrm{Pa}$ depending on the experiment.

#### 3.1. Parameter Estimation in the Product/Chamber Subsystem

#### 3.2. Parameter Estimation for the Condenser Model

## 4. Real time Optimization of the Freeze-Drying Process

#### 4.1. Off-Line Dynamic Optimization

#### 4.2. Real Time Optimization

- An optimal profile for the shelf temperature is computed as in previous section.
- Such profile is sent to the freeze-drying plant as the set point for the PI control.
- As the process is running, measurements of relevant variables are being recorded. In this case, we measure, shelf temperature, chamber temperature, Pirani pressure and product temperature.
- After a given period of time, 1 $\mathrm{h}$ in this case, measured information is introduced in the model. Then, a new optimal profile is computed by solving the optimization problem defined in Equation (24) and taking into account the new available information.
- Steps 2–4 are repeated till the end of the process.

- Product freezing temperature differs from the expected one, i.e., from the one used in Figure 6a,c. This was because the refrigerant group was not able to reach −50 ${}^{\xb0}\mathrm{C}$.
- Initial chamber pressure was around 60 $\mathrm{Pa}$ and it was supposed to be 10 $\mathrm{Pa}$. Besides, chamber pressure controller did not perform as expected and most of the time it remains below the set point (10 $\mathrm{Pa}$).
- Shelf temperature controller is not able to exactly follow the optimal profiles.

- We do not use available plant measurements and force the system to follow the off-line profile previously computed (dashed black line in Figure 7a).

- Unknown disturbances cause lower product temperatures than the predicted ones in the off-line scheme. In this case, final product water content might be larger than the targeted one, i.e., product quality would be lower than required. The RTO scheme can either increase process time or increase shelf temperature to reach the required quality.
- Unknown disturbances cause higher product temperatures than the predicted ones in the off-line scheme. In this case, safety constraints might be not fulfilled which can cause product collapse. The RTO scheme can reduce shelf temperature to avoid product collapse.

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. The Landau Transform

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**Figure 2.**Schematic representation of the freeze drying process. Two subsystems are considered, the condenser and the product. The arrows entering the subsystems represent the process control variables whereas output arrows represent the state variables.

**Figure 3.**Predictive capabilities of the freeze-drying model from the product temperature point of view. Blue line corresponds to the shelf temperature. Continuous black line and black dots represent, respectively, the model predictions and experimental measurements at the bottom of the product. Dashed black lines correspond to the model prediction at other spatial points in the product. Vertical lines represent the end of primary drying and the beginning of secondary drying.

**Figure 4.**Predictive capabilities of the freeze-drying model for the chamber vapor pressure. Black circles represent the chamber vapor pressure obtained from measurements of Pirani and total chamber pressure whereas continuous black lines correspond to the model predictions.

**Figure 5.**Evolution of the product temperature and shelf temperature (top figures) as well as water content (bottom figures). Figures on the left correspond with experiment 1 whereas the optimal design results are represented on the figures on the right. Horizontal line on the bottom figures represent the constraint on the final water content.

**Figure 6.**Offline and real time optimization (RTO) scheme results. Subfigures (

**a**,

**c**,

**e**) represent the evolution of shelf temperature (decision variable) and chamber pressure. Black and gray parts of the profiles correspond with the optimal solution. Blue and red parts correspond with the plant measurements. Subfigures (

**b**,

**d**,

**f**) represent the evolution of product mean water content. Dashed line indicates the quality objective to reach.

**Figure 7.**Comparison, in terms of process duration, between the RTO approach (continuous black lines) and the off-line control profile (dashed black lines). (

**a**) Shelf temperature and chamber pressure. (

**b**) Mean water content evolution. Blue and red segments in subplot (

**a**) correspond with the real plant measurements for shelf temperature and chamber pressure, respectively.

**Table 1.**Parameters involved in the freeze-drying model. Parameters to be estimated are indicated in the “Value” column as t.b.e.

