Optimal Control of the Primary Drying Stage of Freeze Drying of Solutions in Vials Using Variational Calculus

The problem of operating freeze drying of pharmaceutical products in vials placed in trays of a freeze dryer to remove free water (in frozen state) at a minimum time was formulated as an optimal control problem. Two different types of freeze dryer designs were considered. In type I freeze dryer design, upper and lower plate temperatures were controlled together, while in type II freeze dryer design, upper and lower plate temperatures were controlled independently. The heat input to the material being dried and the drying chamber pressure were considered as control variables. Constraints were placed on the system state variables by the melting and scorch temperatures during primary drying stage. Necessary conditions of optimality for the primary drying stage of freeze drying process in vials are derived and presented. Furthermore, an approach for constructing the optimal control policies that would minimize the drying time for the primary drying stage was given. In order to analyze optimal control policy for the primary drying stage of the freeze-drying process in vials, a rigorous multi-dimensional unsteady state mathematical model was used. The theoretical approach presented in this work was applied in the freeze drying of skim milk. Significant reductions in the drying times of primary drying stage of freeze drying process in vials were obtained, as compared to the drying times obtained from conventional operational policies.     

Optimal Control of the Primary Drying Stage of Freeze Drying of Solutions in Vials Using Variational Calculus

Abstract

The problem of operating freeze drying of pharmaceutical products in vials placed in trays of a freeze dryer to remove free water (in frozen state) at a minimum time was formulated as an optimal control problem. Two different types of freeze dryer designs were considered. In type I freeze dryer design, upper and lower plate temperatures were controlled together, while in type II freeze dryer design, upper and lower plate temperatures were controlled independently. The heat input to the material being dried and the drying chamber pressure were considered as control variables. Constraints were placed on the system state variables by the melting and scorch temperatures during primary drying stage. Necessary conditions of optimality for the primary drying stage of freeze drying process in vials are derived and presented. Furthermore, an approach for constructing the optimal control policies that would minimize the drying time for the primary drying stage was given. In order to analyze optimal control policy for the primary drying stage of the freeze-drying process in vials, a rigorous multi-dimensional unsteady state mathematical model was used. The theoretical approach presented in this work was applied in the freeze drying of skim milk. Significant reductions in the drying times of primary drying stage of freeze drying process in vials were obtained, as compared to the drying times obtained from conventional operational policies.     

Research and Development Needs and Opportunities in Freeze Drying

Experimental and theoretical research and development studies are preposed and presented in important areas of the freeze drying process. The practical importance of the synergistic integration of advanced theoretical models and experiments is indicated, and the potential contributions of the proposed     

Research and Development Needs and Opportunities in Freeze Drying

ABSTRACT

Experimental and theoretical research and development studies are preposed and presented in important areas of the freeze drying process. The practical importance of the synergistic integration of advanced theoretical models and experiments is indicated, and the potential contributions of the proposed     

Spray Freeze Drying in a Fluidized Bed at Normal and Low Pressure

The aim of this study is to develop the spray freeze drying process and its hardware and to investigate its capabilities to dry thermosensible substances such as pharma-proteins at normal and low pressures. As the result, the spray freeze fluidized–bed dryer was constructed. During the study, the drying kinetic comparison between classical and spray freeze–drying technologies was done. Spray freeze drying has shown short process times and allows advanced control, product particle shape and size uniformity, and high solubility. This shows that the fluidized-bed freeze-drying process could be an alternative for classical freeze-drying processes. Identified problems are the low yield of the primary drying phase and the strong electrostatic effects during the secondary drying step. However, the innovative process has shown an excellent capability to dry and stabilize the thermosensitive substances, such as pharma-proteins.     

Spray Freeze Drying in a Fluidized Bed at Normal and Low Pressure

The aim of this study is to develop the spray freeze drying process and its hardware and to investigate its capabilities to dry thermosensible substances such as pharma-proteins at normal and low pressures. As the result, the spray freeze fluidized-bed dryer was constructed. During the study, the drying kinetic comparison between classical and spray freeze-drying technologies was done. Spray freeze drying has shown short process times and allows advanced control, product particle shape and size uniformity, and high solubility. This shows that the fluidized-bed freeze-drying process could be an alternative for classical freeze-drying processes. Identified problems are the low yield of the primary drying phase and the strong electrostatic effects during the secondary drying step. However, the innovative process has shown an excellent capability to dry and stabilize the thermosensitive substances, such as pharma-proteins.     

