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 Secondary Drying Stage of Freeze Drying of Solutions in Vials Using Variational Calculus

Abstract:

The problem of operating freeze drying of pharmaceutical products in vials loaded on trays of freeze dryer to obtain a desired final bound water content in minimum time is formulated as an optimal control problem. Two different types of freeze dryer designs were considered. In the type I freeze dryer design, upper and lower plate temperatures were controlled together, while in the 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. Only the scorch temperature was considered as a constraint on the system state variables during the secondary drying stage, because all the free water content (frozen water) is removed from the solid matrix during the primary drying stage of freeze drying. Necessary conditions of optimality for the secondary drying stage of freeze drying process in vials were derived and presented by using rigorous multidimensional unsteady-state mathematical models. The theoretical approach presented in this work was applied in the freeze drying of skim milk. Significant reductions in drying times of the secondary drying stage of the freeze drying process in vials were observed and more uniform bound water and temperature distributions in the material being dried were obtained compared to the conventional operational policies.     

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.     

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.     

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.     

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     

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     

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。     

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.     

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.     

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.     

模型预测控制及在粮食干燥过程的应用研究

从模型预测、滚动优化及反馈校正三方面简要介绍了模型预测控制（MPC）方法,归纳了MPC的优点;概述了粮食干燥控制的发展动态、我国粮食干燥控制研究概况以及国外的粮食干燥MPC研究状况;通过具体实例,详细介绍了美国研发的适用于横流式谷物干燥杌的MPC系统,以期为我国的同类研究提供借鉴和参考。     

Comparison of the Retention of 6-Gingerol in Drying of Ginger Under Modified Atmosphere Heat Pump Drying and other Drying Methods

A heat pump dryer using normal air, nitrogen, and carbon dioxide was selected to dry sliced West Indian ginger (Zingiber officinale Roscoe) rhizome (3 mm in thickness). The drying characteristics were compared with each other and inert gas heat pump drying showed an improved effective diffusivity. Quantities of the main pungent principle (6-gingerol) of ginger, extracted from these dried samples, were determined by high-pressure liquid chromatography (HPLC). The evaluation included dried samples obtained by heat pump, modified atmosphere heat pump, freeze drying, and vacuum drying. Retention of 6-gingerol increased in the order of normal air drying, freeze drying, nitrogen drying, carbon dioxide drying, and vacuum drying. From this point of view, inert gas also showed a better retention of flavor compared to most other types of drying.     

Mathematical Modelling of the Primary and Secondary Drying Stages of Bulk Solution Freeze-Drying in Trays: Parameter Estimation and Model Discrimination by Comparison of Theoretical Results With Experimental Data

A mathematical model was constructed and solved in order to describe quantitatively the dynamic behavior of the primary and secondary drying stages of the freeze-drying of pharmaceuticals in trays. The theoretical results were compared with the experimental data of the freeze-drying of skim milk, and the agreement between the experimental data and the theoretical results is good. Detailed model calculations have indicated that the contribution of the removal of hound (unfrozen) water to the total mass flux of the water removed during primary drying, is not significant. For this reason, it was found that one could not     

Mathematical Modelling of the Primary and Secondary Drying Stages of Bulk Solution Freeze-Drying in Trays: Parameter Estimation and Model Discrimination by Comparison of Theoretical Results With Experimental Data

ABSTRACT

A mathematical model was constructed and solved in order to describe quantitatively the dynamic behavior of the primary and secondary drying stages of the freeze-drying of pharmaceuticals in trays. The theoretical results were compared with the experimental data of the freeze-drying of skim milk, and the agreement between the experimental data and the theoretical results is good. Detailed model calculations have indicated that the contribution of the removal of hound (unfrozen) water to the total mass flux of the water removed during primary drying, is not significant. For this reason, it was found that one could not     

涂装涂层系统使用阶段的质量控制

论述了涂装涂层系统在使用阶段的质量控制问题,详细分析了涂层体系在使用过程中的质量控制点,叙述了预防涂层质量事故的方法和措施。     

硝酸氧化煤系硫铁矿生产聚铁工艺条件及氧化机理初探

对HNO3氧化煤系硫铁矿以及放出NO和引入O2 催化氧化FeS2 和FeSO4 生产聚铁进行了试验 ,得出较佳工艺条件 ,并通过试验结果来分析了有关FeS2 氧化机理     

形成针对太湖流域三类水源的综合解决方案实现典型和新型污染物的协同控制

太湖流域是我国典型的水质缺水型地区,饮用水安全保障工作面临较大挑战."十三五"水专项"太湖流域饮用水安全保障工程技术与综合管理技术集成研究"课题梳理了太湖流域饮用水专项实施以来的课题及关键技术,针对太湖流域水源水质特征和区域供水特征,通过水源水质评估、关键技术遴选、工艺协同优化和示范工程验证,形成了针对太湖流域三类典型...