An Efficient Calculation of Multidimensional Freeze-Drying Problems Using Fixed Grid Method

A calculation procedure based on a fixed grid method streamlined for multidimensional freeze-drying problems was developed, considering coupled heat and mass transfer, sublimation of ice, and motion of ice front. Efficiency of the fixed grid calculation in solving one- and two-dimensional freeze-drying problems was evaluated by comparing its accuracy and calculation cost with those of a conventional moving grid calculation. The numerical investigation showed that the fixed grid method can produce accurate results comparable to those obtained by the moving grid method, with almost equal calculation cost in case of one-dimensional freeze-drying problems. But the fixed grid method required about 70% more computation time than the moving grid method in case of two-dimensional freeze-drying problems. It is thought that the fixed grid method may be advantageous over conventional moving grid methods when freeze-drying problems involving cumbersome moving interface geometries have to be considered.     

Freeze-Drying of Skim Milk in a Cylindrical Container

Abstract

An experimental study on freeze-drying of skim milk in a cylindrical container was conducted to provide fundamental data essential for numerical studies on the process. Temperature histories at different positions were carefully measured during each experimental run and the measured temperature histories were combined to produce instantaneous temperature fields inside the container. In addition, mass reduction history by sublimation of ice was also measured with the same drying conditions. The obtained temperature fields clearly indicated multi-dimensional drying characteristics of the freeze-drying in the cylindrical container. Along with the experiments, the process was simulated with an analysis program developed for this study, based on a finite volume method with a moving grid system. The numerical and experimental results were in good agreement.     

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.     

Freeze-Drying of Skim Milk in a Cylindrical Container

An experimental study on freeze-drying of skim milk in a cylindrical container was conducted to provide fundamental data essential for numerical studies on the process. Temperature histories at different positions were carefully measured during each experimental run and the measured temperature histories were combined to produce instantaneous temperature fields inside the container. In addition, mass reduction history by sublimation of ice was also measured with the same drying conditions. The obtained temperature fields clearly indicated multi-dimensional drying characteristics of the freeze-drying in the cylindrical container. Along with the experiments, the process was simulated with an analysis program developed for this study, based on a finite volume method with a moving grid system. The numerical and experimental results were in good agreement.     

Diffuse interface model of the freeze-drying process of individually frozen products

Abstract

Vacuum freeze-drying (VFD) is a dehydration method based on the sublimation of the liquid phase contained in a certain product, previously frozen, at low pressure and temperature. Since it is a time and energy consuming process, it is crucial to select the best processing conditions to minimize drying duration, thus reducing the energy requirement. Additionally, product temperature must be monitored since it plays an important role in preserving product quality. The aim of this study was to develop a Diffuse Interface Model (DIM) for in-silico simulation of the freeze-drying process of individually frozen products. Due to the geometrical features of the samples, and to the role of radiation in the heat transfer to the product, the usual one-dimensional approach is inappropriate. Using a DIM, each cell of the computational domain can be described as a porous solid matrix filled by ice and vapor with a time-varying composition, thus allowing the use of a fixed computational grid and making the computation effort less demanding in comparison to moving interface-based models. Drying of eggplant cubic samples was considered as case study: model parameters were estimated by fitting the experimentally measured product temperature and drying time to the calculated ones. The model was proven to be reliable in providing an accurate estimate of both the drying time and the product temperature. Therefore, it can be used for off-line process design and optimization, minimizing the experimental effort required to design and optimize the process.     

Theoretical Study on Microwave Freeze-Drying of an Aqueous Pharmaceutical Excipient with the Aid of Dielectric Material

Abstract

A mathematic model of simultaneous heat and mass transfer for the dielectric material assisted microwave freeze-drying was derived and solved numerically using the finite-deference technique with two moving boundaries. Lactose, a typical pharmaceutical excipient, was used as the representative solid material in the aqueous solution to be freeze-dried. Silicon carbide (SiC) was selected as the dielectric material. Numerical results show that the dielectric material can significantly enhance the microwave freeze-drying process. Under typical operating conditions, the drying time is 43% shorter than that of ordinary microwave freeze-drying. Temperature variations at sublimation fronts were examined in order to determine the appropriate microwave power input. Profiles of temperature, ice saturation, vapor concentration, and pressure during freeze-drying are presented, and rate-controlling mechanisms are discussed.     

ANALYTICAL SOLUTIONS FOR THE MOVING INTERFACE PROBLEM IN FREEZE-DRYING WITH OR WITHOUT BACK HEATING

ABSTRACT

A mathematical model for freeze-drying without back surface heating is established first in this paper by a combination of URIF (uniformly retreating ice front) and Sheng/Peck's theories. Explicit analytical expressions are obtained for the rates of interface movement and drying. The predicted drying rate and temperature profile are in good agreement with the Sheng/Peck's model results. Furthermore, the model is extended to the process of freeze-drying with back heating in which the temperature at the interface is assumed to increase along the path of movement. It is shown that the average rates for drying and interface moving may be obtained by assuming a quadric distribution for the interface temperature.     