瓶装物料的冷冻干燥二维模型的建立与求解

本文对瓶装物料的冻干建立了一个考虑因素较为全面的二维数学模,该模型了侧面传热、结合水解吸和水蒸气温升对冻干过程的影响。根据冻干过程和模型方程的特点,本文把二循环对称分裂格式、预测－校正格式等算法联立起来对模型进行了数值求解,这些工作比目前的“只建模,不求解”或“虽求解,但模型简单、适用范围窄”的研究进行了一步。     

Freeze-Drying of Pharmaceutical Crystalline and Amorphous Solutes in Vials: Dynamic Multi-Dimensional Models of the Primary and Secondary Drying Stages and Qualitative Features of the Moving Interface

ABSTRACT

Dynamic and spatially multi-dimensional mathematical models of the primary and secondary drying stages of the freeze-drying of pharmaceutical crystalline and amorphous solutes in vials, are constructed and presented in this work. The models account for the removal of free and bound water and could also provide the geometric shape of the moving interface and its position. It is proved that the temperature of the moving interface can not be constant if the flux of heat flow to the sides of the vial is not zero. It is also proved that the slope of the free surface (moving interface) at the edge of the vial is always curved downward.

The numerical solution of the nonlinear partial differential equations of the models would allow model simulations that could indicate design conditions, operating conditions, and control strategies that could provide high drying rates and could lead to a series of novel experiments in freeze-drying.     

Freeze-Drying of Pharmaceutical Crystalline and Amorphous Solutes in Vials: Dynamic Multi-Dimensional Models of the Primary and Secondary Drying Stages and Qualitative Features of the Moving Interface

Dynamic and spatially multi-dimensional mathematical models of the primary and secondary drying stages of the freeze-drying of pharmaceutical crystalline and amorphous solutes in vials, are constructed and presented in this work. The models account for the removal of free and bound water and could also provide the geometric shape of the moving interface and its position. It is proved that the temperature of the moving interface can not be constant if the flux of heat flow to the sides of the vial is not zero. It is also proved that the slope of the free surface (moving interface) at the edge of the vial is always curved downward.

The numerical solution of the nonlinear partial differential equations of the models would allow model simulations that could indicate design conditions, operating conditions, and control strategies that could provide high drying rates and could lead to a series of novel experiments in freeze-drying.     

瓶装物料的冻干规律研究

以脱脂牛奶为物料,进行了瓶装物料的冷冻干燥试验,探索了加热温度,辐射温度与冻干室压强对冻干过程的影响规律,分析了所得冻干曲线的特征,并对冻干机理和物料温度的监测井行阐述,得到在加热温度与辐射温度相同时恒温恒压操作的最佳参数,并提出了一些强经冻干过程的措施。本研究对实验冻干生产有一定的指导作用,也为瓶装物料的二维理论模型研究提供了基础。     

Experimental Investigation of the Recovery of Soaked Paper Using Evaporative Freeze Drying

The aim of this article is to develop an experiment and a procedure to investigate the restoration of water-damaged paper and archival materials using freeze drying in order to allow a reproducible test and comparison of the influence of different operating conditions on drying time and restored paper quality. Firstly, a reproducible method for the preparation of soaked samples simulating water-damaged paper has been developed. Then, the samples have been freeze-dried in a laboratory-scale apparatus that allowed monitoring the temperature as well as the weight of the samples. The technique of evaporative freezing, which reduces the drying time required, has been used in this case. An innovative procedure for the visualization of the progress of the drying process has been validated, thus allowing the validation of a simple phenomenological model of the time evolution of the ice core volume; in addition, data on the residual moisture of the dried paper sheets in different zones have been given. Finally, optimization of this particular drying process by using simple or more sophisticated approaches has been discussed.     

