Weibull函数及其在干燥动力学研究中的应用

简要介绍了Weibull函数的基本理论,系统评析了干燥动力学研究中的Weibull函数及其应用。     

Modification of the Weibull Distribution for Modeling Atmospheric Freeze-Drying of Food

An empirical physical model was derived from the Weibull distribution and investigated for its ability to describe the moisture content for common atmospheric (or convective) freeze-drying processes (AFD). A set of experiments was performed for different products: peas, apple, pineapple, cod, and zooplankton. The effect of drying temperatures (?6°C, ?3°C, 0°C, 10°C and 20°C), approach velocities (1 m sec^?1, 1.8 m sec^?1, 2.6 m sec^?1, 3.1 m sec^?1, and 4.7 m sec^?1), and particle sizes (8.7 mm, 15.7 mm, and 28.8 mm) was investigated using the selected products. Non-linear regression analyses showed good agreement between the model and experimental data. The coefficient of determination was at least 99.9% (R² > 0.999) and the chi-square lower than 0.0001 (χ² < 0.0001) for all investigations. The shape parameter β in the modified Weibull model varied in a narrow range from 0.661 to 0.937, which indicates that AFD is controlled by internal mass transfer (=diffusivity). The diffusivity (D_calc) ranged from 1.554 to 8.681 10^?9 m² sec^?1, depending on the product and drying conditions. The modification of the Weibull distribution can be used to describe AFD processes based on a simple empirical but highly accurate model and for the determination of the effective diffusion (Fick's law).     

Freeze Drying of Apple Slices with and without Application of Microwaves

In this study, Fuji apple slices were dehydrated using freeze drying (FD) combined with microwave assisted with vacuum drying (VMD). The optimal parameter for the diversion point of moisture content from FD to VMD process was at the moisture level of 21%, and for VMD the optimal parameter for vacuum pressure was at 9.15 kPa and microwave power density was at a level of 3.18 w/g. The results show that the two-step technique can significantly reduce total FD time required by up to 40%, while the nutritional value of the dried apple chips remained unchanged compared to FD used alone.     

Ceramics Reliability: Statistical Analysis of Multiaxial Failure Using the Weibull Approach and the Multiaxial Elemental Strength Model

Methodology for designing reliable ceramic components requires a precise evaluation and correlation of strengths in different stress states. The present paper compares the merits of the Weibull approach and the multiaxial elemental strength model on an experimental case involving mixed-mode failure in the presence of bimodal flaw populations (surface and volume flaws). The experimental data were obtained using flexure specimens of Si₃N₄ tested at various spans, with the purpose of enhancing shearing effects. The analysis of data was refined by developing an advanced postprocessor program to finite-element codes for failure probability determination based upon the Barnett-Freudenthal approximation of the Weibull approach and the multiaxial elemental strength model. In a second step, the strengths of the specimens exhibiting failures from the two concurrent populations of flaws (intermediate span) were predicted using both approaches from data obtained with different span lengths (long and short spans). Comparison with experimental data showed that the multiaxial elemental strength model is an improvement over the Weibull approach. It also allowed the short-span bending test to be assessed. Finally, important implications for structural design with ceramics are discussed.     

Process-Based Drying Temperature and Humidity Integration Control Enhances Drying Kinetics of Apricot Halves

The effects of drying temperature (50, 60, and 70°C) and absolute humidity (65, 90, 115, 140 g/kg at the initial stage) on drying kinetics and color attributes of apricot halves under process-based drying temperature and humidity integration control (PDTHIC) were investigated. Results indicated that appropriate PDTHIC could reduce the drying time by 18.75% compared to the control group. The absolute humidity parameter should be controlled well rather than continuously dehumidify for traditional practices. The moisture effective diffusivity (D_eff) at 70°C presented drastic fluctuations with increasing absolute humidity parameters, calculated using the Weibull distribution model. The activation energy (E_a) of samples treated by continuous dehumidification and PDTHIC were 31.40 and 74.18 kJ/mol, respectively. The variation trend of color parameters was different from the conventional hot air drying probably due to the dehydration mechanism. Observation of the microstructure of the dried samples indicated that the PDTHIC process can enhance the drying rate of apricot through generating a larger pore network compared to continuous dehumidification at the same temperature. The findings of the current work clearly indicated that process-based drying temperature and humidity integration control enhances the drying kinetics of apricot halves, leading to a promising technology for energy savings.     