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.     

Simulation of melting of ice in a porous media under multiple constant temperature heat sources using a combined transfinite interpolation and PDE methods

A numerical study is made of the melting of ice in a rectangular cavity filled with a porous medium subjected to multiple constant temperature heat sources. Focus is placed on establishing a computationally efficient approach for solving moving boundary heat transfer problem in a two-dimensional structured grids. Specific application to multidimensional melting problem with a complicated moving boundary condition is considered. Preliminary grids are first generated by an algebraic method, based on a transfinite interpolation method, with subsequent refinement using a PDE mapping (parabolic grid generation) method. A preliminary case study indicates successful implementation of the numerical procedure. A two-dimensional melting model is then validated against available analytical solution and experimental results and subsequently used as a tool for efficient computational prototyping.     

基于四叉树直角坐标网格的运动界面追踪方法

运动界面的追踪是两相流模拟中需要解决的主要问题之一。基于四叉树直角坐标网格实现了Level set界面追踪方法和网格的局部自适应控制。通过模拟3个给定速度场（平移流场、旋转流场、剪切流场）的运动界面追踪问题,证明了自适应四叉树直角坐标网格与Level set方法结合的优势。     

An Analogy between Countercurrent and Two-Dimensional Separation Cascades

Abstract

The relationships between countercurrent fixed bed and two-dimensional cascades are explored for continuous contact and for staged systems. With moving feed and product ports, fixed beds will simulate countercurrent operation. Two-dimensional cascades will simulate countercurrent operation if slanted feed and product lines are employed. The relationships are also extended to three-phase systems for one-, two-, and three-dimensional cascades. Fixed bed systems simulating countercurrent systems are in commercial operation. The two-dimensional cascade offers an alternative geometry for obtaining the same result. This alternative may be attractive for some separations such as electrophoresis.     

Numerical Investigation on Dielectric Material Assisted Microwave Freeze-Drying of Aqueous Mannitol Solution

Abstract

The dielectric material assisted microwave freeze-drying was investigated theoretically in this study. A coupled heat and mass transfer model was developed considering distributions of the temperature, ice saturation and vapor mass concentration inside the material being dried, as well as the vapor sublimation-desublimation in the frozen region. The effects of temperature and saturation on the effective conductivities were analyzed based on heat and mass flux equations. The model was solved numerically by the variable time-step finite-deference technique with two movable boundaries in an initially unsaturated porous sphere frozen from an aqueous solution of mannitol. The sintered silicon carbide (SiC) was selected as the dielectric material. The results show that dielectric material can significantly enhance microwave freeze-drying process. For case of the dielectric field strength, E = 4000 V/m under typical operating conditions, the drying time is 2081 s, 30.1% shorter and 47.2% longer, respectively, than those for E = 2000V/m and E = 6000 V/m. The heat and mass transfer mechanisms during the drying process were discussed.     

Solution of hyperbolic PDEs using a stable adaptive multiresolution method

An efficient adaptive multiresolution numerical method is described for solving systems of partial differential equations. The grid is dynamically adapted during the integration procedure so that only the relevant information is stored. The convection terms are discretised with high-resolution methods, thus ensuring boundedness. The proposed method is general, but is particularly useful for highly convective problems involving sharp moving fronts, a situation that frequently occurs in many chemical engineering problems, and where standard procedures may lead to unphysical oscillations in the computed solution.Numerical results for five test problems are presented to illustrate the efficiency and robustness of the method. The adaptive strategy is found to significantly reduce the computation time and memory requirements, as compared to the fixed grid approach.     

Moving mesh generation for tracking a shock or steep moving front

For tracking a shock or steep moving front in the numerical solution of Partial Differential Algebraic Equations (PDAEs), an accurate spatial discretization method, Weighted Essentially Non-Oscillatory (WENO) scheme, is combined with moving grid techniques so that spacing of moving meshes is smoothed locally and globally. Several monitor functions, as metric criteria of node concentration, are examined. While the fixed grid method (uniform grid size) needs many mesh points to obtain enough solution accuracy, the moving grid method (non-uniform grid size) enhances accuracy even at small mesh numbers but it may be prohibitive owing to the addition of complex and non-linear mesh equations into physical PDAEs. The combination of the WENO scheme (based on an adaptive stencil idea) with the moving grid techniques improves stability and accuracy in the numerical solution over the commonly used moving grid method of central discretization. To locate adequate grid position in the moving mesh method, suitable monitor function according to problems must be selected.     

