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.     

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.     

Prediction of Transient Temperature Distribution During Freeze Drying of Yoghurt

A one dimensional heat and mass transfer analysis is employed to predict the temperature distribution in the dry layer of a frozen food sample in the form of a slab undergoing freeze drying. The method avoids the use of the effective diffusivity and the Knudsen diffusivity, both parameters dependent upon the stnrcture of the food product. The accu acy of the numerical solution is validated with published data as well as data of present study obtained for the freeze drying of yoghurt. The predicted surface temperature of the food sample is in good agreement with the experimental data. Energy supplied to product surface through conduction across evacuated head space was found to be significant. The model is useful for the analysis of freeze drying of a food product.     

Effect of Dielectric Material on Microwave Freeze Drying of Skim Milk

Abstract:

A simultaneous heat and mass transfer model of the dielectric material–assisted microwave freeze drying was derived in this study considering the vapor sublimation-desublimation in the frozen region. The mathematical model was solved numerically by using the finite-difference technique with two moving boundaries. Silicon carbide (SiC) was selected as the dielectric material, and the skim milk was used as the representative solid material in the aqueous solution to be freeze-dried. The results show that the dielectric material can significantly enhance the microwave freeze drying process. The drying time is greatly reduced compared to cases without the aid of the dielectric material. Profiles of the temperature, ice saturation, vapor concentration, and pressure during freeze drying were presented. Mechanisms of the heat and mass transfer inside the material sphere were analyzed. For an initially unsaturated frozen sample of 16 mm in diameter with a 4-mm-diameter dielectric material core, the drying time is 288.2 min, much shorter than 380.1 min of ordinary microwave freeze drying and 455.0 min of conventional vacuum freeze drying, respectively, under typical operating conditions.     

Prediction of Transient Temperature Distribution During Freeze Drying of Yoghurt

ABSTRACT

A one dimensional heat and mass transfer analysis is employed to predict the temperature distribution in the dry layer of a frozen food sample in the form of a slab undergoing freeze drying. The method avoids the use of the effective diffusivity and the Knudsen diffusivity, both parameters dependent upon the stnrcture of the food product. The accu acy of the numerical solution is validated with published data as well as data of present study obtained for the freeze drying of yoghurt. The predicted surface temperature of the food sample is in good agreement with the experimental data. Energy supplied to product surface through conduction across evacuated head space was found to be significant. The model is useful for the analysis of freeze drying of a food product.     

Freeze‐drying of aqueous solution frozen with prebuilt pores

To save drying time and increase productivity, a novel idea was proposed for freeze‐drying of liquid materials by creating an initially unsaturated frozen structure. An experimental investigation was carried out aiming at verifying the idea using a multifunctional freeze‐drying apparatus. Mannitol was selected as the primary solute in aqueous solution. Liquid nitrogen ice‐cream making method was used to prepare the frozen materials with different initial porosities. Results show that freeze‐drying can be significantly enhanced with the initially unsaturated frozen material, and substantial drying time can be saved compared with conventional freeze‐drying of the initially saturated one. Drying time was found to decrease with the decrease in the initial saturation. The drying time for the initially unsaturated frozen sample (S₀ = 0.28 or 0.69 of initial porosity) can be at best 32% shorter than that required for the saturated one (S₀ = 1.00 or zero porosity). This unique technique is easy to implement and improves the freeze‐drying performance of liquid materials. © 2015 American Institute of Chemical Engineers AIChE J, 61: 2048–2057, 2015     

MICROWAVE FREEZE DRYING CHARACTERISTICS OF BEEF

Microwave freeze drying characteristics of unsaturated raw beef were studied experimentally for various levels of electric field strength, vacuum pressure, sample thickness and initial saturation. The results show that drying time is proportional to initial saturation and inversely proportional to the electric field strength and sample thickness. The effect of vacuum pressure on drying time is negligible. Some advantages of microwave freeze-drying over conventional freeze drying with radiant heating are pointed out     

Atmospheric Freeze Drying—A Review

This review article focuses on the development of atmospheric freeze drying (AFD): technological aspects, product possibilities, physical properties of products, drying kinetics, modeling, and simulation. The main motivation for developing atmospheric freeze drying as a new drying technology is the desire to reduce the energy consumption compared to vacuum freeze drying while maintaining a high product quality. One technical solution of atmospheric freeze drying is a combined atmospheric freeze drying and heat pump system with new environmentally friendly refrigerants. Temperature programs make it possible to customize products with desired qualities and properties, like retained color, instant properties, aroma, and nutritional value.     