Parameter | Value | Units | Description |
---|---|---|---|

${\rho}_{dr}$ | 200.31 | $\mathrm{k}\mathrm{g}\text{}{\mathrm{m}}^{-3}$ | Dried region density |

${\rho}_{fr}$ | 1001.6 | $\mathrm{k}\mathrm{g}\text{}{\mathrm{m}}^{-3}$ | Frozen region density |

${c}_{p,dr}$ | 1254 | $\mathrm{J}\text{}\mathrm{k}{\mathrm{g}}^{-1}\text{}{\mathrm{K}}^{-1}$ | Dried region heat capacity |

${c}_{p,fr}$ | 1818.8 | $\mathrm{J}\text{}\mathrm{k}{\mathrm{g}}^{-1}\text{}{\mathrm{K}}^{-1}$ | Frozen region heat capacity |

${\kappa}_{dr}$ | t.b.e. | $\mathrm{W}\text{}{\mathrm{m}}^{-1}\text{}{\mathrm{K}}^{-1}$ | Dried region heat conductivity |

${\kappa}_{fr}$ | 2.4 | $\mathrm{W}\text{}{\mathrm{m}}^{-1}\text{}{\mathrm{K}}^{-1}$ | Frozen region heat conductivity |

$\sigma $ | 5.6704 × 10^{−8} | $\mathrm{W}\text{}{\mathrm{m}}^{-2}\text{}{\mathrm{K}}^{-4}$ | Stefan–Boltzmann constant |

${e}_{p}$ | 0.78 | - | Thermal emissivity at the product top |

${f}_{p}$ | 0.99 | - | Geometrical correction factor |

${K}_{clap}$ | 1.6548 × 10^{−4} | ${\mathrm{K}}^{-1}$ | Constant in the Clapeyron equation |

$\Delta {H}_{s}$ | 2791.2 × 10^{−3} | $\mathrm{J}\text{}\mathrm{k}{\mathrm{g}}^{-1}$ | Sublimation heat |

R | 8314 | $\mathrm{Pa}{\mathrm{m}}^{3}\text{}{\mathrm{K}}^{-1}\text{}{\mathrm{kmol}}^{-1}$ | Ideal gas constant |

${L}_{x}$ | 5.75 × 10^{−3} | $\mathrm{m}$ | Food product height |

${L}_{z}$ | 0.242 | $\mathrm{m}$ | Food product length |

${L}_{y}$ | 0.307 | $\mathrm{m}$ | Food product width |

${M}_{w}$ | 18 | $\mathrm{k}\mathrm{g}\text{}{\mathrm{koml}}^{-1}$ | Water molecular mass |

${h}_{L,1}$ | 3.3 | $\mathrm{W}\text{}{\mathrm{m}}^{-2}\text{}{\mathrm{K}}^{-1}$ | Heat transfer coefficient constant |

${h}_{L,2}$ | t.b.e. | ${\mathrm{Pa}}^{-1}$ | Heat transfer coefficient constant |

${h}_{L,3}$ | 34.4 | $\mathrm{Pa}$ | Heat transfer coefficient constant |

${k}_{1}$ | 430.0 | $\mathrm{s}\text{}\mathrm{m}\text{}\mathrm{k}{\mathrm{g}}^{-1}$ | Mass transfer coefficient constant in Darcy’s equation |

${k}_{2}$ | t.b.e. | $\mathrm{s}\text{}\mathrm{Pa}\text{}\mathrm{k}{\mathrm{g}}^{-1}$ | Mass transfer coefficient constant in Darcy’s equation |

$\beta $ | t.b.e. | $\mathrm{k}\mathrm{g}\text{}{\mathrm{s}}^{-1}\text{}{\mathrm{K}}^{-1}$ | Mass transfer coefficient constant in |

chamber/condenser flux | |||

${T}_{ch}$ | 293.15 | $\mathrm{K}$ | Chamber temperature |

${V}_{ch}$ | 0.202 | ${\mathrm{m}}^{3}$ | Chamber volume |

${\tau}_{A}^{ref}$ | 2.689 × 10^{4} | $\mathrm{s}$ | Compartment A reference time constant |