Microwave Freeze Drying of Apple Slices Based on the Dielectric Properties

Freeze drying (FD) yields the best quality of dried apple slices but requires a long drying time and is not cost-effective. To mitigate these problems, a microwave freeze drying (MFD) technique was developed to dry apple slices. The relationship between corona discharge and microwave power at various pressures and initial moisture content conditions was studied to avoid the possibility of corona discharge during MFD. It was found that with change of moisture content and temperature of samples during MFD, the dielectric property also changed, which resulted in dynamic microwave dissipation. Based on the dielectric property of samples, a changed microwave loading scheme can lead to perfect product quality and greatly reduce the drying time; thus, MFD can be used to replace the traditional FD technique.     

CFD Model of Apple Atmospheric Freeze Drying at Low Temperature

An Internet-controlled atmospheric freeze dryer was designed, built, and tested in the Department of Heat and Mass Transfer, Technical University of ?ód?. A film sublimation-based CFD model was developed and verified using Fluent 6.1 commercial CFD software. The model enables the simulation of phase change and water vapor diffusion process within porous media. Transport of non-condensable species can be calculated with species transport inbuilt model of Fluent 6.1 and used to predict sublimation rate under given conditions. Results were compared with AFD experimental data of 10-mm Idared apple cubes. The viability of applying the film sublimation model to atmospheric freeze-drying process was demonstrated. Higher mass flux was found on the leading edge, relatively uniform mass flux within the porous zone illustrates that vapor diffusion dominates atmospheric freeze drying process at low temperature (below 0°C). CFD results for apple cubes show a predomination of diffusional resistance of porous tissue.     

Drying of Apple Slices in Atmospheric and Vacuum Freeze Dryer

Atmospheric freeze drying (AFD) and vacuum freeze drying (VFD) of Fuji apples was investigated. A factorial design was used in each case, with particle size, freezing rate, infrared (IR) radiation, and air temperature as factors. The effect of these factors on moisture content, duration of drying, rehydration properties, color, and texture were determined. The drying curves were fitted with both the simplified constant diffusivity model (SCDM) and the modified Page model, and rehydration curves were tested with the Peleg model, resulting in R² higher than 0.96. Antioxidant capacity, polyphenols, and ascorbic acid content in the final product were determined and compared with those obtained in a tunnel dryer. A sensory evaluation was performed.

The drying times obtained for VFD were shorter than the drying times obtained for AFD. The particle size and IR application were found to be the significant parameters that affect duration of drying for both AFD and VFD (duration of drying was proportional to particle size and inversely proportional to IR application). Air temperature affected drying time only during the secondary drying stage. Estimated D_eff values were on the order of 10^?10 (m²/s). In AFD, R² higher than 0.81 was obtained for the SCDM, and R² higher than 0.96 was obtained for the Page model. In VFD, a better fit was obtained (R² > 0.97) for both models. AFD produced nutritional alterations similar to convective drying. Sensory quality was not altered by AFD or VFD.     

Modeling Microwave Vacuum Drying Kinetics and Moisture Diffusivity of Carrot Slices

Carrot slices of 3.5 mm thickness were dried in a laboratory microwave vacuum dryer at five different microwave power density levels of 2, 4.66, 7.33, 10, and 12.66 W/g and at three vacuum chamber pressure levels of 6.66, 19.98, and 33.3 kPa to 4–6% d.b. moisture content. Inside the dryer the sample holding plate was rotated with the speed of 4 rpm for uniform microwaves application. The drying rates were increased with the increase in microwave power density at all pressure levels and the Page model was found to be the most suitable model to predict the drying behavior of carrot slices at all process conditions. The Page model drying rate constant (k, min^?1) showed high correlation with microwave power density at constant pressure by a power law equation and showed a logarithmic relationship with the microwave power density and pressure. Similar to the drying rate constant, the average moisture diffusivity at constant pressure was found to be function of microwave power density by power law equation as well as was also dependent on the power density and pressure by a logarithmic relationship.     