Atmospheric Freeze Drying—A Review

This review article focuses on the development of atmospheric freeze drying (AFD): technological aspects, product possibilities, physical properties of products, drying kinetics, modeling, and simulation. The main motivation for developing atmospheric freeze drying as a new drying technology is the desire to reduce the energy consumption compared to vacuum freeze drying while maintaining a high product quality. One technical solution of atmospheric freeze drying is a combined atmospheric freeze drying and heat pump system with new environmentally friendly refrigerants. Temperature programs make it possible to customize products with desired qualities and properties, like retained color, instant properties, aroma, and nutritional value.     

MICROWAVE FREEZE DRYING CHARACTERISTICS OF BEEF

ABSTRACT

Microwave freeze drying characteristics of unsaturated raw beef were studied experimentally for various levels of electric field strength, vacuum pressure, sample thickness and initial saturation. The results show that drying time is proportional to initial saturation and inversely proportional to the electric field strength and sample thickness. The effect of vacuum pressure on drying time is negligible. Some advantages of microwave freeze-drying over conventional freeze drying with radiant heating are pointed out     

Effects of different drying methods on the extraction rate and qualities of oils from demucilaged flaxseed

Effects of different drying methods, i.e., oven drying, freeze drying, vacuum drying, and microwave drying, on the extraction rate and qualities of oil extracted from demucilaged flaxseed were investigated. The results show that microwave drying and freeze drying have higher extraction rate and yield (46.36 and 46.16% vs. 33.90%) of oil compared with native flaxseed. The oil extracted from dried flaxseed, especially the one freeze dried, had higher peroxide value (21.63 vs. 6.15?meq/kg) and specific ultraviolet extinction at 232?nm (4.56 vs. 3.67) and 268?nm (1.23 vs. 1.12), as well as a shorter induction time (1.8 vs. 3.3?h) than the one from native sample. Demucilaging and drying have no significant effect on fatty acid composition of flaxseed oil. However, these processes increased the content of total phenols and total sterols while reduced the content of chlorophyll and carotenoid pigment. In addition, the oil extracted from vacuum dried flaxseed contained more tocols when compared with native sample (447.29 vs. 370.99?mg/kg).     

Dehydration of Garlic Slices by Combined Microwave-Vacuum and Air Drying

Combination of microwave-vacuum drying and air drying was investigated as a potential mean for drying garlic slices. The sample was dried by microwave-vacuum until the moisture content reached 10% (wet basis), and then by conventional hot-air drying at the temperature of 45°C to final moisture content less than 5% (wet basis). Pungency, color, texture, and rehydration ratio of garlic slices dried by this method were evaluated and compared with those dried by freeze drying and conventional hot-air drying. The comparison showed that the quality of garlic slices dried by the current method was close to that of freeze dried garlic slices and much better than that of conventional hot-air dried ones. The lab microwave-vacuum dryer which the materials to be dried could be rotated in the cavity was developed by the authors.     

Dehydration of Garlic Slices by Combined Microwave-Vacuum and Air Drying

Abstract

Combination of microwave-vacuum drying and air drying was investigated as a potential mean for drying garlic slices. The sample was dried by microwave-vacuum until the moisture content reached 10% (wet basis), and then by conventional hot-air drying at the temperature of 45°C to final moisture content less than 5% (wet basis). Pungency, color, texture, and rehydration ratio of garlic slices dried by this method were evaluated and compared with those dried by freeze drying and conventional hot-air drying. The comparison showed that the quality of garlic slices dried by the current method was close to that of freeze dried garlic slices and much better than that of conventional hot-air dried ones. The lab microwave-vacuum dryer which the materials to be dried could be rotated in the cavity was developed by the authors.     