A Direct Characterization Method of the Ice Morphology. Relationship Between Mean Crystals Size and Primary Drying Times of Freeze-Drying Processes

Abstract

In the freeze-drying process, the freezing step is one of the most important steps which determines the texture of the frozen material and, consequently, the final morphological characteristics of the freeze-dried material and its biological activity and its stability. As a matter of fact, the parameters of the freezing protocol have a direct effect on the pore size distribution and on the pore connectivity of the porous network of the freeze-dried matrix. Thus, the ice crystal morphology determines indirectly the mass and the heat transfer rates through the dry layer and, consequently, the freezing parameters have a strong influence on the total duration of the primary and secondary sublimation steps. The main objective of this study was to adapt and to develop a new optical direct microscopy method, based on the reflected flux differences, with episcopic axial lighting to characterize the structure of the different phases of a standard pharmaceutical matrix used for pharmaceutical proteins freeze-drying. First, the results obtained have been validated by another independent method, the scanning electron microscopy, carried out with freeze-dried samples. Finally, this technique has been principally used to investigate the effects of the freezing conditions on the ice crystal structure characterized by the distribution of the ice crystals mean sizes. Moreover, the influence of annealing treatment on ice crystal mean diameter and primary drying times has been also investigated.     

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.     

Simulation of Moving Feed Port Chromatography by Rate Model with Mass Transfer Effect

Abstract

Moving feed port chromatography (MFPC) was simulated by a rate model with mass transfer effect, and two criteria are presented for selection of the feed port velocity to obtain the best separation. When the moving feed port velocity simultaneously satisfied these two criteria, the resulting concentration profiles for MFPC had narrower bandwidth, higher concentration, and improved resolution than those of conventional chromatography. In addition, there existed an optimum number of feed ports (or column length for feed injection) to obtain the best resolution at a fixed total feed time.     

Freeze-Drying of Pharmaceutical Proteins in Vials: Modeling of Freezing and Sublimation Steps

The commercial finite element code FEMLAB was used to perform two-dimensional axisymmetric simulations of the temperature profiles and the moving front velocities of standard BSA (bovine serum albumin)-based formulations used to stabilize pharmaceutical proteins during the freeze-drying process. The simulations were validated with both experimental and numerical approaches.

In an initial step, the heat transfer phenomena taking place during the cooling of liquid solutions was studied in commercial size glass vials without freezing or sublimation.

Then, this model was extended and validated for the freezing process of aqueous BSA-based solutions encountered in the industrial freeze-drying processes with the same vials in order to confirm the identified values of the different thermal conductances between the product and the shelf and between the product and the surroundings.

Finally, the conductances between the product and the shelf and between the vial and the surroundings thus determined were used in a dynamic sublimation model with two zones and a moving sublimation front similar to the ones previously proposed in the literature.

The simulations showed a satisfactory agreement between experimental and simulated data.

The results of this study demonstrated that the freeze-drying process of pharmaceutical proteins in glass vials for standard industrial operating conditions was mainly controlled by the heat transfer from the shelf and the surroundings to the product sublimation front.     

Effects of different drying methods on the physical characteristics and flavor of dried hawthorns (Crataegus spp.)

ABSTRACT

The objective of this study was to compare the physical characteristics and flavor of dried hawthorns obtained by freeze-drying (FD), microwave freeze-drying (MFD), atmospheric freeze-drying (AFD), and heat pump drying (HPD). The parameters including moisture content, product temperature, rehydration ratio, chroma, hue angle, yellowness index, total color difference, and energy consumption were investigated. The use of fuzzy reasoning for the sensory evaluation of hawthorn quality was also performed. As expected, AFD, FD, and MFD produced better accepted dried hawthorn products than did HPD. Nevertheless, FD consumed the highest energy and had a long drying time, but its product was the best; AFD had a similar energy cost as HPD, but its drying time was the longest; and MFD had a higher energy cost and longer drying time than HPD, but its product quality was similar to that of FD products. As a result, MFD and AFD had potential to replace FD to yield dried hawthorns with high product quality and relatively low cost.     

Optimal moving and fixed grids for the solution of discretized population balances in batch and continuous systems: droplet breakage

The numerical solution of droplet population balance equations (PBEs) by discretization is known to suffer from inherent finite domain errors (FDE). Tow approaches that minimize the total FDE during the solution of discrete droplet PBEs using an approximate optimal moving (for batch) and fixed (for continuous systems) grids are introduced. The optimal grids are found based on the minimization of the total FDE, where analytical expressions are derived for the latter. It is found that the optimal moving grid is very effective for tracking out steeply moving population density with a reasonable number of size intervals. This moving grid exploits all the advantages of its fixed counterpart by preserving any two pre-chosen integral properties of the evolving population. The moving pivot technique of Kumar and Ramkrishna (Chem. Eng. Sci. 51 (1996b) 1333) is extended for unsteady-state continuous flow systems, where it is shown that the equations of the pivots are reduced to that of the batch system for sufficiently fine discretization. It is also shown that for a sufficiently fine grid, the differential equations of the pivots could be decoupled from that of the discrete number density allowing a sequential solution in time. An optimal fixed grid is also developed for continuous systems based on minimizing the time-averaged total FDE. The two grids are tested using several cases, where analytical solutions are available, for batch and continuous droplet breakage in stirred vessels. Significant improvements are achieved in predicting the number densities, zero and first moments of the population.