Modeling Microwave Vacuum Drying Kinetics and Moisture Diffusivity of Carrot Slices

Carrot slices of 3.5 mm thickness were dried in a laboratory microwave vacuum dryer at five different microwave power density levels of 2, 4.66, 7.33, 10, and 12.66 W/g and at three vacuum chamber pressure levels of 6.66, 19.98, and 33.3 kPa to 4-6% d.b. moisture content. Inside the dryer the sample holding plate was rotated with the speed of 4 rpm for uniform microwaves application. The drying rates were increased with the increase in microwave power density at all pressure levels and the Page model was found to be the most suitable model to predict the drying behavior of carrot slices at all process conditions. The Page model drying rate constant (k, min^-1) showed high correlation with microwave power density at constant pressure by a power law equation and showed a logarithmic relationship with the microwave power density and pressure. Similar to the drying rate constant, the average moisture diffusivity at constant pressure was found to be function of microwave power density by power law equation as well as was also dependent on the power density and pressure by a logarithmic relationship.     

Modeling the Dehydration of Apple Slices by Deep Fat Frying

One of the objectives of immersion frying is to remove water from the food. Thus, predicting moisture loss is important when developing a model for that process. With the aim to model moisture transfer during apple frying, Granny Smith apple slabs were processed at 140, 150, 160, and 170°C. The moisture diffusivity was computed, ranging between 10.7 · 10^?9 and 17.7 · 10^?9 m² · s^?1. There was a close fit between the model and the experimental data (average %var 99.0). Model validation was carried out considering different slice thickness. Temperature influence was interpreted by the Arrhenius relationship with an activation energy of 25.4 kJ · mol^?1.     

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.     

Infrared Drying: From Process Modeling to Advanced Process Control

The article surveys the drying of solids materials and polymer solutions when infrared radiation (IR) is employed as the main heating source. The study reviews the current research trends of IR drying of specific applications. A case study similar to an industrial setting is presented to illustrate a model development and control scheme of an IR drying unit. The design and online implementation of an internal model controller (IMC) is discussed. The study demonstrates the controller capabilities to suppress random variations of the moisture content in the material entering the dryer. Simulation results also showed the success of model predictive control (MPC) multivariable controller ability, while handling process interactions and process constraints, to track setpoint changes in the humidity and temperature of the material exiting the dryer and to reject unmeasured stochastic disturbances in the inlet humidity stream.     

Infrared Drying: From Process Modeling to Advanced Process Control

The article surveys the drying of solids materials and polymer solutions when infrared radiation (IR) is employed as the main heating source. The study reviews the current research trends of IR drying of specific applications. A case study similar to an industrial setting is presented to illustrate a model development and control scheme of an IR drying unit. The design and online implementation of an internal model controller (IMC) is discussed. The study demonstrates the controller capabilities to suppress random variations of the moisture content in the material entering the dryer. Simulation results also showed the success of model predictive control (MPC) multivariable controller ability, while handling process interactions and process constraints, to track setpoint changes in the humidity and temperature of the material exiting the dryer and to reject unmeasured stochastic disturbances in the inlet humidity stream.     