Drying of Apple Slices in Atmospheric and Vacuum Freeze Dryer

Atmospheric freeze drying (AFD) and vacuum freeze drying (VFD) of Fuji apples was investigated. A factorial design was used in each case, with particle size, freezing rate, infrared (IR) radiation, and air temperature as factors. The effect of these factors on moisture content, duration of drying, rehydration properties, color, and texture were determined. The drying curves were fitted with both the simplified constant diffusivity model (SCDM) and the modified Page model, and rehydration curves were tested with the Peleg model, resulting in R² higher than 0.96. Antioxidant capacity, polyphenols, and ascorbic acid content in the final product were determined and compared with those obtained in a tunnel dryer. A sensory evaluation was performed.

The drying times obtained for VFD were shorter than the drying times obtained for AFD. The particle size and IR application were found to be the significant parameters that affect duration of drying for both AFD and VFD (duration of drying was proportional to particle size and inversely proportional to IR application). Air temperature affected drying time only during the secondary drying stage. Estimated D_eff values were on the order of 10^?10 (m²/s). In AFD, R² higher than 0.81 was obtained for the SCDM, and R² higher than 0.96 was obtained for the Page model. In VFD, a better fit was obtained (R² > 0.97) for both models. AFD produced nutritional alterations similar to convective drying. Sensory quality was not altered by AFD or VFD.     

Development of a High Gain Observer for In-Line Monitoring of Sublimation in Vial Freeze Drying

In-line monitoring of the primary drying of the lyophilization process of a pharmaceutical product in vials is addressed in this article. A soft sensor (observer) based on the high gain concept is designed to this purpose. The proposed observer combines the physical measure of the frozen product temperature at the bottom of the vial with a simplified mathematical model of the process. This approach allows the continuous estimation of the entire temperature profile of the product and the monitoring of the state of progression of the primary drying phase. The main issues arising in the design of this observer are discussed. A detailed and experimentally validated mono-dimensional model is used as a source of measured data, in order to compare and validate by means of numerical simulations the results provided by the soft sensor. Finally, the performance of the high gain observer is compared with that of a Kalman filter–based observer previously proposed for the same process, evidencing the advantages of the high gain approach.     

Heating Policies during the Primary and Secondary Drying Stages of the Lyophilization Process in Vials: Effects of the Arrangement of Vials in Clusters of Square and Hexagonal Arrays on Trays

The dynamic behavior of the primary and secondary drying stages of the lyophilization process were studied when (a) single vials located at different positions on the tray were individually being dried, and (b) the vials on the tray are arranged in clusters of square and hexagonal arrays and all the vials on the tray are simultaneously being dried. For both cases (a) and (b), fast drying times and relatively more uniform distributions of temperature and concentration of bound water at the end of the secondary drying stage are obtained by heat input control that runs the lyophilization process close to the melting and scorch temperature constraints. The heating control policies for the systems of case (b) are found to be more conservative and significantly more complex than those for the systems of case (a), because in case (b) there are vials on the tray that are in their secondary drying stage while other vials on the same tray have not yet completed their primary drying stage. Furthermore, the analysis and synthesis of the results presented in this work (i) indicate the minimum number of vials and their relative locations on the tray that have to be monitored by sensors so that the dynamic drying state of all the vials being dried simultaneously on the tray, could be satisfactorily determined in real time and appropriate, if necessary, control action could be applied, and (ii) suggest changes in the design of the freeze drying equipment so that the production rate of the freeze dryer could be improved and the product could also have enhanced properties of stability and quality at the end of the lyophilization process.     

Heating Policies during the Primary and Secondary Drying Stages of the Lyophilization Process in Vials: Effects of the Arrangement of Vials in Clusters of Square and Hexagonal Arrays on Trays

Abstract

The dynamic behavior of the primary and secondary drying stages of the lyophilization process were studied when (a) single vials located at different positions on the tray were individually being dried, and (b) the vials on the tray are arranged in clusters of square and hexagonal arrays and all the vials on the tray are simultaneously being dried. For both cases (a) and (b), fast drying times and relatively more uniform distributions of temperature and concentration of bound water at the end of the secondary drying stage are obtained by heat input control that runs the lyophilization process close to the melting and scorch temperature constraints. The heating control policies for the systems of case (b) are found to be more conservative and significantly more complex than those for the systems of case (a), because in case (b) there are vials on the tray that are in their secondary drying stage while other vials on the same tray have not yet completed their primary drying stage. Furthermore, the analysis and synthesis of the results presented in this work (i) indicate the minimum number of vials and their relative locations on the tray that have to be monitored by sensors so that the dynamic drying state of all the vials being dried simultaneously on the tray, could be satisfactorily determined in real time and appropriate, if necessary, control action could be applied, and (ii) suggest changes in the design of the freeze drying equipment so that the production rate of the freeze dryer could be improved and the product could also have enhanced properties of stability and quality at the end of the lyophilization process.     

