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.     

OPTIMAL CONTROL OF THE PRIMARY AND SECONDARY DRYING STAGES OF BULK SOLUTION FREEZE DRYING IN TRAYS

The problem of operating a tray freeze dryer to obtain a desired final bound water content in minimum time is formulated as an optimal control problem with the use of the rigorous unsteady state mathematical model of Sadikoglu and Liapis [9] that has been found to describe satisfactorily the experimental dynamic behavior of the primary and secondary drying stages of bulk solution freeze drying of pharmaceuticals in trays. The heat input to the material being dried and the drying chamber pressure are considered to be control variables. Constraints are placed on the system state variables by the melting and scorch temperatures during primary drying, and by the scorch temperature during secondary drying. Necessary conditions of optimality for both the primary and secondary drying stages are derived and presented, and an approach for constructing the optimal control policies that would minimize the drying times for both the primary and secondary drying stages, is presented. The theoretical approach presented in this work was applied in the freeze drying of skim milk, and significant reductions in the drying times of primary and secondary drying were obtained, when compared with the drying times obtained using the operational policies reported in the literature, by using the optimal control policies constructed from the theory presented in this work. Furthermore, it is shown that the optimal control policy leads to the desired in practice result of having at the end of secondary drying temperature and bound water concentration profiles (in the dried layer) whose gradients are very small. It is also shown that by using the optimal control policy and an excipient capable of increasing the melting temperature without affecting product quality, one can significantly reduce the drying time of the primary drying stage.     

OPTIMAL CONTROL OF THE PRIMARY AND SECONDARY DRYING STAGES OF BULK SOLUTION FREEZE DRYING IN TRAYS

Abstract

The problem of operating a tray freeze dryer to obtain a desired final bound water content in minimum time is formulated as an optimal control problem with the use of the rigorous unsteady state mathematical model of Sadikoglu and Liapis [9] that has been found to describe satisfactorily the experimental dynamic behavior of the primary and secondary drying stages of bulk solution freeze drying of pharmaceuticals in trays. The heat input to the material being dried and the drying chamber pressure are considered to be control variables. Constraints are placed on the system state variables by the melting and scorch temperatures during primary drying, and by the scorch temperature during secondary drying. Necessary conditions of optimality for both the primary and secondary drying stages are derived and presented, and an approach for constructing the optimal control policies that would minimize the drying times for both the primary and secondary drying stages, is presented. The theoretical approach presented in this work was applied in the freeze drying of skim milk, and significant reductions in the drying times of primary and secondary drying were obtained, when compared with the drying times obtained using the operational policies reported in the literature, by using the optimal control policies constructed from the theory presented in this work. Furthermore, it is shown that the optimal control policy leads to the desired in practice result of having at the end of secondary drying temperature and bound water concentration profiles (in the dried layer) whose gradients are very small. It is also shown that by using the optimal control policy and an excipient capable of increasing the melting temperature without affecting product quality, one can significantly reduce the drying time of the primary drying stage.     

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.     

Drying of Seabuckthorn (Hippophae rhamnoides L.) Berry: Impact of Dehydration Methods on Kinetics and Quality

Convective hot air drying and freeze drying were investigated as potential dehydration processes to obtain powders of seabuckthorn fruit pulp. Halved seabuckthorn fruits were placed in a hot air dryer and dried at 1 m/s and at 50 or 60°C or freeze dried at less than 30 mTorr and at 20 or 50°C shelf plate temperature. An initial characterization of the seabuckthorn pulp (moisture, pH, soluble solid content, vitamins C and E, total phenolics, and carotenoids) was performed. Water loss, total phenolic compounds, total carotenoids, and vitamin C were determined at different processing times. Vitamin E was determined before and at the end of drying.

Freeze-drying kinetics were faster than air drying, probably due to lower moisture diffusion in the compact, sugary, and oily structure of the air-dried tissue. The temperature had an important impact on hot air–drying and freeze-drying kinetics. Drying method and processing times affected the remaining phenolic, carotenoid, and vitamin contents of seabuckthorn berries. Freeze drying was revealed as a superior method to obtain seabuckthorn powders because of the lower residual moisture content, the ease of grinding, as well as the better nutritional retention.     

A FINITE VOLUME ANALYSIS OF VACUUM FREEZE DRYING PROCESSES OF SKIM MILK SOLUTION IN TRAYS AND VIALS

A numerical code that can predict vacuum freeze drying processes in trays and vials was developed using a finite volume method to discretize the governing partial differential equations. Along with the finite volume method, a moving grid system was adopted to handle irregular and continuously changing physical domains encountered during the primary drying stage. To show the validity of the present calculation scheme, freeze drying in a tray was simulated and the results were compared with available experimental data. After successful validation, freeze drying processes in vials with different operation policies were simulated to show the capability of the present calculation tool in handling multi-dimensional freeze drying problems.     

A FINITE VOLUME ANALYSIS OF VACUUM FREEZE DRYING PROCESSES OF SKIM MILK SOLUTION IN TRAYS AND VIALS

ABSTRACT

A numerical code that can predict vacuum freeze drying processes in trays and vials was developed using a finite volume method to discretize the governing partial differential equations. Along with the finite volume method, a moving grid system was adopted to handle irregular and continuously changing physical domains encountered during the primary drying stage. To show the validity of the present calculation scheme, freeze drying in a tray was simulated and the results were compared with available experimental data. After successful validation, freeze drying processes in vials with different operation policies were simulated to show the capability of the present calculation tool in handling multi-dimensional freeze drying problems.     