春笋微波干燥动力学模型研究

在不同微波功率下研究春笋（5mm厚度，200g）干燥过程中失水速率和颜色的变化规律。失水速率随微波功率增大而增大，用拟和性较好的Page方程建立了水分比变化模型。色泽变化参数用亮度L、红色度a和黄色度b表示。红色度a值和黄色度b值上升而亮度L下降，笋片微波干燥后总体色泽加深。色泽各参数变化用食品中零级和一级反应模型来拟和，其中L和b的变化适合一级反应模型，a的变化适合用零级反应模型表达。     

Mathematical Modeling of Drying Kinetics for Apricots: Influence of the External Resistance to Mass Transfer

The drying curves of halved and deseeded apricots obtained during convective drying at different temperatures (from 50 to 90°C) have been examined, and a diffusional model, solved by a finite elements method, has been proposed to simulate the drying kinetics. The importance of taking into account both the internal and the external resistances to mass transfer when modeling the drying curves is discussed. Due to the geometry of halved apricots as a hemisphere losing water only through the flat section, the mass transfer coefficient (k_c) was not correctly estimated through an empirical correlation. Only the identification of this coefficient from the experimental results allowed an accurate simulation, decreasing the mean relative error from 12.3±3.8% when k_c was calculated through the Pasternak and Gauvin correlation to 2.9±1.0% when k_c was identified.     

Mathematical Modeling of Drying Kinetics for Apricots: Influence of the External Resistance to Mass Transfer

The drying curves of halved and deseeded apricots obtained during convective drying at different temperatures (from 50 to 90°C) have been examined, and a diffusional model, solved by a finite elements method, has been proposed to simulate the drying kinetics. The importance of taking into account both the internal and the external resistances to mass transfer when modeling the drying curves is discussed. Due to the geometry of halved apricots as a hemisphere losing water only through the flat section, the mass transfer coefficient (k_c) was not correctly estimated through an empirical correlation. Only the identification of this coefficient from the experimental results allowed an accurate simulation, decreasing the mean relative error from 12.3±3.8% when k_c was calculated through the Pasternak and Gauvin correlation to 2.9±1.0% when k_c was identified.     

Mathematical Modeling of Kiln Drying of Softwood Timber: Model Development, Validation, and Practical Application

Mathematical modeling of wood drying is a powerful tool to better understand and quantify the effects of wood properties as well as the effects of drying and post-drying treatment conditions on drying and thus the wood drying models can be used to improve drying quality. The models that have been developed can be divided into three categories: models for drying a single board, models for drying a kiln-wide stack, and models for drying stress and deformation. The single-board drying model employs comprehensive heat and moisture mass transfer equations and can be used to investigate the influence of wood variability. The kiln-wide drying model, which is based on the transfer processes between wood and the drying medium, is able to examine the influence of drying schedules and wood properties. The stress model can predict stress development in drying and stress relief in final steam conditioning and post-kiln treatment. An integrated model can be used to optimize drying schedules and develop strategies for high-quality dried timber.     

Modeling of Diffusion in Ellipsoidal Solids: A Comparative Study

Abstract

Most food materials are usually measured in three mutually perpendicular dimensions (length, width, and thickness) and considered approximately as general ellipsoids. In this article, we briefly reviewed the available studies on how to deal with the diffusion problems in ellipsoidal solids. A three-dimensional (3-D) model was proposed to accurately represent the kernel geometry and rigorously interpret the diffusion phenomena. By the finite difference method, the numerical solution of the diffusion equation for kernels of different shape factors was obtained and compared with those predicted by the one-dimensional (sphere) model and two-dimensional (axi-symmetrical body) model. Meanwhile, the effectiveness of the latter two models was evaluated.     

Influence of Osmotic Pretreatment on Kinetics of Convective Drying and Quality of Apples

This article examines the influence of apple osmotic dehydration (OD) on the kinetics of convective drying and the quality of the products obtained. Fresh apples (Malus domestica), variety Ligol, were used as the experimental material. Samples were first dehydrated in three osmotic agents (glucose, fructose, and sucrose), each at a concentration of 40%, for 30, 60, and 120 min, and then dried convectively. Efficiency of osmotic dewatering was assessed on the basis of water loss (WL) and solids gain (SG). The influence of OD on convective drying kinetics was analyzed by comparing the drying curves and the basic kinetic parameters (time and drying rate). Quality of products was assessed on the basis of colorimetric measurements and sensory evaluation. It was found that OD significantly improved the quality of dried products through better retention of the original color, shape, texture, aroma, and taste, but it did not remarkably affect the rate of convective drying.