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.     

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.     

液体物料冷冻干燥的热点问题

液体物料冷冻干燥广泛应用于制药工业。其过程工艺的重要性正赢得世界范围的研究关注。文献已证实科学方法能够用最少的试差来提高产品质量。过程工艺的建立和规划需要系统地理解冷冻、冷冻干燥、材料科学和质热传递等物理化学过程。本文就液体物料冷冻干燥过程做了简要评述,并对其过程的热点问题做了分析。     

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。     

瓶装物料的冷冻干燥过程模拟与实验验证

对于瓶装物料的冷冻干燥,在前面工作＾「１」的基础上,以脱脂牛奶为物料对冻干过程进行实验研究和模拟计算,结果表明物料的温度分布近瓶壁处的温度较中心部分为高,因此说明瓶壁的热传导不可忽略：同时对模拟值与实验值进行比较,其结果吻合良好。     

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.     

Role of Freeze Drying in Nanotechnology

Governments worldwide have spent more money on nanotechnology between 1997 and 2005 than on the Apollo project. The spending trend is still going upwards. Drying is an indispensable operation in the fabrication of nanosized materials. Hence, it is no surprise to find a large number of papers published in the past decade on drying and nano-related materials. This article reviews the literature and highlights the opportunities and challenges of freeze drying in nanotechnology. Freeze drying has found application in the production of nanoparticles for electrochemical, environmental, engineered materials, and pharmaceutical industries. The retention of the homogenous properties typically found in a solution, the small size of particles produced, and the long shelf life obtained for pharmaceutical applications are the primary reasons for choosing freeze drying. The relatively cheap operation cost compared to supercritical fluid extraction is another reason. Freezing was found to be a very important step in obtaining desired particle size and properties. Primary drying of solvent sublimation should be carried out at a temperature below the collapse temperature. Cryoprotectants are frequently necessary in preserving the original properties of active pharmaceutical ingredients. Spray freezing into liquid was found to be an optimal operation in order to minimize the air-liquid interfacial loss of bioactivity. A continuous freeze-drying process for production of granules of nanoparticles would be in demand, a dream that freeze-drying researchers have had for over a decade. This freeze drying may be carried out under vacuum or at atmospheric pressure using proper gases.     

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.     

预冻在冻干过程中的作用简

预冻环节中降温速度、冻结温度、冻结时间都是重要的工艺参数,它们互相作用,影响着冻结体的结晶情况,最终会对冻干过程形成决定性的影响。本文通过实例,介绍了预冻过程中冻结速率对冻干结果的影响。说明适当的预冻参数可大大提高冻干效率和产品质量。     

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.     

Trends in Microwave-Assisted Freeze Drying of Foods

Microwave-assisted freeze drying (MFD) can be accomplished in two distinct ways: freeze drying assisted concurrently with microwave application (MFD-1) and freeze drying and assisted microwave/vacuum microwave drying in two consecutive separate drying stages (MFD-2). MFD is a rapid dehydration technique that can be applied to certain foods, particularly to seafoods, solid soup, and fruits and vegetables. MFD involves much less drying time and energy consumption than conventional freeze-drying methods. Currently, this technology has been successfully used to dry many food materials and has potential in the food industry. Increasing concerns over product quality, energy savings, and production costs have motivated researchers and the industry to adopt MFD technologies. The advantages of MFD include shorter drying time, energy savings, improved product quality, and flexibility in producing a wide variety of dried products. However, current applications are limited to small categories of foods due to high startup costs and relatively complex technology compared to conventional freeze drying. This article presents a concise review of recent progress in MFD R&D and makes recommendations for future research to bridge the gap between laboratory research and industrial applications.     

Non-Invasive Temperature Monitoring in Freeze Drying: Control of Freezing as a Case Study

In this study, a new technology, based on the use of an array of thin-film thermocouples, is proposed for the real-time and non-invasive monitoring of product temperature during a freeze-drying cycle. This array of thin-film thermocouples was demonstrated to give quantitative accurate results without altering the freezing and drying behavior of the monitored vials, making this sensor a good alternative to conventional wire thermocouples. Furthermore, unlike conventional thermocouples, the array of thin-film thermocouples can measure the entire temperature profile within the product being dried and can also measure the position of the interface of sublimation. This information was then used to estimate the heat and mass transfer coefficient of the system under investigation, showing a fairly good agreement with data obtained by conventional, time-consuming procedures. Finally, it is shown how these parameters can be used to guide the lyophilization professional during the cycle development.     