Effect of Processing Conditions on Drying Kinetics and Particle Microstructure of Carrot

The drying of carrot particles (6 mm × 6 mm × 12 mm) was studied in a tunnel dryer; a vacuum-freeze dryer, either with or without infrared radiation; a pulsed fluidized-bed dryer assisted by microwave radiation; and combinations of these methods. The effect of two freezing rates (quick freezing in liquid nitrogen and slow freezing in a household freezer) was also studied.

The drying kinetics for these drying methods were determined and modeled, and the dried products were subjected to texture (hardness), color, and rehydration analysis, as well as 2D and 3D image analysis of pictures from scanning electron microscope.

The combination of freeze drying with other dehydration techniques reduces the drying time by 6–70%, although, in general, the structural damage increases with respect to freeze drying alone. The hybrid drying systems did not show significant differences in drying times either for quick- or slow-frozen samples. The combination of freeze drying followed by conventional drying reduces the drying time between 23 and 40% on average.

The Page empirical model represents adequately the entire drying process for combined methods, with specific parameters for each drying zone. The values of effective diffusivity calculated with the simplified constant diffusivity model agree with those reported in the literature.     

How to Adapt a Lab-Scale Freeze Dryer for Assessing Dehydrating Curves at Different Heating Conditions

A laboratory-scale freeze dryer was adapted to allow control of the heat supply and on-line monitoring of sample weight during drying under vacuum condition. Several tests were carried out to verify the reliability of the developed system. The system was also tested with 10-mm-thick banana slices, obtaining the fruit drying rate, at different sample holder plate temperatures including 20 and 30°C as well as an unheated plate. The system presented in this work represents a low-cost, flexible, and easy-to-assemble piece of equipment that allows the study and optimization of the freeze drying of foods.     

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.     

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     

Freeze Drying of Pharmaceutical Excipients Close to Collapse Temperature: Influence of the Process Conditions on Process Time and Product Quality

The aim of this article is to investigate the characteristics of a vial freeze-dried product as a function of the temperature at which the primary drying is carried out. The effect of the drying temperature and composition on the product appearance and final characteristics (apparent density and rehydratability) of pure substances (mannitol, sucrose, and dextran) has been studied: rehydratability is faster for the product freeze dried at lower temperature, when the structure of the product is more porous and softer; higher rehydration rates were also observed for samples obtained from lower initial concentration. In case of binary mixtures (mannitol and dextran, sucrose and dextran) the rehydratability gets worse when higher drying temperatures are used, whereas the relative composition has a smaller effect. Nevertheless, high drying temperatures are required to shorten the process, both in case of pure substances and in case of binary mixtures; thus, it is important to consider the process time–product quality relationship to choose the optimal drying temperature. Finally, we show that the freeze drying of mixtures can be easier and faster than that of one of the components alone; this is partly a consequence of the higher temperatures allowed, but it is evident that in some cases the drying kinetics can be higher even with the same process temperature.     

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     

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.     

Freeze Drying of Saffron (Crocus sativus L.)

Saffron obtained from the dried stigmas of a flower scientifically known as Crocus sativus L. is considered to be the most precious and expensive agricultural product due to its labor-intensive harvest and post-harvest processing. The post-harvest processing such as dehydration and storage conditions determine stability, quality, and economical value of the final product. The contents of crocin (degraded carotenoids) and safranal (carotenoid oxidation products) are the key components that characterize color, taste, and aroma characteristics of saffron. In this work, the quality parameters such as crocin and safranal contents of commercial saffron that were obtained by using the freeze-drying method and natural sun drying were studied. The sarfanal contents of the samples dried in a freeze dryer were found to be five times higher than the safranal contents of the samples dried naturally under the sun, while crocin contents of the samples dried in a freeze dryer were about 40% higher than the crocin contents of the samples dried naturally under the sun. These encouraging results indicate that the freeze-drying process can be used effectively for dehydration of saffron by minimal loss of safranal and crocin contents.     

Freeze Drying of Aluminum Nanoflakes from Cyclohexane Suspensions

Hydrophobic aluminum nanoflake slurries were freeze dried using varying freezing conditions, aluminum concentrations, and shelf temperatures. The effectiveness of using cyclohexane as a carrier to produce dry, de-agglomerated aluminum powders was determined from light scattering measurements. Significantly less aggregation was observed in the dried aluminum powders when cyclohexane was used as opposed to water, with most aggregation being prevented by freeze drying dilute aluminum slurries at low shelf temperatures. A significant increase in the primary rate of freeze drying was also observed when particles were dried from cyclohexane compared to particles dried from water.     

Optimal control of a freeze dryer—I Theoretical development and quasi steady state analysis

A mathematical model which describes the nonsteady state heat and mass transfer operations during the freeze drying process is presented. With the use of this model the problem of operating a freeze dryer to obtain a fixed final moisture content in minimum time is formulated as an optimal control problem. Radiator energy output and total pressure in the drying chamber are considered control variables. Constraints are placed on the system state variables by the scorch point of the dried product and the melting point of the frozen interface. A quasi steady state analysis is performed which provides general guidelines about the optimal control policy at the beginning of the drying process (when neither of the state constraints is active), as well as during operation when the process may be heat and/or mass transfer limited.