How to Meet the Freeze Drying Standard in Combined Drying Processes: Pre and Finish Drying of Carrot Dice

Freeze-drying (FD) processes are well known to produce high-quality consumer products. Major problems are long drying times, high investment costs, and resulting maintenance and operating costs that make FD a very costly process. In this study, the potential of air drying (AD) and microwave vacuum drying (MVD) is tested to reduce freeze-drying times and make better utilization of the investment by combining FD with alternative lower cost drying processes using a pre or finish drying step while preserving the qualwity of the final product. Using carrot dice as a sample product, predrying did not lead to the desired qualities and reduction in drying time, whereas finish drying produced promising results. FD/MVD was 40% faster than pure FD and achieved freeze-drying standards regarding volume retention, shape, color, and rehydration. The good performance of microwave vacuum finish drying can directly be traced back to a time effect that is possibly linked to a characteristic collapse time of the cellular matrix. In order to use this effect, a stable skeleton must have developed at the point of process changeover, changeover has to be immediate, and finish drying has to be quick.     

Parametric Study of High-Intensity Ultrasound in the Atmospheric Freeze Drying of Peas

Atmospheric freeze drying (AFD) is a dehydration process that can be used to produce high-end products for the food, pharmaceutical, and biological industries. Evaporation or sublimation at the drying temperatures used for these processes is generally low. Airborne ultrasound can be used to increase drying rates. This parametric study investigates the influence of the drying temperature, drying time, and ultrasonic power for atmospheric freeze drying in the presence of an airborne ultrasonic field. Accelerated effective diffusion of up to 14.8% was obtained for atmospheric freeze drying with a fluid bed. The faster drying in ultrasonic-assisted atmospheric freeze drying is assumed to be due to a higher mass transfer rate at the solid-gas interface, caused by a reduced boundary layer due to a higher turbulent interface. Thus high intensity, airborne ultrasound used with modern drying systems has great potential to accelerate drying, reduce investment and production costs, and improve product quality.     

红外辅助冷冻干燥装置摔制系统的研发

真空泠冻干燥产品质量优异,广受青睐,但其干燥速率慢、耗能较大,制约其在普通产品干燥的推广应用。本文基于红外辅助加热和搁板传导的结合,提高对物料加热的有效性。并研发了红外辅助冷冻干燥装置的控制系统,以实现对物料冻结和加热的可控性和用户友好操作。     

Studies on the Microwave Freeze Drying Technique and Sterilization Characteristics of Cabbage

The vacuum freeze-drying (FD) technique used in the food industry can yield a high-quality product, but it is very expensive and requires a long processing time. Besides, the quantity of microorganisms in FD products can often exceed the required standard. As a result, it will be important to develop a new freeze-drying technique. In this article, cabbage was used as a model material, and the microwave field was used as a heat source to supply sublimation heat so that the drying time was shortened greatly. The effect of the microwave sterilization during the drying process was evaluated. Effects of the pressure, thickness of material being dried, and the input microwave power on such indices as drying time and the microorganism number were studied. Compared with the method of ordinary freeze drying, microwave freeze drying (MFD) can greatly reduce the drying time and has a notable sterilization effect.     

Studies on the Microwave Freeze Drying Technique and Sterilization Characteristics of Cabbage

The vacuum freeze-drying (FD) technique used in the food industry can yield a high-quality product, but it is very expensive and requires a long processing time. Besides, the quantity of microorganisms in FD products can often exceed the required standard. As a result, it will be important to develop a new freeze-drying technique. In this article, cabbage was used as a model material, and the microwave field was used as a heat source to supply sublimation heat so that the drying time was shortened greatly. The effect of the microwave sterilization during the drying process was evaluated. Effects of the pressure, thickness of material being dried, and the input microwave power on such indices as drying time and the microorganism number were studied. Compared with the method of ordinary freeze drying, microwave freeze drying (MFD) can greatly reduce the drying time and has a notable sterilization effect.