Combined Microwave-Vacuum and Freeze Drying of Carrot and Apple Chips

A combination of microwave-vacuum (MWV) drying and freeze drying was investigated as potential means for drying carrot and apple chips. The sample was first dried by microwave-vacuum to dehydrate some amount of internal free water and then by freeze drying to a final moisture content of less than 7% (wet basis). Chemical properties (carotene and vitamin C retention) and physical properties (shrinkage, color, texture, and rehydration ratio) of carrot and apple slices dried by this method were evaluated and compared with those dried by freeze drying alone, MWV drying alone, and conventional hot air drying, respectively. The comparison showed that the carotene retention of carrot slices and the vitamin C retention of apple slices dried by the current method were close to those of freeze-dried carrot and apple slices and much better than those of conventional hot air–dried ones. The samples prepared by the current method exhibited very close rehydration capacity, color retention, and texture with those of the freeze-dried ones but with a little higher shrinkage. However, the samples still showed the attractive external appearance without marked warp.     

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.     

In Situ Determination of the Physical State of Biological Samples during Freeze Drying

The proceeding change of water content in a freeze-drying process, as well as the physical state of cell–sugar suspensions with Lactobacillus paracasei ssp. paracasei, was monitored by means of an in situ weighing system. Due to the pressure and temperature variations in the drying chamber, the weighing system was found to be influenced. This deviation of the weighing system was quantified by means of drying simulations with drying-inert Teflon dummies. This quantified deviation could be used for correction of the following freeze-drying processes. The applicability of this correction method was verified by drying model sugar solutions with known dry matter. Taking the quantified drift of the balance into account, the measured water contents for the model solution were proved to deviate less than 5% from the actual value. Finally, this verified correction method was applied to freeze drying of cell–sugar suspensions. As a result, the drying course of the cell–sugar samples could be depicted in the appropriate state diagram. Thus, the weighing system could be used to monitor the state transitions, which the bacterial suspension experienced during freeze drying, and qualified for specific investigations on the glassy state during freeze drying.     

Spray and freeze drying of human milk on the retention of immunoglobulins (IgA,IgG, IgM)

Several freeze-drying and spray-drying methods were investigated in relation to the retention of immunoglobulins (Ig) A, IgG, and IgM. Spray drying produced human milk powders with 2% humidity and a good retention of IgG (>88%) and IgM (～70%). However, only 38% of IgA remained after spray drying. For freeze drying, only the highest heating plate temperature used in this study (40°C) brought IgA content down to 55% in powder with 1.75% residual humidity, whereas milk samples undergoing lower temperatures had higher preservation rates (75% for IgA and 80% for IgG and IgM) and higher residual moisture contents. From these results, it can be concluded that IgA is the most sensitive Ig lost during drying processing of human milk. The best method to generate human milk powders without a significant loss of Ig was thus freeze drying at 30°C heating plate temperature, which accelerated the process compared to lower processing temperatures, but still had good overall Ig retention.     

Foam-Mat Freeze Drying of Egg White—Mathematical Modeling Part II: Freeze Drying and Modeling

Foam-mat freeze drying is one of the promising methods of drying, which utilizes advantages of both freeze drying and foam-mat drying. Egg white with its excellent foaming properties makes a suitable candidate for foam-mat freeze drying. Experiments were conducted to study foam-mat freeze drying of egg white, in an effort to determine the suitability of this method. Xanthan gum (XG) at 0.125% concentration was used as stabilizer for foaming. The results showed that the addition of xanthan gum during foaming has a positive impact in reducing the total drying time and also produces excellent quality egg white powder. The addition of stabilizer also plays an important role in improving drying. Simple models were applied for determining drying time and diffusion coefficients during freeze drying.     

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.