${\tau}_{B}^{ref}$ | 6.493 × 10^{5} | $\mathrm{s}$ | Compartment B reference time constant |

${E}_{a}$ | 4.271 × 10^{4} | $\mathrm{k}\mathrm{J}\text{}\mathrm{k}{\mathrm{g}}^{-1}$ | Activation energy in desorption model |

${T}_{ref}$ | 273.15 | $\mathrm{K}$ | Reference temperature in desorption model |

${\alpha}_{A}$ | 0.669 | - | Compartment A ratio between equilibrium water content |

${\alpha}_{B}$ | 0.331 | - | Compartment B ratio between equilibrium water content |

${M}_{g}$ | 0.0434 | $\mathrm{k}\mathrm{g}$-water $\text{}\mathrm{k}{\mathrm{g}}^{-1}$-total | Constant of the GAB equation |

${C}_{g}$ | 7.4789 | - | Constant of the GAB equation |

${K}_{g}$ | 0.9827 | - | Constant of the GAB equation |

${K}_{T,g}$ | 8.2 | - | Constant of the glass transition temperature |

${T}_{g,l}$ | −135 | °C | Constant of the glass transition temperature |

${T}_{g,s}$ | 75.58 | °C | Constant of the glass transition temperature |

$\alpha $ | 1.6 | - | Ratio of molecular heat conductivities of vapor |

and nitrogen |

**Table 2.**Cross validation results for the product model. Five estimations of the parameters were performed, each one excluding a different experiment. $RMS{E}_{e}$ and $RMS{E}_{v}$ denote, respectively, the Root Mean Square Error (RMSE) values obtained for the estimation and validation experiments.

Parameter | Excluded Experiment | ||||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |

${h}_{L,2}$ | 1.96 | 1.86 | 1.93 | 1.82 | 2.03 |

${k}_{2}$ | 8.60 × 10^{7} | 7.86 × 10^{7} | 6.52 × 10^{7} | 9.41 × 10^{7} | 9.51 × 10^{7} |

${\kappa}_{dr}$ | 2.45 × 10^{−4} | 4.13 × 10^{−4} | 2.41 × 10^{−4} | 2.85 × 10^{−4} | 2.85 × 10^{−4} |

$RMS{E}_{e}$ | 2.15 | 1.64 | 2.11 | 2.10 | 2.00 |

$RMS{E}_{v}$ | 1.62 | 3.10 | 2.01 | 2.49 | 2.54 |

**Table 3.**Cross validation results for the product/condenser model. Five estimations of parameter $\beta $ were carried out, each one excluding a different experiment. $RMS{E}_{e}$ and $RMS{E}_{v}$ denote, respectively, the RMSE values obtained for the estimation and validation experiments.

Parameter | Excluded Experiment | ||||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |

$\beta $ | 2.31 × 10^{3} | 2.17 × 10^{3} | 2.31 × 10^{3} | 2.37 × 10^{3} | 2.37 × 10^{3} |

$RMS{E}_{e}$ | 1.6 | 1.7 | 2.2 | 2.0 | 1.8 |

$RMS{E}_{v}$ | 2.8 | 2.1 | 0.6 | 0.6 | 2.3 |

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**MDPI and ACS Style**

Vilas, C.; A. Alonso, A.; Balsa-Canto, E.; López-Quiroga, E.; Trelea, I.C.
Model-Based Real Time Operation of the Freeze-Drying Process. *Processes* **2020**, *8*, 325.
https://doi.org/10.3390/pr8030325

**AMA Style**

Vilas C, A. Alonso A, Balsa-Canto E, López-Quiroga E, Trelea IC.
Model-Based Real Time Operation of the Freeze-Drying Process. *Processes*. 2020; 8(3):325.
https://doi.org/10.3390/pr8030325

**Chicago/Turabian Style**

Vilas, Carlos, Antonio A. Alonso, Eva Balsa-Canto, Estefanía López-Quiroga, and Ioan Cristian Trelea.
2020. "Model-Based Real Time Operation of the Freeze-Drying Process" *Processes* 8, no. 3: 325.
https://doi.org/10.3390/pr8030325