SiC多孔预制体的定向冷冻干燥制备工艺研究

通过定向冷冻、冷冻干燥和高温烧结制备了具有定向层状孔隙的SiC多孔预制体,研究了浆料球磨时间、SiC粉料粒径、定向冷冻时间和冷冻干燥过程对预制体制备工艺的影响,确定了合适的定向冷冻干燥制备工艺.结果表明:SiC粉料在液相中充分地分散、润湿和形成水化膜后可以获得性能稳定的浆料,三种SiC粉料(初始粒径1.8μm、9.0μm和17.4μm)的球磨时间应分别大于12 h、14 h和20 h;定向冷冻过程中冰晶的生长和陶瓷颗粒的重排与SiC颗粒大小有着密切的关系,采用粒径1.8μm粉料制备的预制体中层状孔隙定向性和连通性最好;固相含量较低的浆料要采用较长的冷冻时间;虽然冷冻干燥温度较高时失重速率较高,但低温更有利于保持冰晶形貌,因此可选用定向冷冻温度作为冷冻干燥温度,冷冻干燥时间应不少于24 h.     

奶粉的喷雾冷冻干燥研究

研究开发了实现喷雾冷冻干燥工艺的装置,并以奶粉为物料,进行了试验研究。结果表明,料液可以在-40～-30℃实现快速冷冻,且采用该喷雾冷冻干燥装置可将干燥时间从原来真空冷冻干燥的48h缩短为10h。最后,对获得的干燥产品进行了粉体形态的测试和分析。     

Analysis of Heat and Mass Transfer in Freeze Drying

The main objective of this work is to build a mathematical model that describes heat and mass transfer in freeze-drying when both plate heating and radiation heating are applied and also to provide further understanding of the mechanism of the drying process. The model, unlike other models, may be used for situations in which sublimation occurs within a temperature range, i.e., the non-existence of a sharp interface and also for cases in which more than one interface may form. The developed model has been tested against experimental measurements of freeze-drying of milk under different operating conditions. Measurements were done using Virtis BT3.3ES freeze dryer with vertical manifolds. The milk was contained in a glassware, specially designed for this project. Four thermocouples were fixed at different positions to track the drying progress. The experimental measurements show no significant shrinkage in the frozen milk when dried, leaving the milk highly porous in structure. In this experimental work, the low thermal conductivity of the dried layer was found to control the process without any significant mass transfer resistance. This includes plate heating where drying was found to progress from the heating surface similar to radiation heating. This is unlike what has been reported in some of literature that drying starts always from the top surface. The model, which was based on heat transfer control, showed a reasonable agreement with the experimental measurements of both plate heating and radiation heating.     

真空冷冻干燥技术在生物制药种的应用

近些年来,我国生物制药行业的发展已经达到了顶峰,生物制药水平也越来越高,真空冷冻干燥技术在生物制药中也得到了广泛应用。就真空冷冻干燥技术进行分析,探讨出真空冷冻技术在我国生物制药当中的应用。对真空冷冻干燥的过程进行探讨,以此来提升我国的真空冷冻干燥技术,把真空冷冻干燥技术科学合理地融入到我国的生物制药中,发挥真空冷冻干燥技术的最大功效,从而提高我国的生物制药水平,保障人们的生活用药,解决好我国的民生问题,促进我国经济发展。     

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.     

浅谈大型不锈钢冻干机化学抛光

给出了不锈钢在室温条件下化学抛光液配方和抛光工艺条件。工艺了抛光液中各种组成的作用。详细讨论了不锈钢材料品种,质量和加工工艺以及抛光液注排和搅拌方法等对化学抛质量的影响。分析了化学抛光处理中进行挂件实验的作用。